Quantum Skew Divergence

Abstract

In this paper we study the quantum generalisation of the skew divergence, which is a dissimilarity measure between distributions introduced by L. Lee in the context of natural language processing. We provide an in-depth study of the quantum skew divergence, including its relation to other state distinguishability measures. Finally, we present a number of important applications: new continuity inequalities for the quantum Jensen-Shannon divergence and the Holevo information, and a new and short proof of Bravyi's Small Incremental Mixing conjecture.

Supersedes 1102:3041, as it contains many new results and applications. v2: minor modifications, including a streamlining of the proofs. 32 1/4 pages

References in corpus (5)

• Properties of Classical and Quantum Jensen-Shannon Divergence
• Chain Rules for Smooth Min- and Max-Entropies
• Entanglement rates and area laws
• Upper bounds on entangling rates of bipartite Hamiltonians
• On the asymmetry of the relative entropy

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• On a generalization of the Jensen-Shannon divergence and the JS-symmetrization of distances relying on abstract means
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• Speed limits and locality in many-body quantum dynamics
• Tight continuity bounds for the quantum conditional mutual information, for the Holevo quantity and for capacities of quantum channels
• Bounds in Nonequilibrium Quantum Dynamics
• Maximal entanglement velocity implies dual unitarity
• Logarithmic coherence: Operational interpretation of ${\ell }_1$-norm coherence
• Entropic bounds on information backflow
• Provably-secure quantum randomness expansion with uncharacterised homodyne detection
• Entropy power inequalities for qudits
• Spin-phase-space-entropy production
• Entropic and trace distance based measures of non-Markovianity
• Complexity of quantum circuits via sensitivity, magic, and coherence
• Continuity of quantum entropic quantities via almost convexity
• Quantum Rényi and $f$-divergences from integral representations
• Advantages and limitations of quantum routing
• Holevo skew divergence for the characterization of information backflow
• On Relations Between the Relative entropy and ${\chi }^2$-Divergence, Generalizations and Applications
• Dependence of integrated, instantaneous, and fluctuating entropy production on the initial state in quantum and classical processes
• Irreversibility at zero temperature from the perspective of the environemnt
• The Differential Entropy of Mixtures: New Bounds and Applications
• Entanglement rates for Renyi, Tsallis and other entropies
• Non-equilibrium quantum thermodynamics of a particle trapped in a controllable time-varying potential
• Characterizing irreversibility in open quantum systems
• Quantifying protocol efficiency: a thermodynamic figure of merit for classical and quantum state-transfer protocols
• The dynamical $\alpha$-Rényi entropies of local Hamiltonians grow at most linearly in time
• A lower bound on the fidelity between two states in terms of their trace-distance and max-relative entropy
• On the role of initial coherence in the spin phase-space entropy production rate
• Strongly Convex Divergences
• Transmission distance in the space of quantum channels
• Classicality of the heat produced by quantum measurements
• Nearly optimal time-independent reversal of a spin chain
• Coherence and Imaginarity as Resources in Quantum Circuit Complexity
• Entanglement rates for bipartite open systems
• $\alpha$-Geodesical Skew Divergence
• Quantum Statistical Complexity Measure as a Signalling of Correlation Transitions
• Oracle problems as communication tasks and optimization of quantum algorithms
• Quantum Routing and Entanglement Dynamics Through Bottlenecks