{"task_id": "OQP-1", "split": "test", "title": "All the Bell Inequalities", "statement": "Find all those linear inequalities characterizing the existence of joint probability distributions for all variables in a correlation experiment. The title was taken from a recent exposition by A. Peres [1]. More specifically, suppose that measurements are made on systems, which are decomposed into N subsystems. On each of these subsystems one out of M observables is measured, producing K outcomes each. Thus we consider different experimental setups, each of which may lead to different outcomes, so all in all probabilities are measured. Classically (in a \"realistic local theory\") these numbers would be generated by specifying probabilities for each \"classical configuration\", i. e. every assi", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum nonlocality / Bell inequalities", "source_year": 1999, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-2; FOP-4", "split": "test", "title": "NPT bound entanglement / undistillability implies PPT", "statement": "A state on a bipartite quantum system is called distillable, if from sufficiently many pairs prepared in that state one can obtain a close approximation of a maximally entangled singlet state, using only local quantum operations and classical communication (LOCC). It is well-known that states with positive partial transpose (PPT) are not distillable. The problem is to decide the converse.", "aliases": "NPT bound entanglement; undistillability implies PPT; Undistillability implies PPT; Undistillability implies ppt?; NPT bound entanglement", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2000, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": true, "verifiability_class": ""}}
{"task_id": "Simon-2", "split": "test", "title": "Anderson localization in two dimensions", "statement": "Localization in two dimensions. Prove that for , the spectrum of the Anderson model is dense pure point for all values of .", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "spectral theory / Schrodinger operators", "source_year": 2000, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Simon-1", "split": "test", "title": "Anderson-model extended states", "statement": "Extended states. Prove for and suitable values of that the Anderson model has purely absolutely continuous spectrum in some energy range.", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "spectral theory / Schrodinger operators", "source_year": 2000, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Simon-12", "split": "test", "title": "First-principles justification of molecular-configuration methods", "statement": "Is there a mathematical sense in which one can justify from first principles current techniques for determining molecular configurations?", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "molecular / atomic mathematical physics", "source_year": 2000, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Simon-8", "split": "test", "title": "Infinite-multiplicity absolutely continuous spectrum for Schrodinger operators", "statement": "Let be a function on which obeys Prove that has a.c. spectrum of infinite multiplicity on if .", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "spectral theory / Schrodinger operators", "source_year": 2000, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Simon-15", "split": "test", "title": "Lieb-Thirring constants", "statement": "Prove the Lieb-Thirring conjecture on their constants for and .", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "spectral theory / Schrodinger operators", "source_year": 2000, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Simon-11", "split": "test", "title": "Mathematical shell model of the atom", "statement": "Make mathematical sense of the shell model of an atom.", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "molecular / atomic mathematical physics", "source_year": 2000, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Simon-13", "split": "test", "title": "Periodic limit for neutral molecular/electron systems", "statement": "Prove that the ground state of some neutral system of molecules and electrons approaches a periodic limit as the number of nuclei goes to infinity.", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "molecular / atomic mathematical physics", "source_year": 2000, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Simon-3", "split": "test", "title": "Quantum diffusion", "statement": "Quantum diffusion. Prove that for and values of where there is a.c. spectrum that grows as as .", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "mathematical physics", "source_year": 2000, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-12", "split": "test", "title": "Bell Inequalities for long range vacuum correlations", "statement": "It is well known [1] that vacuum fluctuations maximally violate the CHSH-Bell inequalities for suitable spacelike separated observables, and that this violation goes to zero as the two localization regions are moved apart. Decide whether some (necessarily small) violation of the inequalities is possible for regions arbitrarily far apart. For definiteness, consider a massive scalar free relativistic Bose field.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum nonlocality / Bell inequalities", "source_year": 2002, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-16", "split": "test", "title": "Complexity of product preparations", "statement": "What can be said about the algorithmic complexity of preparing , asymptotically, as a function of and the algorithmic complexity of preparing ? Take to be a state of qubits. By algorithmic complexity I mean the number of gates required to prepare the state from . This depends on the gate set used so the question concerns asymptotics. For the present purposes, one can take as a gate set all roations where is a product of Pauli matrices. The complexity of this gate is . It might be useful to consider a version of this question involving an approximation parameter also.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum complexity / computation", "source_year": 2003, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-13; FOP-2; Bandeira-6.2.A; RS-22", "split": "dev", "title": "Mutually unbiased bases in dimension 6", "statement": "Determine the maximal number of orthonormal bases in a -dimensional Hilbert space, which are mutually unbiased in the following sense: If denotes the th vector of the th basis, all scalar products with have the same absolute value (namely ). It is known that if d is the power of a prime, can be achieved, but this is not known for any other composite number. So the problem is already to decide whether there exist mutually unbiased bases in dimensions (up to now only a set of three MUBs is known).", "aliases": "Mutually unbiased bases; MUBs; MUB(6)<7; MUBs; Mutually unbiased bases; Mutually Unbiased Bases; MUBs in dimension 6", "category": "Quantum information", "subcategory": "quantum measurements and designs", "source_year": 2003, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": true, "verifiability_class": ""}}
{"task_id": "OQP-8", "split": "test", "title": "Qubit formula for Relative Entropy of Entanglement", "statement": "The relative entropy of entanglement is an entanglement monotone that quantifies to what extent a given state can be operationally distinguished (in the sense of Stein's Lemma) from the closest state which is either separable or has a positive partial transpose (PPT). For a state it is defined as [1] where stands for the convex sets of separable or PPT states, and is the quantum relative entropy. The problem is to find a closed formula for this quantity for systems consisting of two qubits.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2003, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-15", "split": "test", "title": "Separability from spectrum", "statement": "For a mixed state on a bipartite -dimensional Hilbert space, are there any factorizations of the space into an -dimensional tensor an -dimensional space with respect to which the state is entangled? The answer to this question depends only on the spectrum of , and the problem is to characterize the spectra for which the answer is \"no\". Another reformulation of the problem is: for which spectra does it happen that is separable whenever it has spectrum ?", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2003, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-14", "split": "test", "title": "Tough error models", "statement": "An error model is an -dimensional vector space of operators acting on an -dimensional Hilbert space . A quantum code is a subspace , and is said to correct , if the projector onto satisfies for all , and suitable scalars . *Given and , find the largest such that we can assert the existence of a code of dimension without further information about . *Find \"tough error models\" for which this bound is (nearly) tight.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2003, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-29", "split": "test", "title": "Entanglement of formation for Gaussian states", "statement": "Entanglement of formation is defined as a minimum over all convex decompositions of a bipartite state into pure states (see Problem 7). It has been shown that for certain two-mode Gaussian states this minimum can be taken over decompositions of the given state into pure states, all of which are translates of the same squeezed Gaussian state with Gaussian weights. Show (or disprove) that this is true for all Gaussian states.