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arXiv:2401.09554 · Communications in Mathematical Physics

Entanglement Cost for Infinite-Dimensional Physical Systems

Hayata Yamasaki, Kohdai Kuroiwa, Patrick M. Hayden, Ludovico Lami

Published Oct 3, 2025• 8 citations
arXivHTMLPDF

Abstract

. 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-dimensional systems is nontrivial because the conventional tools for establishing both the direct and converse bounds, i.e., strong typicality, 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.

DOI
10.1007/s00220-025-05431-1

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Abstract A renormalized version of the von Neumann quantum entropy (which is finite and continuous in general, infinite dimensional case) which obeys several of the natural physical demands (as expected for a “good” measure of entanglement in the case of general quantum states describing bipartite and infinite-dimensional systems) is proposed. The renormalized quantum entropy is defined by the explicit use of the Fredholm determinants theory. To prove the main results on continuity and finiteness of the introduced renormalization, the fundamental Grothendick approach, which is based on the infinite dimensional Grassmann algebra theory, is applied. Several features of majorization theory are preserved under the introduced renormalization as it is proved in this paper. This fact allows to extend most of the known (mainly, in the context of two-partite, finite-dimensional quantum systems) results of the LOCC comparison theory to the case of genuine infinite-dimensional, two-partite quantum systems.
A long-standing open question is to characterize the classical communication cost of simulating the correlations arising from measurements on finite-dimensional quantum systems. We consider this question in the prepare-and-measure scenario. First we construct a classical simulation model involving a finite average number of bits of one-way communication that simulates exactly the correlations arising from arbitrary ddd-dimensional quantum states and arbitrary measurements. Second, we show that no classical simulation protocol involving a finite average message dimension can exist for d≥4d\geq 4d≥4, hence strengthening a recent result of de Gois, Santos and Vieira. Our first result relies on a more general simulation theorem which can be applied to certain continuous variable (i.e. infinite-dimensional) quantum systems, and even beyond quantum theory, e.g. to generalized probabilistic theories. In particular, we show that multimode Gaussian states of bounded average energy and general Gaussian measurements admit an exact classical simulation with a finite expected number of bits. Overall, our results show that the classical simulation cost of quantum communication is essentially as high as it could be.
An universal approximation technique for analysis of different characteristics of states of composite infinite-dimensional quantum systems is proposed and used to prove general results concerning the properties of correlation and entanglement measures in such systems. Then these results are applied to the study of three important characteristics: the relative entropy of πππ-entanglement, the Rains bound (the unregularized and regularized versions of both characteristics are considered) and the conditional entanglement of mutual information. In particular, we analyse continuity and convexity properties of the above entanglement measures, prove several results simplifying their definitions and establish a finite-dimensional approximation property for these characteristics that allows us to generalize to the infinite-dimensional case the results proved in the finite-dimensional settings.