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215 records covering 215 distinct questions. Filter counts refer to records.
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Entanglement cost of an amplitude-damping-channel Choi state
What is the entanglement cost of the Choi state of the qubit amplitude-damping channel
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Quantum capacity of a qubit Pauli channel
What is the unassisted quantum capacity of a general qubit Pauli channel?
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Quantum validity of the Zhang–Yeung inequality
Does the four-party Zhang–Yeung inequality hold for the von Neumann entropies of every finite-dimensional quantum state?
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Many-copy Bell nonlocality of every entangled state
Does every entangled bipartite state become Bell nonlocal under collective local measurements on some finite number of copies?
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Square-root distance bound for weight-four stabilizer codes
Is there a universal constant \(C>0\) such that every qubit stabilizer code that encodes at least one logical qubit, and whose stabilizer group is generated by Pauli operators of weight at most four, has distance at most \(C\sqrt n\) on \(n\) physical qubits?
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Wigner entropy conjecture
Does every finite-energy single-mode state with a nonnegative Wigner function have Wigner entropy at least \(1+\ln\pi\)?
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Asymptotic nonclassical-state conversion by linear optics
What is the optimal asymptotic conversion rate between finite-energy bosonic states under passive linear-optical protocols?
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POVM steering threshold of higher-dimensional Werner states
What is the exact steering threshold for arbitrary POVMs on a higher-dimensional Werner state?
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Secret key from every Bell-nonlocal behavior
Does every finite-alphabet Bell-nonlocal behavior have a strictly positive asymptotic secret-key rate against arbitrary individual nonsignalling attacks?
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SIC-POVM existence in every dimension
Does a symmetric informationally complete positive-operator-valued measure exist in every finite dimension \(d\ge2\)?
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Finite-alphabet nonsignalling simulation of entangled qubits
For every partially entangled two-qubit pure state, does some fixed finite-alphabet nonsignalling resource give an exact noncommunicating simulation of all local projective measurements?
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Universal simulation with two PR boxes
Can shared randomness and at most two Popescu–Rohrlich boxes exactly simulate every pair of local projective measurements on every two-qubit state, without communication?
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Quantum LDPC codes at the Pauli hashing bound
For which probability vectors \(\mathbf p=(p_I,p_X,p_Y,p_Z)\) can quantum low-density parity-check (LDPC) stabilizer codes reliably attain the exact hashing rate over the Pauli channel?
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One-bit simulation of partially entangled qubits
Can shared randomness and one classical bit exactly simulate every pair of local projective measurements on every pure entangled two-qubit state?
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Long-range vacuum CHSH violation
Does the vacuum of a massive free scalar Bose field violate the CHSH inequality between bounded localization regions at arbitrarily large spacelike separation?
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Weyl–Heisenberg-covariant SICs in every dimension
Does every finite dimension admit a symmetric informationally complete measurement that is a single Weyl–Heisenberg orbit?
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Statistical strength of CGLMP measurements
For every \(d\geq3\), do the standard CGLMP Fourier–phase measurements maximize the relative-entropy statistical strength against local realism among all projective \(d\)-outcome measurements on the fixed state \(\lvert\Phi_d\rangle=d^{-1/2}\sum_{j=0}^{d-1}\lvert j,j\rangle\), when the setting distribution is also optimized?
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Extensible causality and process purification
Is every finite-dimensional extensibly causal process matrix purifiable?
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Semialgebraicity of closed quantum correlations
Is the closure of the finite-dimensional tensor-product correlation set semialgebraic in every fixed bipartite Bell scenario?
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Maximum number of mutually unbiased bases
For each integer \(d\geq2\), determine the maximum number \(\mu(d)\) of pairwise mutually unbiased orthonormal bases of \(\mathbb C^d\).
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Optimal CGLMP measurements for a maximally entangled state
For every \(d\geq3\), do the standard Fourier–phase measurements optimize the CGLMP violation of the maximally entangled state among all projective \(d\)-outcome measurements?
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Quantum violations of bipartite Bell facets
Must every nontrivial facet Bell inequality in a finite bipartite scenario have a quantum violation?
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Complete facet descriptions for Bell polytopes
Determine complete facet descriptions for local-behavior polytopes beyond the presently solved Bell scenarios, either for a specified unresolved finite scenario or for a nontrivial infinite family with additional structure.
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Zauner-symmetric Weyl–Heisenberg SIC fiducials
Does every finite dimension admit a Weyl–Heisenberg SIC fiducial that is an eigenvector of a Zauner Clifford unitary?
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Anticoncentration of independent complex Gaussian hafnians
Do independent complex Gaussian hafnians satisfy a polynomial lower-tail bound at their root-mean-square scale?
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Optimal precision dependence of diamond-norm channel tomography
Does tomography of an arbitrary \(d\)-dimensional quantum channel in diamond norm require and suffice with \(\Theta(d^4/\varepsilon^2)\) channel uses?
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One-way state generation from private Hamiltonian phase states
Do private-architecture Hamiltonian phase states yield a one-way state generator for explicit polynomial parameters?
