Covariance matrix entanglement criterion for an arbitrary set of operators

Abstract

A criterion for entanglement detection based on covariance matrices for an arbitrary set of observables is formulated. The method generalizes the covariance matrix entanglement criterion by Simon (2000 Phys. Rev. Lett. 84 2726) to a more general set of operators using the positive partial transpose test for the covariance matrix. The relation is found by starting from the generalized uncertainty relation for multiple operators, and taking the partial transpose on the bipartition. The method is highly efficient and versatile in the sense that the set of measurement operators can be freely chosen, and there is no constraint on the commutation relations. The main restriction on the chosen set of measurement operators is that the correlators and expectation values of the partially transposed observable operators can be evaluated. 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. We illustrate the approach by examining the entanglement between two spin ensembles, and show that it detects entanglement in a basis independent way.

Stronger EPR-steering criterion based on inferred Schrödinger–Robertson uncertainty relation

“…It is shown that a stricter entanglement condition is provided by the SRUR, using positive partial transpose for non-Gaussian entangled states 42 , 43 and using covariance matrix entanglement criterion for Gaussian entangled states 44 . This has been later generalized by V Tripathi et al 45 .…”

Certification of non-Gaussian Einstein–Podolsky–Rosen steering

“…For k = l = 1 we recover the Gaussian CM and the correlations between standard quadratures. Our HO CM fulfills the following compact form of the uncertainty principle for arbitrary chosen orders k and l [63][64][65]:…”

Training variational quantum circuits with CoVaR: covariance root finding with classical shadows

“…Note that the above covariance matrix C(ψ) expresses fundamental uncertainty relations between the observables O k in the operator pool via the matrix inequation C ≥ 0 [18]. We remark that our definition above of a covariance matrix has complex entries: While this definition will simplify our following arguments, it is worth noting that in the literature other conventions are also commonly used [16][17][18]. For example, ref.…”

“…In contrast to usual variational minimisation of a single cost function, we leverage the following observation: in order to find an eigenstate a large number of properties of the variational quantum state must satisfy certain uncertainty relations with respect to observable measurements. We define these properties as covariances [16][17][18] between the problem Hamiltonian and elements of an operator pool of our choice. This definition leaves us great flexibility in choosing our operator pools and the ability to pose the problem of finding eigenstates as joint roots of all possible covariances.…”