Faithful Squashed Entanglement

Abstract

Squashed entanglement is a measure for the entanglement of bipartite quantum states. In this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states. This implies that squashed entanglement is faithful, that is, strictly positive if and only if the state is entangled. We derive the bound on squashed entanglement from a bound on quantum conditional mutual information, which is used to define squashed entanglement and corresponds to the amount by which strong subadditivity of von Neumann entropy fails to be saturated. Our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality. The proof is based on two recent results from quantum information theory: the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing. The distance to the set of separable states is measured by the one-way LOCC norm, an operationally-motivated norm giving the optimal probability of distinguishing two bipartite quantum states, each shared by two parties, using any protocol formed by local quantum operations and one-directional classical communication between the parties. A similar result for the Frobenius or Euclidean norm follows immediately. The result has two applications in complexity theory. The first is a quasipolynomial-time algorithm solving the weak membership problem for the set of separable states in one-way LOCC or Euclidean norm. The second concerns quantum Merlin-Arthur games. Here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations thereby providing a new characterisation of the complexity class QMA.

24 pages, 1 figure, 1 table. Due to an error in the published version, claims have been weakened from the LOCC norm to the one-way LOCC norm

References in corpus (17)

• Entanglement detection
• Distillation of secret key and entanglement from quantum states
• Coding Theorem and Strong Converse for Quantum Channels
• A complete family of separability criteria
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Aspects of generic entanglement
• Quantifying Entanglement with Witness Operators
• One-and-a-half quantum de Finetti theorems
• Entanglement Theory and the Second Law of Thermodynamics
• The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography
• Separable Multipartite Mixed States - Operational Asymptotically Necessary and Sufficient Conditions
• Hiding Quantum Data
• Testing product states, quantum Merlin-Arthur games and tensor optimisation
• Construction of extremal local positive operator-valued measures under symmetry
• Highly Entangled States With Almost No Secrecy
• A paradigm for entanglement theory based on quantum communication
• Distinguishing sets of quantum states

Cited by in corpus (112)

