diff --git "a/E9FQT4oBgHgl3EQfRDbD/content/tmp_files/load_file.txt" "b/E9FQT4oBgHgl3EQfRDbD/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/E9FQT4oBgHgl3EQfRDbD/content/tmp_files/load_file.txt" @@ -0,0 +1,755 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf,len=754 +page_content='Do entangled states correspond to entangled measurements under local transformations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Florian Pimpel,1, ∗ Martin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Renner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ∗ and Armin Tavakoli4 1Atominstitut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Technische Universität Wien,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Stadionallee 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 1020 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Austria 2University of Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Vienna Center for Quantum Science and Technology (VCQ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Boltzmanngasse 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 1090 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Austria 3Institute for Quantum Optics and Quantum Information - IQOQI Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Austrian Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Boltzmanngasse 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 1090 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Austria 4Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Lund University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Box 118,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 22100 Lund,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Sweden We investigate whether pure entangled states can be associated to a measurement basis in which all vectors are local unitary transformations of the original state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We prove that for bipartite states with a local dimension that is either 2, 4 or 8, every state corresponds to a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Via numerics we strongly evidence the same conclusion also for two qutrits and three qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' However, for some states of four qubits we are unable to find a basis, leading us to conjecture that not all quantum states admit a corresponding measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Furthermore, we investigate whether there can exist a set of local unitaries that transform any state into a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' While we show that such a state-independent construction cannot exist for general quantum states, we prove that it does exist for real-valued n-qubit states if and only if n = 2, 3, and that such constructions are impossible for any multipartite system of an odd local dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Our results suggest a rich relationship between entangled states and iso-entangled measurements with a strong dependence on both particle numbers and dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Entanglement is a fundamental, broadly useful and an in- tensely studied feature of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' However, in spite being of arguably similar foundational significance, much less is known about the entanglement of joint quan- tum measurements than the entanglement of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Entangled measurements are crucial for seminal quantum in- formation protocols such as teleportation [1], dense coding [2] and entanglement swapping [3], which are instrumen- tal for various quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Typically, they are based on the paradigmatic Bell basis, which is composed of the four maximally entangled states (|00⟩ ± |11⟩)/ √ 2 and (|01⟩ ± |10⟩)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In the same way that the Bell basis may be thought of as the measurement corresponding to the max- imally entangled state, it is natural to ask whether entangled states in general can be associated with a corresponding en- tangled measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Studying the relationship between en- tangled states and entangled measurements is not only inter- esting for understanding quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' It is also an invitation to explore, in the context of quantum information applications, the largely uncharted terrain of entangled mea- surements beyond the Bell basis and its immediate general- isations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Most notably, entangled measurements beyond the Bell basis are also increasingly interesting for topics such as network nonlocality [4] and entanglement-assisted quantum communication [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Consider that we are given a pure quantum state |ψ⟩ com- prised of n subsystems, each of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Is it possible to find a measurement, namely an orthonormal basis of the global dn-dimensional Hilbert space, in which all basis states have the same degree of entanglement as |ψ⟩?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Specifically, we want to decide the existence of dn strings, {Vj}dn j=1, of local unitary transformations, Vj = n � k=1 U (j) k (1) ∗ These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' where U (j) k is a d-dimensional unitary operator, such that the set of states |ψj⟩ ≡ Vj |ψ⟩ form a basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' | ⟨ψj|ψj′⟩ | = δjj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' If affirmative, we say that |ψ⟩ admits a basis and we call the set of basis vectors {|ψj⟩}dn j=1 a |ψ⟩-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Known examples of entangled measurements can be ac- commodated in this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For example, the Bell basis can be obtained from operating on |ψ⟩ = (|00⟩ + |11⟩)/ √ 2 with the four strings of local unitaries {Vj}4 j=1 = {11 ⊗ 11, 11 ⊗ X, Z ⊗ 11, Z ⊗ X}, where X and Z are bit-flip and phase- flip Pauli operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' A well-known generalisation of the Bell basis to n systems of dimension d can be thought of as a |GHZn,d⟩-measurement where the relevant state is the higher- dimensional GHZ state |GHZn,d⟩ = 1 √ d �d−1 k=0 |k⟩⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The corresponding strings of local unitaries are Vj = Zj1 d ⊗Xj2 d ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ⊗ Xjn d |GHZn,d⟩ where j = j1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' jn ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , d − 1}n and where Zd = �d−1 l=0 e 2πi d l |l⟩⟨l| and Xd = �d−1 l=0 |l + 1⟩⟨l| are generalised Pauli operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' More generally, any state that is locally maximally entanglable (for example graph states) is known to admit a basis via suitable unitaries of the form Vj = U j1 1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ⊗ U jn n [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' These states are characterised by the property that if each qubit is supplemented with a qubit ancilla and controlled unitary gates are performed on the state-ancilla pairs, then a maximally entangled bipartite state can be constructed between the collection of state-qubits and the collection of ancilla-qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' However, this