Typical bipartite steerability and generalized local quantum measurements

Abstract

Recently proposed correlation-matrix based sufficient conditions for bipartite steerability from Alice to Bob are applied to local informationally complete positive operator valued measures (POVMs) of the $(N,M)$-type. These POVMs allow for a unified description of a large class of local generalized measurements of current interest. It is shown that this sufficient condition exhibits a peculiar scaling property. It implies that all types of informationally complete $(N,M)$-POVMs are equally powerful in detecting bipartite steerability from Alice to Bob and, in addition, they are as powerful as local orthonormal hermitian operator bases (LOOs). In order to explore the typicality of steering numerical calculations of lower bounds on Euclidean volume ratios between steerable bipartite quantum states from Alice to Bob and all quantum states are determined with the help of a hit-and-run Monte-Carlo algorithm. These results demonstrate that with the single exception of two qubits this correlation-matrix based sufficient condition significantly underestimates these volume ratios. These results are also compared with a recently proposed method which reduces the determination of bipartite steerability from Alice's qubit to Bob's arbitrary dimensional quantum system to the determination of bipartite entanglement. It is demonstrated that in general this method is significantly more effective in detecting typical steerability provided entanglement detection methods are used which transcend local measurements.