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Channels superactivated by antidegradable helpers
What necessary and sufficient structural conditions characterize the finite-dimensional channels of zero quantum capacity, and separately those of zero private capacity, that acquire a positive capacity of the same kind when used in parallel with some antidegradable…
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Maximal length of stabilizer quantum MDS codes
Is there a stabilizer quantum maximum-distance-separable code \([[n,k,d]]_q\) with \(k\geq1\) and \(d\geq3\) whose length \(n\) exceeds the conjectured limit \(b(q,d)\) of Eq. (2)?
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Average-case additive hardness of squared complex Gaussian hafnians
Is additive estimation of squared complex Gaussian hafnians \(\#\mathrm P\)-hard under randomized polynomial-time oracle reductions?
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Second-level collapse of the exact PPT entanglement-cost hierarchy
Does the semidefinite hierarchy for exact PPT entanglement cost collapse at its second level for every finite-dimensional bipartite density operator \(\rho_{AB}\)?
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Aaronson–Ambainis conjecture
Must every bounded low-degree real polynomial on the Boolean cube with non-negligible variance have a variable of inverse-polynomial influence?
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Quantum query complexity of Welded Tree path-finding
Does finding an explicit ENTRANCE-to-EXIT path in the standard Welded Tree oracle problem require exponentially many quantum queries?
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Polynomial-time quantum algorithm for approximate Shortest Vector Problem
Does the polynomial-factor approximate Shortest Vector Problem admit a polynomial-time quantum algorithm?
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Polynomial-time quantum algorithm for Learning With Errors
Does the Learning With Errors problem in its standard worst-case-hard parameter regime admit a polynomial-time quantum algorithm?
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Parity is not in QAC0
Can polynomial-size constant-depth \(\mathsf{QAC}^0\) circuits compute the parity function, or equivalently, is \(\mathrm{PARITY}\notin\mathsf{QAC}^0\)?
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Average-case approximation hardness of random-circuit output probabilities
Does there exist a fixed family of \(n\)-qubit circuit layouts with \(m=\operatorname{poly}(n)\) one- and two-qubit gates for which the following task is \(\#\mathrm{P}\)-hard?
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Permanent-of-Gaussians Conjecture
Is the following Gaussian permanent estimation task \(\#\mathrm P\)-hard under randomized polynomial-time oracle reductions?
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Quantum query complexity of Triangle Finding
What is the bounded-error quantum query complexity of finding a triangle in an \(n\)-vertex graph given oracle access to its adjacency matrix?
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Polynomial-time quantum algorithm for Graph Isomorphism
Does the Graph Isomorphism problem admit a polynomial-time quantum algorithm?
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Squashed entanglement of the qubit depolarizing channel
What is the squashed entanglement \(E_{\mathrm{sq}}(\Lambda_p)\) of the qubit depolarizing channel, defined for a mixing parameter \(p\in[0,1]\) by
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Private capacity of the qubit depolarizing channel
What is the private classical capacity \(P(\Lambda_p)\) of the qubit depolarizing channel, defined for a mixing parameter \(p\in[0,1]\) by
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Additivity of the relative entropy of entanglement
Does the relative entropy of entanglement of every bipartite state equal its regularization, or is regularization genuinely necessary?
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QMA(2) versus QMA
Is every promise problem verifiable by a quantum Merlin-Arthur protocol with two unentangled witnesses also verifiable by a protocol with a single arbitrary witness, that is, is \(\mathsf{QMA}(2) = \mathsf{QMA}\) in the standard, unrelativized setting?
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Asymptotic growth of the stabilizer rank of T-state tensor powers
Does the exact stabilizer rank of the tensor powers of the single-qubit magic state \(|T\rangle\) grow polynomially in the number of copies, or is it not polynomially bounded?
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Additivity of the entanglement of purification
Is the entanglement of purification additive on tensor products?
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Smallest output dimension violating minimum output entropy additivity
What is the smallest output dimension in which the minimum output von Neumann entropy of quantum channels fails to be additive?
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Closed-form nonadditivity of the Holevo capacity and entanglement of formation
Construct a simple closed-form or practically computable counterexample to additivity of the Holevo capacity or the entanglement of formation, together with a rigorous certificate of a strict violation.
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Multi-slot overhead of virtual channel conjugation
What is the optimal quasiprobability overhead of implementing the complex conjugate of an unknown quantum channel from \(n\) queries?
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Closed-form exact PPT distillable entanglement
What computable expression, if any, equals the regularized exact PPT distillable entanglement of a bipartite state?
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Universal purification with classically simulable operations
Can classically simulable operations purify an unknown depolarized pure state from any number of copies?
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Amortization collapse for superchannel divergences
Does amortization collapse for the geometric Rényi divergence of arbitrary finite-dimensional quantum superchannels?
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Ordinary-Petz recovery bound for conditional mutual information
Does the ordinary, unrotated Petz map universally recover a tripartite state with fidelity controlled by its conditional mutual information?
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Parallel versus nested-adaptive superchannel discrimination
Can nested-adaptive strategies improve the Stein exponent for discriminating two finite-dimensional quantum superchannels?