• Fundamental rate-loss tradeoff for optical quantum key distribution
• Holographic Entanglement of Purification
• Holographic Mutual Information is Monogamous
• Computing quantum discord is NP-complete
• Quantum conditional mutual information and approximate Markov chains
• Exponential Decay of Correlations Implies Area Law
• The squashed entanglement of a quantum channel
• Should Entanglement Measures be Monogamous or Faithful?
• Principles of Quantum Communication Theory: A Modern Approach
• Quantum Conditional Mutual Information, Reconstructed States, and State Redistribution
• The Church of the Symmetric Subspace
• Negativity and strong monogamy of multi-party quantum entanglement beyond qubits
• Fidelity of recovery, geometric squashed entanglement, and measurement recoverability
• Semidefinite programming relaxations for quantum correlations
• Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps
• Limitations to sharing entanglement
• Catalytic Transformations of Pure Entangled States
• The Fidelity of Recovery is Multiplicative
• High-Dimensional Entanglement in States with Positive Partial Transposition
• General polygamy inequality of multi-party quantum entanglement
• Bounds for Entanglement via an Extension of Strong Subadditivity of Entropy
• Amortized entanglement of a quantum channel and approximately teleportation-simulable channels
• On a gap in the proof of the generalised quantum Stein's lemma and its consequences for the reversibility of quantum resources
• Rényi squashed entanglement, discord, and relative entropy differences
• Distinguishing multi-partite states by local measurements
• An improved semidefinite programming hierarchy for testing entanglement
• Rényi generalizations of quantum information measures
• Product-state Approximations to Quantum Ground States
• Multipartite quantum correlations and local recoverability
• Detection of Multiparticle Entanglement: Quantifying the Search for Symmetric Extensions
• Disentanglement Cost of Quantum States
• Semidefinite programming hierarchies for constrained bilinear optimization
• Energy-constrained two-way assisted private and quantum capacities of quantum channels
• Asymptotic state transformations of continuous variable resources
• Approximate reversal of quantum Gaussian dynamics
• Bounds on entanglement distillation and secret key agreement for quantum broadcast channels
• Squashed entanglement in infinite dimensions
• Independence of work and entropy for equal energetic finite quantum systems: Passive state energy as an entanglement quantifier
• Quantum Shannon Theory
• Exponential clustering of bipartite quantum entanglement at arbitrary temperatures
• From log-determinant inequalities to Gaussian entanglement via recoverability theory
• Asymptotic State Discrimination and a Strict Hierarchy in Distinguishability Norms
• Short Multi-Prover Quantum Proofs for SAT without Entangled Measurements
• Generalized Entropy and Global Quantum Discord in Multi-party Quantum systems
• Non-classical correlations in quantum mechanics and beyond
• Extendibility of bosonic Gaussian states
• Conditional mutual information and self-commutator
• Approximate quantum Markov chains
• Operational meaning of quantum measures of recovery
• A lower bound of quantum conditional mutual information
• Testing symmetry on quantum computers
• Quantum Information Complexity and Amortized Communication
• Strong monogamy of multi-party quantum entanglement for partially coherently superposed states
• Limitations of semidefinite programs for separable states and entangled games
• Ancilla dimension in quantum channel discrimination
• Markovianizing Cost of Tripartite Quantum States
• Improved Soundness for QMA with Multiple Provers
• Emergent classicality in general multipartite states and channels
• Distillable entanglement under dually non-entangling operations
• Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states
• Highly entangled stationary states from strong symmetries
• Causal and Non-Causal Revivals of Information: A New Regime of Non-Markovianity in Quantum Stochastic Processes
• No-go theorem for entanglement distillation using catalysis
• The black hole information puzzle and the quantum de Finetti theorem
• Entanglement manipulation and distillability beyond LOCC
• On Dimension Bounds for Auxiliary Quantum Systems
• Modulus of convexity for operator convex functions
• Approximate orthogonality of permutation operators, with application to quantum information
• Entanglement monogamy via multivariate trace inequalities
• $k$-extendibility of high-dimensional bipartite quantum states
• Two-message quantum interactive proofs and the quantum separability problem
• Communication Cost for Non-Markovianity of Tripartite Quantum States: A Resource Theoretic Approach
• More on the upper bound of holographic n-partite information
• Estimating operator norms using covering nets
• Zonoids and sparsification of quantum measurements
• On the equivalence of separability and extendability of quantum states
• Optimized Correlation Measures in Holography
• A fermionic de Finetti theorem
• Universal Spreading of Conditional Mutual Information in Noisy Random Circuits
• Probabilistic state synthesis based on optimal convex approximation
• A strengthened monotonicity inequality of quantum relative entropy: A unifying approach via Rényi relative entropy
• Learning Properties of Quantum States Without the I.I.D. Assumption
• Computable lower bounds on the entanglement cost of quantum channels
• Locally-Measured Rényi Divergences
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• On the Power of Quantum Distributed Proofs
• Heralded Channel Holevo Superadditivity Bounds from Entanglement Monogamy
• Unifying Entanglement with Uncertainty via Symmetries of Observable Algebras
• On discontinuity of information characteristics of quantum systems and channels
• The squashed entanglement of the noiseless quantum Gaussian attenuator and amplifier
• Maximum Violation of Monogamy of Entanglement for Indistinguishable Particles by Measures that are Monogamous for Distinguishable Particles
• Strong polygamy of quantum correlations in multi-party quantum systems
• Gaussian intrinsic entanglement for states with partial minimum uncertainty
• Operational Resource Theory of Non-Markovianity
• Squashed Entanglement from Generalized Rindler Wedge
• Percolation of secret correlations in a network
• Fundamental limitations to key distillation from Gaussian states with Gaussian operations
• Reexamination of strong subadditivity: A quantum-correlation approach
• Generating random Gaussian states
• Trade-off between Squashed Entanglement and Concurrence in Bipartite Quantum States
• A de Finetti theorem for quantum causal structures
• The multimode conditional quantum Entropy Power Inequality and the squashed entanglement of the multimode extreme bosonic Gaussian channels
• The Dynamical Additivity And The Strong Dynamical Additivity Of Quantum Operations
• Upper bounds on the leakage of private data and operational approach to markovianity
• Quantum PCPs: on Adaptivity, Multiple Provers and Reductions to Local Hamiltonians
• Squashed quantum non-Markovianity: a measure of genuine quantum non-Markovianity in states
• Entanglement cost hierarchies in quantum fragmented mixed states
• Topological squashed entanglement: nonlocal order parameter for one-dimensional topological superconductors
• Shorter unentangled proofs for Ground State Connectivity
• Testing quantum satisfiability
• Unveiling NPT bound problem: From Distillability Sets to Inequalities and Multivariable Insights
• Tapping into Permutation Symmetry for Improved Detection of k-Symmetric Extensions