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arXiv:0908.0265 · Physical Review A

Characterizing entanglement sources

Pavel Lougovski, S. J. van Enk

Published Nov 18, 2009• 11 citations
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Abstract

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.

DOI
10.1103/physreva.80.052324

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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.
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.
Suppose an experimentalist wishes to verify that his apparatus produces entangled quantum states. A finite amount of data cannot conclusively demonstrate entanglement, so drawing conclusions from real-world data requires statistical reasoning. We propose a reliable method to quantify the weight of evidence for (or against) entanglement, based on a likelihood ratio test. Our method is universal in that it can be applied to any sort of measurements. We demonstrate the method by applying it to two simulated experiments on two qubits. The first measures a single entanglement witness, while the second performs a tomographically complete measurement.
Entanglement verification and measurement is essential for experimental tests of quantum mechanics and also for quantum communication and information science. Standard methods of verifying entanglement in a bipartite mixed state require detection of both particles and involve coincidence measurement. We present a method that enables us to verify and measure entanglement in a two-photon mixed state without detecting one of the photons, i.e., without performing any coincidence measurement or postselection. We consider two identical sources, each of which can generate the same two-photon mixed state but they never emit simultaneously. We show that one can produce a set of single-photon interference patterns, which contain information about entanglement in the two-photon mixed state. We prove that it is possible to retrieve the information about entanglement from the visibility of the interference patterns. Our method reveals a distinct avenue for verifying and measuring entanglement in mixed states.
The statistical analysis of measurement data has become a key component of many quantum engineering experiments. As standard full state tomography becomes unfeasible for large dimensional quantum systems, one needs to exploit prior information and the 'sparsity' properties of the experimental state in order to reduce the dimensionality of the estimation problem. In this paper we propose model selection as a general principle for finding the simplest, or most parsimonious explanation of the data, by fitting different models and choosing the estimator with the best trade-off between likelihood fit and model complexity. We apply two well established model selection methods—the Akaike information criterion (AIC) and the Bayesian information criterion (BIC)—two models consisting of states of fixed rank and datasets such as are currently produced in multiple ions experiments. We test the performance of AIC and BIC on randomly chosen low rank states of four ions, and study the dependence of the selected rank with the number of measurement repetitions for one ion states. We then apply the methods to real data from a four ions experiment aimed at creating a Smolin state of rank 4. By applying the two methods together with the Pearson χ 2 test we conclude that the data can be suitably described with a model whose rank is between 7 and 9. Additionally we find that the mean square error of the maximum likelihood estimator for pure states is close to that of the optimal over all possible measurements.