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2005, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-31", "split": "test", "title": "Individual measurement strategies on geometrically uniform states", "statement": "We call a set of quantum states geometrically uniform if there is a unitary operator that transforms into for all , with indices read mod . Suppose now that copies of those geometrically uniform states are given, i.e., . If we have a quantum memory, and can do collective measurements on the systems, the square-root collective measurement will be the optimal strategy and provide the minimum error [1]. Otherwise, we must rely on measurements performed on individual copies. Does there always exist a suitably designed individual measurement strategy that asymptotically reaches the minimum error of the collective measurement?", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum measurements and designs", "source_year": 2005, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-25", "split": "test", "title": "Lockable entanglement measures", "statement": "Are two-way distillible entanglement and secret key rate lockable?", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2005, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-20", "split": "test", "title": "Reversible entanglement manipulation", "statement": "Are PPT operations sufficient to ensure asymptotically reversibly interconversion of all, i.e. pure and mixed, bipartite entangled states? What is the smallest non-trivial class of operations that permits asymptotically reversible interconversion of all, i.e. pure and mixed, bipartite entangled states?", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2005, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-24", "split": "test", "title": "Secret key from all entangled states", "statement": "Can all bipartite entangled states be used to generate secret keys?", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2005, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-23; FOP-1; Bandeira-6.3.A; RS-24", "split": "test", "title": "SIC-POVM / Zauner's conjecture", "statement": "We will give three variants of the problem, each being stronger than its predecessor. The terminology of problems 1 and 2 is taken mainly from [1]. For problem 3 see [2] and [3]. Problem 1: SIC-POVMs A set of normed vectors in a Hilbert space of dimension constitutes a set of <strong>equiangular lines</strong> if their mutual inner products are independent of the choice of . It can be shown [1] that *the associated projection operators sum to a multiple of unity and thus induce a POVM (up to normalization) and that *these operators are linearly independent and hence any quantum state can be reconstructed from the measurement statistics of the POVM. A POVM that arises in this way is called sy", "aliases": "Zauner's conjecture; SIC POVMs; complex equiangular lines; SIC POVMs; Zauner's conjecture; SIC POVMs and Zauner's Conjecture; SIC-POVMs in infinitely many dimensions; Zauner's Conjecture (SIC-POVM); Zauner's Conjecture", "category": "Quantum information", "subcategory": "quantum measurements and designs", "source_year": 2005, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": true, "verifiability_class": ""}}
{"task_id": "OQP-27", "split": "test", "title": "The power of CGLMP inequalities", "statement": "In the setting of [[Bell inequalities holding for all quantum states|Problem 26]] , consider especially the case . Problem 27. A Show that every face of the local polytope , which is not already contained in a face of the no-signalling polytope is of CGLMP type, i.e., an inequality of the form first written out in [1], but possibly lifted from lower dimensions by fusing together some outcomes. Problem 27. B Numerically, the observables maximally violating the CGLMP inequality on a maximally entangled state are of a very specific form [2], involving measurements in computational basis, transformed by only discrete Fourier transformation and diagonal unitaries [1]. Show that this is necessaril", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2006, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-26", "split": "test", "title": "Bell inequalities holding for all quantum states", "statement": "Below are two connected problems, both asking for a better description of the possible correlation polytopes.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum nonlocality / Bell inequalities", "source_year": 2010, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Bandeira-7.2.A", "split": "test", "title": "Boolean Classification and Annulus Conjecture", "statement": "Source excerpt: Open Problem 7.2 It is conjectured in [AABS15] (Conjecture 3 in [AABS15]) that the optimal rate in this case is given by R A(n,n,n ) = + (1 -) R A(1,n, (1 -)) + o(1), where o(1) goes to zero as n goes to infinity. This is established in [AABS15] for 2 but open in general. 7.2.2 The proof of Theorem 7.3 Reed-Solomon codes[RS60] are [ n,m,n -m+ 1]q codes, for m n q. They meet the Singleton bound, the drawback is that they have very large q (q > n). We'll use their existence to prove Theorem 7.3 Proof. [of Theorem 7.3] We will construct a family A of sets achieving the upper bound in Theorem 7.3.", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "general", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium-high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-3.2.A", "split": "test", "title": "Certifying positive-semidefiniteness", "statement": "Is there a quasi-linear-time randomized procedure that, given a symmetric matrix M with a condition-number promise, outputs a certificate of M >= 0 (a la Cholesky) with zero false-positive probability?", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "SDP / SoS / optimization", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "verified", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-6.5.A", "split": "test", "title": "Constructive Kadison-Singer", "statement": "Source excerpt: Open Problem 6.5 Give a (polynomial time) construction of the tight frame partition satisfying the properties required in the Kadison-Singer problem (or the related Weaver's conjecture). 30We note that the quadratic residues are known to have pseudorandom properties, and indeed have been leveraged to reduce the randomness needed in certain RIP constructions [BFMM14] 98 7 Group Testing and Error-Correcting Codes 7.1 Group Testing During the Second World War the United States was interested in weeding out all syphilitic soldiers called up for the army. However, syphilis testing back then was exp", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "compressed sensing / frames", "source_year": 2016, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-5.1.A", "split": "test", "title": "Deterministic Restricted Isometry Property matrices", "statement": "Construct deterministic matrices A in C^(M x N) satisfying (s, 1/3)-RIP for s >= M^0.6 / polylog(N).", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "general", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium-high", "resolution_year": null, "statement_quality": "verified", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-2.2.A", "split": "test", "title": "Erdos-Hajnal Conjecture", "statement": "Prove or disprove the induced Erdos-Hajnal conjecture: for every finite graph H, every n-vertex graph with no induced copy of H has a clique or independent set of size n^c(H).", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "random graphs / combinatorics", "source_year": 2016, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "verified", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-7.1.A", "split": "test", "title": "Gilbert-Varshamov bound", "statement": "Source excerpt: Open Problem 7.1 1. Construct an explicit (deterministic) binary code ( q = 2 ) satisfying the GV bound (67). 2. Is the GV bound tight for binary codes ( q= 2)? 7.2.1 Boolean Classification A related problem is that of Boolean Classification [AABS15]. Let us restrict our attention to In error-correcting codes one wants to build a linear codebook that does not contain a codeword with weight d-1. In other words, one wants a linear codebook C that does intersect B(d-1) = {x {0,1}n : 0 <(x) d-1}the pinched Hamming ball of radius d (recall that (d) is the Hamming weight of x, meaning the number of ", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "general", "source_year": 2016, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-0.1.A; RS-10", "split": "test", "title": "Komlos conjecture", "statement": "Prove or refute a dimension-free discrepancy bound for signed sums of bounded Euclidean vectors.", "aliases": "Komlos Conjecture; Komlos discrepancy; Komlos Conjecture", "category": "Mathematical statistics / data science", "subcategory": "discrepancy theory", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": true, "verifiability_class": ""}}
{"task_id": "Bandeira-4.3.A; RS-11", "split": "test", "title": "Matrix six deviations suffice", "statement": "Source excerpt: Open Problem 4.3 Prove or disprove Conjecture 4.23. Note that, when the matrices Hk are diagonal, this problem corresponds to Spencer's Six Standard Deviations Suffice Theorem [Spe85]. Remark 4.24 Also, using Theorem 4.22, it is easy to show that if one picks i as i.i.d. Rademacher random variables, then with positive probability (via the probabilistic method) the inequality will be satisfied with an extra log n term. In fact one has E n k=1 kHk log n n k=1 H2 k log n n k=1 Hk2 log nn. Remark 4.25 Remark 4.24 motivates asking whether Conjecture 4.23 can be strengthened to ask for 1,..., n such", "aliases": "Matrix Six Deviations Suffice; How many deviations are needed after all?; Matrix Six deviations Suffice; How many deviations are needed afterall?", "category": "Mathematical statistics / data science", "subcategory": "discrepancy theory", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": true, "verifiability_class": ""}}
