Abstract
We develop a scheme for the detection of entanglement in any continuous variable system, by constructing an optimal entanglement witness from random homodyne measurements. To this end, we introduce a set of linear constraints that guarantee the necessary properties of a witness and allow for its optimisation via a semidefinite program. We test our method on the class of squeezed vacuum states and study the efficiency of entanglement detection in general unknown covariance matrices. The results show that we can detect entanglement, including bound entanglement, in arbitrary continuous variable states with fewer measurements than in full tomography. The statistical analysis of our method shows a good robustness to statistical errors in experiments.
Highlights
The most valuable characteristics of quantum systems are quantum correlations such as entanglement, which represents a useful resource for applications unattainable in the framework of classical theory, such as quantum teleportation, quantum cryptography and dense coding [1, 2]
The evident difference in the efficiency of entanglement detection in random covariance matrix (CM) compared to squeezed vacuum states may reside in the fact that highly squeezed states look classical in random measurement directions, which does not have to be the case for random states
Since the proposed semidefinite program (SDP) algorithm can be generalized to multi-mode continuous variable (CV) states, we provide an example of a four-mode CM with 12 independent parameters, mentioned in Ref. [28], which has bipartite bound entanglement:
Summary
The most valuable characteristics of quantum systems are quantum correlations such as entanglement, which represents a useful resource for applications unattainable in the framework of classical theory, such as quantum teleportation, quantum cryptography and dense coding [1, 2]. First a full tomography is required for completely unknown states, in order to reconstruct the entire covariance matrix This method may be a very resource-consuming and demanding experimental procedure, especially for quantum states with a high number of modes. The best strategy in this case would be to perform random measurements, serving as building blocks for the construction of an entanglement witness by means of a semidefinite optimization algorithm This idea is inspired by an analogous method for the discrete-variable case, which was developed in [16].
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