Quantifying Unextendibility via Virtual State Extension

TL;DR

This paper introduces a new framework to quantify the unshareability of entanglement through virtual state extension, deriving exact costs for isotropic states and connecting it to quantum broadcasting and entanglement measures.

Contribution

It develops an operational framework for quantifying entanglement unextendibility, derives exact formulas for isotropic states, and constructs explicit protocols for quantum broadcasting.

Findings

Exact closed-form expression for virtual extension cost of isotropic states.

Constructed an explicit quantum circuit for optimal virtual quantum broadcasting.

Established the virtual extension cost as an entanglement measure bounding distillable entanglement.

Abstract

Monogamy of entanglement, which limits how entanglement can be shared among multiple parties, is a fundamental feature underpinning the privacy of quantum communication. In this work, we introduce a novel operational framework to quantify the unshareability or unextendibility of entanglement via a virtual state-extension task. The virtual extension cost is defined as the minimum simulation cost of a randomized protocol that reproduces the marginals of a $k$-extension. For the important family of isotropic states, we derive an exact closed-form expression for this cost. Our central result establishes a tight connection: the virtual extension cost of a maximally entangled state equals the optimal simulation cost of universal virtual quantum broadcasting. Using the algebra of partially transposed permutation matrices, we obtain an analytical formula and construct an explicit quantum circuit for the optimal broadcasting protocol, thereby resolving an open question in quantum broadcasting. We further relate the virtual extension cost to the absolute robustness of unextendibility, providing it with a clear operational meaning, and show that the virtual extension cost is an entanglement measure that bounds distillable entanglement and connects to logarithmic negativity.