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Faithful Squashed Entanglement

Fernando G. S. L. Brandao, Matthias Christandl, Jon Yard

TL;DR

This work proves that squashed entanglement is faithful by deriving a quantitative lower bound on conditional mutual information in terms of a distance to the set of separable states, measured in a one-way LOCC (and Frobenius) norm. The bound I(A;B|E) ≥ (1/(8 ln 2)) ||ρAB − SAB||^2, together with a bound for E_sq, implies that any entangled state has strictly positive squashed entanglement. The authors connect state redistribution and the regularised relative entropy of entanglement to obtain a monogamy-like inequality and a dimension-independent separation from separable states, which in turn yields practical consequences: a quasipolynomial-time weak-membership algorithm for separability in LOCC/Euclidean norms and a new QMA characterization with a constant number of unentangled proofs. The results illuminate the structure of near-Markov states, data-hiding phenomena, and broader complexity implications in quantum information theory.

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.

Faithful Squashed Entanglement

TL;DR

This work proves that squashed entanglement is faithful by deriving a quantitative lower bound on conditional mutual information in terms of a distance to the set of separable states, measured in a one-way LOCC (and Frobenius) norm. The bound I(A;B|E) ≥ (1/(8 ln 2)) ||ρAB − SAB||^2, together with a bound for E_sq, implies that any entangled state has strictly positive squashed entanglement. The authors connect state redistribution and the regularised relative entropy of entanglement to obtain a monogamy-like inequality and a dimension-independent separation from separable states, which in turn yields practical consequences: a quasipolynomial-time weak-membership algorithm for separability in LOCC/Euclidean norms and a new QMA characterization with a constant number of unentangled proofs. The results illuminate the structure of near-Markov states, data-hiding phenomena, and broader complexity implications in quantum information theory.

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.

Paper Structure

This paper contains 9 sections, 15 theorems, 91 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Results
  3. Proof of Theorem
  4. Proof of Lemma \ref{['nonlockability']}
  5. Proof of Lemma \ref{['almostmonogamy']}
  6. Proof of Lemma \ref{['lowerboundnorm']}
  7. Proof of Theorem
  8. Proofs of Corollaries
  9. Acknowledgments

Key Result

Theorem 1

For every tripartite finite-dimensional state $\rho_{ABE}$,

Figures (1)

  • Figure 1: General protocol for redistributing the $B$ parts of a pure state $|\psi\rangle_{ABEE'}^{\otimes n}$ where the sender holds $E$ and the receiver holds $E'$. The protocol consumes maximal entanglement on $DD'$, transmits the quantum system $G$, and produces maximal entanglement on $FF'$. The dimensions of $D$, $F$ and $G$ typically depend exponentially on $n$. The protocol is asymptotically reversible.

Theorems & Definitions (27)

  • Theorem
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Definition
  • Corollary 4
  • Definition
  • Corollary 5
  • Lemma 1
  • Lemma 2
  • ...and 17 more