On the comparison of volumes of quantum states

          @misc{ pirsa_PIRSA:10070010,
            doi = {10.48660/10070010},
            url = {https://pirsa.org/10070010},
            author = {Ye, Deping},
            keywords = {},
            language = {en},
            title = {On the comparison of volumes of quantum states},
            publisher = {Perimeter Institute},
            year = {2010},
            month = {jul},
            note = {PIRSA:10070010 see, \url{https://pirsa.org}}
          }
          

Abstract

Entangled (i.e., not separable) quantum states play fundamental roles in quantum information theory; therefore, it is important to know the ''size'' of entanglement (and hence separability) for various measures, such as, Hilbert-Schmidt measure, Bures measure, induced measure, and $\alpha$-measure. In this talk, I will present new comparison results of $\alpha$-measure with Bures measure and Hilbert-Schmidt measure. Employing these comparison results to the subsets of separable states and of states with positive partial transpose, we show that the probability of separability is very small, and the well-known Peres-Horodecki PPT Criterion as a tool to detect separability is imprecise for (even moderate) large dimension of Hilbert space. This talk is based on my papers: J. Math. Phys. 50 (2009) 083502, and J. Phys. A: Math. Theor. in press.