Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints

2015/07/31 by Andreas Winter · 9 citations
Computer Science · Mathematics · Physics and Astronomy · #Bounded function #Conditional entropy #Entropy (arrow of time) #Entropy rate #Generalized relative entropy #Hilbert space #Joint quantum entropy #Kullback–Leibler divergence #Mathematical analysis #Mathematics #Min entropy #Physics #Principle of maximum entropy #Pure mathematics #Quantum #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum discord #Quantum entanglement #Quantum many-body systems #Quantum mechanics #Quantum mutual information #Quantum relative entropy #Quantum system #Statistical physics #Upper and lower bounds #Von Neumann entropy #cs.IT #math-ph #math.IT #math.MP #quant-ph

paper · pdf · doi:10.1007/s00220-016-2609-8

published as Commun. Math. Phys 347(1):291-313 (2016) · 12pp, lots of lemmas but no theorem. V2, v3 v4, v5, v6 have ever more references and more accurate attribution of previous results. In v6 correct proof of the asymptotic continuity of entanglement cost, which had been claimed already in v4 and been retracted in v5

arxiv created 2016/01/12 · openalex publication_date 2016/03/22 · openalex created_date 2016/06/24 · arxiv updated 2016/09/06 · openalex updated_date 2026/08/05

Abstract

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, E_R, and its regularization E_R^∞, as well as of the entanglement of formation, E_F. 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^∞. Second, 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.

Citations

• The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography
• On Quantum Communication Channels with Constrained Inputs
• Gaussian quantum information
• A continuity property of the entropy density for spin lattice systems
• A sharp continuity estimate for the von Neumann entropy
• Discussion of Probability Relations between Separated Systems
• Private classical capacity with a symmetric side channel and its application to quantum cryptography
• Quantifying Entanglement

Cited by

• Optimal uniform continuity bound for conditional entropy of classical--quantum states
• Approximate Degradable Quantum Channels
• Sharp Continuity of Petz and Sandwiched Rényi Conditional Entropies
• Quantum pseudoresources imply cryptography
• Convergence Rates for the Quantum Central Limit Theorem
• Analytic semi-device-independent entanglement quantification for bipartite quantum states
• Relative entropy for von Neumann subalgebras
• Quantum resource theories
• Conditional Quantum One-Time Pad
• Asymptotic security of discrete-modulation protocols for continuous-variable quantum key distribution
• Continuity of the relative entropy of resource

Related

• Continuity bounds for quantum entropies arising from a fundamental entropic inequality
• Purity based continuity bounds for quantum information measures
• Sharp continuity of quantum conditional entropy
• Optimal continuity bound for the von Neumann entropy under energy constraints
• Continuity of Relative Entropy of Entanglement