{"task_id": "Bandeira-1.2.A", "split": "test", "title": "Monotonicity of eigenvalues", "statement": "Source excerpt: Open Problem 1.2 (Monotonicity of singular values [BKS13a]) Consider the setting above but with p= n, then X Rnn is a matrix with iid N(0,1) entries. Let i ( 1nX ) , denote the i-th singular value 4 of 1nX, and define R(n) := E [ 1 n n i=1 i ( 1nX )] , as the expected value of the average singular value of 1nX. The conjecture is that, for every n1, R(n+ 1) R(n). Moreover, for the analogous quantity C(n) defined over the complex numbers, meaning simply that each entry of X is an iid complex valued standard gaussian CN(0,1) the reverse inequality is conjectured for all n1: C(n+ 1) C(n). 4The i-t", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "general", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-3.3.A", "split": "test", "title": "Multy-way Cheeger's inequality", "statement": "Source excerpt: Open Problem 3.3 Let G= (V,E,W ) be a graph and k a positive integer, is the following true? G(k) polylog(k) k. (32) We note that (32) is known not to hold if we ask that the subsets form a partition (meaning that every vertex belongs to at least one of the sets) [LRTV12]. Note also that no dependency on k would contradict the Small-Set Expansion Hypothesis above. 55 4 Concentration Inequalities, Scalar and Matrix Versions 4.1 Large Deviation Inequalities Concentration and large deviations inequalities are among the most useful tools when understanding the performance of some algorithms. In a ", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "general", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium-high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-3.1.A", "split": "test", "title": "Optimality of Cheeger's inequality", "statement": "Source excerpt: Open Problem 3.1 Does there exists a constant c >0 such that it is NP-hard to, given , and G distinguis between the cases 1. hG , and 2. hG c? It turns out that this is a consequence [RST12] of an important conjecture in Theoretical Computer Science (see [BS14] for a nice description of it). This conjecture is known [RS10] to imply the Unique- Games Conjecture [Kho10], that we will discuss in future lectures. Conjecture 3.10 (Small-Set Expansion Hypothesis [RS10]) For every > 0 there exists >0 such that it is NP-hard to distinguish between the cases 1. There exists a subset S V with vol(S) = v", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "general", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium-high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-6.4.A; RS-29", "split": "test", "title": "Paley ETF RIP beyond the square-root bottleneck", "statement": "Show whether the Paley equiangular tight frame gives RIP beyond the square-root bottleneck.", "aliases": "Paley ETF Conjecture; RIP beyond square-root bottleneck; The Paley ETF Conjecture", "category": "Mathematical statistics / data science", "subcategory": "compressed sensing / frames", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": true, "verifiability_class": ""}}
{"task_id": "Bandeira-8.4.A; RS-25", "split": "test", "title": "Paley graph clique number", "statement": "Determine the asymptotic clique number of Paley graphs and whether stronger SoS/theta-style certificates can prove it.", "aliases": "Paley Clique Problem; Paley graph clique number; The Paley Clique Problem", "category": "Mathematical statistics / data science", "subcategory": "random graphs / combinatorics", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": true, "verifiability_class": ""}}
{"task_id": "Bandeira-2.3.A", "split": "dev", "title": "Planted Clique Problems", "statement": "Source excerpt: Open Problem 2.3 (The planted clique problem) Let Gbe a random graph constructed by tak- ing a G ( n,1 2 ) and planting a clique of size . 1. Is there a polynomial time algorithm that is able to find the largest clique of G (with high prob- ability) for n. For example, for n log n. 11There is an amplification technique that allows one to find the largest clique for cn for arbitrarily small c in polynomial time, where the exponent in the runtime depends on c. The rough idea is to consider all subsets of a certain finite size and checking whether the planted clique contains them. 29 2. Is there ", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "random graphs / combinatorics", "source_year": 2016, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-7.3.A", "split": "test", "title": "Shannon Capacity of 7 cycle", "statement": "Source excerpt: Open Problem 7.3 What is the Shannon Capacity of the 7 cycle? 7.3.2 The deletion channel In many applications the erasures or errors suffered by the messages when sent through a channel are random, and not adversarial. There is a beautiful theory understanding the amount of information that can be sent by different types of noisy channels, we refer the reader to [CT] and references therein for more information. A particularly challenging channel to understand is the deletion channel. The following open problem will envolve a particular version of it. Say we have to send a binary string \"10010\"", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "random graphs / combinatorics", "source_year": 2016, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-7.4.A", "split": "test", "title": "The Deletion Channel", "statement": "Determine asymptotics and hardest pairs for the binary deletion channel problem.", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "general", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium-high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-8.3.A", "split": "dev", "title": "The Grothendieck Constant", "statement": "Source excerpt: Open Problem 8.3 What is the real Grothendieck constant KG? 8.5 The Paley Graph Let p be a prime such that p = 1 mod 4. The Paley graph of order p is a graph on p nodes (each node associated with an element of Zp) where ( i,j) is an edge if i-j is a quadratic residue modulo p. In other words, ( i,j) is an edge is there exists a such that a2 = i-j mod p. Let (p) denote the clique number of the Paley graph of order p, meaning the size of its largest clique. It is conjectured that (p) pollywog(n) but the best known bound is (p) p(which can be easily obtained). The only improvement to date is that", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "SDP / SoS / optimization", "source_year": 2016, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-8.1.A", "split": "test", "title": "The Unique Games Conjecture", "statement": "Resolve the Unique Games Conjecture, including the associated hardness threshold for the Unique Games problem and the extent to which constant-degree Sum-of-Squares refutes it.", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "SDP / SoS / optimization", "source_year": 2016, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "verified", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-9.3.A", "split": "test", "title": "Tightness of k-median LP", "statement": "Source excerpt: Open Problem 9.3 Is the LP relaxation for k-medians tight for a natural (random) generative model of points even without a clustering planted structure (such as, say, gaussian independent points)? Ideally, one would like to show that these relaxations (both the k-means SDP and the k-medians LP) are integral in instances that have clustering structure and not necessarily arising from generative random models. It is unclear however how to define what is meant by \"clustering structure\". A particularly interesting approach is through stability conditions (see, for example [AJP13]), the idea is tha", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "high-dimensional statistics", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium-low", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-10.3.A", "split": "test", "title": "Tightness of the Multireference Alignment SDP", "statement": "Determine SDP tightness, consistency, or sample complexity thresholds for multireference alignment.", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "SDP / SoS / optimization", "source_year": 2016, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "OQP-41", "split": "test", "title": "All rank inequalities for reduced states of quadripartite quantum states", "statement": "Given any tripartite density matrix and let denote the rank of the respective marginals, prove or find a counterexample to the following hypothesis [1]: (1)", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "compressed sensing / frames", "source_year": 2017, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-43", "split": "test", "title": "Are all extensibly causal processes purifiable?", "statement": "The most general situation compatible with the assumption that the operations performed in each local laboratory are described by the quantum formalism without assuming a global causal order between the operations can be represented in the \"process matrix\" formalism [1]. The probability that parties observe the outcomes , for a choice of operations is a multilinear function of the corresponding CP maps as given by where are the CJ representations of the CP maps and is a process matrix with and being the input and the output Hilbert spaces of party . The set of process matrices is defined by requiring that probabilities are well-defined [1,2]. The condition for a bipartite probability distrib", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum thermodynamics / causality", "source_year": 2017, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-36", "split": "test", "title": "Composition of decoherence functionals", "statement": "Consider a physical system where a number of (possibly incompatible) experiments can be in principle conducted. Each experiment allows estimating a