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All-code exponential strong converse for degradable channels
Does every finite-dimensional degradable quantum channel satisfy an all-code exponential strong converse for quantum communication at its quantum capacity?
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Diamond-smoothed max-relative-entropy AEP for quantum channels
Does the max-relative entropy of finite-dimensional quantum channels satisfy an asymptotic equipartition property under uniform diamond-norm smoothing?
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Asymptotic metrology with quantum-controlled causal order
Does quantum control of causal order yield a persistent asymptotic metrological advantage over parallel access for some regular, finite-dimensional channel family?
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Nontrivial mutually degradable channel pairs
Does there exist an integer \(d\geq2\) and a pair of distinct channels \(\mathcal M,\mathcal N:\mathcal L(A)\to\mathcal L(B)\), with \(A\simeq B\simeq\mathbb C^d\), that both have Choi rank exactly \(d\), are mutually degradable, and are each nondegradable?
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Strict inclusion of degradable channels in the less-noisy class
Do there exist finite-dimensional quantum channels that are less noisy in Watanabe’s regularized sense but are not degradable?
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Transpose degradability beyond degradability
Does there exist a finite-dimensional transpose-degradable quantum channel that is not degradable?
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Minimal dimensions for strict transpose degradability
What are the componentwise-minimal dimension triples \((d_A,d_B,d_E)\) that admit a transpose-degradable but nondegradable channel?
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Fixed-error parallel Stein lemma for quantum channels
Let \(\mathcal N,\mathcal M:\mathcal L(A)\to\mathcal L(B)\) be quantum channels on finite-dimensional systems.
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Gaussian entanglement of formation beyond bisymmetry
Does Gaussian entanglement of formation equal unrestricted entanglement of formation for every non-bisymmetric multimode Gaussian state?
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Achievability of the Rains bound under completely PPT-preserving channels
Is the Rains bound asymptotically achievable by completely positive-partial-transpose-preserving (completely PPT-preserving) channels for every full-rank two-qubit Bell-diagonal state, and what channel family achieves it?
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The PPT-squared conjecture
Must the composition of any two compatible PPT completely positive maps be entanglement breaking?
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NPT bound entanglement and the rank-five frontier
Does there exist a finite-dimensional bipartite state with non-positive partial transpose (NPT) that is undistillable by local operations and classical communication (LOCC)?
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Quantum capacity of a bosonic thermal attenuator
What is the unassisted quantum capacity of a single-mode thermal attenuator at every strictly positive finite environment temperature?
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Polynomial shared-resource lower bounds for routing
Does an explicit total Boolean family require polynomial shared-state cost for bounded-error one-round \(f\)-routing?
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Entanglement cost of a qubit Bell-diagonal state
What is the entanglement cost of a qubit Bell-diagonal state for an arbitrary probability vector \(\mathbf p=(p_I,p_X,p_Y,p_Z)\)?
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Eight-ququart AME state and the ((7,4,4)) ququart MDS code
Does an absolutely maximally entangled state of eight ququarts exist?
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Distillable entanglement of Bell-diagonal states
What is the distillable entanglement \(D(\rho_{\mathbf p})\) under local operations and classical communication (LOCC) for the Bell-diagonal state
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Two-way quantum capacity: amplitude-damping channel
What is the two-way quantum capacity \(\mathcal{Q}_2(\mathcal A_p)\) of the qubit amplitude-damping channel
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Three-dimensional self-correcting quantum memory
Does there exist a passive self-correcting quantum memory in three spatial dimensions?
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High-dimensional crossover of optimized port-based teleportation
What is the high-dimensional crossover fidelity of optimized deterministic port-based teleportation?
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Hilbert–Schmidt inradius of the multiqubit stabilizer polytope
Is every \(n\)-qubit state of purity at most \(1/(2^n-1/2)\) a convex mixture of stabilizer states?
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Quantum erasure-channel upper bound for information combining
Does the quantum erasure-channel upper information-combining bound hold for two independent uniform bits with arbitrary quantum side information?
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Weight-four generators for an [[11,3,3]] stabilizer code
Does there exist a qubit stabilizer code with parameters \([[11,3,d]]\) and \(d\geq3\) whose stabilizer group is generated by eight independent commuting Pauli operators of weight at most four?
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Two-error perfect quantum codes over non-prime-power local dimensions
Does a nontrivial pure perfect quantum code correcting arbitrary errors on two qudits exist for some local dimension \(q\) with at least two distinct prime factors?
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Semi-Clifford structure of two-qudit hierarchy gates above the third level
Is every two-qudit gate in every level \(k\geq4\) of the Clifford hierarchy semi-Clifford when the local dimension is an odd prime?
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Squashed entanglement of qubit Bell-diagonal states
Is there a closed-form formula for the squashed entanglement \(E_{\mathrm{sq}}\) of an arbitrary two-qubit Bell-diagonal state, or a certified approach to determining it numerically?
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Universality of every fixed non-Gaussian polynomial generator
Does every fixed non-Gaussian polynomial Hamiltonian become universal when all Gaussian controls are available?
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Gaussian-measurement equality with the Holevo bound
When does the optimal single-copy Gaussian-measurement cost equal the Holevo bound?
