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.
Highlights
The method is 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
For systems with small Hilbert space dimension, one standard approach is to reconstruct the density matrix and perform a positive partial transpose (PPT) test to check for separability [6–10]
This is again due to the complexity of the optimization which scales with the Hilbert space dimension D, rather than the number of operators N chosen
Summary
Entanglement is a key resource for many quantum information technologies and detecting its presence is a fundamental experimental task. For systems with small Hilbert space dimension, one standard approach is to reconstruct the density matrix and perform a positive partial transpose (PPT) test to check for separability [6–10]. This requires full tomography of the density matrix, which for systems with large Hilbert space may be impractical or even impossible to measure. In this case what is most desirable (e.g. from an experimental point of view) are simple criteria that can be evaluated based on a small number of observables. This is in contrast to alternative entanglement detection methods which can require computations scaling with the Hilbert space dimension D ≫ N , which can potentially be large
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