topic stringlengths 3 149 | describe stringlengths 0 3.02k | questions listlengths 1 9 | date stringdate 2011-01-12 00:00:00 2022-12-20 00:00:00 | author stringclasses 5
values | license stringclasses 4
values | level stringclasses 2
values | tags listlengths 0 5 | source dict |
|---|---|---|---|---|---|---|---|---|
The Pretty-Good-Measurement is not Optimal | The pretty-good-measurement is useful when we have an ensemble that we don’t understand very well and we need to distinguish the states in the ensemble with some success probability. (Due to De Huang) | [
{
"question": "### (a) Suppose Alice sends Bob one of the three states \\( \\rho_0 = |0\\rangle \\langle 0|, \\rho_1 = \\frac{1}{2} I, \\rho_2 = |1\\rangle \\langle 1| \\) with equal probability. Bob wants to figure out which state Alice sent. Compute the success probability achieved by Bob if he uses the prett... | 18.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"pretty-good-measurement",
"state-discrimination",
"POVM",
"quantum-measurement"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Properties of the Pretty-Good-Measurement | This problem is adapted from a StackExchange answer by Norbert Schuch. [^1] That post is not an allowed resource for this problem. When we make a pretty-good measurement to distinguish the ensemble \( \rho = \sum_i p_i \rho_i \), we associate to each \( \rho_i \) a measurement operator \( M_i = \rho^{-rac{1}{2}} p_i \... | [
{
"question": "### (a) Prove inequality 2 for the case where the ensemble has only two states.",
"solution": "Suppose \\( \\rho = p_0 \\rho_0 + p_1 \\rho_1 \\), \\( p_0, p_1 \\geq 0 \\), \\( p_0 + p_1 = 1 \\), then the pretty-good-measurement is given by\n\\[ M_0 = \\rho^{-\frac{1}{2}} p_0 \\rho_0 \\rho^{-\... | 18.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"pretty-good-measurement",
"state-discrimination",
"POVM"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Deterministic Extractors on Bit-Fixing Sources | We saw in the edX lecture notes that no deterministic function can serve as an extractor for all random sources of a given length. However, this doesn’t rule out the possibility that a deterministic extractor can work for some restricted class of sources. (Due to Bolton Bailey) | [
{
"question": "### (a) Fix an even integer \\( n \\) and integer \\( t < n \\). Consider the following sources.\n- \\( X_0 \\) is all 1s on the first \\( t \\) bits and uniformly random on the last \\( n - t \\) bits.\n- \\( X_1 \\) is uniformly random over the set of strings with an even number of 0s.\n- \\( X... | 18.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"randomness-extractors",
"bit-fixing-sources",
"information-theory"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
No Chain Rule for Conditional Min-Entropy | Recall the definition of conditional Shannon entropy. \[ H(Y \mid X) = \sum_x \Pr[X = x] H(Y \mid X = x) \] (3). (Due to De Huang) | [
{
"question": "### (a) Prove that conditional Shannon entropy satisfies the chain rule: \\[ H(Y \\mid X) = H(XY) - H(X). \\] (4)",
"solution": "\\[ H(Y|X) = \\sum_x \\Pr[X = x] H(Y|X = x) \\] \\[ = \\sum_x \\Pr[X = x] \\sum_y \\Pr[Y = y|X = x] \\log \\left( \\frac{1}{\\Pr[Y = y|X = x]} \\right) \\] \\[ = \\... | 18.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"min-entropy",
"conditional-entropy",
"chain-rule",
"quantum-information"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Optimal qubit strategies in the CHSH game | Questions (a), (b) and (d) of this problem are worth one point each. The others are worth zero points and are optional. You should still read the problem to its end, as the conclusion is used in the following problem. The goal of this problem is to evaluate the maximum success probability that can be achieved in the CH... | [