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Entropy photon-number inequality
Does the entropy photon-number inequality hold for every pair of independent finite-energy bosonic inputs?
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Optimal precision dependence of low-energy Hamiltonian simulation
What is the tight precision dependence of worst-case query complexity for low-energy simulation in the regime (2)?
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Simultaneously optimal queries to both quantum linear-system oracles
Can one quantum linear-system algorithm attain both query bounds in (2) simultaneously?
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Strong converse for the quantum capacity of a general channel
Does every finite-dimensional memoryless quantum channel obey the strong converse at its unassisted quantum capacity?
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Polynomial-time local-unitary equivalence of graph states
Does a deterministic classical algorithm running in \(n^{O(1)}\) time decide local-unitary equivalence of arbitrary graph states on \(n\) labelled qubits?
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Loss of mixed unitarity after a positive time in a quantum dynamical semigroup
Does there exist a norm-continuous semigroup \((\Phi_t)_{t\geq0}\) of unital completely positive trace-preserving maps on \(\mathcal L(\mathbb C^d)\), for some finite integer \(d\geq3\), such that \(\Phi_s\) is mixed unitary and \(\Phi_t\) is not mixed unitary for some \(0<s<t\)?
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Candidate pure-loss second-order converse
Does the pure-loss bosonic channel admit the following candidate second-order classical converse under a maximum-photon-number occupation constraint?
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Convergence of the JRF iteration for mixed-state discrimination
Does the Ježek–Řeháček–Fiurášek (JRF) iteration, initialized by the uniform POVM, converge to a globally optimal minimum-error measurement for every finite ensemble containing mixed quantum states?
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Entanglement of formation of generalized Bell-diagonal states
For every local dimension \(d\geq3\), determine the entanglement of formation of an arbitrary Weyl–Bell-diagonal state.
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Exponential strong converse for transpose-degradable channels
Does every finite-dimensional transpose-degradable channel satisfy an exponential strong converse for quantum communication at its single-letter quantum capacity?
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Relative entropy of entanglement for two qubits
Find a closed formula for the relative entropy of entanglement of every two-qubit density operator \(\rho\), including an explicit closest separable state.
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Sample complexity of low-rank tomography with Pauli measurements
How many copies of an unknown low-rank \(N\)-qubit state does tomography with Pauli measurements need?
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Linear-size exact quantum Fourier transform
Can the exact quantum Fourier transform on \(n\) qubits be implemented with \(O(n)\) one- and two-qubit gates?
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Unrestricted quantum time–space tradeoff for collision finding
Does every unrestricted quantum collision-finding algorithm satisfy \(T^2S=\Omega(N\log N)\)?
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Subquadratic total quantum gate complexity of RSA factoring
Can every balanced RSA semiprime be factored with polynomially subquadratic total quantum gate complexity?
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Quantum Mrs. Gerber lower bound for information combining
Does the quantum Mrs.
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Sparse Hamiltonian learning without short-time control
Can every polynomially sparse Hamiltonian be learned with Heisenberg-limited total evolution time when every oracle call has a fixed minimum duration?
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Sublinear logical quantum space for RSA factoring
Can every balanced RSA semiprime be factored in polynomial time using sublinear logical quantum space?
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Sampling weakly depolarized constant-depth random circuits
Can weak final-layer depolarizing noise make constant-depth two-dimensional Haar-random circuits classically samplable to inverse-polynomial total-variation error?
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Optimal dimension-free copy complexity of shadow tomography
Can list-dependent shadow tomography always achieve dimension-free copy complexity \(O(\log M/\varepsilon^2)\)?
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Multiplicative sparse-access lower bound for quantum linear systems
Does sparse-oracle quantum linear-system solving require \(\Omega(\kappa\sqrt{s}\log(1/\varepsilon))\) queries in the worst case?
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Distillation of every Wigner-negative depolarized Strange state
Can every Wigner-negative state in the depolarized qutrit Strange-state family be distilled arbitrarily accurately by catalyst-free stabilizer operations?
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Conditional-correlation decay in amplitude-damped random circuits
Do computational-basis outputs of one-dimensional Haar-random circuits with fixed amplitude damping have conditional mutual information that decays exponentially with separation, uniformly in circuit depth?
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Unknown-structure Hamiltonian learning from Gibbs states at all temperatures
Can every bounded-degree local Hamiltonian with unknown interaction support be learned efficiently from copies of its Gibbs state at any fixed inverse temperature?
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Optimal single-copy tomography of fermionic Gaussian states
Can arbitrary mixed fermionic Gaussian states be learned to trace distance \(\varepsilon\) from \(\Theta(m^2/\varepsilon^2)\) copies without measurements across copies?
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Uniform modified log-Sobolev constant for one-dimensional Gibbs samplers
Do normalized one-dimensional Chen–Kastoryano–Gilyén Gibbs samplers have a positive modified log-Sobolev constant uniformly in system size at every fixed finite temperature?
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Qutrit magic-distillation thresholds beyond the ternary Golay code
Does any finite qutrit stabilizer-code protocol distill depolarized Strange states above the ternary Golay code’s threshold?
