Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints
By Andreas J. Winter. arXiv.org, 2015
334 citations recorded by the source corpus.
We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: first, a tight analysis of the Alicki–Fannes continuity bounds for the conditional von Neumann entropy, reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications, we give new proofs, with tighter bounds, of the asymptotic continuity of the relative entropy of entanglement, ER, and its regularization $${E_R^{\infty}}$$ER∞, as well as of the entanglement of formation, EF. Using a novel “quantum coupling” of density operators, which may be of independent interest, we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation, aka entanglement cost, $${E_C=E_F^{\infty}}$$EC=EF∞. Second, we derive analogous continuity bounds for the von Neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint, most importantly systems of multiple quantum harmonic oscillators. While without an energy bound the entropy is discontinuous, it is well-known to be continuous on states of bounded energy. However, a quantitative statement to that effect seems not to have been known. Here, under some regularity assumptions on the Hamiltonian, we find that, quite intuitively, the Gibbs entropy at the given energy roughly takes the role of the Hilbert space dimension in the finite-dimensional Fannes inequality.
Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics
View Paper: https://www.semanticscholar.org/paper/e001ec44698ebde9fa6ce33d68a462ce9d092896
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In Artificial Intelligence
In Atomic and Molecular Physics, and Optics
Citations in the corpus
- A Tight Uniform Continuity Bound for Equivocation (2019, background): “On the other hand, uniform bounds, which are not tight but are independent of the size of the conditioning system, were proven for the conditional Shannon and von Neumann entropies in [10], [8].”