{
"question": "### (a) Let \\( O \\) be a single-qubit observable such that \\( O \\) is non-degenerate (\\( O \\neq \\pm I \\)). Show that there exists real numbers \\( \\alpha, \\beta, \\gamma \\) such that \\( \\alpha^2 + \\beta^2 + \\gamma^2 = 1 \\) and \\( O = \\alpha X + \\beta Y + \\gamma Z \\), with \\(... | 18.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"CHSH-game",
"nonlocal-games",
"quantum-strategy",
"Tsirelson-bound"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Trading success probability for randomness in the CHSH game | The goal of this problem is to show that, if players succeed with higher and higher probability in the CHSH game then Alice's outputs in the game must contain more and more randomness. (Due to De Huang) | [
{
"question": "### (a) Suppose that Alice and Bob play the CHSH game using a two-qubit entangled state \\( |\\psi \rangle_{AB} \\) as in (5). Let \\( p_0(a|x) \\) be the probability that, in this strategy, Alice returns answer \\( a \\in \\{0, 1\\} \\) to question \\( x \\in \\{0, 1\\} \\). Show that \\( \\max_... | 18.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"CHSH-game",
"nonlocal-games",
"quantum-randomness"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Thinking adversarially | (Due to De Huang) | [
{
"question": "### Let’s imagine that we are playing the role of the eavesdropper Eve. We observe two parties, Alice and Bob, trying to implement certain QKD protocols. Because QKD is hard, Alice and Bob might try to cut corners in the implementation of their protocols. Here are two suggested protocols that Ali... | 2.12.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"BB84",
"QKD",
"eavesdropping",
"quantum-key-distribution"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
BB’84 fails in the device-independent setting | Consider the purified variant of the BB’84 protocol. Suppose that Eve prepares the state \(\rho_{ABE}\) in the following form:
\[
\rho_{ABE} = \frac{1}{2} \sum_{x,z=0}^{1} |xz\rangle \langle xz|_A \otimes |xz\rangle \langle xz|_B \otimes |xz\rangle \langle xz|_E,
\]
where \(|xz\rangle\) is short-hand notation for \(|... | [
{
"question": "### (a) Alice and Bob put blind faith in their hardware and attempt to implement BB’84. They want to check that their state is an EPR pair, so Alice asks her box to measure in the standard basis. The box returns a measurement outcome of 0. Determine the post-measurement state.",
"solution": "... | 2.12.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Commuting observables are compatible | Consider \( X \otimes X \) and \( Z \otimes Z \). Each of these is a \( 4 \times 4 \) Hermitian matrix which squares to identity, so it has \( \pm 1 \) eigenvalues. Moreover, since \( X \otimes X \) and \( Z \otimes Z \) mutually commute, they have a simultaneous eigenbasis. It turns out it consists of the Bell states
... | [
{
"question": "### (a) Suppose we measure an arbitrary two-qubit state \\(\\lvert \\phi \\rangle \\) using the observable \\( X \\otimes X \\) and obtain the outcome \\(-1\\). To which two-dimensional eigenspace does the post-measurement state belong? (Specify the subspace using two of the Bell states above.) N... | 2.12.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"quantum-measurement",
"observables",
"compatibility"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
A coherent attack on a nonlocal game | In video 7.5-2 on EdX, you saw a nonlocal game where a coherent attack allowed the players to do just as well when playing two parallel copies of the game as they did when playing just one copy.