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Pseudorandom unitaries from ordinary random two-qubit circuits
Does a polynomial-length random walk generated by ordinary two-qubit gates form a pseudorandom-unitary ensemble?
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Exact spectral gap of random Pauli rotations
Is the exact Haar-mixing spectral gap of uniformly random Pauli rotations equal to \(2^n(2^n-3)/(8(4^n-1))\) for every \(n\geq4\)?
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Optimal asymptotic distillation yield of dephased T states
What is the optimal catalyst-free asymptotic \(T\)-state yield of a dephased single-qubit \(T\) state?
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Single-copy identity testing of local Gibbs states
What is the minimax copy complexity of identity testing two unknown \(k\)-local Gibbs states using only measurements on individual copies?
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Optimal universal eight-copy concentration to a CCZ state
What is the optimal success probability for a fixed catalyst-free stabilizer protocol that converts eight copies of an arbitrary unknown pure qubit into an exact \(|CCZ\rangle\) state?
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CNOT-count and depth frontier for quantum Golay state preparation
What is the exact CNOT-count and depth frontier for ancilla-free Clifford preparation of the logical zero state of the \([[23,1,7]]_2\) quantum Golay code?
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Endpoint quantum KKL inequality for Boolean observables
Does the Montanaro–Osborne \(L^2\) quantum KKL inequality hold for every Boolean quantum observable?
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Exact constant-bond PEPS mixtures for two-dimensional Gibbs states
Is every fixed-temperature Gibbs state of a bounded finite-range qubit Hamiltonian on the square lattice an exact convex combination of PEPS with system-size-independent bond dimension?
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Scalable pseudorandom unitaries with an independent security parameter
Can pseudorandom-unitary security scale independently of Hilbert-space dimension while construction uses only polynomially many oracle queries?
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Asymptotically good quantum locally testable codes
Does there exist an infinite family of qubit CSS codes with linear rate and linear distance whose check matrices have uniformly bounded row and column weights and constant soundness?
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LOCC entanglement cost of Werner states
What is the entanglement cost of the Werner state \(\rho_{d,p}\) for every integer \(d\geq2\) and every antisymmetric weight \(1/2<p\leq1\), and in particular, does it equal the entanglement of formation?
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Optimal dimension of nine-qubit distance-three codes
Is the known non-additive \(((9,12,3))_2\) code of largest possible dimension, or does a nine-qubit code of dimension thirteen and distance three exist?
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Good quantum LDPC codes with a non-Clifford transversal CCZ gate
Do there exist triples of qubit CSS LDPC codes with linear combined rate and linear distance on which the transversal CCZ gate induces a non-Clifford logical operation?
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Superactivation of bipartite classical secret-key rates
Can two classical sources, each with zero secret-key rate between two honest parties, yield a positive secret-key rate when used jointly?
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LOCC entanglement cost of isotropic states
What is the entanglement cost of the isotropic state \(\sigma_{d,F}\) for every integer \(d\geq2\) and every fidelity \(1/d<F<1\)?
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Minimal block length of permutation-invariant qubit codes
Is the least number of physical qubits on which a two-dimensional permutation-invariant code corrects arbitrary errors on \(t\) qubits equal to \(3t^2+3t+1\) for every integer \(t\geq1\)?
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Superactivation of distillable secret key
Can two bipartite quantum states with zero distillable secret key yield positive distillable secret key when used jointly?
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Efficient determination of the Clifford-hierarchy level of bounded-degree permutation gates
For each fixed degree bound \(d\geq2\), is there a deterministic polynomial-time algorithm that computes the Clifford-hierarchy level of an \(n\)-qubit permutation gate from degree-at-most-\(d\) algebraic normal forms of the permutation and its inverse, or reports…
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Superactivation of two-way secret-key capacity
Can two quantum channels with zero two-way-assisted secret-key capacity have positive secret-key capacity when used jointly?
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Smallest qubit number for a non-semi-Clifford third-level gate
What is the smallest number of qubits on which the third level of the Clifford hierarchy contains a gate that is not semi-Clifford?
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Optimal dimension of seven-qubit distance-two codes
For an exact seven-qubit code that detects every single-qubit error, what is the largest possible code-space dimension: is \(K_{\max}^{(2)}(7,2)\) equal to \(24\), \(25\), or \(26\)?
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Linear-size logarithmic-depth encoders for good quantum LDPC codes
Does some family of asymptotically good qubit CSS LDPC codes admit unitary encoding circuits with linearly many gates and logarithmic depth?
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Finite-energy superactivation of Gaussian amplifiers and additive-noise channels
Is there a single-mode phase-insensitive Gaussian amplifier or additive-noise channel of zero finite-energy quantum capacity that acquires positive finite-energy quantum capacity when used in parallel with a two-mode positive-partial-transpose Gaussian channel…
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Distance of constant-rate quantum XYZ product codes
Do there exist quantum XYZ product codes, built from classical parity-check matrices with uniformly bounded row and column weights, that have constant rate and minimum distance of order \(N^{2/3}\) on \(N\) physical qubits?
