Covariance matrix entanglement criterion for an arbitrary set of operators

Abstract

We generalize entanglement detection with covariance matrices for an arbitrary set of observables. A generalized uncertainty relation is constructed using the covariance and commutation matrices, then a criterion is established by performing a partial transposition on the operators. The method is highly efficient and versatile in the sense that the set of measurement operators can be freely chosen, do not need to be complete, and there is no constraint on the commutation relations. The method is particularly suited for systems with higher dimensionality since the computations do not scale with the dimension of the Hilbert space rather they scale with the number of chosen observables which can always be kept small. We illustrate the approach by examining the entanglement between two spin ensembles, and show that it detects entanglement in a basis independent way.

Characterizing coherence with quantum observables

“…The contribution due to correlations is found to be of a form that is the difference of two covariance matrices [33][34][35][36][37][38][39][40]. The same set of measurements that are used to construct the coherence measure can also be used to construct the covariance matrix, which has been used as an effective way of detecting entanglement [33][34][35][36][37][38][39][40][41][42]. Our approach in characterizing coherence with observables extends the toolbox for characterizing the features of a quantum state.…”

“…This quantity can be written in an illuminating way by defining the covariance matrix [33][34][35][36][37][38][39][40][41][42]…”