Now we’ll see another example of a game with such an attack, and we’ll prove that this attack is the best strategy for the g... | [
{
"question": "### (a) We begin by describing the single-shot game. Eve starts by generating a pair \\((s, t) \\in \\{(0,0), (0,1), (1,0), (1,1)\\}\\) uniformly at random. She gives \\(s\\) to Alice and \\(t\\) to Bob. Alice and Bob generate output bits \\(a, b \\in \\{0,1\\}\\), respectively. They win if \\(a ... | 2.12.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"nonlocal-games",
"quantum-strategy",
"coherent-attacks"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
A dual formulation for the conditional min-entropy | In the notes, the conditional min-entropy of a cq state \( \rho_{XE} = \sum_{x \in \mathcal{X}} |x\rangle \langle x| \otimes \rho_x^E \) (where \( \mathcal{X} \) is any finite set of outcomes) is defined through the guessing probability, \( H_{\min}(X|E) = -\log P_{\text{guess}}(X|E) \) where
\[ P_{\text{guess}}(X|E) ... | [
{
"question": "### (a) Suppose \\( \\{N_x\\} \\) is a valid POVM. Show that the matrix \\( Z = \\sum_x |x\\rangle \\langle x| \\otimes N_x \\) satisfies \\( Z \\geq 0 \\), and compute \\( \\Phi(Z) \\) (the result should be a matrix defined on system \\( E \\) only).",
"solution": "For any \\( |\\Psi\\rangle... | 11.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"min-entropy",
"conditional-entropy",
"SDP",
"quantum-information"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Computing the min-entropy | How much can a quantum register \( E \) help us guess \( X \)? In the following, you will show that \( H_{\min}(X|E) \geq H_{\min}(X) - \log |E| \), where \( E \) denotes the dimension of the associated Hilbert space (so \( \log |E| \) is just the number of qubits of \( E \)). (Due to Mandy Huo) | [
{
"question": "### (a) Write out what we want to show in terms of the guessing probability \\( P_{\\text{guess}}(X|E) \\) using the definition of the min-entropy.",
"solution": "By definition \\( H_{\\text{min}}(X | E) = -\\log P_{\\text{guess}}(X | E) \\) and \\( H_{\\text{min}}(X) = -\\log \\max_x p_x = -... | 11.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"min-entropy",
"guessing-probability",
"quantum-information"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Bounding the winning probability in the CHSH game | The goal of this problem is to demonstrate that no quantum strategy, however large a quantum state it uses, can succeed with probability larger than \( \cos^2(\pi/8) \approx 0.85 \) in the CHSH game. The first step consists in having an accurate model for what a “quantum strategy” is. The players, Alice and Bob, should... | [
{
"question": "### (a) Show that if \\( \\{A_a^0, A_a^1\\} \\) is a valid POVM then \\( \\|A_x\\| \\leq 1 \\) (where as usual \\( \\|\\cdot\\| \\) is the operator norm, the largest singular value). Similarly for \\( B_y \\).",
"solution": "Since \\(\\{A_x^0, A_x^1\\}\\) is a valid POVM, we have\n\n\\[ 0 \\l... | 11.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"CHSH-game",
"nonlocal-games",
"Bell-inequalities",
"quantum-strategy"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
A guessing game | Imagine Alice and Eve play a guessing game where they share some state \( \rho_{AE} \) and Alice produces a random bit \( \theta \), measures her qubit in the standard basis if \( \theta = 0 \) and measures in the Hadamard basis if \( \theta = 1 \). In both cases she obtains a bit \( x \) as measurement outcome. She th... | [
{
"question": "### (a) However, Alice wants to foil Eve so, before measuring, she first applies some unitary \\( U \\) to her qubit, and then measures. Of course Eve, being really smart, gets wind of this so she will know what unitary Alice has used before measuring. So they share the state\n\\[ |\\Phi_U\\rangl... | 11.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"guessing-probability",
"min-entropy",