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Minimal permutation-invariant qubit code for two errors
Is nineteen the least number of physical qubits on which a two-dimensional code contained in the fully symmetric subspace corrects every error on at most two qubits?
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Non-additive ((6,2,3)) qubit code with transversal T
Does there exist a genuinely non-additive two-dimensional six-qubit code of distance three on which a tensor product of single-qubit unitaries implements the logical \(T\) gate exactly?
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Highest Clifford-hierarchy level of controlled gates with higher-level targets
What is the highest Clifford-hierarchy level that a controlled gate can reach when its target lies in a given level \(k\geq3\) and has a given Pauli periodicity, and is that level bounded independently of the number of target qubits?
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Zero two-way quantum capacity of endpoint Holevo–Werner channels
Is the two-way assisted quantum capacity of the Holevo–Werner channel \(\mathcal W_d\) zero for every integer \(d\geq3\)?
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Geelen’s simulation conjecture for vertex-minor-closed graph classes
Does Geelen’s simulation conjecture hold for every nonempty proper class \(\mathcal C\subsetneq\mathcal G_{\mathrm{fin}}\) of finite simple graphs that is closed under isomorphism, local complementation, and vertex deletion?
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Holevo random-coding exponent for classical–quantum channels
Does every memoryless classical–quantum channel with a finite input alphabet and finite-dimensional output satisfy the random-coding lower bound in Eq. (5), including channels with mixed, noncommuting output states?
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Secret-key capacity of thermal attenuators and amplifiers
What is the secret-key capacity of a thermal attenuator for every transmissivity and thermal noise level?
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Bounded witnesses for every continuous-variable entangled state
Does every entangled bosonic state admit a bounded entanglement witness?
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LOCC entanglement cost of PPT-entangled Gaussian states
What is the exact LOCC entanglement cost of a finite-energy PPT-entangled Gaussian state?
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Additivity of entanglement of formation for two-mode Gaussian states
Is unrestricted entanglement of formation additive on tensor powers of every finite-energy two-mode Gaussian state?
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Gaussian optimality of two-mode entanglement of formation
Does Gaussian entanglement of formation equal unrestricted entanglement of formation for every finite-energy two-mode Gaussian state?
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Visible quantum compression with free shared randomness at the Holevo rate
Does free shared classical randomness make the Holevo information achievable for visible compression of every finite mixed-state ensemble?
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Deterministic quadratic-size vertex-minor-universal graphs
Does there exist an absolute constant \(C>0\) and a deterministic algorithm that, for each integer \(k\geq2\) supplied in unary, runs in time \(k^{O(1)}\) and outputs a \(k\)-vertex-minor-universal graph with at most \(Ck^2\) vertices?
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Exact LOCC entanglement cost of the elegant joint measurement
Is the exact deterministic LOCC entanglement cost of the two-qubit elegant joint measurement \(\mathcal M_{\mathrm{EJM}}\) equal to one ebit when arbitrary finite-dimensional pure entangled resources are allowed?
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Minimum output entropy additivity for qubit channels
Is minimum output von Neumann entropy additive whenever one channel is a qubit channel?
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Existence of a ((7,3,3)) qubit code
Does a qubit quantum error-correcting code with parameters \(((7,3,3))_2\) exist, that is, a three-dimensional subspace of seven qubits with minimum distance at least three?
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Low-energy Holevo additivity for a squeezed thermal attenuator
Is energy-constrained Holevo information additive for a squeezed thermal attenuator below its water-filling threshold?
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Universality of arbitrary self-adjoint polynomial Hamiltonians
Can every physical bosonic unitary be approximated by a single self-adjoint polynomial Hamiltonian evolution at any finite input energy?
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Compact-time validity of the rotating wave approximation
Does the Jaynes–Cummings evolution approximate the Rabi evolution strongly and uniformly on each compact time interval in the high-frequency limit?
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Two-way distillable entanglement of mixed two-mode Gaussian states
What is the exact two-way distillable entanglement of a finite-energy mixed two-mode Gaussian state?
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Optimal success probability for Gaussian processing of non-Gaussian entanglement
What is the maximal success probability for preparing an approximate two-mode squeezed vacuum from finitely many copies of a non-Gaussian state?
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Two-way quantum capacity of a thermal amplifier
What is the two-way quantum capacity of a thermal amplifier for every gain and thermal noise level?
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Gaussian-input preservation under a Gaussian reference extension
Does preservation of Gaussian inputs by a trace-decreasing operation imply preservation when a Gaussian reference is attached?
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Strict stationary squeezing limit for one-mode thermal diffusion
Does every stable one-mode thermal Gaussian diffusion obey the sharp stationary squeezing bound \(\lambda_{\min}(V_\infty)>\nu/2\)?
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Single-seed post-processing of Gaussian POVM densities
Is every POVM with Gaussian density a classical post-processing of one displaced Gaussian seed?
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Unassisted quantum capacity of a thermal amplifier
What is the unassisted quantum capacity of a thermal amplifier for every gain and thermal noise level?
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Optimal output monitoring for parametric-oscillator squeezing
Can any causal output measurement improve the mean conditional position squeezing achieved by ideal homodyne detection?
