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arXiv:2412.18586 · IEEE Transactions on Information Theory

Uniform Additivity of Tripartite Optimized Correlation Measures

Joshua Levin, Ariel Shlosberg, Vikesh Siddhu, Graeme Smith

Published Nov 18, 2025• 0 citations
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Abstract

Information theory provides a framework for answering fundamental questions about the optimal performance of many important quantum communication and computational tasks. In many cases, the optimal rates of these tasks can be expressed in terms of regularized formulas that consist of linear combinations of von Neumann entropies optimized over state extensions. However, evaluation of regularized formulas is often intractable, since it involves computing a formula’s value in the limit of infinitely many copies of a state. To find optimized linear entropic functions of quantum states whose regularized versions are tractable to compute, we search for examples which are additive. We use the method of cross2017uniform, which considers bipartite formulas, to identify convex polyhedral cones of tripartite correlation measures which satisfy a stronger a form of additivity called uniform additivity. We rely only on strong subadditivity of the von Neumann entropy and use these cones to prove that three previously established tripartite optimized correlation measures are additive.

DOI
10.1109/tit.2025.3632803

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Information theory provides a framework for answering fundamental questions about the optimal performance of many important quantum communication and computational tasks. In many cases, the optimal rates of these tasks can be expressed in terms of regularized formulas that consist of linear combinations of von Neumann entropies optimized over state extensions. However, evaluation of regularized formulas is often intractable, since it involves computing a formula’s value in the limit of infinitely many copies of a state. To find optimized linear entropic functions of quantum states whose regularized versions are tractable to compute, we search for examples which are additive. We use the method of cross2017uniform, which considers bipartite formulas, to identify convex polyhedral cones of tripartite correlation measures which satisfy a stronger a form of additivity called uniform additivity. We rely only on strong subadditivity of the von Neumann entropy and use these cones to prove that three previously established tripartite optimized correlation measures are additive.
Information theory quantifies the optimal rates of resource interconversions, usually in terms of entropies. However, nonadditivity often makes evaluating entropic formulas intractable. In a few auspicious cases, additivity allows a full characterization of optimal rates. We study uniform additivity of formulas, which is easily evaluated and captures all known additive quantum formulas. Our complete characterization of uniform additivity exposes an intriguing new additive quantity and identifies a remarkable coincidence-the classical and quantum uniformly additive functions with one auxiliary variable are identical.
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The strong subadditivity inequality of von Neumann entropy relates the entropy of subsystems of a tripartite state ${\ensuremath{\rho}}_{ABC}$ to that of the composite system. Here, we define ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$ as the extent to which ${\ensuremath{\rho}}_{ABC}$ fails to satisfy the strong subadditivity inequality $S({\ensuremath{\rho}}_{B})+S({\ensuremath{\rho}}_{C})\ensuremath{\le}S({\ensuremath{\rho}}_{AB})+S({\ensuremath{\rho}}_{AC})$ with equality and investigate its properties. In particular, by introducing auxiliary subsystem EEE, we consider any purification $|{\ensuremath{\psi}}_{ABCE}\ensuremath{\rangle}$ of ${\ensuremath{\rho}}_{ABC}$ and formulate ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$ as the extent to which the bipartite quantum correlations of ${\ensuremath{\rho}}_{AB}$ and ${\ensuremath{\rho}}_{AC}$, measured by entanglement of formation and quantum discord, change under the transformation $B\ensuremath{\rightarrow}BE$ and $C\ensuremath{\rightarrow}CE$. Invariance of quantum correlations of ${\ensuremath{\rho}}_{AB}$ and ${\ensuremath{\rho}}_{AC}$ under such transformation is shown to be a necessary and sufficient condition for vanishing ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$. Our approach allows one to characterize, intuitively, the structure of states for which the strong subadditivity is saturated. Moreover, along with providing a conservation law for quantum correlations of states for which the strong subadditivity inequality is satisfied with equality, we find that such states coincide with those that the Koashi-Winter monogamy relation is saturated.
We consider a class of correlation measures for quantum states called optimized correlation measures defined as a minimization of a linear combination of von Neumann entropies over purifications of a given state. Examples include the entanglement of purification EP{E}_{P}EP​ and squashed entanglement Esq{E}_{\mathrm{sq}}Esq​. We show that when evaluating such measures on ``nice" holographic states in the large-NNN limit, the optimal purification has a semiclassical geometric dual. We then apply this result to confirm several holographic dual proposals, including the nnn-party squashed entanglement. Moreover, our result suggests two new techniques for determining holographic duals: holographic entropy inequalities and direct optimization of the dual geometry.
In one-shot and zero-error information theory, the conditional min-entropy is a fundamental tool. It may be expressed as a conic program over the positive semidefinite cone. Recently, Chitambar et al. showed that the same conic program altered to be over the separable cone is a measure of transmitting classical communication over a quantum channel called the ‘communication value.’ In this work, we extend this idea to a broad class of convex cones to induce new families of entropic quantities. We show this methodology has operational relevance by characterizing a generalized notion of communication value and relating a class of cone-restricted entropies to a partial ordering on converting quantum channels via bistochastic preprocessing. We also show regularized smooth versions of these entropic quantities do not in general converge to the von Neumann entropy, which shows tasks characterized by these quantities are not equivalent even in an asymptotic i.i.d. fashion.