"information-theory"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Decoherence | A quantum state can naturally be exposed to a phenomenon called decoherence, due to its interaction with the surrounding environment. Suppose the state of a qubit and the surrounding environment is initially \( |\Psi\rangle |E\rangle \), where \( |\Psi\rangle = \alpha |0\rangle + \beta |1\rangle \), and \( |E\rangle \)... | [
{
"question": "### Suppose that this state undergoes “decoherence”, as described by the CPTP map\n\n\\[ |0\\rangle |E\\rangle \\mapsto |0\\rangle |E_0\\rangle , \\]\n\\[ |1\\rangle |E\\rangle \\mapsto |1\\rangle |E_1\\rangle , \\]\n\nwhere the states \\( |E\\rangle \\), \\( |E_0\\rangle \\) and \\( |E_1\\rangle... | 11.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"decoherence",
"quantum-noise",
"quantum-channels"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Classical one-time pad | We meet up again with our favourite protagonists, Alice and Bob. As you’ve seen in class, Alice and Bob have an adversary named Eve who is intent on listening in on all the conversations Alice and Bob have. In order to protect themselves, they exchange a classical key \( k = k_1 k_2 \ldots k_n \) which they can use to ... | [
{
"question": "### (a) How many bits of key will Alice use on average with the new protocol?",
"solution": "(Due to Daniel Gu)\nLet \\( X \\) be the random variable which is the number of bits that Alice uses in total, and \\( X_i \\) be the number of bits that Alice uses at step \\( i \\) in the protocol. ... | 25.10.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"one-time-pad",
"perfect-secrecy",
"symmetric-crypto"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Superpositions and mixtures | Alice wants to send the state \( |0\rangle \) to Bob. But 50% of the time, her (noisy) device outputs the state \( |1\rangle \) instead. | [
{
"question": "### (a) Give the density matrix \\( \\rho_0 \\) describing Bob’s state.",
"solution": "(Due to Alex Meiburg)\n\\[\n\\frac{1}{2} |0\\rangle \\langle 0| + \\frac{1}{2} |1\\rangle \\langle 1| = \\begin{bmatrix} \\frac{1}{2} & 0 \\\\ 0 & \\frac{1}{2} \\end{bmatrix}\n\\]",
"difficulty": "easy"... | 25.10.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"quantum-states",
"superposition",
"mixed-states",
"density-matrices"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Quantum one-time pad | In the lecture notes, you saw that two classical bits of key suffice to encrypt one quantum bit. On an intuitive level, our scheme needed to use both the \( X \) and \( Z \) gates because the \( X \) operation has no effect on the \( |+\rangle \) state and the \( Z \) operation has no effect on the \( |0\rangle \) stat... | [
{
"question": "### (a) Is this protocol a correct encryption scheme?",
"solution": "(Due to Anish Thilagar)\nThis protocol is correct. Bob will receive the state \\(H^k |\\psi\\rangle\\). He can then apply \\(H^k\\) again, to get the qubit \\(H^{2k} |\\psi\\rangle = |\\psi\\rangle\\) because \\(H^{2k} = (H^... | 25.10.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"quantum-one-time-pad",
"quantum-encryption",
"perfect-secrecy"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Unambiguous quantum state discrimination | (adapted from Nielsen and Chuang)
In this problem we explore an essential practical advantage that comes with general POVMs rather than strictly projective measurements. Consider the following scenario: Bob sends Alice a qubit prepared in one of the two non-orthogonal states \( |0\rangle \) and \( |+\rangle \). Alice w... | [
{
"question": "### (a) Suppose Alice measures in the basis \\( \\{|0\\rangle , |1\\rangle \\} \\). She identifies the state as \\( |0\\rangle \\) if she gets the outcome \\( |0\\rangle \\) and as \\( |+\\rangle \\) if she gets the outcome \\( |1\\rangle \\). What is her probability of misidentifying the state g... | 25.10.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"state-discrimination",
"POVM",
"quantum-measurement"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Robustness of GHZ and W States | In this problem we explore two classes of \(N\)-qubit states that are especially useful for cryptography and communication, but behave very differently under tracing out a single qubit. Let’s first define them for \(N = 3\):