is far from a complete characterisation of the states that admit a basis, which is seen already in the restrictive form of the strings of unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For example, the three-qubit W-state, |W3⟩ = (|001⟩+|010⟩+|100⟩)/ √ 3, is not locally maximally entanglable but is neverthelss known to admit a basis [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In what follows, we set out to systematically explore whether entangled states admit a corresponding basis and then, as we will introduce later, whether such bases can be constructed even without prior knowledge of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Let us begin with considering the simplest situation, namely when |ψ⟩ is a state of two qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We constructively show that every such state admits a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' To this end, we first apply the state-dependent local unitaries W A ψ ⊗ W B ψ that map |ψ⟩, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='13285v1 [quant-ph] 30 Jan 2023 2 via a Schmidt decomposition, into the computational basis, |ψS⟩ = λ |00⟩+ √ 1 − λ2 |11⟩ for some coefficient 0 ≤ λ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Then, we consider the action of the following four strings of local unitaries � � � � � 11 ⊗ 11 11 ⊗ XZ XZ ⊗ Z XZ ⊗ X � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (2) One can verify that this transforms |ψS⟩ into a |ψ⟩-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' No- tice that once the state has been rotated into the Schmidt form |ψS⟩, the subsequent unitaries (2) do not depend on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' This construction can be extended to bipartite (n = 2) states of local dimension d = 4 and d = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Again via Schmidt de- composition, we can find state-dependent local unitaries that transform |ψ⟩ into |ψS⟩ = �d−1 l=0 λl |ll⟩ for some Schmidt co- efficients � l λ2 l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In Appendix A, we show that there is a set of local unitaries that indeed leads to a |ψ⟩-basis indepen- dently of the specific Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' It is natural to consider also the simplest case that is not of the above convenient form, namely that of two qutrits, (n, d) = (2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' This appears to be considerably different be- cause we fail to find strings of local unitaries that bring the Schmidt decomposition |ψS⟩ into a basis without explicit de- pendence on the Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Nevertheless, a basis might still be possible to construct by taking the Schmidt co- efficients into account when choosing the local unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Ac- tually, this seems to always be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' To arrive at this, we have used a numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Let {|φj⟩}m j=1 be a set of states in a given Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' These states are pairwise orthogonal if and only if they realise the global minimum (zero) of the following objective function f({φj}) ≡ � j̸=j′ | ⟨φj|φj′⟩ |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (3) For a given state |ψ⟩, we numerically minimise f({ψj}) over all possible strings {Vj}dn j=1 of local unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' To this end, we parameterise the local unitaries U (j) k using the scheme of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For the two-qutrit case, we have randomly chosen 1000 pairs of Schmidt coefficients (λ1, λ2) which (up to local unitaries) fully specifies the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In each case we numerically minimise f({ψj}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Without exception, we find strings of lo- cal unitaries that yield a result below our selected precision threshold of f ≤ 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Furthermore, we have also numerically investigated the case of three qubits, (n, d) = (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' This scenario requires a different approach than the previous cases since multipar- tite states have no Schmidt decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Instead, for any given three-qubit state |ψ⟩, there exists local unitary transfor- mations that map it onto the canonical form a |000⟩+b |011⟩+ c |101⟩ + d |110⟩ + e |111⟩ where (b, c, d, e) are real num- bers and a is a complex number [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Hence, up to lo- cal unitaries, the state space (after normalisation) is charac- terised by five real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Later, we will provide an analyt- ical construction of a |ψ⟩-basis for the four-parameter family corresponding to restricting a to be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' However, we have not found an analytical basis construction for general three- qubit states, but we nevertheless conjecture that it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' To evidence this, we have employed the previously introduced numerical search method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Again, we have randomly chosen 1000 normalised sets of coefficients (a, b, c, d, e) and searched for the minimal value of f over all the strings of local qubit unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In all cases, we find that f vanishes up to our se- lected precision of f ≤ 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Given the above case studies, one might suspect that ev- ery pure quantum state admits a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Interestingly, this seems not to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' While some states of four qubits, (n, d) = (4, 2), are found to admit a basis, for example a W state and doubly-excited Dicke state [23], it appears that most four-qubit states do not admit a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We have sampled many different four-qubit states and repeatingly at- tempted to numerically find a basis via the minimisation of (3), also using several different search algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' It was reg- ularly found that the estimated minimum is multiple orders of magnitude above our given precision threshold for a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For example, we searched for the minimum of f for the state 2 √ 6 |W⟩ + √ 2 √ 6 |GHZ4,2⟩, with 100 randomised initial points, and never reached below f = 10−1, five orders of magnitude above our precision threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We have attempted to prove that no basis exists by employing semidefinite outer relax- ations of f over the set of dimensionally-restricted quantum correlations [12] combined with a modified sampling of the state and measurement space [13] and symmetrisation tech- niques [14] to efficiently treat the large number of single-qubit unitaries featured in this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' However, the conjecture has resisted our efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' A guiding intuition for the impossi- bility of a basis is to note that the number of free parameters is 3n(2n − 1) whereas the number of orthogonality constraints (counting both the real and imaginary part) is 22n − 2n, and the latter is larger than the former only when n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Furthermore, if an n-qubit state |ψ⟩ does not admit a ba- sis, then the (n + 1)-qubit state |ψ′⟩ = |ψ⟩ ⊗ |0⟩ also does not admit a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' By contradiction, suppose there are 2n+1 unitaries V ′ j = Vj ⊗ U (j) n+1 such that |⟨ψ′|(V ′ j )†V ′ k|ψ′⟩| = δjk ∀j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=', 2n+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Divide the 2n+1 states U (j) n+1 |0⟩ into two sets such that two orthogonal vectors are not in the same set (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' the northern and southern hemisphere of the Bloch ball).