Maximum and minimum entropy states yielding local continuity bounds

Abstract

Given an arbitrary quantum state ($\sigma$), we obtain an explicit construction of a state ${\rho }_{\epsilon }^{\ast }(\sigma )$ (resp. ${\rho }_{\ast ,\epsilon }(\sigma )$) which has the maximum (resp. minimum) entropy among all states which lie in a specified neighbourhood ($\epsilon$-ball) of $\sigma$. Computing the entropy of these states leads to a local strengthening of the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannes inequality. Our bound is local in the sense that it depends on the spectrum of $\sigma$. The states ${\rho }_{\epsilon }^{\ast }(\sigma )$ and ${\rho }_{\ast ,\epsilon }(\sigma )$ depend only on the geometry of the $\epsilon$-ball and are in fact optimizers for a larger class of entropies. These include the Rényi entropy and the min- and max- entropies. This allows us to obtain local continuity bounds for these quantities as well. In obtaining this bound, we first derive a more general result which may be of independent interest, namely a necessary and sufficient condition under which a state maximizes a concave and Gâteaux-differentiable function in an $\epsilon$-ball around a given state $\sigma$. Examples of such a function include the von Neumann entropy, and the conditional entropy of bipartite states. Our proofs employ tools from the theory of convex optimization under non-differentiable constraints, in particular Fermat's Rule, and majorization theory.

38 pages; v2: added an application, streamlined proofs of Lem. 6.8-6.11, corrected typos, corrected figure 1, updated the style of figure 2

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• A Tight Uniform Continuity Bound for Equivocation
• Tight uniform continuity bound for a family of entropies
• Quantum Causal Structure and Quantum Thermodynamics
• Smooth entropy in axiomatic thermodynamics
• Extremal elements of a sublattice of the majorization lattice and approximate majorization
• Transcendental properties of entropy-constrained sets
• Entropy dynamics for a harmonic propeller-shaped planar molecular quantum Brownian rotator
• Adaptive Online Learning of Quantum States
• A continuity bound for the expected number of connected components of a random graph: a model for epidemics
• Maximal extension on converse monogamy of entanglement for tripartite pure states
• Universal proofs of entropic continuity bounds via majorization flow