property with value , and any vector of possible values , with is called a history . The above is called a joint measurement framework [1]. Given the set of all histories , define a Hilbert space by assigning to each an element of an orthonormal basis . For any , define the vector . To model experiments on this system within the framework of Sorkin's quantum measure theory [2], we assume that there exists a complex-valued operator , dubbed decoherence functional , with the following properties: Hermiticity: . Weak positivity : , for all . Normali", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum thermodynamics / causality", "source_year": 2017, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-42", "split": "test", "title": "Reversible dynamics on composite systems", "statement": "Quantum theory allows to create entanglement from product states in a purely reversible way: for example, there is a unitary (in fact, many) that maps the pure product state to Surprisingly, this possibility might be a characteristic feature of quantum theory - which is what we conjecture here. To state the conjecture, we need the basic machinery of generalized probabilistic theories [1]. Consider a -dimensional compact convex set of (normalized) states , affinely embedded into a real vector space of dimension (in standard complex quantum theory, would be the density matrices over some , where , and would be the space of Hermitian matrices). Similarly, consider another state space embedded a", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2017, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-34", "split": "test", "title": "The geometry of quantum nonlocality", "statement": "Given a nonlocality scenario, with parties, settings and outcomes, consider the set of probability distributions of the form , s.t. This is the multipartite analog of the set of quantum correlations defined in Problem 33. The question is whether the closure of in each nonlocality scenario is a semi-algebraic set, or, in other words, whether, for any , there is a finite set of polynomials such that iff for .", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum nonlocality / Bell inequalities", "source_year": 2017, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-38", "split": "test", "title": "The PPT-squared conjecture", "statement": "Let and be quantum states with positive partial transpose (i.e., they are \"PPT-states\"), and let be a positive operator decribing a yes/no measurement on the BC-system. Then consider the state on AD, conditional on the result of being `yes', i.e. , where is a normalization factor. The conjecture [1] by Matthias Christandl states that all such are separable. Of course, the problem is to prove or disprove this.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2017, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "FOP-5", "split": "dev", "title": "2-copy distillability of a two-ququart Werner state", "statement": "Determine whether the highlighted two-ququart Werner state is distillable using two copies.", "aliases": "two-ququart state; Werner state rho(4,-1/2)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2020, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "OQP-44", "split": "test", "title": "Complexity of the separability problem", "statement": "At a high level, the separability problem is to decide whether a bipartite mixed state is separable or far from separable. More precisely, let denote the convex set of separable density matrices acting on . The -weak membership problem for is to decide whether a given bipartite density matrix acting on (when given as matrix entries) is in or -far in trace distance from any state in , promised that one is the case. Question : what is the computational complexity of solving the -weak membership problem for as a function of and ?", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2023, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-47", "split": "test", "title": "Is there bound information?", "statement": "Consider the following cryptographic scenario, known as secret key agreement [1]. A source distributes an i.i.d. stream of correlated triples of random variables with distribution , which are accessible to Alice, Bob, and Eve, respectively. It is the task for Alice and Bob by public discussion to convert their symbols with high probability into a rate of perfectly random bits that are unknown to Eve. The rate at which this is possible is called the secret key rate of . Conversely, one may also define the rate of secret key cost (a.k.a. information of formation), by which we mean the rate of secret bits required for Alice and Bob to create, with high probability, a stream of symbols and , res", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2023, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-46", "split": "test", "title": "Thermodynamic implementation of Gibbs-Preserving Maps", "statement": "Determine the minimal resources required to implement a general Gibbs-preserving map in a one-shot thermodynamic framework whose free operations can reasonably be implemented physically (e.g., thermal operations).", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum thermodynamics / causality", "source_year": 2023, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-5", "split": "test", "title": "Density threshold for Global Synchrony", "statement": "Source excerpt: Conjecture 5 (Density threshold for Global Synchrony) . For any > 0 \\varepsilon>0 , there exists n > 0 n>0 and a graph G G on n n nodes such that the minimum degree of G G is at least ( 3 4 - ) n \\left(\\frac{3}{4}-\\varepsilon\\right)n and G G is not globally synchronizing.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "general", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-12", "split": "test", "title": "Discrepancy lower-bound conjecture", "statement": "Source excerpt: Conjecture 12 . Given the definitions above, we have lim sup n sup A { 1 } n n 1 n disc ( A ) > 1 . \\limsup_{n\\to\\infty}\\sup_{A\\in\\{\\pm 1\\}^{n\\times n}}\\frac{1}{\\sqrt{n}}\\mathrm{disc}(A)>1.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "discrepancy theory", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-6", "split": "test", "title": "Ellipsoid fitting", "statement": "Analyze the SDP/random-geometry threshold for fitting ellipsoids to the specified random point model.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "convex geometry", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-4", "split": "test", "title": "Global Synchrony with negative edges", "statement": "Source excerpt: Conjecture 4 (Global Synchrony with negative edges) . Given any > 0 \\varepsilon>0 , the n n n\\times n random matrix A A with zero diagonal and off-diagonal entries given by A i j = { 1 with probability 1 2 + - 1 with probability 1 2 - , A_{ij}=\\left\\{\\begin{array}[]{ccl}1&\\text{with probability}&\\frac{1}{2}+\\delta\\\\ -1&\\text{with probability}&\\frac{1}{2}-\\delta,\\end{array}\\right. with ( 1 + ) log n 2 n \\delta\\geq(1+\\varepsilon)\\sqrt{\\frac{\\log n}{2n}} , is globally synchronizing with high probability.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "general", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-3", "split": "test", "title": "Globally Synchronizing Regular Graphs", "statement": "Source excerpt: Conjecture 3 (Globally Synchronizing Regular Graphs) . A uniform random 3 3 -regular graph is globally synchronizing with high probability (probability going to 1 1 as n n\\to\\infty ).", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "random graphs / combinatorics", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-2", "split": "test", "title": "Group Spencer", "statement": "Find Spencer-type discrepancy guarantees when signs are replaced by group-valued choices.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "discrepancy theory", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-14", "split": "test", "title": "Hadamard Conjecture", "statement": "Source excerpt: Conjecture 14 (Hadamard Conjecture) . For any positive n n a multiple of 4 4 , there exists an n n n\\times n Hadamard matrix.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "random graphs / combinatorics", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-9", "split": "test", "title": "Kikuchi Spectral Threshold", "statement": "Identify the spectral threshold at which Kikuchi-type algorithms detect the planted tensor/PCA structure.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "statistical mechanics", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-1", "split": "test", "title": "Matrix Spencer", "statement": "Extend Spencer-type discrepancy bounds to matrix-valued inputs with spectral-norm control.