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Consider the set that contains at least as many elements as the other one, hence, at least 2n el- ements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' By construction, these states cannot be distin- guished on the last qubit, |⟨0|U (j)† n+1U (k) n+1|0⟩| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Since |⟨ψ′|(V ′ j )†V ′ k|ψ′⟩| = |⟨ψ|V † j Vk|ψ⟩| · |⟨0|U (j)† n+1U (k) n+1|0⟩|, we must have |⟨ψ|V † j Vk|ψ⟩| = δjk for all of those pairs, which contradicts that |ψ⟩ does not admit a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' By induction, this argument shows that if our above conjecture holds, namely that some four-qubit states do not admit a basis, then the same holds for any number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Since not all pure quantum states admit a basis, and this seems to be typical rather than exceptional for four qubits, it is interesting to ask whether some distinguished families of n-qubit states can nevertheless admit a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' This is well- known to be the case for n-qubit GHZ-states and graph-states 3 since they are locally maximally entanglable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' More interest- ingly, a positive answer is also possible for states that are not of this kind: we construct a basis for the n-qubit W-state, |Wn⟩ = 1 √n � σ σ(|0⟩⊗n−1 |1⟩) where σ runs over all permu- tations of the position of “1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Note that |W1⟩ = |1⟩ and that a |W1⟩-basis is obtained from the unitaries {11, X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Now we ap- ply induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Consider that the strings {V (n) j }2n j=1 generate a |Wn⟩-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' One can then construct a basis for n+1 qubits as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For half of the basis elements, namely j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , 2n, define V (n+1) j = V (n) j ⊗ 11 and for the other half, namely j = 2n +1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , 2n+1, define V (n+1) j = �n k=1 U (j) k Z ⊗X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' As we detail in Appendix B, one can verify that {V (n+1) j |Wn+1⟩}j is a W-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We note that for the purpose of entanglement distillation, a different construction of a W-basis was given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' So far, we have considered whether a specific state can be associated to a specific measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In other words, the uni- tary constructions have been state-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We now go fur- ther and introduce a complementary concept, namely whether there exist strings of local unitaries {Vj} that can transform any state in a space of states S into a basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' strings of local unitaries that satisfy ∀ψ ∈ S, |⟨ψ|V † j Vj′|ψ⟩| = δjj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (4) Naturally, this state-independent notion of basis construc- tion is much stronger than the previously considered state- dependent notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In the most ambitious case, when we choose the space S to be the entire Hilbert space of n sub- systems of dimension d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' S ≃ (Cd)⊗n, then a state- independent construction cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In fact, not even two orthogonal vectors can be state-independently constructed for the full quantum state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' To show this, we can w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' set V1 = 11 and assume that there exists local unitaries {Uk} such that |ψ1⟩ = |ψ⟩ and |ψ2⟩ = �n k=1 Uk |ψ⟩ are orthog- onal for all |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Focus now on the particular state |ψ⟩ = �n k=1 |µk⟩ where |µk⟩ is some eigenvector of the unitary Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Since the eigenvalues of a unitary are complex phases, writ- ten eiϕk for Uk and |µk⟩, we obtain |ψ1⟩ = �n k=1 |µk⟩ and |ψ2⟩ = ei �n k=1 ϕk �n k=1 |µk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' These two states are evidently not orthogonal and hence we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Interestingly, the situation changes radically if we limit our state-independent investigation to all quantum states in a real- valued Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' That is, S ≃ (Rd)⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Such real quan- tum systems have also been contrasted in the literature with their complex counterparts [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Let us momentarily ig- nore the n-partition structure of our Hilbert space and sim- ply consider two real states |ψ1⟩ = |ψ⟩ and |ψ2⟩ = U |ψ⟩ obtained from a given real target state |ψ⟩ and a fixed (ψ- independent) unitary U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' It holds that ψ1 and ψ2 are or- thogonal if and only if U is skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' To prove this, assume first the skew-symmetry property U = −U T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Since for real states ⟨ψ1|ψ2⟩ = ⟨ψ2|ψ1⟩∗ is equivalent to ⟨ψ|U|ψ⟩ = ⟨ψ|U †|ψ⟩∗ = ⟨ψ|U T |ψ⟩, skew-symmetry im- plies that ⟨ψ1|ψ2⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Conversely, assume that ⟨ψ|U|ψ⟩ = 0 for all real-valued ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Choosing in particular |ψ⟩ = |k⟩ for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , d − 1, it follows that all diagonal elements of U must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Then, choose |ψ⟩ = 1 √ 2(|i⟩ + |j⟩) for any pair i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' This yield Uii+Ujj+Uij+Uji = 0, but since we know that the diagonals vanish we are left with just Uij = −Uji which defines a skew-symmetric operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Returning to our n-partitioned real Hilbert space, and still w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' taking V1 = 11, the above result demands that we find local unitaries such that U1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ⊗ Un = −U T 1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ⊗ U T n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (5) This is only possible if U T k = ±Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Hence, all local unitaries must be either symmetric or