"Squashed Entanglement" - An Additive Entanglement Measure

• Existing entanglement measures have struggled to satisfy all desired axioms, including LOCC monotonicity, vanishing on separable states, convexity, additivity, and continuity.Source: §I · p. 1
• Squashed entanglement is convex, additive on tensor products, and superadditive in general.Source: §I · p. 1
• The paper motivates its definition by analogy with intrinsic information from classical cryptography.Source: §I · p. 1
• The authors prove continuity only on the interior of the state space and provide evidence for general continuity through a conjectured Fannes-type inequality for conditional von Neumann entropy.Source: §I · p. 1

Definition

• Squashed entanglement is the infimum of 1/2 I(A; B|E) over all extensions ρABE of ρAB.The conditional mutual information is I(A; B|E) := S(AE) + S(BE) − S(ABE) − S(E).Source: §II · p. 1

Normalization and separability

• For pure states, squashed entanglement equals the entropy-based entanglement E(|ψ⟩), while it vanishes for separable states.The vanishing result uses a zero-conditional-mutual-information extension for separable states; finite-dimensional zero conditional mutual information also implies separability.Source: §II · pp. 2–4

Monotonicity and convexity

• Squashed entanglement is an entanglement monotone: it does not increase under LOCC and is convex.The proof uses auxiliary systems, unitary evolution, strong subadditivity, the chain rule, and the validity of the resulting extensions.Source: §II · p. 2

Additivity

• Squashed entanglement is superadditive in general and additive on tensor-product states.Superadditivity follows from strong subadditivity and the infimum over extensions; subadditivity on tensor products completes additivity.Source: §II · p. 3

Bounds

• Squashed entanglement is upper bounded by entanglement of formation and entanglement cost, and lower bounded by distillable entanglement.The upper bound on entanglement of formation follows because the minimization ranges over all extensions, including those induced by pure-state ensembles.Source: §II · pp. 3–4

Continuity and limitations

• Continuity is the only stated property without a proof; evidence includes continuity on the interior and a conjectured Fannes-type inequality that is proved classically.The paper also notes that unbounded extension dimension prevents the finite-dimensional zero-conditional-mutual-information result from directly proving strict positivity for all entangled states.Source: §II · p. 4

Example

• For a two-qutrit totally antisymmetric state, EF = EC = 1 while the paper obtains a smaller squashed-entanglement value of 1/2 log 3 ≈ 0.792.The best known Rains-bound upper bound cited there is log 5/3 ≈ 0.737, and it remains open whether squashed entanglement can be smaller than that bound.Source: §II · p. 4

Distillation bound

• The lower bound on distillable entanglement is established by applying the measure to arbitrary LOCC distillation protocols and taking n → ∞ and δ → 0.The argument uses continuity near maximally entangled states together with additivity and normalization on Bell states.Source: §II · p. 4

Classical motivation

• A proposed quantum analog using conditional mutual information helped motivate the current squashed-entanglement construction.The earlier proposal had several desirable entanglement-measure properties and initiated the discussion developed here.Source: §III · p. 5

Definition correspondence

• Every extension of ρAB can be generated from a purification by applying a quantum operation to the purifying system, linking the two definitions structurally.The construction uses a purification |Ψ⟩ABC, an extension purification |Φ⟩ABEE′, partial trace over E′, and a quantum operation Λ.Source: §III · p. 5

Continuity contrast

• The minimization differs in continuity behavior: unlike intrinsic information, squashed entanglement lacks an easy continuity proof because the relevant restriction technique does not apply.Intrinsic information can restrict the minimizing auxiliary variable to a bounded domain, whereas the analogous restriction is unavailable here.Source: §III · p. 5

Classical motivation

• Intrinsic information measures correlations in secret-key agreement and upper-bounds the secret-key rate S = S(X; Y ||Z).The analogous quantum operational quantity is distillable entanglement.Source: §III · p. 5

Operational analogy

• Squashed entanglement is an upper bound for distillable entanglement, paralleling intrinsic information’s upper bound on the secret-key rate.The analogy identifies distillable entanglement as the quantum counterpart of secret-key rate.Source: §III · p. 5

Formation analogy

• The intrinsic-information analogy also extends to formation bounds through a classical lower bound on the formation cost of correlations from secret correlations.The paper connects this counterpart to its quantum formation results.Source: §III · p. 5

Obstruction

• A universal dimension bound for the extension system E would make the infimum a minimum and yield continuity by compactness arguments.The paper does not establish such a bound; it notes that this would also make Esq(ρAB) = 0 imply separability using.Source: §IV · p. 6

Proposed route

• The authors conjecture a Fannes-type inequality for conditional von Neumann entropy whose validity would imply continuity of squashed entanglement.The conjecture concerns nearby bipartite density operators measured in trace distance.Source: §IV · p. 6
• The conjectured continuity bound is notable because it depends only on the dimension of system A, not the dimension of system B.A direct application of Fannes’ inequality would introduce additional logarithmic dependence on dB.Source: §IV · p. 6

Status

• The conjectured inequality is proved in the classical case, while the general quantum case remains unresolved.The paper also notes that the measure is upper semicontinuous as an infimum of continuous functions under bounded extension dimension.Source: §IV · p. 6

Established continuity

• Squashed entanglement is continuous on the interior of the state set and on the relative interior of every face.These are the faithful states and their corresponding relative-interior subsets.Source: §IV · p. 6
• Continuity near pure states additionally implies continuity on all rank-2 density operators.This combines the pure-state continuity result with the broader continuity proposition.Source: §IV · p. 6

• Squashed entanglement is lower bounded by distillable entanglement and upper bounded by entanglement cost.Source: §V · p. 7
• Its definition enables simple proofs of LOCC monotonicity, convexity, tensor-product additivity, and general superadditivity through strong subadditivity of quantum entropy.Source: §V · p. 7
• Continuity remains unproved in general, although continuity is established near pure states and in the relative interior of state-space faces.Source: §V · p. 7
• A conjectured Fannes-type inequality for conditional von Neumann entropy would imply general continuity, and proving it remains a central open challenge.Source: §V · p. 7
• Whether nonseparable states can have zero squashed entanglement remains open, as does its general relationship to other entanglement measures.Source: §V · p. 7

• The classical-case proof treats states whose components are diagonal in a common basis, so the bipartite states represent joint probability distributions.Source: Appendix A · p. 7
• The proof proceeds by estimating the relevant entropy differences for the specified state forms, using the triangle inequality in successive steps.Source: Appendix A · p. 7
• The estimates apply Fannes inequality to component states, use concavity of its upper bound, and establish the required average bound P_k p_k ε_k ≤ 2ε.Source: Appendix A · p. 7
• For pure states, the conjecture follows directly from Fannes inequality because S(AB) = 0 and S(A) = S(B).Source: Appendix A · p. 8