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Permanent anticoncentration for complex Gaussian matrices
Do complex Gaussian permanents satisfy a polynomial lower-tail bound at their root-mean-square scale?
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Fully polynomial sampling of boson sampling with constant photon transmission
For every fixed rational transmission \(0<\eta<1\), is boson sampling with independent photon loss classically samplable in fully polynomial time?
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Exact covariance criterion for bipartite Gaussian separability
What necessary and sufficient condition on a covariance matrix characterizes bipartite Gaussian separability?
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LOCC discrimination of three maximally entangled states
Can every set of three orthogonal maximally entangled pure states on \(\mathbb C^d\otimes\mathbb C^d\), for every integer \(d\geq4\), be distinguished exactly from one copy using unrestricted two-way local operations and classical communication (LOCC)?
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Exponential entanglement cost with simultaneous classical communication
Does there exist a family of bipartite unitaries requiring exponentially many shared Bell pairs for fixed-error implementation when classical communication is limited to one simultaneous exchange?
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Quasi-linear graph-state resources for Pauli pairability
Does there exist an absolute constant \(a\geq0\) such that \(N_{\mathrm P}(k)=O(k[\log_2(k+1)]^a)\) as the integer \(k\geq2\) tends to infinity?
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Distributed compression of bipartite pure-state ensembles
Determine the achievable rate region for independent local compression of an arbitrary finite ensemble of bipartite pure states.
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Exact privacy-amplification exponent under relative-entropy security
What is the optimal exponential rate of decay of relative-entropy insecurity achievable by two-universal hashing against quantum side information at every positive extraction rate below the conditional entropy?
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Quantum capacity of higher-dimensional depolarizing channels
What is the unassisted quantum capacity of a depolarizing channel in every finite dimension \(d\geq3\)?
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Chernoff exponents under separable and LOCC tests across copies
Determine the optimal symmetric discrimination exponents for two faithful noncommuting states when measurements must be separable across copies or implemented by local operations and classical communication (LOCC) among the copies.
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Optimal random-unitary decomposition of symmetric Werner–Holevo channels
For every odd integer \(d\geq5\), do there exist \(r=d(d+1)/2\) unitary operators \(U_1,\ldots,U_r\) on \(\mathbb C^d\) such that
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Two-way secret-key capacity of the qubit amplitude-damping channel
What is the exact secret-key capacity \(K(\mathcal A_p)\) of the qubit amplitude-damping channel for every damping probability \(p\in[0,1]\)?
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Qubit code distance bound
Does there exist a sequence of qubit quantum codes \(Q_j\subseteq(\mathbb C^2)^{\otimes n_j}\), with \(n_j\to\infty\), \(\dim Q_j\geq2\), and minimum distances \(d_j\), satisfying
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Polynomial-time quantum algorithm for the Dihedral Hidden Subgroup Problem
Does the Dihedral Hidden Subgroup Problem admit a quantum algorithm whose running time is polynomial in the input length?
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Computability of ordinary quantum capacity
Is the ordinary unassisted quantum capacity a computable function of a finite description of a finite-dimensional quantum channel?
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Ordinary-Petz fidelity remainder for relative-entropy data processing
Does the ordinary Petz recovery map give a universal fidelity remainder for monotonicity of quantum relative entropy?
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Positivity threshold for thermal-attenuator quantum capacity
For \(0<\eta<1\) and \(0<\nu<\infty\), let the single-mode bosonic thermal attenuator be
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Unclassified existence parameters for homogeneous AME states
For which of the parameter pairs specified below does an absolutely maximally entangled state exist?
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Energy-constrained quantum capacity of a thermal attenuator
For \(0<\eta<1\) and \(0<\nu<\infty\), let the single-mode bosonic thermal attenuator be
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Distillable entanglement of the antisymmetric state
What is the exact LOCC distillable entanglement \(E_D(\rho_{\mathrm{as}}^{(d)})\) of the \(d\times d\) antisymmetric state, for each integer \(d\ge3\)?
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Universal strong converse for the classical capacity
Does every finite-dimensional quantum channel satisfy the strong converse property at its unassisted classical capacity, or does some channel admit a sequence of codes above capacity whose average success probability stays bounded away from zero?
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Is bipartite Quantum Max-Cut in BPP?
Is the following bipartite Quantum Max-Cut promise problem in \(\mathrm{BPP}\)?
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Average-case approximation hardness of random Ising partition functions
Is it \(\#\mathrm{P}\)-hard to approximate \(|Z_R|^2\) to relative multiplicative error \(a+o(1)\) on a \(b\) fraction of random Ising instances?
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Average-case approximation hardness of squared normalized gaps of random cubic polynomials
Is it \(\#\mathrm{P}\)-hard to approximate \(\operatorname{ngap}(f)^2\) to relative multiplicative error \(a+o(1)\) on a \(b\) fraction of uniformly random degree-3 polynomials over \(\mathbb{F}_2\)?
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Bidirectional classical-communication cost of bipartite channel simulation
What is the asymptotic classical-communication cost of simulating a bipartite quantum channel with bidirectional classical communication and non-signalling assistance?
