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International Journal of Theoretical Physics

Bell’s Inequalities and Methods of Quantifying Measures of Entanglement Correlations

Paul Bracken

Published Apr 7, 2014• 3 citations
Publisher
DOI
10.1007/s10773-014-2079-8

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Abstract. Entanglement is a critical resource used in many current quantum information schemes. As such entanglement has been extensively studied in two qubit systems and its entanglement nature has been exhibited by violations of the Bell inequality. Can the amount of violation of the Bell inequality be used to quantify the degree of entanglement? What do Bell inequalities indicate about the nature of entanglement? Entanglement was recognized early as one of the key features of quantum mechanics [1, 2]. Entanglement can be described as the correlation between distinct subsystems which cannot be created by local actions on each subsystem separately. The advantage o ered by quantum entanglement relies on the crucial premise that it cannot be reproduced by any classical theory [3-5]. Despite the fact that the possibility of quantum entanglement was acknowledged almost as soon as quantum theory was discovered, it is only in recent years that consideration has been given to nding methods to quantify it [6-17]. Historically the Bell inequal-ities were seen as a means of determining whether a two qubit system is entangled.
We find a single parameter family of genuinely entangled three-qubit pure states, called the maximally Bell-inequality violating states (MBV), which exhibit maximum Bell-inequality violation by the reduced bipartite system for a fixed amount of genuine tripartite entanglement quantified by the so-called tangle measure. This in turn implies that there holds a complementary relation between the Bell-inequality violation by the reduced bipartite systems and the tangle present in the three-qubit states, not necessarily pure. The MBV states also exhibit maximum Bell-inequality violation by the reduced bipartite systems of the three-qubit pure states with a fixed amount of genuine tripartite correlation quantified by the generalized geometric measure, a genuine entanglement measure of multiparty pure states, and the discord monogamy score, a multipartite quantum correlation measure from information-theoretic paradigm. The aforementioned complementary relation has also been established for three-qubit pure states for the generalized geometric measure and the discord monogamy score, respectively. The complementarity between the Bell-inequality violation by the reduced bipartite systems and the genuine tripartite correlation suggests that the Bell-inequality violation in the reduced two-qubit system comes at the cost of the total tripartite correlation present in the entire system.
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.