Optimal Extensions of Resource Measures and their Applications

Abstract

We develop a framework to extend resource measures from one domain to a larger one. We find that all extensions of resource measures are bounded between two quantities that we call the minimal and maximal extensions. We discuss various applications of our framework. We show that any relative entropy (i.e. an additive function on pairs of quantum states that satisfies the data processing inequality) must be bounded by the min and max relative entropies. We prove that the generalized trace distance, the generalized fidelity, and the purified distance are optimal extensions. And in entanglement theory we introduce a new technique to extend pure state entanglement measures to mixed bipartite states.

arXiv comment6 pages (main text) + 16 pages (supplemental material), v3, added references and a closed formula for the sandwiched or minimal quantum Renyi divergence with alpha in [0,1/2)

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Cited by in corpus (18)

• Many-body quantum magic
• Continuity of quantum entropic quantities via almost convexity
• Quantum Rényi and $f$-divergences from integral representations
• Monotones in General Resource Theories
• Uniqueness and Optimality of Dynamical Extensions of Divergences
• Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond
• Quantum Bell Nonlocality is Entanglement
• Complete Characterization of Entanglement Embezzlement
• An Extention of Entanglement Measures for Pure States
• What is Entropy? A new perspective from games of chance
• Inevitability of knowing less than nothing
• Multivariate Fidelities
• Locally-Measured Rényi Divergences
• Continuity bounds for quantum entropies arising from a fundamental entropic inequality
• Chain rules for quantum channels
• Inevitable Negativity: Additivity Commands Negative Quantum Channel Entropy
• Single-shot entanglement manipulation of states and channels revisited
• Extending Resource Monotones using Kan Extensions