\[
\text{GHZ state: } |GHZ_3\rangle = \frac{1}{\sqrt{2}} (|000\rangle + |111\rangle)
\]
\[
\t... | [
{
"question": "### (a) Calculate the overlaps\n(i) \\(\\text{Tr}(|GHZ_2\\rangle \\langle GHZ_2| \\text{Tr}_3 |GHZ_3\\rangle \\langle GHZ_3|)\\) and\n(ii) \\(\\text{Tr}(|W_2\\rangle \\langle W_2| \\text{Tr}_3 |W_3\\rangle \\langle W_3|)\\).\n\nNow we generalize to the \\(N\\)-qubit case. As you might expect, \\(... | 25.10.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"GHZ-state",
"W-state",
"entanglement",
"quantum-error"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Universal Cloning | In this problem we analyze a single-qubit universal cloner. | [
{
"question": "### (a) Consider the map which takes as input a pure single-qubit state \\(\\rho = |\\psi\\rangle \\langle \\psi|\\), and returns \\(T_1(\\rho) = \\rho \\otimes \\frac{1}{2} I\\), where \\(\\frac{1}{2} I\\) is the maximally mixed state of a single qubit.\n(i) Show that this map is a valid quantum... | 25.10.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"quantum-cloning",
"universal-cloning-machine",
"no-cloning-theorem"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Superdense Coding | In Homework 1, you were introduced to the idea of “quantum teleportation”. By sending just two bits of classical information, Alice was able to “teleport” her single-qubit quantum state to Bob, provided they shared a pair of maximally entangled qubits to begin with. In this problem, Alice instead wants to share two cla... | [
{
"question": "### (a) The first idea she has is to encode her two classical bits into her preparation of one of four states in \\(\\{ |0\\rangle, |1\\rangle, |+\\rangle, |-\\rangle \\}\\), and then send this qubit to Bob. Suppose that the a priori distribution of Alice’s two classical bits is uniform. What is ... | 3.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"superdense-coding",
"entanglement",
"quantum-communication"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Semidefinite programming | A **semidefinite program** (SDP) is a triple \((\Phi, A, B)\), where
- \(\Phi : M_d(\mathbb{C}) \rightarrow M_d(\mathbb{C})\) is a linear map of the form \(\Phi(X) = \sum_{i=1}^k K_i X K_i^\dagger\), for \(K_i\) arbitrary \(d' \times d\) matrices with complex entries, and
- \(A \in M_d(\mathbb{C}), B \in M_d(\mathbb{C... | [
{
"question": "### (a) Show that it is always the case that \\(\\alpha \\leq \\beta\\). This condition is called **weak duality**.",
"solution": "We first prove a Lemma: If \\( X, Y \\in M_d(\\mathbb{C}), X \\geq 0, Y \\geq 0 \\), then \\( \\text{tr}(XY) \\geq 0 \\).\n\n**Proof:** Consider the eigenvalue de... | 3.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"SDP",
"semidefinite-programming",
"optimization",
"duality"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
Maximally entangled properties | Let A and B be quantum systems of the same dimension d. Let |φ⁺⟩ = \(\frac{1}{\sqrt{d}}\) ∑₀≤i≤d-1 |i⟩ₐ ⊗ |i⟩B. This is referred to as a maximally entangled pair of qudits. (Due to Bolton Bailey) | [
{
"question": "### (i) What is the reduced state on subsystem A?",
"solution": "We have a maximally entangled pair of qubits\n\n\\[ |\\Phi^+\\rangle = \\sum_{0 \\leq i \\leq d-1} \\frac{1}{\\sqrt{d}} |i\\rangle_A \\otimes |i\\rangle_B \\]\n\nTo find the reduced state on \\( A \\) we trace out the \\( B \\) ... | 3.11.2016 | Thomas Vidick, Andrea Coladangelo, Jalex Stark, Charles Xu | Caltech | graduate | [
"entanglement",
"maximally-entangled",
"Bell-states"
] | {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials by Thomas Vidick et al.",
"url": "http://www.theory.caltech.edu/~vidick/teaching/120_qcrypto/"
} |
End of preview. Expand in Data Studio
Open Problem Exams: Cryptography and Security (LaTeX)
A curated dataset of open-ended exam problems (with solutions) in cryptography and computer security, formatted in LaTeX. The dataset is sourced from university courses at three institutions.
