Detection of typical bipartite entanglement by local generalized measurements

Motivated by the need for efficient local entanglement detection for applications in quantum information processing, sufficient conditions for arbitrary-dimensional local bipartite entanglement detection based on correlation matrices and joint probability distributions are investigated. In particular, their dependence on the nature of different classes of local measurements is explored for generalized measurements based on informationally complete $(N,M)$ positive-operator-valued measures (POVMs) [Siudzinska, Phys. Rev. A 105, 042209 (2022)]. It is shown that symmetry properties of $(N,M)$ POVMs necessarily imply that these sufficient conditions for bipartite entanglement detection exhibit characteristic scaling properties relating equivalent sufficient conditions. Based on these general scaling properties, the efficiency of different classes of local quantum measurement detecting typical bipartite entanglement is investigated quantitatively. For this purpose Euclidean volume ratios between locally detectable bipartite entangled states and all bipartite quantum states are determined numerically with the help of a Monte Carlo algorithm. Our results demonstrate that physically realizable $(N,M)$ POVMs are sufficient for optimal local entanglement detection. In particular, this implies that for this purpose the construction of optimal $(N,M)$ POVMs is not necessary. As questions concerning the existence and construction of optimal $(N,M)$ POVMs are still largely open, this may offer interesting perspectives for practical applications in quantum information processing.

Figure 1

Dependence of volume ratios $R_{NM}({\tilde{x}}_A,{\tilde{x}}_B)$ between detected entangled and all bipartite quantum states on the scaled parameters ${\tilde{x}}_A$ and ${\tilde{x}}_B$ of Alice's and Bob's informationally complete local $(N,M)$ POVMs as obtained from a violation of inequality (37) for $(d_A,d_B)=(2,3)$. The horizontal dashed line indicates the upper bound for ${\tilde{x}}_B$ for $(N,M)$ POVMs with $M_B=2$. All $(N,M)$ POVMs with these values of $d_A$ and $d_B$ yield the same results.

Figure 2

Dependence of volume ratios $R_{NM}({\tilde{x}}_A,{\tilde{x}}_B)$ between detected entangled and all bipartite quantum states on the scaled parameters ${\tilde{x}}_A$ and ${\tilde{x}}_B$ of Alice's and Bob's informationally complete local $(N,M)$ POVMs as obtained from a violation of inequality (37) for $(d_A,d_B)=(3,3)$. The vertical and horizontal dashed lines indicate the upper bounds for ${\tilde{x}}_A$ and ${\tilde{x}}_B$ for $(N,M)$ POVMs with $M_A=2$ and $M_B=2$. All $(N,M)$ POVMs with these values of $d_A$ and $d_B$ yield the same results.