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Strong Subadditivity Inequality and Entropic Uncertainty Relations

Li Gao, Marius Junge, Nicholas D. LaRacuente

Published Oct 27, 2017• 8 citations
OpenAlex

Abstract

We prove a generalization of the strong-subadditivity (SSA) inequality in the context of finite dimensional von Neumann algebras. Our generalized SSA inequality subsumes various entropic uncertainty relations, such as the Maassen-Uffink inequality in the presence of quantum memory. Motivated by the generalization of SSA, we introduce an analog of conditional mutual information and discuss superadditivity and monotonicity under certain operations. Our proof uses complex interpolation and noncommutative mixed $L_p$ spaces.

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We prove a generalization of the strong-subadditivity (SSA) inequality in the context of finite dimensional von Neumann algebras. Our generalized SSA inequality subsumes various entropic uncertainty relations, such as the Maassen-Uffink inequality in the presence of quantum memory. Motivated by the generalization of SSA, we introduce an analog of conditional mutual information and discuss superadditivity and monotonicity under certain operations. Our proof uses complex interpolation and noncommutative mixed LpL_pLp​ spaces.
Strong subadditivity goes beyond the tensored subsystem and commuting operator models. As previously noted by Petz and later by Araki and Moriya, two subalgebras of observables satisfy a generalized SSA-like inequality if they form a commuting square. We explore the interpretation and consequences in finite dimensions, connecting various entropic uncertainty relations for mutually unbiased bases with the positivity of a generalized conditional mutual information (CMI), and with inequalities on relative entropies of coherence and asymmetry. We obtain a bipartite resource theory of operations under which the two subalgebras are respectively invariant and covariant, with CMI as a monotone, and generalized non-classical monotones based on squashed entanglement and entanglement of formation. Free transformations support conversion between entanglement and uncertainty-based configurations, as "EPR 2UCR." Our theory quantifies the common non-classicality in entanglement and uncertainty, implying a strong conceptual link between these fundamentally quantum phenomena.
We prove a generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum Gaussian systems. Such generalization determines the minimum values of linear combinations of the entropies of subsystems associated to arbitrary linear functions of the quadratures, and holds for arbitrary quantum states including the scenario where the entropies are conditioned on a memory quantum system. We apply our result to prove new entropic uncertainty relations with quantum memory, a generalization of the quantum Entropy Power Inequality, and the linear time scaling of the entanglement entropy produced by quadratic Hamiltonians.
We prove a generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum Gaussian systems. Such generalization determines the minimum values of linear combinations of the entropies of subsystems associated to arbitrary linear functions of the quadratures, and holds for arbitrary quantum states including the scenario where the entropies are conditioned on a memory quantum system. We apply our result to prove new entropic uncertainty relations with quantum memory, a generalization of the quantum Entropy Power Inequality, and the linear time scaling of the entanglement entropy produced by quadratic Hamiltonians.
The entropic uncertainty principle in the form proven by Maassen and Uffink yields a fundamental inequality that is prominently used in many places all over the field of quantum information theory. In this work, we provide a family of versatile generalizations of this relation. Our proof methods build on a deep connection between entropic uncertainties and interpolation inequalities for the doubly stochastic map that links probability distributions in two measurements bases. In contrast to the original relation, our generalization also incorporates the von Neumann entropy of the underlying quantum state. These results can be directly used to bound the extractable randomness of a source independent QRNG in the presence of fully quantum attacks, to certify entanglement between trusted parties, or to bound the entanglement of a system with an untrusted environment.