Measures of Quantum Entanglement

Vlatko Vedral, Martin Bodo Plenio

We improve previously proposed conditions each measure of entanglement has to satisfy. We present a class of entanglement measures that satisfy these conditions and show that the quantum relative entropy and Bures metric generate two measures of this class. We calculate the measures of entanglement for a number of mixed two spin-1/2 systems using the quantum relative entropy, and provide an efficient numerical method to obtain the measures of entanglement in this case. In addition, we prove a number of properties of our entanglement measure that have important physical implications. We briefly explain the statistical basis of our measure of entanglement in the case of the quantum relative entropy. We then argue that our entanglement measure determines an upper bound to the number of singlets that can be obtained by any purification procedure.

Leah Henderson

Jean-Luc Brylinski

Zhi‐Xiang Jin, Xianqing Li‐Jost, Shao-Ming Fei +1 more

Quantum entanglement has been verified experimentally and applied in quantum computing, quantum sensing, and quantum networks. It is of great significance to find measures to characterize the quantum entanglement faithfully. In this work, by exploiting the Schmidt decomposition of bipartite states, we first establish a one-to-one correspondence between the characteristic polynomial of the reduced state of a bipartite pure state and the trace of the reduced state. We introduce a family of entanglement measures based on the eigenvalues of the reduced density matrices. Specific measures called informationally complete entanglement measures (ICEMs) are presented. It is demonstrated that such ICEMs can characterize better the entanglement than the existing well-known entanglement measures. The ICEMs also give rise to criteria of state transformations under local operation and classical communication. Moreover, it is shown that the ICEMs can be efficiently estimated on a quantum computer. The full separability, entanglement, and genuine multipartite entanglement can be detected faithfully on quantum devices.

Prashanna Simkhada, Jean-François S. Van Huele

Danko Dimchev Georgiev, Stanley Gudder

Entanglement measures quantify the amount of quantum entanglement that is contained in quantum states. Typically, different entanglement measures do not have to be partially ordered. The presence of a definite partial order between two entanglement measures for all quantum states, however, allows for meaningful conceptualization of sensitivity to entanglement, which will be greater for the entanglement measure that produces the larger numerical values. Here, we have investigated the partial order between the normalized versions of four entanglement measures based on Schmidt decomposition of bipartite pure quantum states, namely, concurrence, tangle, entanglement robustness and Schmidt number. We have shown that among those four measures, the concurrence and the Schmidt number have the highest and the lowest sensitivity to quantum entanglement, respectively. Further, we have demonstrated how these measures could be used to track the dynamics of quantum entanglement in a simple quantum toy model composed of two qutrits. Lastly, we have employed state-dependent entanglement statistics to compute measurable correlations between the outcomes of quantum observables in agreement with the uncertainty principle. The presented results could be helpful in quantum applications that require monitoring of the available quantum resources for sharp identification of temporal points of maximal entanglement or system separability.