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arXiv:2107.08537 · IEEE Transactions on Information Theory

Asymptotic Continuity of Additive Entanglement Measures

Péter Vrana

Published Jan 18, 2022• 4 citations
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Abstract

We study rates of asymptotic transformations between entangled states by local operations and classical communication and a sublinear amount of quantum communication. It is known that additive asymptotically continuous entanglement measures provide upper bounds on the rates that are achievable with asymptotically vanishing error. We show that for transformations between pure states, the optimal rate between any pair of states can be characterized as the infimum of such upper bounds provided by fully additive asymptotically continuous entanglement measures.

DOI
10.1109/tit.2022.3143845

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We study rates of asymptotic transformations between entangled states by local operations and classical communication and a sublinear amount of quantum communication. It is known that additive asymptotically continuous entanglement measures provide upper bounds on the rates that are achievable with asymptotically vanishing error. We show that for transformations between pure states, the optimal rate between any pair of states can be characterized as the infimum of such upper bounds provided by fully additive asymptotically continuous entanglement measures.
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The trade-off relation between the rate and the strong converse exponent for probabilistic asymptotic entanglement transformations between pure multipartite states can in principle be characterised in terms of a class of entanglement measures determined implicitly by a set of strong axioms. A nontrivial family of such functionals has recently been constructed, but their previously known characterisations have so far only made it possible to evaluate them in very simple cases. In this paper we derive a new regularised formula for these functionals in terms of a subadditive upper bound, complementing the previously known superadditive lower bound. The upper and lower bounds evaluated on tensor powers differ by a logarithmically bounded term, which provides a bound on the convergence rate. In addition, we find that on states satisfying a certain sparsity constraint, the upper bound is equal to the value of the corresponding additive entanglement measure, therefore the regularisation is not needed for such states, and the evaluation is possible via a single-letter formula. Our results provide explicit bounds on the success probability of transformations by local operations and classical communication and, due to the additivity of the entanglement measures, also on the strong converse exponent for asymptotic transformations.
We consider distillation of ebits between a specified pair of subsystems from pure tripartite states by local operations and classical communication. It is known that, allowing an asymptotically vanishing error, the maximal rate is the minimum of the von Neumann entropies of the two corresponding marginals, and under asymptotic stochastic local operations and classical communication the maximal rate is given by a minimization over a one-parameter family of entanglement measures. In this paper, we determine the direct and strong converse error exponents, and the optimal rate for deterministic transformations.
We investigate the asymptotic equipartition property (AEP) in the context of multipartite entanglement measures on pure states. In particular, we formulate AEP for subadditive entanglement measures that admit certain weak conditions. This is motivated by the uniqueness of the entanglement entropy in the asymptotic limit in the bipartite case. From an operational perspective, the result is relevant in the LOCCqscenario (asymptotic local operations and classical communication with a sublinear amount of quantum communication). Analogously to the classical AEP, we show that the regularization of smooth, weakly additive entanglement measures - under mild additional assumptions -yields entanglement measures, that are weakly additive and asymptotically continuous. We then evaluate the mentioned regularization and smoothing procedure for a known family of Rényi type multipartite entanglement measures, showing that the resulting entanglement measures reduce to convex combinations of bipartite entanglement entropies.
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It is known that general convertibility of bipartite entangled states is not possible to arbitrary error without some classical communication. While some tradeoffs between communication cost and conversion error have been proven, these bounds can be very loose. In particular, there are many cases in which tolerable error might be achievable using zero-communication protocols. In this work we address these cases by deriving the optimal fidelity of pure quantum state conversions under local unitaries as well as local operations and shared randomness. We also use these results to explore catalytic conversions between pure states using zero communication.