skew-symmetric, and the number of the latter must be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' When extended from two orthogonal states to a whole basis, we require that this property holds for every pair of distinct labels (j, j′) in the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In other words, we require that every string (Vj)†Vj′ with j ̸= j′ is skew- symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The question becomes whether the above condition can be satisfied for a given scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Consider it first for qubit systems (d = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In Appendix C we show that the set of complex qubit unitaries that are either symmetric or skew- symmetric and whose products are again either symmetric or skew-symmetric, must obey a simple structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' they are essentially equivalent to the four Pauli-type operators P ≡ {11, X, Z, XZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Thus, if a state-independent construction ex- ists, we can restrict to selecting one of these four operators for each of our local unitaries U (j) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Interestingly, for the case of two qubits, (n, d) = (2, 2), a state-independent construction is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' It is in fact given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' One can straightfor- wardly verify that the above criterion is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' all local unitaries are selected from P and all pairs of products of uni- tary strings in (2) are skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Alternatively, one can easily verify that (2) maps every state � i,j=0,1 αij |ij⟩ into a basis, for any real coefficients αij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Furthermore, by the same token, a state-independent basis is also possible for every real state of three qubits, (n, d) = (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' One explicit construc- tion that satisfies our necessary and sufficient criterion is the following set of eight strings of local unitaries � � � � � � � � � � � � � � � � � � � � � 11 ⊗ 11 ⊗ 11 Z ⊗ Z ⊗ XZ Z ⊗ XZ ⊗ 11 XZ ⊗ 11 ⊗ 11 Z ⊗ X ⊗ XZ X ⊗ 11 ⊗ XZ X ⊗ XZ ⊗ Z X ⊗ XZ ⊗ X � � � � � � � � � � � � � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Again, one may easily verify that every real state � i,j,k=0,1 αijk |ijk⟩ is mapped into a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Two- and three-qubits are interesting cases because they are exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' As we now show, there exists no state- independent construction for real states of four or more qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We first prove this for n = 4 and then show that this im- plies impossibility also for n > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The four-qubit case con- tains 16 strings of unitaries and we know that each local uni- tary can w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' be selected from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Since we seek a state- independent construction, we can momentarily consider only the state |0000⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In order for it to be mapped into a basis, we 4 (2,2,R) (2,2,C) (3,2,R) (3,2,C) (4,2,R) (2,3,C) (2,4 or 8,C) (n, 2m + 1,R) State-dependent construction \x13 \x13 \x13 (\x13) (\x17) (\x13) \x13 − − − State-independent construction \x13 \x17 \x13 \x17 \x17 \x17 \x17 \x17 TABLE I: Overview of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The first row indicates the scenario: (n, d, S) gives particle number, dimension and the type of state space respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The symbol \x13indicates the existence of a basis under local unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The symbol \x17indicates that there in general can be no basis under local unitaries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' at least one state admits no basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Paranthesis indicates that the result is obtained from numerical search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The symbol − − − indicates that no investigation was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' see that Z acts trivially on every register and therefore each one of the 16 combinations of bit-flip or identity operators, {Xc1 ⊗ Xc2 ⊗ Xc3 ⊗ Xc4} for c1, c2, c3, c4 ∈ {0, 1}, must be featured in exactly one of the 16 unitary strings {Vj}16 j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Let us now look only at six of these strings, namely those corresponding to having zero bit-flips (1 case), one bit-flip (4 cases) and four bit-flips (1 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' fixing V1 = 11 (zero bit-flips), the strings take the form V1 11 ⊗ 11 ⊗ 11 ⊗ 11 V2 XZr11 ⊗ Zr12 ⊗ Zr13 ⊗ Zr14 V3 Zr21 ⊗ XZr22 ⊗ Zr23 ⊗ Zr24 V4 Zr31 ⊗ Zr32 ⊗ XZr33 ⊗ Zr34 V5 Zr41 ⊗ Zr42 ⊗ Zr43 ⊗ XZr44 V6 XZr51 ⊗ XZr52 ⊗ XZr53 ⊗ XZr54 , (6) where rij ∈ {0, 1} represent our freedom to insert a Z operator and thus realise the two relevant elements of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Since every row must be skew-symmetric and the only skew- symmetric element in P is XZ, we must have r11 = r22 = r33 = r44 = 1 and r51 + r52 + r53 + r54 = 1 where ad- dition is modulo two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Moreover, every product of two rows must be skew-symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' the product must have an odd number of XZ operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For the four middle rows, this im- plies rij + rji = 1 for distinct indices i, j ∈ {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For the products V † 6 Vj for j = 2, 3, 4, 5, the conditions for skew- symmetry respectively become r12 + r13 + r14 + r52 + r53 + r54 = 1 r21 + r23 + r24 + r51 + r53 + r54 = 1 r31 + r32 + r34 + r51 + r52 + r54 = 1 r41 + r42 + r43 + r51 + r52 + r53 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (7) Summing these four equations and using the previously es- tablished skew-symmetry conditions, one can cancel out all degrees of freedom rij and arrive at the contradiction 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Hence, we conclude that the state-independent basis construc- tion for four qubits is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For the case of five qubits, we can again assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' that the 32 combinations of bit-flip or identity operators, {Xc1 ⊗ Xc2⊗Xc3⊗Xc4⊗Xc5} for c1, c2, c3, c4, c5 ∈ {0, 1} must be featured in exactly one of the 32 unitary strings since the state |00000⟩ has to be mapped into an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Suppose there is a state-independent construction that maps every real- valued five-qubit state into a basis, in especially any state of the form |ψ⟩ ⊗ |0⟩, where |ψ⟩ is an arbitrary real-valued four qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Now consider the 16 strings where c5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Since the fifth qubit is always mapped to itself, it has to hold that the first four qubits are pairwise distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' However, this implies a state-independent construction for four qubits which is in contradiction to