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arXiv:1603.07134 · Physical Review Letters

Recovering Quantum Properties of Continuous-Variable States in the Presence of Measurement Errors

Evgeny V Shchukin, Peter van Loock

Published Sep 28, 2016• 6 citations
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Abstract

We present two results which combined enable one to reliably detect multimode, multipartite entanglement in the presence of measurement errors. The first result leads to a method to compute the best (approximated) physical covariance matrix given a measured nonphysical one assuming that no additional information about the measurement is available except the standard deviations from the mean values. The other result states that a widely used entanglement condition is a consequence of negativity of partial transposition. Our approach can quickly verify the entanglement of experimentally obtained multipartite states, which is demonstrated on several realistic examples. Compared to existing detection schemes, ours is very simple and efficient. In particular, it does not require any complicated optimizations.

DOI
10.1103/physrevlett.117.140504

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We present two results which combined enable one to reliably detect multimode, multipartite entanglement in the presence of measurement errors. The first result leads to a method to compute the best (approximated) physical covariance matrix given a measured nonphysical one assuming that no additional information about the measurement is available except the standard deviations from the mean values. The other result states that a widely used entanglement condition is a consequence of negativity of partial transposition. Our approach can quickly verify the entanglement of experimentally obtained multipartite states, which is demonstrated on several realistic examples. Compared to existing detection schemes, ours is very simple and efficient. In particular, it does not require any complicated optimizations.
We demonstrate how to efficiently derive a broad class of inequalities for entanglement detection in multimode continuous variable systems. The separability conditions are established from partial transposition (PT) in combination with several distinct necessary conditions for a quantum physical state, which include previously established inequalities as special cases. Remarkably, our method enables us to support Peres' conjecture to its full generality within the framework of Cavalcanti-Foster-Reid-Drummond multipartite Bell inequality [Phys. Rev. Lett. 99, 210405 (2007)] that the nonlocality necessarily implies negative PT entangled states.
Multipartite entanglement detection is crucial for the develop of quantum information science and quantum computation, communication, simulation and metrology tasks. In contrast to experiments, where several handreds of qubits have been entangled to build a quantum computer or a quantum simulator, theoretically only for lower dimensional quantum system such as two qubit system or two-mode Gaussian state and some special quantum states, efficient criteria have been developed to detect the entanglement. An efficient and practical entanglement detection method is anticipated to appear for current scalable quantum system. Based on the matched entanglement witness scheme and integrated uncertainty relations, we propose a quite general entanglement detection method for all kinds of multipartite entanglement of multimode continuous variable systems. With the covariance matrix of any randomly generated multimode continuous variable quantum state, our criterion can detect entanglement, genuine entanglement and other kinds of inseparabilities almost imidiately.
We derive a hierarchy of continuous-variable multipartite entanglement conditions in terms of second-order moments of position and momentum operators that generalize existing criteria. Each condition corresponds to a convex optimization problem which, given the covariance matrix of the state, can be numerically solved in a straightforward way. The conditions are independent of partial transposition and thus are also able to detect bound entangled states. Our approach can be used to obtain an analytical condition for genuine multipartite entanglement. We demonstrate that even a special case of our conditions can detect entanglement very efficiently. Using multiobjective optimization, it is also possible to numerically verify genuine entanglement of some experimentally realizable states.
We formulate an entanglement criterion using Peres-Horodecki positive partial transpose operations combined with the Schr\"odinger-Robertson uncertainty relation. We show that any pure entangled bipartite and tripartite state can be detected by experimentally measuring mean values and variances of specific observables. Those observables must satisfy a specific condition in order to be used, and we show their general form in the $2\ifmmode\times\else\texttimes\fi{}2$ (two qubits) dimension case. The criterion is applied on a variety of physical systems including bipartite and multipartite mixed states and reveals itself to be stronger than the Bell inequalities and other criteria. The criterion also work on continuous variable cat states and angular momentum states of the radiation field.