Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints
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- 作者：Andreas Winter
- 刊名：Communications in Mathematical Physics
- 出版年：2016
- 出版时间：October 2016
- 年：2016
- 卷：347
- 期：1
- 页码：291-313
- 全文大小：568 KB
- 刊物类别：Physics and Astronomy
- 刊物主题：Physics
Mathematical and Computational Physics
Quantum Physics
Quantum Computing, Information and Physics
Complexity
Statistical Physics
Relativity and Cosmology - 出版者：Springer Berlin / Heidelberg
- ISSN：1432-0916
- 卷排序：347
文摘
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 \({E_R^{\infty}}\), 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, \({E_C=E_F^{\infty}}\). 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.