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arXiv:0806.4165

Detecting and Quantifying Entanglement via Bayesian Updating

Pavel Lougovski, S. J. van Enk

Published Jun 25, 2008• 1 citation
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Abstract

We show how a straightforward Bayesian updating procedure allows one to detect and quantify entanglement from any finite set of measurement results. The measurements do not have to be tomographically complete, and may consist of POVMs rather than von Neumann measurements. One obtains a probability that one's state is entangled and an estimate of any desired entanglement measure, including their error bars. As an example we consider (tomographically incomplete) spin correlation measurements on both 2-qubit and 3-qubit states. As byproducts we obtain estimates of the volume of entangled states vs. states that violate a given Bell inequality for both pure and mixed states, and an inequality that relates the expectation value of the Bell operator to the negativity.

DOI
10.48550/arxiv.0806.4165

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We show how a straightforward Bayesian updating procedure allows one to detect and quantify entanglement from any finite set of measurement results. The measurements do not have to be tomographically complete, and may consist of POVMs rather than von Neumann measurements. One obtains a probability that one's state is entangled and an estimate of any desired entanglement measure, including their error bars. As an example we consider (tomographically incomplete) spin correlation measurements on both 2-qubit and 3-qubit states. As byproducts we obtain estimates of the volume of entangled states vs. states that violate a given Bell inequality for both pure and mixed states, and an inequality that relates the expectation value of the Bell operator to the negativity.
We discuss how to characterize entanglement sources with finite sets of measurements. The measurements do not have to be tomographically complete and may consist of POVMs rather than von Neumann measurements. Our method yields a probability that the source generates an entangled state as well as estimates of any desired calculable entanglement measures, including their error bars. We apply two criteria, namely, Akaike's information criterion and the Bayesian information criterion, to compare and assess different models (with different numbers of parameters) describing entanglement-generating devices. We discuss differences between standard entanglement-verification methods and our present method of characterizing an entanglement source.
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Bell inequalities were meant to test quantum mechanics vs local hidden variable models, but can also be used to verify entanglement. For entanglement verification purposes one assumes the validity of quantum mechanics as well as quantum descriptions of one's measurements. With the help of these assumptions it is possible to derive a strengthened Bell inequality whose violation implies entanglement. We generalize known examples of such inequalities by relating the expectation value of the Bell operator to a particular quantitative measure of entanglement, namely, the negativity. Moreover, we obtain statistics illustrating the fact that violating a given (strengthened or not) Bell inequality is a much more rare feat for a quantum state of two qubits than it is to be entangled.
A complete chapter on quantum entanglement measures and detection of entanglement is given because there are a wide variety of entanglement measures for pure and mixed states which are used very often. The interesting feature of these routines is the fact that, these operations can be done between any random set of qubits not necessarily contiguous qubits. Key recipes to perform operations like partial tracing and partial transpose are disscussed in detail. After that, it was decided to devote several routines to calculate the entanglement measures like concurrence and its variants, von Neumann entropy, block entropy, Renyi entropy, Q measure, negativity and log negativity, entanglement spectrum and residual entanglement (three tangle). Last but not least, certain entanglement detection procedures like Positive Partial Transpose (the Peres-Horodecki criteria) and the reduction criteria are also explained via examples.
Recently, V\'ertesi and Bene [Phys. Rev. A 82, 062115 (2010)] derived a two-qubit Bell inequality ICH3{I}_{\mathrm{CH}3}ICH3​, which they show to be maximally violated only when more general positive operator-valued measures (POVMs) are used instead of the usual von Neumann measurements. Here we consider a general parametrization for the three-element POVM involved in the Bell test and obtain a higher quantum bound for the ICH3{I}_{\mathrm{CH}3}ICH3​ inequality. With a higher quantum bound for ICH3{I}_{\mathrm{CH}3}ICH3​, we investigate whether there is an experimental setup that can be used for observing that POVMs give higher violations in Bell tests based on this inequality. We analyze the maximum errors supported by the inequality to identify a source of entangled photons that can be used for the test. Then we study whether POVMs are also relevant in the more realistic case where partially entangled states are used in the experiment. Finally, we investigate the required efficiencies of the ICH3{I}_{\mathrm{CH}3}ICH3​ inequality, and the type of measurements involved, for closing the detection loophole. We obtain the result that POVMs allow for the lowest threshold detection efficiency, and that it is comparable to the minimal required detection efficiency (in the case of two qubits) of the Clauser-Horne-Bell inequality.