Typical bipartite steerability and generalized local quantum measurements

Abstract

Abstract Recently proposed correlation-matrix-based sufficient conditions for bipartite steerability from Alice to Bob for arbitrary measurements are applied to local informationally complete positive operator valued measures (POVMs) of the (N,M)-type. These POVMs include a large class of local generalized measurements of current interest [Siudzińska K 2022 Phys. Rev. A 105 042209]. It is shown that the trace norm of correlation matrices with local (N,M)-POVMs is proportional to that of local orthonormal hermitian operator bases (LOOs). This 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 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 to a recently proposed method that determines bipartite steerability from Alice’s qubit to Bob’s arbitrary dimensional quantum system by bipartite entanglement detection. It is demonstrated that in general this method is significantly more effective in detecting typical steerability provided non local entanglement detection methods are used which transcend local measurements.