k-Extendibility of high-dimensional bipartite quantum states
Cécilia Lancien · Random Matrices Theory and Application · 2016
The idea of detecting the entanglement of a given bipartite state by searching for symmetric extensions of this state was first proposed by Doherty et al. The complete family of separability tests it generates, often referred to as the hierarchy of [Formula: see text]-extendibility tests, has already proved to be most promising. The goal of this paper is to try and quantify the efficiency of this separability criterion in typical scenarios. For that, we essentially take two approaches. First, we compute the average width of the set of [Formula: see text]-extendible states, in order to see how it scales with the one of separable states. And second, we characterize when random-induced states are, depending on the ancilla dimension, with high probability violating or not the [Formula: see text]-extendibility test, and compare the obtained result with the corresponding one for entanglement vs separability. The main results can be precisely phrased as follows: on [Formula: see text], when [Formula: see text] grows, the average width of the set of [Formula: see text]-extendible states is equivalent to [Formula: see text], while random states obtained as partial traces over an environment [Formula: see text] of uniformly distributed pure states are violating the [Formula: see text]-extendibility test with probability going to [Formula: see text] if [Formula: see text]. Both statements converge to the conclusion that, if [Formula: see text] is fixed, [Formula: see text]-extendibility is asymptotically a weak approximation of separability, even though any of the other well-studied separability relaxations is outperformed by [Formula: see text]-extendibility as soon as [Formula: see text] is above a certain (dimension independent) value.