[Paper Review] Reduction criterion of separability and limits for a class of protocols of entanglement distillation

Michał Horodecki, Paweł Horodecki|ArXiv.org|Aug 7, 1997

Quantum Information and Cryptography2 references20 citations

TL;DR

This paper introduces the reduction criterion of separability—a necessary and sufficient condition for separability in 2×2 and 2×3 systems—using a positive map that generalizes the transposition map. It proves that any quantum state violating this criterion can be distilled into a maximally entangled state via a generalized entanglement distillation protocol, while states satisfying the criterion cannot be distilled, thus establishing a fundamental limit on distillability for higher-dimensional systems.

ABSTRACT

We analyse the problem of distillation of entanglement of mixed states in higher dimensional compound systems. Employing the positive maps method [M. Horodecki et al., Phys. Lett. A 223 1 (1996)] we introduce and analyse a criterion of separability which relates the structures of the total density matrix and its reductions. We show that any state violating the criterion can be distilled by suitable generalization of the two-qubit protocol which distills any inseparable two-qubit state. Conversely, all the states which can be distilled by such a protocol must violate the criterion. The proof involves construction of the family of states which are invariant under transformation $\varrho o U\otimes U^*\varrho U^\dagger\otimes U^{*\dagger}$ where $U$ is a unitary transformation and star denotes complex conjugation. The states are related to the depolarizing channel generalized to non-binary case.

Motivation & Objective

• To address the fundamental question of which mixed quantum states can be distilled into maximally entangled states using local operations and classical communication (LOCC).
• To develop a physically motivated separability criterion based on positive maps that generalizes the Peres criterion for higher-dimensional systems.
• To establish a necessary and sufficient condition for distillability in N×N systems by linking violation of the reduction criterion to effective distillation protocols.
• To clarify the physical meaning of positive but not completely positive maps in the context of entanglement detection and distillation.

Proposed method

• Proposes a positive map Λ(σ) = I Tr(σ) - σ, which acts on density matrices and defines the reduction criterion for separability.
• Uses the equivalence between positive maps and positive operators in tensor product spaces (via the Jamiołkowski isomorphism) to analyze the map’s structure.
• Demonstrates that the map is decomposable (specifically, Λ = T ∘ Γ with Γ completely positive), linking it to physical operations like time reversal (transposition).
• Constructs a family of states invariant under the transformation ϱ → U⊗U*ϱU†⊗U*†, which generalizes the depolarizing channel to non-binary systems.
• Applies the reduction criterion to show that any state violating it can be distilled via a generalized version of the two-qubit distillation protocol.
• Uses spectral decomposition and operator representations to prove that the map’s positivity under partial transposition implies complete positivity of its transposed counterpart.

Experimental results

Research questions

• RQ1 Can all inseparable mixed states in higher-dimensional systems be distilled into maximally entangled states via LOCC protocols?
• RQ2 What is the operational criterion that separates distillable from non-distillable states in N×N systems?
• RQ3 How does the reduction criterion relate to the structure of entanglement and the action of positive maps on density matrices?
• RQ4 Can the reduction criterion be physically interpreted, and does it correspond to a measurable or implementable operation in quantum information protocols?
• RQ5 What is the role of unitary invariance and symmetry under U⊗U* transformations in characterizing distillable states?

Key findings

• The reduction criterion is necessary and sufficient for separability in 2×2 and 2×3 systems, generalizing the Peres criterion.
• Any quantum state violating the reduction criterion can be distilled into a maximally entangled state using a generalized version of the two-qubit distillation protocol.
• States satisfying the reduction criterion cannot be distilled by any protocol based on the same structure as the two-qubit protocol, establishing a fundamental limit.
• The positive map Λ(σ) = I Tr(σ) - σ is decomposable and physically meaningful, with Λ = T ∘ Γ where Γ is completely positive, linking it to time reversal and physical operations.
• The family of states invariant under U⊗U* transformations generalizes the depolarizing channel to non-binary systems and plays a key role in proving the distillation threshold.
• The reduction criterion is equivalent to the positivity of the operator D = (P_+)_{A}⊗I - P_+ under partial transposition, confirming its physical consistency and operational relevance.