Dataset Overview
| Institution | Files | Topics | Questions |
|---|---|---|---|
| Caltech & TU Delft | 8 | 38 | 145 |
| EPFL | 6 | 19 | 86 |
| ETH Zurich | 1 | 14 | 37 |
| MIT | 3 | 33 | 79 |
| Total | 18 | 104 | 347 |
Difficulty Distribution
| Institution | Easy | Medium | Hard | Total |
|---|---|---|---|---|
| Caltech & TU Delft | 28 | 82 | 35 | 145 |
| EPFL | 16 | 45 | 25 | 86 |
| ETH Zurich | 0 | 18 | 19 | 37 |
| MIT | 11 | 65 | 3 | 79 |
| Total | 55 | 210 | 82 | 347 |
Problem Type Distribution
| Type | Count |
|---|---|
| Proof | 107 |
| Mixed | 97 |
| Explanation | 88 |
| Computation | 37 |
| Construction | 18 |
Subject Coverage
- Caltech & TU Delft — Quantum cryptography and quantum information theory: BB84, quantum money, entanglement, CHSH game, secret sharing, semidefinite programming, state discrimination, teleportation, superdense coding, decoherence.
- EPFL — Classical cryptography and cryptanalysis: DES/3DES attacks, PKCS#1 signature attacks, hash collisions, MACs, Pedersen commitments, ECDSA, DSA, hard disk encryption, EKE protocols.
- ETH Zurich — Cryptographic protocols: zero-knowledge proofs, interactive proofs, commitment schemes, oblivious transfer, secure multi-party computation, secret sharing, Byzantine agreement, player elimination.
- MIT — Broad computer security: encryption schemes, authentication, isolation, symbolic execution, browser security, timing attacks, TLS, collision resistance, public-key encryption, differential privacy, software security, privilege separation.
Data Format
Each JSON file contains an array of problem sets. Every problem set follows this schema:
{
"topic": "Problem title",
"level": "graduate",
"describe": "Problem description and context (LaTeX)",
"tags": ["POVM", "state-discrimination", "quantum-measurement"],
"source": {
"institution": "Caltech & TU Delft",
"course": "CS/Ph 120 — Quantum Cryptography",
"license": "Academic use — original course materials",
"url": "https://..."
},
"questions": [
{
"question": "Question text (LaTeX)",
"solution": "Solution/answer (LaTeX)",
"difficulty": "hard",
"problem_type": "proof",
"prerequisite_knowledge": ["linear algebra", "density matrices", "POVM"],
"tags": ["pretty-good-measurement", "POVM"]
}
],
"date": "DD.MM.YYYY",
"author": "Author name(s)",
"license": "Institution name"
}
All mathematical notation is written in LaTeX.
Field Reference
| Field | Level | Values | Description |
|---|---|---|---|
level |
topic | undergraduate · graduate |
Academic level of the course |
tags |
topic | list of strings | Subtopic keywords for filtering |
source |
topic | object | Institution, course, license, URL |
difficulty |
question | easy · medium · hard |
Estimated question difficulty |
problem_type |
question | proof · computation · true_false · explanation · construction · mixed |
Nature of the question |
prerequisite_knowledge |
question | list of strings | Concepts needed to solve the question |
tags |
question | list of strings | Specific subtopic tags |
Labels were assigned automatically via rule-based heuristics and may need manual review for edge cases.
Directory Structure
data/
├── caltech/ # Quantum cryptography (Caltech CS/Ph 120 & TU Delft)
├── epfl/ # Cryptography and security (COM-401)
├── eth_zurich/ # Cryptographic protocols (SS 2021)
└── mit/
└── basic/ # Computer security (6.858 / 6.1600)
License
BSD 3-Clause — see individual file license fields for per-source attribution.
Citation
@misc{natnitaract2026openexamscrypto,
author = {natnitaract},
title = {Open Problem Exams: Cryptography and Security (LaTeX)},
year = {2026},
publisher = {Hugging Face},
url = {https://huggingface.co/datasets/natnitaract/open-problem-exams-cryptography-and-security-latex}
}
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