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arXiv:1510.08859 · Annals of Physics

Average subentropy, coherence and entanglement of random mixed quantum states

Lin Zhang, Uttam Singh, Arun Kumar Pati

Published Dec 19, 2016• 24 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2016.12.024

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The generic aspects of the entanglement for random pure states are known to be established via the powerful phenomena of concentration of measure. Here, we find analytical expressions for the average subentropy over the set of random mixed states generated via various probability measures on it. The main ingredients in these results are Selberg's integrals. Surprisingly, our results show that the average subentropy of random mixed states approaches to the maximum value of the subentropy which is attained for the maximally mixed state as we increase the dimension. As an application, we find the average coherence of random mixed quantum states sampled from various probability measures, analytically. In the special case of the random mixed states sampled from the induced measure via the partial tracing of the random bipartite pure states, we establish the typicality of the relative entropy of coherence for random mixed states, numerically. In particular, we show that almost all random mixed states have relative entropy of coherence equal to the average relative entropy of coherence.
Abstract The Wishart ensemble is a useful and important random matrix model used in diverse fields. By realizing induced random mixed quantum states as a Wishart ensemble with fixed unit trace, using matrix integral technique we give a fast track to the average coherence for random mixed quantum states induced via partial-tracing of the Haar-distributed bipartite pure states. As a direct consequence of this result, we get a compact formula for the average subentropy of random mixed states. These compact formulae extend our previous work.
We propose to quantify the entanglement of pure states of $N\ifmmode\times\else\texttimes\fi{}N$ bipartite quantum systems by defining its Husimi distribution with respect to $\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(N)$ coherent states. The Wehrl entropy is minimal if and only if the analyzed pure state is separable. The excess of the Wehrl entropy is shown to be equal to the subentropy of the mixed state obtained by partial trace of the bipartite pure state. This quantity, as well as the generalized (R\'enyi) subentropies, are proved to be Schur concave, so they are entanglement monotones and may be used as alternative measures of entanglement.