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Multipartite entanglement for informationally complete quantum measurements

Jakub Czartowski, Dardo Goyeneche, Karol Życzkowski

Published Jan 24, 2018• 0 citations
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Abstract

We present a comprehensible introduction to tight informationally complete measurements for arbitrarily large multipartite systems and study their allowed entanglement configurations, focusing on those requiring simplest possible quantum resources. We demonstrate that tight measurements cannot be exclusively composed neither by fully separable nor maximally entangled states, according to a selected measure of entanglement. We establish an upper bound on the maximal number of fully separable states allowed by tight measurements and we study the distinguished case in which every measurement operator has the same amount of entanglement. Furthermore, we introduce the notion of nested tight measurements, i.e. multipartite tight informationally complete measurements such that every reduction to a certain number of parties induces a lower dimensional tight measurement, proving that they exist for any number of parties and internal levels.

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Journal of Physics Conference Series•Mar 1, 2016•54 citations
We present a comprehensible introduction to tight informationally complete measurements for arbitrarily large multipartite systems and study their allowed entanglement configurations, focusing on those requiring simplest possible quantum resources. We demonstrate that tight measurements cannot be exclusively composed neither by fully separable nor maximally entangled states, according to a selected measure of entanglement. We establish an upper bound on the maximal number of fully separable states allowed by tight measurements and we study the distinguished case in which every measurement operator has the same amount of entanglement. Furthermore, we introduce the notion of nested tight measurements, i.e. multipartite tight informationally complete measurements such that every reduction to a certain number of parties induces a lower dimensional tight measurement, proving that they exist for any number of parties and internal levels.
Abstract We analyze tight informationally complete measurements for arbitrarily large multipartite systems and study their configurations of entanglement. We demonstrate that tight measurements cannot be exclusively composed neither of fully separable nor maximally entangled states. We establish an upper bound on the maximal number of fully separable states allowed by tight measurements and investigate the distinguished case in which every measurement operator carries the same amount of entanglement. Furthermore, we introduce the notion of nested tight measurements, i.e. multipartite tight informationally complete measurements such that every reduction to a certain number of parties induces a lower dimensional tight measurement, proving that they exist for any number of parties and internal levels.
The entanglement of superpositions [Linden et al., Phys. Rev. Lett. 97, 100502 (2006)]is generalized to the multipartite scenario: an upper bound to the multipartite entanglement of a superposition is given in terms of the entanglement of the superposed states and the superposition coefficients. This bound is proven to be tight for a class of states composed of an arbitrary number of qubits. We also extend the result to a large family of quantifiers, which includes the negativity, the robustness of entanglement, and the best separable approximation measure.
Heterogeneous bipartite quantum pure states, composed of two subsystems with a different number of levels, cannot have both reductions maximally mixed. In this work, we demonstrate the existence of a wide range of highly entangled states of heterogeneous multipartite systems consisting of N>2N>2N>2 parties such that every reduction to one and two parties is maximally mixed. Two constructions of generating genuinely multipartite maximally entangled states of heterogeneous systems for an arbitrary number of subsystems are presented. Such states are related to quantum error correction codes over mixed alphabets and mixed orthogonal arrays. Additionally, we show the advantages of considering heterogeneous systems in practical implementations of multipartite steering.
The relation between entanglement and nonlocality is discussed in the case of multipartite quantum systems. We show that, for any number of parties, there exist genuinely multipartite entangled states that admit a fully local hidden variable model, i.e., where all parties are separated. Hence, although these states exhibit the strongest form of multipartite entanglement, they cannot lead to Bell inequality violation considering general nonsequential local measurements. Then, we show that the nonlocality of these states can nevertheless be activated using sequences of local measurements, thus revealing genuine multipartite hidden nonlocality.
Despite multipartite entanglement being a global property of a quantum state, a number of recent works have made it clear that it can be quantified using only local measurements. This is appealing because local measurements are the easiest to implement on current quantum hardware. However, it remains an open question what protocol one should use in order to minimize the resources required to estimate multipartite entanglement from local measurements alone. In this work, we construct and compare several estimators of multipartite entanglement based solely on the data from local measurements. We first construct statistical estimators for a broad family of entanglement measures using local randomized measurement (LRM) data before providing a general criterion for the construction of such estimators in terms of projective 2-designs. Importantly, this allows us to derandomize the multipartite estimation protocol based on LRMs. In particular, we show how local symmetric, informationally complete positive operator-valued measures enable multipartite entanglement quantification with only a single measurement setting. For all estimators, we provide both the classical postprocessing cost and rigorous performance guarantees in the form of analytical upper bounds on the number of measurements needed to estimate the measures to any desired precision.