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "discrepancy theory", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-7", "split": "test", "title": "Multi-frequency synchronization polynomial-time detection", "statement": "Source excerpt: Open Problem 7 . Consider synchronization model over S O ( 2 ) SO(2) or synchronization model over a finite group, where the signal x x is sampled uniformly over group elements. Find a scaling L = L n L=L_{n} of number of frequencies such that there exists a polynomial-time algorithm that can detect the signal reliably for all > comp , L \\lambda>\\lambda_{\\text{comp},L} for some comp , L < 1 \\lambda_{\\text{comp},L}<1 as n n\\to\\infty .", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "high-dimensional statistics", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-8", "split": "test", "title": "Multi-frequency synchronization sub-exponential detection", "statement": "Source excerpt: Open Problem 8 . Consider synchronization model over S O ( 2 ) SO(2) or synchronization model over a finite group, where the signal x x is sampled uniformly over group elements. Fix number of frequencies L L that does not depend on n n . In the computationally hard regime, i.e., when ( stat , 1 ) \\lambda\\in(\\lambda_{\\text{stat}},1) , does there exist a sub-exponential algorithm, i.e., an algorithm running in time exp n \\exp{n^{\\delta}} for some < 1 \\delta<1 , that can detect the signal reliably as n n\\to\\infty ?", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "high-dimensional statistics", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-15", "split": "test", "title": "SK-model Glauber dynamics fast mixing up to the replica-symmetry-breaking threshold", "statement": "Source excerpt: Open Problem 15 . Consider the SK-model with an arbitrary external field. Does Glauber Dynamics fast mix (meaning in polynomial time) up to the replica symmetry breaking threshold = 1 \\beta_{\\ast}=1 ?", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "statistical mechanics", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-13", "split": "test", "title": "Sylvester Hadamard discrepancy for odd cases", "statement": "Source excerpt: Conjecture 13 . Let H k H_{k} be the 2 k 2 k 2^{k}\\times 2^{k} Hadamard matrix obtained by the Sylvester construction (see above). For odd k k , we have disc ( H k ) = 2 2 k . \\mathrm{disc}(H_{k})=\\sqrt{2}\\sqrt{2^{k}}.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "discrepancy theory", "source_year": 2025, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-48", "split": "test", "title": "The entanglement cost of f-routing", "statement": "-routing is a distributed task involving two players, Alice and Bob, who co-operate to redirect a quantum system based on the value of classical inputs . Given a Boolean function , an -routing protocol has the following communication pattern: The diagram should be read from bottom to top. The lower curved wire represents an initial shared entangled state held between Alice and Bob: the logarithm of its local dimension is the entanglement cost of the protocol. After receiving input , Alice performs the quantum operation on both system and her part of the shared entangled state. This operation outputs two quantum systems, one of which she sends to Bob. Similarly, upon receiving input , Bob imp", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2026, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-20", "split": "test", "title": "Balan-Wang stability conjecture for phase retrieval", "statement": "Pin down injectivity or stability thresholds for phase retrieval in the stated finite-dimensional random model.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "high-dimensional statistics", "source_year": 2026, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-28", "split": "test", "title": "Degree-4 SoS bound for Paley clique number", "statement": "Determine whether a degree-4 Sum-of-Squares certificate can prove the claimed finite-dimensional nonexistence or upper-bound statement.", "aliases": "degree-4 SoS Paley clique bound", "category": "Probability / statistical mechanics / random structures", "subcategory": "random graphs / combinatorics", "source_year": 2026, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-23", "split": "test", "title": "Degree-4 SoS proof for no MUBs in dimension 6", "statement": "Decide whether a complete set of seven mutually unbiased bases exists in complex dimension 6, and clarify maximal MUB families in non-prime-power dimensions.", "aliases": "degree-4 SoS no-MUB certificate; no seven MUBs in C^6", "category": "Probability / statistical mechanics / random structures", "subcategory": "quantum measurements and designs", "source_year": 2026, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-21", "split": "test", "title": "Gaussian-row stability quantity in phase retrieval", "statement": "Pin down injectivity or stability thresholds for phase retrieval in the stated finite-dimensional random model.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "high-dimensional statistics", "source_year": 2026, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-30", "split": "test", "title": "KLS Conjecture", "statement": "Resolve the Kannan-Lovasz-Simonovits isoperimetric conjecture or tractable special cases of it.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "convex geometry", "source_year": 2026, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-17", "split": "test", "title": "Lovasz number asymptotics for Erdos-Renyi random graphs", "statement": "Prove that for G distributed as G(n, 1/2), E[theta(G)] = (1 + o(1)) sqrt(n), where theta is the Lovasz theta number (Randomstrasse101 Conjecture 17).", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "random graphs / combinatorics", "source_year": 2026, "answer": {"status": "open", "status_confidence": "high", "resolution_year": null, "statement_quality": "verified", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "RS-18", "split": "test", "title": "Lovasz number asymptotics for random dense circulant graphs", "statement": "Determine asymptotics of the Lovasz theta number for the specified random graph model.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "random graphs / combinatorics", "source_year": 2026, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-26", "split": "test", "title": "Paley 1-localization Lovasz-number bound", "statement": "Source excerpt: Conjecture 26 . ( G p , 1 ) p / 2 \\vartheta(\\overline{G_{p,1}})\\sim\\sqrt{p/2} (for p 1 ( mod 4 ) p\\equiv 1\\pmod{4} prime) where G p , 1 G_{p,1} is the 1-localization of the Paley Graph described above.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "random graphs / combinatorics", "source_year": 2026, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-27", "split": "test", "title": "Paley 2-localization Lovasz-number bound", "statement": "Source excerpt: Conjecture 27 . For p 1 ( mod 4 ) p\\equiv 1\\pmod{4} prime and large enough we have ( G p , 2 ) 2 3 p \\vartheta(\\overline{G_{p,2}})\\leq\\frac{2}{3}\\sqrt{p} where G p , 2 G_{p,2} is the 2-localization of the Paley Graph described above.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "random graphs / combinatorics", "source_year": 2026, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-19", "split": "test", "title": "Phase retrieval injectivity threshold", "statement": "Pin down injectivity or stability thresholds for phase retrieval in the stated finite-dimensional random model.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "high-dimensional statistics", "source_year": 2026, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-31", "split": "test", "title": "Sharp graph matrix bounds", "statement": "Prove the conjectured sharp graph-matrix spectral bounds, with finite matrix tests and SoS/certificate-relevant formulations.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "matrix/tensor inequalities", "source_year": 2026, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "RS-16", "split": "test", "title": "Type-2 constant of tensors", "statement": "Prove the conjectured finite-dimensional tensor concentration/type-constant inequality or identify extremal tensor examples.", "aliases": "(none listed)", "category": "Probability / statistical mechanics / random structures", "subcategory": "matrix/tensor inequalities", "source_year": 2026, "answer": {"status": "open", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Simon-14", "split": "test", "title": "Continuity of integrated density of states", "statement": "Prove that the integrated density of states k(E) of ergodic Schrodinger operators is continuous in the energy. Continuity is known in one dimension and in the discrete case; the higher-dimensional continuum case is the open one (Simon 2000, Problem 14).", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "mathematical physics", "source_year": 2000, "answer": {"status": "partially_resolved", "status_confidence": "high", "resolution_year": null, "statement_quality": "verified", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-10.1.A", "split": "dev", "title": "Angular Synchronization via Projected Power Method", "statement": "Determine sharp algorithmic or SDP tightness thresholds for angular synchronization under the stated noise model.", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "high-dimensional statistics", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-5.2.A", "split": "test", "title": "Certifying the Restricted Isometry Property", "statement": "Design a polynomial-time algorithm certifying, with high probability, that a Gaussian random matrix satisfies s-RIP (equivalently has no s-sparse near-null vectors) for s well beyond sqrt(M).", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "general", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "high", "resolution_year": null, "statement_quality": "verified", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-10.4.A", "split": "test", "title": "Consistency and sample complexity of Multireference Alignment", "statement": "Determine SDP tightness, consistency, or sample complexity thresholds for multireference alignment.", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "quantum complexity / computation", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-1.3.B", "split": "test", "title": "Cut rank constrained SDP Spike Model problem", "statement": "Determine the limiting value or threshold for the rank-constrained semidefinite spike model.", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "SDP / SoS / optimization", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-9.1.A", "split": "test", "title": "Detection Threshold for SBM for three of more communities", "statement": "Source excerpt: Open Problem 9.1 Consider the balanced Stochastic Block Model for k> 3 (constant) communities with inner probability p = a n and outer probability q = b n, what is the threshold at which it becomes possible to make an estimate that correlates with the original partition is open (known as the par- tial recovery or detection threshold). We refer the reader to [DKMZ11, ZMZ14, GZC +15] for more information on this and many other interesting conjectures often motivated from statistical physics. 