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Advantage of fully general superchannel-discrimination strategies
Can a fully general adaptive strategy attain a larger Stein exponent than every nested-adaptive strategy for discriminating two quantum superchannels?
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Uniformly efficient HSW pretty-good decoding
Can Holevo–Schumacher–Westmoreland codebooks operating at every rate below their ensemble Holevo information be chosen so that their square-root, or pretty-good, measurements have uniform quantum implementations whose cost is polynomial in the blocklength and…
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Multimode constrained output entropy of a pure-loss channel
Is Guha’s multimode Strong Conjecture 2 true for every correlated input state?
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Universal finite truncation of quantum and private capacities
Do there exist channel-independent finite integers \(m_Q\) and \(m_P\) that determine, respectively, the quantum capacity and the private classical capacity of every finite-dimensional quantum channel?
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Exactly solvable nondegradable quantum channels
Does there exist a finite-dimensional quantum channel that is neither degradable nor antidegradable and whose unassisted quantum capacity is known exactly?
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Structural criterion for classical–entanglement trade-off advantage
Characterize the finite-dimensional quantum channels for which joint classical–entanglement coding strictly outperforms time sharing between unassisted and unlimited-entanglement classical communication.
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Strong converse for general mixed-state quantum compression
Does general finite-dimensional i.i.d.
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Quantum capacity of the Gaussian random-displacement channel
What is the unconstrained, unassisted quantum capacity of the single-mode Gaussian random-displacement channel?
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Minimum LU–LC counterexample for graph states
What is the least number of qubits for which two graph states can be locally unitary equivalent without being locally Clifford equivalent?
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Equal-weight low-Choi-rank decompositions of quantum channels
Can every finite-dimensional quantum channel be written as the uniform mixture of \(d_B\) channels whose Choi ranks are at most the input dimension?
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Umegaki relative entropy of local recovery
Does the conditional mutual information of every finite-dimensional tripartite state dominate its Umegaki relative entropy of local recovery?
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Multiplicativity for polarized Werner–Holevo channels
For every integer \(d\geq3\), every \(x\in(0,1)\), and every \(1<p<2\), is the maximal output Schatten \(p\)-norm of the polarized Werner–Holevo channel multiplicative on two identical copies?
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The quantum PCP conjecture
Is the constant-relative-gap local Hamiltonian problem QMA-hard?
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Delayed-onset additivity violation for minimum output Rényi entropy
Does there exist a finite-dimensional quantum channel whose minimum output Rényi entropy is additive for every tensor power below some order and first becomes strictly subadditive at that order?
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Finite nontrivial LU moduli of AME states
Do there exist integers \(N,d\geq 2\) for which the absolutely maximally entangled states of \(N\) qudits of local dimension \(d\) form finitely many, but more than one, local-unitary equivalence classes?
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Generalized Stein lemma for fully quantum channel resources
Let \(\mathcal N:\mathcal L(A)\to\mathcal L(B)\) be a finite-dimensional quantum channel.
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Unconditional classical verification with one quantum prover
Does every language \(L\in\mathsf{BQP}\) admit a single-prover interactive proof with a fully classical verifier, an efficient quantum honest prover, and information-theoretic soundness?
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Resources for implementing Gibbs-preserving channels
What are the exact one-shot coherence and work costs of implementing an arbitrary Gibbs-preserving channel by thermal operations?
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Collective cost of tensor-power state preparation
Determine the asymptotic weighted circuit cost of preparing tensor powers of a known pure state, and characterize when collective preparation is cheaper per copy than independent preparation.
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Trace-exponential lower bound for matrix-word averages
Is the normalized trace average of all words in two positive-definite matrices always bounded below by the corresponding trace exponential?
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Minimal-support frontier for absolutely maximally entangled states
Determine, for every integer \(d\geq 2\), the largest number \(\mathcal N(d)\) of \(d\)-level parties that admit an absolutely maximally entangled state of minimal computational-basis support.
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Lockability of two-way distillable entanglement
Can discarding one local qubit reduce two-way distillable entanglement by an arbitrarily large amount?
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Absolute separability from spectra
Characterize the spectra of bipartite states that remain separable under every global unitary, and decide whether absolute separability equals absolute positivity under partial transpose in higher local dimensions.
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Tough error models
Determine \(c(e,n)\) and construct tough error models in the sense of [KW05].
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Secret key from every entangled state
Does every finite-dimensional entangled bipartite state have positive asymptotic distillable secret key?
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Square-root remainder in generalized quantum equipartition
Let \(H\) be a \(d\)-dimensional Hilbert space.
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Constant trace-distance separability testing
What is the computational complexity of testing bipartite separability with a constant trace-distance promise gap?
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Honest-party lockability of distillable key
Can loss of one qubit held by an honest party reduce the two-way distillable secret key by an arbitrarily large amount?
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Real four-quhex absolutely maximally entangled state
Does there exist an \(\operatorname{AME}(4,6)\) state whose coefficients are real in a product basis?
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Capacity-achieving codes for amplitude damping
What is the constructive quantum code for achieving the quantum capacity \(\mathcal{Q}(\mathcal A_p)\) of the qubit amplitude-damping channel
Solved
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