$k$-extendibility of high-dimensional bipartite quantum states

The idea of detecting the entanglement of a given bipartite state by searching for symmetric extensions of this state was first proposed by Doherty, Parrilo and Spedialeri. The complete family of separability tests it generates, often referred to as the hierarchy of $k$-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 $k$-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 $k$-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 $\mathbf{C}^d\otimes\mathbf{C}^d$, when $d$ grows, the average width of the set of $k$-extendible states is equivalent to $(2/\sqrt{k})/d$, while random states obtained as partial traces over an environment $\mathbf{C}^s$ of uniformly distributed pure states are violating the $k$-extendibility test with probability going to $1$ if $s<((k-1)^2/4k)d^2$. Both statements converge to the conclusion that, if $k$ is fixed, $k$-extendibility is asymptotically a weak approximation of separability, even though any of the other well-studied separability relaxations is outperformed by $k$-extendibility as soon as $k$ is above a certain (dimension independent) value.