9.4 Exact recovery We now turn our attention to the problem of recovering the cluster membership of every", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "high-dimensional statistics", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-4.6.A", "split": "test", "title": "Feige's Conjecture", "statement": "Source excerpt: Open Problem 4.6 Prove or disprove Conjecture 4.33. 21 21We thank Francisco Unda and Philippe Rigollet for suggesting this problem. 78 5 Johnson-Lindenstrauss Lemma and Gordons Theorem 5.1 The Johnson-Lindenstrauss Lemma Suppose one has n points, X = {x1,...,x n}, in Rd (with d large). If d > n, since the points have to lie in a subspace of dimension n it is clear that one can consider the projection f : Rd Rn of the points to that subspace without distorting the geometry of X. In particular, for every xi and xj, f(xi) -f(xj)2 = xi -xj2, meaning that f is an isometry in X. Suppose now we allow", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "general", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-1.1.A", "split": "test", "title": "Mallat and Zeitouni's problem", "statement": "Source excerpt: Open Problem 1.1 (A. Mallat and Zeitouni [MZ11]) Let g N (0,) be a gaussian random vector in Rp with a known covariance matrix and d < p. Now, for any orthonormal basis V = [v1,...,v p] of Rp, consider the following random variable V: Given a draw of the random vector g, V is the squared l2 norm of the largest projection of g on a subspace generated by d elements of the basis V. The question is: What is the basis V for which E[V] is maximized? 2Note that Tr (n) = p k=1 k(n). 14 The conjecture in [MZ11] is that the optimal basis is the eigendecomposition of . It is known that this is the case f", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "general", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "medium-high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-8.5.A", "split": "test", "title": "Maximum and minimum bisections on random regular graphs", "statement": "Source excerpt: Open Problem 8.5 Prove or disprove Conjecture 8.5. Recently, it was shown that the conjecture holds up to o( d) terms [DMS15]. We also point the reader to this paper [Lyo14], that contains bounds that are meaningful already for d= 3. 117 9 Community detection and the Stochastic Block Model 9.1 Community Detection Community detection in a network is a central problem in data science. A few lectures ago we discussed clustering and gave a performance guarantee for spectral clustering (based on Cheeger's Inequality) that was guaranteed to hold for any graph. While these guarantees are remarkable, ", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "random graphs / combinatorics", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-4.4.A", "split": "test", "title": "OSNAP problem", "statement": "Source excerpt: Open Problem 4.4 (OSNAP [NN]) Let sdmn. 1. Let Rmn be a random matrix with i.i.d. entries ri = riris , where ri is a Rademacher random variable and ri = { 1s with probability s m 0 with probability 1 -s m Prove or disprove: there exist positive universal constants c1 and c2 such that For any U Rnd for which UTU = Idd Prob {(U)T(U) -I } <, for mc1 d+log(1 ) 2 and sc2 log(d ) 2 . 2. Same setting as in (1) but conditioning on m r=1 ri = s, for all i, meaning that each column of has exactly s non-zero elements, rather than on average. The conjecture is then slightly different: Prove or disprove: t", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "general", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-2.1.A", "split": "test", "title": "Ramsey numbers", "statement": "Source excerpt: Open Problem 2.1 Recall the definition of R(r) above, the following questions are open: What is the value of R(5)? What are the asymptotics of R(s)? In particular, improve on the base of the exponent on either the lower bound ( 2) or the upper bound ( 4). 27 Construct a family of graphs G= (V,E) with increasing number of vertices for which there exists > 0 such that9 |V| (1 + )r. It is known that 43 R(5) 49. There is a famous quote in Joel Spencer's book [Spe94] that conveys the difficulty of computing Ramsey numbers: \"Erd os asks us to imagine an alien force, vastly more powerful than us, lan", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "random graphs / combinatorics", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-4.5.A", "split": "test", "title": "Random k-lifts of graphs", "statement": "Source excerpt: Open Problem 4.5 (Random k-lifts of graphs) Give a tight upperbound to E Ak -EAk . Oliveira [Oli10] gives a bound that is essentially of the form log(nk), while the results in [ABG12] suggest that one may expect more concentration for large k. It is worth noting that the case of k= 2 can essentially be reduced to a problem where the entries of the random matrix are independent and the results in [BvH15] can be applied to, in some case, remove the logarithmic factor. 4.8 Another open problem Feige [Fei05] posed the following remarkable conjecture (see also [Sam66, Sam69, Sam68]) Conjecture 4.33", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "random graphs / combinatorics", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "medium-high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-6.1.A", "split": "test", "title": "Random Partial Discrete Fourier Transform", "statement": "Source excerpt: Open Problem 6.1 Consider the random M N matrix obtained by drawing rows uniformly with replacement from the NN discrete Fourier transform matrix. How large does M need to be so that, with high probability, the result matrix satisfies the Restricted Isometry Property (for constant )? 6.5 Coherence and Gershgorin Circle Theorem Last lectures we discussed the problem of building deterministic RIP matrices (building deterministic RIP matrices is particularly important because checking whether a matrix is RIP is computationally 28In these references the sets considered are slightly different than ", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "compressed sensing / frames", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-9.2.A", "split": "test", "title": "Recovery Threshold for SBM for logarithmic many communities", "statement": "Source excerpt: Open Problem 9.2 What is the threshold for exact recovery on the balanced symmetric Stochas- tic Block Model in k log n communities and at what threshold does the SDP succeed at exactly determining the communities? (see [ABKK15]). 9.11 Euclidean Clustering The stochastic block model, although having fascinating phenomena, is not always an accurate model for clustering. The independence assumption assumed on the connections between pairs of vertices may sometimes be too unrealistic. Also, the minimum bisection of multisection objective may not be the most relevant in some applications. One part", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "high-dimensional statistics", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-10.2.A", "split": "test", "title": "Sharp tightness of the Angular Synchronization SDP", "statement": "Determine sharp algorithmic or SDP tightness thresholds for angular synchronization under the stated noise model.", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "SDP / SoS / optimization", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-9.4.A", "split": "test", "title": "Stability conditions for tightness of k-median LP and k-means SDP", "statement": "Source excerpt: Open Problem 9.4 Give integrality conditions to either the k-medians LP or the k-means SDP based on stability like conditions, as described above. 9.12 Probably Certifiably Correct algorithms While the SDP described in this lecture for recovery in the Stochastic Block Model achieves exact recovery in the optimal regime, SDPs (while polynomial time) tend to be slow in practice. There are faster (quasi-linear) methods that are also able to achieve exact recovery at the same threshold. 129 However, the SDP has an added benefit of producing a posteriori certificates. Indeed, if the solution from t", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "SDP / SoS / optimization", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "medium", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Bandeira-8.2.A", "split": "test", "title": "Sum of Squares approximation ratio for Max-Cut", "statement": "Source excerpt: Open Problem 8.2 1. What is the approximation ratio achieved by (or the integrality gap of) the Sum-of-squares degree 4 relaxation of the Max-Cut problem? 2. The relaxation described above (of degree 2) (76) is also known to produce a cut of 1 -O() when a cut of 1 - exists. Can the degree 4 relaxation improve over this? 3. What about other (constant) degree relaxations? Remark 8.4 (triangular inequalities and Sum of squares level 4) A (simpler) natural ques- tion is wether the relaxation of degree 4 is actually strictly tighter than the one of degree 2 for Max-Cut (in the sense of forcing extr", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "SDP / SoS / optimization", "source_year": 2016, "answer": {"status": "partially_resolved", "status_confidence": "high", "resolution_year": null, "statement_quality": "draft", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "OQP-4", "split": "test", "title": "Catalytic majorization", "statement": "By a theorem of Nielsen [1], we have a completely explicit criterion to decide when a single copy of a pure bipartite state can be converted to another such state using only local quantum operations and classical communication. Using Nielsen's criterion, one can show the existence of the phenomenon of catalysis [2]: there exist situations when a state cannot be converted to state , but nevertheless can be converted to for a suitably chosen entangled state , the \"catalyst\". The problem is to give a similarly efficient criterion to decide which pure bipartite states can be converted into each other using a catalyst.