Relative entropy and squashed entanglement

Abstract

We are interested in the properties and relations of entanglement measures. Especially, we focus on the squashed entanglement and relative entropy of entanglement, as well as their analogues and variants. Our first result is a monogamy-like inequality involving the relative entropy of entanglement and its one-way LOCC variant. The proof is accomplished by exploring the properties of relative entropy in the context of hypothesis testing via one-way LOCC operations, and by making use of an argument resembling that by Piani on the faithfulness of regularized relative entropy of entanglement. Following this, we obtain a commensurate and faithful lower bound for squashed entanglement, in the form of one-way LOCC relative entropy of entanglement. This gives a strengthening to the strong subadditivity of von Neumann entropy. Our result improves the trace-distance-type bound derived in [Comm. Math. Phys., 306:805-830, 2011], where faithfulness of squashed entanglement was first proved. Applying Pinsker's inequality, we are able to recover the trace-distance-type bound, even with slightly better constant factor. However, the main improvement is that our new lower bound can be much larger than the old one and it is almost a genuine entanglement measure. We evaluate exactly the various relative entropy of entanglement under restricted measurement classes, for maximally entangled states. Then, by proving asymptotic continuity, we extend the exact evaluation to their regularized versions for all pure states. Finally, we consider comparisons and separations between some important entanglement measures and obtain several new results on these, too.

minor corrections

References in corpus (9)

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• Uncertainty, Monogamy, and Locking of Quantum Correlations
• Relative Entropy of Entanglement and Restricted Measurements
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• On a gap in the proof of the generalised quantum Stein's lemma and its consequences for the reversibility of quantum resources
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• Quantum de Finetti theorem under fully-one-way adaptive measurements
• Multipartite quantum correlations and local recoverability
• Asymptotic state transformations of continuous variable resources
• Squashed entanglement in infinite dimensions
• Exponential clustering of bipartite quantum entanglement at arbitrary temperatures
• Private states, quantum data hiding and the swapping of perfect secrecy
• A lower bound of quantum conditional mutual information
• Markovianizing Cost of Tripartite Quantum States
• Asymptotic local hypothesis testing between a pure bipartite state and the completely mixed state
• Distillable entanglement under dually non-entangling operations
• Dense Coding with Locality Restriction for Decoder: Quantum Encoders vs. Super-Quantum Encoders
• Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states
• Tight asymptotic bounds on local hypothesis testing between a pure bipartite state and the white noise state
• No-go theorem for entanglement distillation using catalysis
• Continuity bounds on observational entropy and measured relative entropies
• Entanglement monogamy via multivariate trace inequalities
• More on the upper bound of holographic n-partite information
• Faithful geometric measures for genuine tripartite entanglement
• Second Law of Entanglement Manipulation with Entanglement Battery
• Continuity of entropies via integral representations
• A strengthened monotonicity inequality of quantum relative entropy: A unifying approach via Rényi relative entropy
• Locally-Measured Rényi Divergences
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• The squashed entanglement of the noiseless quantum Gaussian attenuator and amplifier
• Squashed Entanglement from Generalized Rindler Wedge
• Random private quantum states
• Feynman graphs and the large dimensional limit of multipartite entanglement
• The multimode conditional quantum Entropy Power Inequality and the squashed entanglement of the multimode extreme bosonic Gaussian channels
• Topological squashed entanglement: nonlocal order parameter for one-dimensional topological superconductors