Inequalities for the Schmidt number of bipartite states

Lin Chen, Yang Yu, Wai-Shing Tang

Farid Shähandeh, Jan Sperling, W. Vogel

Hao Chen

Schmidt numbers of bipartite mixed states [B. M. Torhal and P. Horodecki, Phys. Rev. A 61, R040301 (2001)] characterize the minimum Schmidt ranks of pure states that are needed to construct such mixed states. Schmidt number is the minimum number of entangled degrees of freedom of a bipartite mixed state. We give a lower bound for the Schmidt numbers of low-rank bipartite mixed states and conclude that generic low-rank mixed states have relatively high Schmidt numbers and thus entangled.

Xian Shi

Abstract Higher dimensional entangled states demonstrate significant advantages in quantum information processing tasks. The Schmidt number is a quantity of the entanglement dimension of a bipartite state. Here we build families of k-positive maps from the symmetric information complete positive operator-valued measurements and mutually unbiased bases, and we also present the Schmidt number witnesses, correspondingly. At last, based on the witnesses obtained from mutually unbiased bases, we show the distance between a bipartite state and the set of states with a Schmidt number less than k.

Josh Cadney, Marcus Huber, Noah Linden +1 more

Huangjun Zhu, Masahito Hayashi, Lin Chen

We propose a general framework for constructing universal steering criteria that are applicable to arbitrary bipartite states and measurement settings of the steering party. The same framework is also useful for studying the joint measurement problem. Based on the data-processing inequality for an extended Rényi relative entropy, we then introduce a family of steering inequalities, which detect steering much more efficiently than those inequalities known before. As illustrations, we show unbounded violation of a steering inequality for assemblages constructed from mutually unbiased bases and establish an interesting connection between maximally steerable assemblages and complete sets of mutually unbiased bases. We also provide a single steering inequality that can detect all bipartite pure states of full Schmidt rank. In the course of study, we generalize a number of results intimately connected to data-processing inequalities, which are of independent interest.