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2000, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2007, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-3", "split": "test", "title": "Polynomial entanglement invariants", "statement": "We say that two bipartite quantum states and are \"equally entangled\" if they differ only by a choice of bases in Alice's and Bob's subspaces, i.e., if we can find , , such that An \"entanglement invariant\" is by definition any real valued function on the space of bipartite density operators, which assigns the same value to equally entangled density operators. A \"polynomial invariant\" is an entanglement invariant, which can be computed as a polynomial in the matrix elements of . Note that because we only consider hermitian operators, allowing polynomials in the matrix elements and their complex conjugates does not enlarge this class. The basic problem is to decide the following question: *Are ", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2000, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2001, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Simon-7", "split": "test", "title": "Decaying potentials with singular continuous spectrum", "statement": "Do there exist potentials on so that for some and so that has some singular continuous spectrum?", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "spectral theory / Schrodinger operators", "source_year": 2000, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2005, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Simon-6", "split": "test", "title": "Purely absolutely continuous spectrum for irrational subcritical almost-Mathieu", "statement": "Prove that for every irrational frequency alpha and every subcritical coupling |lambda| < 1, the almost Mathieu operator has purely absolutely continuous spectrum (spectral measures absolutely continuous for all phases theta).", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "spectral theory / Schrodinger operators", "source_year": 2000, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2008, "statement_quality": "verified", "famous": false, "verifiability_class": "expert-only"}}
{"task_id": "Simon-4", "split": "test", "title": "Ten Martini problem", "statement": "Ten Martini problem. Prove for all and all irrational that (which is independent) is a Cantor set , that is, that it is nowhere dense .", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "mathematical physics", "source_year": 2000, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2009, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Simon-5", "split": "dev", "title": "Zero-measure spectrum for almost-Mathieu at critical coupling", "statement": "Prove for all irrational and that has measure zero .", "aliases": "(none listed)", "category": "Mathematical physics", "subcategory": "spectral theory / Schrodinger operators", "source_year": 2000, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2006, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-7", "split": "test", "title": "Additivity of Entanglement of Formation", "statement": "The entanglement of formation [1] is one of the standard measures of entanglement. It is defined, for any density operator on a bipartite system, as where denotes the von Neumann entropy and denotes the restriction of a density operator to the \"Alice\" subsystem (partial trace over the other subsystem), the are density operators and the are positive, adding up to one. Since is concave, the infimum is attained at a convex decomposition of into pure states, and the definition is often given as this restricted infimum. Consider now a pair , of bipartite density operators, and their tensor product , which lives on a tensor product of four Hilbert spaces, but can be considered as a bipartite state", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2001, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2009, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-5", "split": "test", "title": "Maximally entangled mixed states", "statement": "Among all density operators of two qubits with the same spectrum one may look for those maximizing some measure of entanglement. It turns out [1] that for the entanglement of formation, the relative entropy of entanglement and the negativity one gets the same maximally entangled states. Is this true for arbitrary entanglement monotones? Obvious variants of this problem are for higher dimensional systems and weaker constraints on the spectrum, e. g., largest eigenvalue or entropy.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2001, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2024, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-6", "split": "test", "title": "Nice error bases", "statement": "There are two special constructions to obtain orthogonal bases of unitaries, i.e., collections of unitary operators , , on a -dimensional Hilbert space, such that [1] [2]. On the one hand one can require in addition that the product of any two unitaries in the basis gives another one up to a phase, i.e., The composition of labels then defines a group, the \"index group\" of the basis. Bases of this kind have been called \"nice error bases\".On the other hand, one may require that, in a suitable basis of the Hilbert space, the unitaries are obtained as the products of a collection of permutation operators and multiplication operators. Bases constructed in this way are called of \"shift and multipl", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2001, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2003, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-9", "split": "dev", "title": "Reduction criterion implies majorization?", "statement": "The density matrix of any separable state is majorized by its reductions (the density matrix reduced to one subsystem, e. g. . This is in fact the strongest separability criterion based on the spectra of a state and one of its reductions. However, it is not known how it is related to other separability criteria like PPT, undistillability or the reduction criterion. The problem is to find out how majorization enters into the known implication chain of separability criteria.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2002, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2003, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-10", "split": "test", "title": "Additivity of classical capacity and related problems", "statement": "For each quantum channel (in the Schrodinger picture), define where the supremum is over all probability vectors , and all collections of input states , and denotes the von Neumann entropy. Show that , or else give a counterexample. The problem can be traced back to [1], see also [2].", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2003, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2009, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-11", "split": "dev", "title": "Continuity of the Quantum channel capacity", "statement": "There are different notions of capacity for a noisy quantum communication channel, e. g. for classical, private classical or quantum communications. Continuity of these is an important property, but from a mathematical point of view it is not at all obvious, because very similar channels can differ a lot given many copies, and the capacity is operationally defined in terms of an asymptotic number of channel uses. Of course this is not a problem when a single-letter formula is available, so that one can reason about it directly, but it becomes quite a challenge, when only a multi-letter formula is available, or even non at all.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2003, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2009, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-18", "split": "dev", "title": "Qubit bi-negativity", "statement": "A little problem introduced in [1] is the bi-negativity on two qubits: Prove that holds for every two-qubit state . Here, denotes the partial transpose with respect to the second system (see also problem 2) and is the operator absolute value, .", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2003, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2005, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-17", "split": "test", "title": "Reversibility of entanglement assisted coding", "statement": "For any two quantum channels and , define the \"entanglement-assisted capacity\" of for -messages as the supremum of all rates such that, for large , parallel copies of may be simulated by copies of , where the simulation involves arbitrary coding and decoding operations using (if necessary) arbitrarily many entangled pairs between sender and receiver, and where the errors go to zero as . Show that . As for other capacities, the \"two-step coding inequality\" is easy to show. Hence . Equality means here, that the two channels are essentially equivalent as a resource for simulating other channels (apart from a constant factor ): (with ). In this case we call and reversible for entanglement-assist", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2003, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2014, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-19", "split": "test", "title": "Stronger Bell Inequalities for Werner states?", "statement": "Find Bell Inequalities which are stronger than the CHSH inequalities in the sense that they are violated by a wider range of Werner states.