Single-letter one-way distillable entanglement for non-degradable states

2026-03-24

AI Summary

This work addresses the absence of a single-letter formula for one-way distillable entanglement in non-degraded quantum states and the unknown conditions for its additivity. Focusing on non-degraded, non-PPT states, the authors introduce a relaxed degradability condition combined with stability under orthogonal flagging to establish a single-letter expression. They propose two new sufficient conditions for additivity—“regularized more noisy” and “informationally degradable”—and demonstrate that these guarantee the additivity of one-way distillable entanglement. The study further reveals the single-letter nature of orthogonally flagged mixed states and develops a generalized spin alignment principle to achieve entropy minimization. By integrating LOCC protocol analysis, Rényi-2 entropy, and the correspondence between Choi states and generalized direct-sum channels, the authors derive single-letter formulas for three classes of non-degraded states and establish the additivity of the quantum capacity for the corresponding generalized direct-sum channels.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

The one-way distillable entanglement is a central operational measure of bipartite entanglement, quantifying the optimal rate at which maximally entangled pairs can be extracted by one-way LOCC. Despite its importance, it is notoriously hard to compute, since it is defined by a regularized optimization over many copies and adaptive one-way protocols. At present, single-letter formulas are only known for (conjugate) degradable and PPT states. More generally, it has remained unclear when one-way distillable entanglement can still be additive beyond degradability and PPT settings, and how such additivity relates to additivity questions of quantum capacity of channels. In this paper, we address this gap by identifying three explicit families of non-degradable and non-PPT states whose one-way distillable entanglement is nevertheless single-letter. First, we introduce two weakened degradability-type conditions--regularized less-noisy and informationally degradable--and prove that each guarantees additivity and hence a single-letter formula. Second, we show a stability result for orthogonally flagged mixtures: when one component has orthogonal support on Alice's system and zero one-way distillable entanglement, the mixture remains single-letter, even though degradability is typically lost under such mixing. Finally, we propose a generalized spin-alignment principle for entropy minimization in tensor-product settings, which we establish in several key cases, including a complete Rényi-2 result. As an application, we obtain additivity results for generalized direct-sum channels and their corresponding Choi states.

Problem

• one-way distillable entanglement
• single-letter formula
• non-degradable states
• additivity
• quantum entanglement

Innovation

• single-letter formula
• one-way distillable entanglement
• non-degradable states
• additivity
• spin-alignment principle