Quantifying Unextendibility via Virtual State Extension

TL;DR

This work develops an operational framework to quantify entanglement unextendibility via virtual state extension, defining the virtual extension cost \eta_k(\rho_{AB}) as the minimum simulation cost to generate a k-extension of a bipartite state. For the maximally entangled case, the authors prove the cost equals the optimal universal virtual broadcasting cost \gamma_k, and they derive a closed-form expression \gamma_k = \log\left(\frac{2kd}{k+d-1}-1\right) using the walled Brauer algebra and mixed Schur-Weyl duality, alongside an explicit circuit construction. They connect the cost to the absolute robustness of EXT_k, establishing the exact relation 2^{\eta_k(\rho_{AB})} = 2{\cal R}_{absolute}^{EXT_k}(\rho_{AB}) + 1, thus giving operational meaning to unextendibility measures and yielding bounds on one-shot distillable entanglement. In the infinite-extension limit, the measure recovers the standard entanglement quantifier, the logarithmic negativity, for pure states, linking the new framework to established theory and providing a robust tool for analyzing finite-copy entanglement tasks and private quantum resources.

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.

Figures (5)

• Figure 1: An element in the walled Brauer algebra ${\cal B}_{3,2}^d$nguyen2023mixed.
• Figure 2: Partial transpose $\sigma^\Gamma$ of a diagram $\sigma$ in ${\cal B}_{3,2}^d$ corresponds to exchanging the last $2$ nodes nguyen2023mixed.
• Figure 3: Schematic diagram of virtual state $k$-extension (left) and virtual state $k$-broadcasting (right). The central grey dot represents the central hub, while the surrounding purple dots denote information stations. In the left diagram, the hub and each station share the same state $\rho_{AB_j}$ statistically. When this state is maximally entangled, i.e. $\Phi_{AB}^d$, the protocol effectively becomes universal virtual broadcasting (right), where the hub broadcasts a state to the stations.
• Figure 4: Quantum circuit for implementing the CPTN map $\Lambda_{A\to B_1B_2B_3}$. The preparation circuit $W$ satisfies $W|0\rangle=\sum_{j=0}^{5}\frac{1}{\sqrt{6}}|j\rangle$, and the entire circuit achieve the linear combination of unitaries $V(\sigma)$, for $\sigma\in\textbf{S}_3$. This serves as an example of $k=3$, wherein analogous symmetric projections for $k$ systems can be similarly achieved.
• Figure S1: Tensor contraction diagrams for computing $\operatorname{Tr}_{23}\left[P^{{{\operatorname{sym}}}_3}(I^{\otimes 2}\otimes |i\rangle\!\langle j|)P^{{{\operatorname{sym}}}_3}\right]$. The line represents identity operator, red box represents the operator $|i\rangle\!\langle j|$.

Theorems & Definitions (14)

• Definition 1: Simulation cost of an HPTP map jiang2021physical
• Definition 2: yao2024optimal
• Definition 3: Robustness of unextendibility
• Definition 4: Virtual state extension
• Definition 5: Virtual extension cost
• Proposition 1
• Theorem 2
• Proposition 3
• Proposition 4
• Theorem 5: Robustness