Converting entanglement into ensemble basis-free coherence

TL;DR

The paper tackles how much ensemble coherence can be generated from a fixed amount of entanglement. It develops two concrete methods: (i) a von Neumann Hadamard measurement on one half of a bipartite state to produce a B92-like ensemble, showing coherence can equal the initial entanglement $E$ for $E\le 0.4$, and (ii) a family of rank-1 symmetric POVMs based on symmetric states that generate BB84/three-state QKD ensembles, with coherence saturating as the number of POVM elements $N$ grows and satisfying $E = \lim_{N\to\infty}(C(\\mathcal{E}_{\\text{sym}})+I_{acc}(\\mathcal{E}_{\\text{sym}}))$. These results connect the entanglement resource to basis-free ensemble coherence and accessible information, highlighting how quantumness in ensembles depends on the measurement strategy and the entanglement budget. The work also outlines limitations (pure bipartite states) and future directions (multipartite settings and broader POVMs).

Abstract

The resource theory of coherence addresses the extent to which quantum properties are present in a given quantum system. While coherence has been extensively studied for individual quantum states, measures of coherence for ensembles of quantum states remain an area of active research. The entanglement-based approach to ensemble coherence - where a partial measurement of an entangled state generates an ensemble - relates the ensemble coherence to both the initial entanglement and the measurement's uncertainty. This paper presents two methods for generating ensemble coherence from a fixed amount of entanglement. The first method involves applying a von Neumann measurement to one part of a non-maximally entangled bipartite state, resulting in a pair of non-orthogonal states whose coherence can equal the initial entanglement. The second method considers a class of symmetric observables capable of generating ensembles used in quantum key distribution (QKD) protocols such as B92, BB84, and three-state QKD. As a result, this work contributes to understanding how much ensemble coherence can be obtained from a given amount of entanglement.

Figures (3)

• Figure 1: Coherence of two non-orthogonal states. Basis-free coherence and the lower bound on the basis-free coherence as functions of the initial entanglement for a pair of pure non-orthogonal equiprobable states. Solid line stands for the coherence values. Dashed line corresponds to the values of the lower bound.
• Figure 2: Scheme of the ensemble generation. (a) Structure of the rank-1 POVM built upon the set of symmetric states in Alice's subsystem. $N=5$. (b) Generated ensemble in Bob's subsystem, $N=5$. When the initial entangled state is the Bell state, the ensemble's structure coincides with the symmetric states on Alice's side. When the entanglement decreases to zero, Bob's ensemble shrinks to the zero basis state $\ket{0}_\text{B}$.
• Figure 3: Coherence generated with symmetric POVM. Basis-free coherence and the gap between the Holevo quantity and accessible information as functions of the initial entanglement resource, for different numbers of POVM elements $N=2,\,3,\,4.$ The line labeled $N\rightarrow \infty$ corresponds to the asymptotic behavior. Different colors correspond to different values of $N$. Solid lines represent coherence values, while dashed lines indicate the corresponding lower bounds. For $N\rightarrow \infty$, the solid and dashed lines coincide.