the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' By induction, this implies that no state-independent construction can exist whenever n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The possibility of state-independent constructions for real- valued bi- and tri-partite systems draws heavily on the sim- ple structure of skew-symmetric qubit unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' If we con- sider real-valued systems of dimension d > 2, the situa- tion changes considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Using our necessary and suffi- cient condition, it follows immediately that state-independent constructions are impossible in all odd dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' when (n, d) = (n, 2m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' This stems from the fact that there exists no skew-symmetric unitary matrix in odd dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' To see that, simply note that if A is skew-symmetric then det(A) = det � AT � = det(−A) = (−1)2m+1 det(A) = − det(A) and hence det(A) = 0, but that contradicts unitarity because the determinant of a unitary has unit modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In summary, we have investigated the correspondence be- tween entangled states and entangled measurements under lo- cal unitary transformations, both when the local transforma- tion can and cannot explicitly depend on the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Per- haps surprisingly, we have found that this problem is not so straightforward and has a strong dependence on both the num- ber of subsystems involved and their dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Our analyt- ical and numerical results and conjectures are summarised in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The conspicuous open problem left by our work is to prove our conjecture that there exists states that do not admit a ba- sis under local unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' An interesting related question is if one can bound the relative volume of four-qubit states that do not admit a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Our numerical investigations suggest that nearly all four-qubit states should belong to this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Fur- thermore, it would be useful to find analytical solutions for the three-qubit and two-qutrit state-dependent cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' More- over, for the state-independent considerations, we focused on real Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' A natural question is whether there ex- ists state-independent basis constructions for other interest- ing spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For example, if one restricts to bipartite states of a known entanglement entropy, can one construct a state- independent basis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The answer is clearly positive for the lim- iting cases of product states and maximally entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Another interesting space to consider is the symmetric sub- space of n-qubit Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 5 Our results may also have prospects in quantum informa- tion as one may now construct entangled measurements asso- ciated to entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Recently there has been proposals of two-qubit entangled projections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' the so-called Elegant Joint Measurements [18, 19] which have also been realised in vari- ous experiments [20?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The Elegant Joint Measurements can be seen as a particular type of |ψ⟩-basis where |ψ⟩ is a partially entangled two-qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' However, the basis addi- tionally has the feature that the collections of reduced states form a tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' This requirement goes beyond our prob- lem formulation, as we do not impose any structure on the reduced states of our bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' However, it suggests an avenue to identifying interesting and highly symmetric measurements by finding the particular |ψ⟩-basis that maximises the Hilbert space volume spanned its collection of reduced states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Finally, one of the notable shortcommings of traditional, GHZ based, multiqubit entanglement swapping protocols is that the loss of one particle renders the measurement separa- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' However, some other states that are inequivalent to GHZ under LOCC can preserve their entanglement under reduc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The existence of an iso-entangled basis composed of such states may constitute an avenue to more noise-resiliant entanglement swapping protocols which have natural quan- tum information applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Note added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='— During the late stage of our work, we be- came aware of the previous work [23] where i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' bases are found for some Dicke states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Hayata Yamasaki, Marcus Huber, Jakub Czartowski and Karol ˙Zyczkowski for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ac- knowledges support from the Wenner-Gren Foundation and from the Wallenberg Centre for Quantum Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' M.' metadata={'source': 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formation 7, 117 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' [23] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Tanaka, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Markham, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Murao, Local encoding of classical information onto quantum states, Journal of Modern Optics 54, 2259 (2007), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='1080/09500340701403301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Appendix A: Basis construction for every bipartite state of local dimension d = 4 and d = 8 Let the local dimension be a power of two, d = 2m, and index the d2 basis elements as (˜j, j) where ˜j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , d − 1 and j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Let W A ψ ⊗ W B ψ be the state-dependent local unitaries that transform the general state |ψ⟩ into the Schmidt basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' |ψS⟩ ≡ W A ψ ⊗ W B ψ |ψ⟩ = �d−1 l=0 λl |l, l⟩, with the Schmidt coefficients λl ∈ R satisfying � l λ2 l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We now further decompose the individual d-dimensional registers as a string of m qubits, writing |l⟩ = |l1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' lm⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Thus, the Schmidt decomposed state reads |ψS⟩ = � l1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=',lm=0,1 λl |l1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' lm, l1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' lm⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (A1) Once the state has been put in the form (A1), we apply a set of local unitaries that is independent of the Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For d = 4 and ˜j = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' the two sets of unitaries read as follows: ˜j j U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 1 U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 2 U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 1 ⊗ U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 2 |ψS⟩ 0 1 11 ⊗ 11 11 ⊗ 11 λ00 |00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 00⟩ + λ01 |01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01⟩ + λ10 |10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 10⟩ + λ11 |11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 11⟩ 0 2 11 ⊗ X 11 ⊗ XZ λ00 |01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01⟩ − λ01 |00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 00⟩ + λ10 |11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 11⟩ − λ11 |10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 10⟩ 0 3 X ⊗ 11 XZ ⊗ Z λ00 |10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 10⟩ − λ01 |11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 11⟩ − λ10 |00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 00⟩ + λ11 |01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01⟩ 0 4 X ⊗ X XZ ⊗ X λ00 |11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 11⟩ + λ01 |10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 10⟩ − λ10 |01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01⟩ − λ11 |00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 00⟩ (A2) In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' we define U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 1 := X ˜j 4 U (˜j=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 1 and U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 2 := U (˜j=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' where Xd is the d-dimensional shift-operator Xd = �d−1 l=0 |l + 1⟩⟨l|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Note that, the unitaries U (˜j,j) 2 coincide with the state-independent set for two qubits given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (2) and do not depend on ˜j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' At the same time, U (˜j=0,j) 1 are the same as U (˜j,j) 2 where the Z gates are left out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We now show that {U (˜j,j) 1 ⊗ U (˜j,j) 2 |ψS⟩}˜j,j is a basis of the bipartite Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' One can check directly that the four states with ˜j = 0 stated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (A2) above are pairwise orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We want to mention that we are exploiting the fact that U (˜j=0,j) 2 are the elements of a state-independent construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' To see the connection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' note that the calculation for the state-independent two-qubit construction reads as follows: (11 ⊗ 11)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (A3) (11 ⊗ XZ)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |01⟩ − λ01 |00⟩ + λ10 |11⟩ − λ11 |10⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (A4) (XZ ⊗ Z)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |10⟩ − λ01 |11⟩ − λ10 |00⟩ + λ11 |01⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (A5) (XZ ⊗ X)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |11⟩ + λ01 |10⟩ − λ10 |01⟩ − λ11 |00⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (A6) Since these states are pairwise orthogonal for arbitrary real coefficients λl1l2, the same holds true for the states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In addition, all of the states where ˜j = 0 are elements of the subspace spanned by |00, 00⟩, |01, 01⟩, |10, 10⟩ and |11, 11⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Hence, they form a basis of this four-dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' By shifting now the first system we obtain a basis for the remaining orthogonal subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' More precisely, since we defined U (˜j,j) 1 = X ˜j 4 U (˜j=0,j) 1 the states where ˜j = 1 are esentially the same states as the ones in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (A2) but with the first system shifted by one l → l ⊕ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For example, λ00 |11, 10⟩ − λ01 |00, 11⟩ − λ10 |01, 00⟩ + λ11 |10, 01⟩ is the state that corresponds to ˜j = 1 and j = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In this way, the four states where ˜j = 1 form a basis of the subspace spanned by |01, 00⟩, |10, 01⟩, |11, 10⟩ and |00, 11⟩ (or all states where |l + 1, l⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Analogously, the four states where ˜j = 2 (˜j = 3) form a basis of the subspaces spanned by the vectors with |l + 2, l⟩ (|l + 3, l⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Altogether, the sixteen states {U (˜j,j) 1 ⊗ U (˜j,j) 2 |ψS⟩}˜j,j form a basis of the entire sixteen dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' A similar construction can be found for d = 8 by using the state-independent construction of three qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Similar as above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 7 the set for ˜j = 0 reads as follows: ˜j j U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 1 U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 2 U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 1 ⊗ U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='j) 2 |ψS⟩ 0 1 11 ⊗ 11 ⊗ 11 11 ⊗ 11 ⊗ 11 +λ000 |000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 000⟩ + λ001 |001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 001⟩ + λ010 |010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 010⟩ + λ011 |011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 011⟩ +λ100 |100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 100⟩ + λ101 |101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 101⟩ + λ110 |110,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 110⟩ + λ111 |111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 111⟩ 0 2 11 ⊗ 11 ⊗ X Z ⊗ Z ⊗ XZ +λ000 |001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 001⟩ − λ001 |000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 000⟩ − λ010 |011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 011⟩ + λ011 |010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 010⟩ −λ100 |101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 101⟩ + λ101 |100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 100⟩ + λ110 |111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 111⟩ − λ111 |110,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 110⟩ 0 3 11 ⊗ X ⊗ 11 Z ⊗ XZ ⊗ 11 +λ000 |010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 010⟩ + λ001 |011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 011⟩ − λ010 |000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 000⟩ − λ011 |001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 001⟩ −λ100 |110,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 110⟩ − λ101 |111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 111⟩ + λ110 |100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 100⟩ + λ111 |101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 101⟩ 0 4 X ⊗ 11 ⊗ 11 XZ ⊗ 11 ⊗ 11 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=') 0 5 11 ⊗ X ⊗ X Z ⊗ X ⊗ XZ (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=') 0 6 X ⊗ 11 ⊗ X X ⊗ 11 ⊗ XZ (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=') 