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum nonlocality / Bell inequalities", "source_year": 2003, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2008, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-22", "split": "test", "title": "Asymptotic cloning is state estimation", "statement": "Fix an arbitrary probability measure on the pure states of a -dimensional quantum system. Let be the optimal single copy fidelity for -to- cloning transformations, averaged with respect to the given probability measure and over all clones. On the other hand, let be the best mean fidelity achievable by measuring on input copies of the state, and repreparing a state according to the measured data. The problem is to decide whether one always gets It is clear that the limit exists, because is non-increasing in . Moreover, the limit will be larger or equal than the right hand side, because estimation with repreparation is a particular cloning method. A weaker, but still interesting version of the", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2005, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2006, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-30", "split": "test", "title": "Asymptotic Version of Birkhoff's Theorem", "statement": "A classical result of Birkhoff states that a doubly stochastic matrix (positive elements, and all rows and columns add to 1) is a convex combination of permutation matrices. In the quantum context, doubly stochastic matrices become doubly stochastic channels, i.e. completely positive maps preserving both the trace and the identity. In the classical case, the permutations are the invertible elements, corresponding in the quantum case to the unitarily implemented channels. It is well-known [1][2] that the analog of Birkhoff's Theorem fails in the quantum case: in other words, there exist doubly stochastic quantum channels which can not be written as a convex combination of unitary channels. Ho", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2005, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2011, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-21", "split": "test", "title": "Bell violation by tensoring", "statement": "Can one find bipartite density operators , neither of which violates any CHSH Bell inequality, with the property that does?", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum nonlocality / Bell inequalities", "source_year": 2005, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2010, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-28", "split": "test", "title": "Local equivalence of graph states", "statement": "Decide whether two graph states, which can be mapped into each other by a local unitary, can also be mapped into each other by a local unitary from the Clifford group.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "random graphs / combinatorics", "source_year": 2005, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2007, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-33", "split": "test", "title": "Bell inequalities and operator algebras", "statement": "Quantum Bell-type inequalities are defined in terms of two (or more) subsystems of a quantum system. The subsystems may be treated either via (local) Hilbert spaces, - tensor factors of the given (global) Hilbert space, or via commuting (local) operator algebras. The latter approach is less restrictive, it just requires that the given operators commute whenever they belong to different subsystems. Are these two approaches equivalent?", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum nonlocality / Bell inequalities", "source_year": 2006, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2020, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "Bandeira-4.2.A", "split": "dev", "title": "Latala-Riemer-Schutt Problem", "statement": "For a symmetric Gaussian matrix with independent entries and arbitrary variance profile, determine whether E||X|| is universally comparable to E max_i ||X e_i||_2.", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "matrix/tensor inequalities", "source_year": 2016, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2017, "statement_quality": "verified", "famous": false, "verifiability_class": "checkable-claim"}}
{"task_id": "Bandeira-0.2.A", "split": "test", "title": "Matrix AM-GM inequality", "statement": "Prove or refute the Recht-Re noncommutative arithmetic-geometric mean conjecture for products of positive semidefinite matrices.", "aliases": "Matrix AM-GM Inequality", "category": "Mathematical statistics / data science", "subcategory": "matrix/tensor inequalities", "source_year": 2016, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2020, "statement_quality": "verified", "famous": false, "verifiability_class": "checkable-claim"}}
{"task_id": "Bandeira-4.1.A", "split": "test", "title": "Non-commutative Khintchine improvement", "statement": "Prove or disprove Conjecture 4.21: for Gaussian series X = sum_k g_k A_k, E||X|| <= C(sigma(X) + sqrt(log d) sigma_*(X)) (weak-variance matrix Khintchine improvement).", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "matrix/tensor inequalities", "source_year": 2016, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2021, "statement_quality": "verified", "famous": false, "verifiability_class": "checkable-claim"}}
{"task_id": "Bandeira-9.5.A", "split": "test", "title": "Positive PCA tightness", "statement": "Determine whether the nonnegative PCA SDP over X >= 0 entrywise, X psd, and Tr(X) = 1 is tight for a GOE/Wigner matrix and has a rank-one optimizer with high probability.", "aliases": "(none listed)", "category": "Mathematical statistics / data science", "subcategory": "high-dimensional statistics", "source_year": 2016, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2020, "statement_quality": "verified", "famous": false, "verifiability_class": "checkable-claim"}}
{"task_id": "OQP-35; FOP-3", "split": "test", "title": "AME(4,6) absolutely maximally entangled state", "statement": "Decide whether an absolutely maximally entangled state AME(4,6), equivalently a four-party six-level perfect tensor / 2-unitary matrix, exists.", "aliases": "Absolutely maximally entangled states; AME(4,6); Existence of absolutely maximally entangled pure states; AME state for four subsystems with six levels each", "category": "Quantum information", "subcategory": "entanglement theory", "source_year": 2017, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2021, "statement_quality": "verified", "famous": true, "verifiability_class": "checkable-claim"}}
{"task_id": "OQP-40", "split": "test", "title": "Refinement of the Bessis-Moussa-Villani conjecture", "statement": "For positive semidefinite matrices A and B, let p_nm(A,B) be the coefficient of t^n s^m in tr(tA+sB)^(n+m) divided by binom(n+m,n), i.e. the average of tr W over all words W with n letters A and m letters B. Show that tr(A^n B^m) >= p_nm(A,B) >= tr exp(n log A + m log B); when A and B commute, equality holds throughout. (Refinement of the Bessis-Moussa-Villani conjecture, due to Daniel Hagele.)", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2017, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2026, "statement_quality": "verified", "famous": false, "verifiability_class": "checkable-claim"}}
{"task_id": "OQP-39", "split": "test", "title": "Steering bound for qubits and POVMs", "statement": "In steering [1] one asks for the possibility to model the correlations in a quantum state by a classical model, in which one party (say Bob) is constrained to be described by quantum mechanics. This is demanded for some class [/latex] of observables for the other party (say Alice). That is, we need (1) a probability space with probability measure , (2) a family of \"hidden\" states , , for Bob's system, and (3) for any observable , a classical observable . That is to every POVM element corresponds a response probability function on , so that and for all . The model is valid if Consider a two-qubit state of the form , where is the swap operator, and . These states are separable for . The proble", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum measurements and designs", "source_year": 2017, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2024, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-37", "split": "test", "title": "Stronger submultiplicativity for the diamond norm", "statement": "Does there exist an absolute constant such that the submultiplicativity statement holds for all quantum channels over any finite dimensional Hilbert space? Here denotes the transposition map, so equals the Hilbert space dimension.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2017, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2026, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}
{"task_id": "OQP-49", "split": "test", "title": "Catalytic entropy conjecture", "statement": "Consider a density matrix on a finite-dimensional system . Let be the von Neumann entropy. Prove or disprove that for any density matrix on such that and , there exists a finite-dimensional system (\"catalyst\") with state and a unitary operator on such that the following holds: If appropriate and can be found, we say that the catalytic transition is possible.", "aliases": "(none listed)", "category": "Quantum information", "subcategory": "quantum information", "source_year": 2019, "answer": {"status": "solved", "status_confidence": "high", "resolution_year": 2022, "statement_quality": "draft", "famous": false, "verifiability_class": ""}}