Entanglement cost for infinite-dimensional physical systems

TL;DR

This paper establishes that the entanglement cost for infinite-dimensional quantum systems equals the regularized entanglement of formation, extending foundational finite-dimensional results to more general quantum states.

Contribution

It introduces a new entanglement dilution protocol for infinite-dimensional states and proves its optimality, advancing the understanding of entanglement measures in infinite-dimensional quantum systems.

Findings

Entanglement cost equals regularized entanglement of formation for infinite-dimensional states.

Developed a new entanglement dilution protocol using typicality arguments.

Derived a novel integral representation for quantum entropy in infinite dimensions.

Abstract

We prove that the entanglement cost equals the regularized entanglement of formation for any infinite-dimensional quantum state $\rho_{AB}$ with finite quantum entropy on at least one of the subsystems $A$ or $B$. This generalizes a foundational result in quantum information theory that was previously formulated only for operations and states on finite-dimensional systems. The extension to infinite dimensions is nontrivial because the conventional tools for establishing both the direct and converse bounds, i.e., strong typically, monotonicity, and asymptotic continuity, are no longer directly applicable. To address this problem, we construct a new entanglement dilution protocol for infinite-dimensional states implementable by local operations and a finite amount of one-way classical communication (one-way LOCC), using weak and strong typicality multiple times. We also prove the optimality of this protocol among all protocols even under infinite-dimensional separable operations by developing an argument based on alternative forms of monotonicity and asymptotic continuity of the entanglement of formation for infinite-dimensional states. Along the way, we derive a new integral representation for the quantum entropy of infinite-dimensional states, which we believe to be of independent interest. Our results allow us to fully characterize an important operational entanglement measure -- the entanglement cost -- for all infinite-dimensional physical systems.