0 7 X ⊗ X ⊗ 11 X ⊗ XZ ⊗ Z (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=') 0 8 X ⊗ X ⊗ X X ⊗ XZ ⊗ X (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=') (A7) Again, we define U (˜j,j) 1 = X ˜j 8 U (˜j=0,j) 1 and U (˜j,j) 2 = U (˜j=0,j) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The proof that this forms a basis of the 64-dimension Hilbert space is completely analogous to the case of d = 4 before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The eight states for ˜j = 0 form a basis of the eight-dimensional subspace spanned by |l1l2l3, l1l2l3⟩ (for li = 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Applying the shift operator X8 to the first system, one obtains bases of the other eight-dimensional orthogonal subspaces spanned by the vectors with ��l + ˜j, l � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' This approach cannot (immediately) be generalized to higher dimensions d = 2n, due to the lack of state-independent constructions for n ≥ 4 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' However, there is in principle no reason to restrict the unitaries on the second system to tensor products of single qubit Pauli gates as we do here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In principle, we could also consider general permutations with suitably chosen signs such that all terms cancel in this pairwise sense as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Even when considering this larger class of possibilities, we made an exhaustive search and could not find any additional construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Due to this, it seems unlikely that a construction exists in which the unitaries do not depend on the Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Appendix B: An n-qubit basis of W-states We define the n-qubit W-state as |W1⟩ ≡ |1⟩ |W2⟩ ≡ 1 √ 2 (|01⟩ + |10⟩) |W3⟩ ≡ 1 √ 3 (|001⟩ + |010⟩ + |100⟩) |W4⟩ ≡ 1 2 (|0001⟩ + |0010⟩ + |0100⟩ + |1000⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (B1) Note that for one and two qubits, the definition is only introduced for sake of convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In general, we write |Wn⟩ ≡ 1 √n � σ σ(|0⟩⊗n−1 |1⟩), (B2) where σ runs over all permutations of the position of “1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' It is also useful to write the state recursively as |Wn+1⟩ = � n n + 1 |Wn⟩ ⊗ |0⟩ + 1 √n + 1 |0⟩n ⊗ |1⟩ (B3) Clearly, if we apply the local unitaries U (1) 1 = 11 and U (2) 1 = X to |W1⟩ we generate the trivial one-qubit W-basis {|0⟩ , |1⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Assume now that the local unitaries {U (j) k } for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' n and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , 2n yield a |Wn⟩-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We will now show that under this assumption we can construct a basis for |Wn+1⟩ and hence it follows from induction that a W-basis exists for any number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 8 We illustrate the induction step as follows, U (1) 1 ⊗ U (1) 2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ⊗ U (1) n ⊗ 11 U (2) 1 ⊗ U (2) 2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ⊗ U (2) n ⊗ 11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' U (2n) 1 ⊗ U (2n) 2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ⊗ U (2n) n ⊗ 11 U (1) 1 Z ⊗ U (1) 2 Z ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ⊗ U (1) n Z ⊗ X U (2) 1 Z ⊗ U (2) 2 Z ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ⊗ U (2) n Z ⊗ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' U (2n) 1 Z ⊗ U (2n) 2 Z ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ⊗ U (2n) n Z ⊗ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (B4) We see that for the first 2n basis elements, we extend the unitaries for n qubits by tensoring with 11 for qubit number n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For the latter 2n basis elements, we extend the unitaries for n qubits by multiplying all of them from the right by Z and finally tensoring with X for qubit number n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' As usual, we now write the string of unitaries associated to each row as V (n+1) j for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , 2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We similarly use V (n) j for the unitary strings for the case of n qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' To see that this yields a basis, we first show that the first 2n basis elements (upper block of table, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , 2n) are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For this purpose, we use the recursion formula (B3) to write for j ̸= j′ ⟨Wn+1|(V (n+1) j′ )†V (n+1) j |Wn+1⟩ = n n + 1⟨Wn0|(V (n) j′ )†V (n) j ⊗ 11|Wn0⟩ + 1 n + 1⟨0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01|(V (n) j′ )†V (n) j ⊗ 11|0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01⟩ + √n n + 1⟨Wn0|(V (n) j′ )†V (n) j ⊗ 11|0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01⟩ + √n n + 1⟨0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01|(V (n) j′ )†V (n) j ⊗ 11|Wn0⟩ = 0 The first term is zero for all j′ ̸= j due to the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The third and fourth terms are zero due to orthogonality in the last qubit register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The second term is zero for every j′ ̸= j there exists at least one qubit register k for which U (j′) k and U (j) k are composed of different numbers of bit-flips (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The latter follows from the initial condition of using {11, X} to construct the |W1⟩-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The same procedure will analogously show that the latter 2n basis elements (lower block of the table, j = 2n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , 2n+1) are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We are left with showing that every overlap between the upper and lower block, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' with any j′ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , 2n and any j = 2n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' , 2n+1, also vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' For this we have ⟨Wn+1|(V (n+1) j′ )†V (n+1) j |Wn+1⟩ = n n + 1⟨Wn0| � (V (n) j′ )†V (n) j ⊗ X � n � k=1 Z ⊗ 11|Wn0⟩ + 1 n + 1⟨0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01| � (V (n) j′ )†V (n) j ⊗ X � n � k=1 Z ⊗ 11|0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01⟩ + √n n + 1⟨Wn0| � (V (n) j′ )†V (n) j ⊗ X � n � k=1 Z ⊗ 11|0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01⟩ + √n n + 1⟨0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01| � (V (n) j′ )†V (n) j ⊗ X � n � k=1 Z ⊗ 11|Wn0⟩ Note that �n k=1 Z ⊗ 11 |Wn0⟩ = − |Wn0⟩ and �n k=1 Z ⊗ 11 |0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01⟩ = |0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 01⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The first and second terms are both zero due to orthogonality in the final qubit register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We thus have ⟨Wn+1|(V (n+1) j′ )†V (n+1) j |Wn+1⟩ = √n n + 1⟨Wn|(V (n) j′ )†V (n) j |0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 0⟩ − √n n + 1⟨0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 0|(V (n) j′ )†V (n) j |Wn⟩ = √n n + 1⟨Wn|(V (n) j′ )†V (n) j − (V (n) j )†V (n) j′ |0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' 0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (B5) The last equality follows from the fact that it is sufficient, for given (j, j′), that there exist some register index k such that (U (j′))† kU (j) k − (U (j))† kU (j′) k = 0 in order for the overlap to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' This is always the case because due to our construction (see initial condition and the table), for every two unitaries there is at least one register k where the single-qubit unitaries differ by X, meaning that either (U (j) k , U (j′) k ) = (11, X)/(Z, XZ), or the same with j ↔ j′ is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' The condition above is satisfied by all of these combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Hence we conclude that the proposed construction satisfies ⟨Wn+1|(V (n+1) j )†V (n+1) j′ |Wn+1⟩ = δjj′ (B6) 9 and therefore yields a W-state basis for any number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Appendix C: The Pauli structure for state-independent qubit unitary constructions We consider the set of local unitaries P that are applied to the i-th qubit in the state-independent construction and show that without loss of generality, the set can be chosen to be the Pauli-type gates P ≡ {11, X, Z, XZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' First, note that the set is finite since there are exactly 2n basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Next, we observe that the identity 11 has to be within the set P since we demand that V1 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Furthermore, we can argue that the gate XZ = � 0 −1 1 0 � has to be within the set as well, since it is the only gate that maps every real qubit state to its orthogonal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' More precisely, if it is not used on the i-th qubit at least once, one can choose a real qubit state |φi⟩ such that none of the gates in P map |φi⟩ to its orthogonal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Hence if we apply the state-independent construction to the real-valued product state |φ⟩ = |0⟩1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' |0⟩i−1 ⊗ |φi⟩ ⊗ |0⟩i+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' ⊗ |0⟩n none of the resulting 2n states are distinguishable on the i-th qubit, which is impossible if these states should form a basis of product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Therefore, the gate XZ has to be within the set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Apart from the gates 11 and XZ we can constrain which other qubit unitaries can be in the set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' We know that if we demand V1 = 11, every string of local unitaries (Vj) and their products (Vj)†Vj′ with j ̸= j′ have to be skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' As a result, the local unitaries on each subsystem (hence, the unitaries in the set P) and also all their products have to be either symmetric or skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' By neglecting a global phase, the general form of a unitary operator can be written as: U = � cos (θ)eiα sin (θ)eiβ − sin (θ)e−iβ cos (θ)e−iα � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (C1) The only skew-symmetric 2×2 unitary is, up to an irrelevant global phase, the Pauli-type operator XZ, which we already found to be necessarily in the set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' All the symmetric matrices of this form can be written as: U = � cos (θ)eiα i sin (θ) i sin (θ) cos (θ)e−iα � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (C2) If the gate U is in P, it is at some point multiplied with the gate XZ since the operator XZ is used at least once on the i-th qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Since we know that the result of this product has to be again either symmetric or skew-symmetric, we obtain that α = π/2, 3π/2 due to: (XZ)†U = � 0 1 −1 0 � � cos (θ)eiα i sin (θ) i sin (θ) cos (θ)e−iα � = � i sin (θ) cos (θ)e−iα − cos (θ)eiα −i sin (θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (C3) The two possibilities for α = π/2, 3π/2 correspond to the two solutions U1 = � cos (θ) sin (θ) sin (θ) − cos (θ) � , U2 = � sin (θ) − cos (θ) − cos (θ) − sin (θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (C4) We left the irrelevant global factor i for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Considering the additional degree of freedom of θ, we can restrict to the first class of solutions U1 since the second class U2 can be obtained by shifting θ by π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Hence, if we add a gate U to the set P, it has to be of the form given by U1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Now if we add two such gates to the set P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' the product of U1 with another valid matrix U ′ 1 is U † 1U ′ 1 = � cos (θ) sin (θ) sin (θ) − cos (θ) � � cos (θ′) sin (θ′) sin (θ′) − cos (θ′) � = = � cos (θ) cos (θ′) + sin (θ) sin (θ′) cos (θ) sin (θ′) − sin (θ) cos (θ′) sin (θ) cos (θ′) − cos (θ) sin (θ′) cos (θ) cos (θ′) + sin (θ) sin (θ′) � = � cos (θ − θ′) − sin (θ − θ′) sin (θ − θ′) cos (θ − θ′) � If both,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' U1 and U ′ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' are in P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' this product has to be again either symmetric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' which is true if θ = θ′ or skew-symmetric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' which is true if θ = θ′ + π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' (Note that, also θ = θ′ + π and θ = θ′ + 3π/2 are possible solutions but we do not have to consider them 10 since they just differ by an irrelevant global factor of (−1) in one of the two unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=') Hence, U ′ 1 is either U1 or the unitary U2 stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Hence, for each single-qubit subsystem, we can only use a set of operators P ≡ {11, U1, U2, XZ} for our basis construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' In a final step, we can show that we can restrict also θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' To see this, suppose a state-independent construction exists where we use the gates from the set P ≡ {11, U1, U2, XZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Now consider the construction where each gate U1 is replaced with W †U1W, each gate U2 with W †U2W, each gate XZ with W †XZW and each gate 11 with W †11W, where: W = � cos (α) − sin (α) sin (α) cos (α) � (C5) for some freely chosen parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' This also has to be a state-independent construction for any state with real coefficients, since W is a map from real states to real states, and all inner products between the basis states remain the same under this local transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Hence, if a state-independent construction exists with the gate set P ≡ {11, U1, U2, XZ}, another state- independent construction with the gate set P′ ≡ {W †11W, W †U1W, W †U2W, W †XZW} has to exist as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'} +page_content=' Choosing α = θ/2, the set P′ ≡ {W †11W, W †U1W, W †U2W, W †XZW} becomes exactly P′ ≡ {11, X, Z, XZ}, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}