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Journal of Physics Conference Series

Quantum Entanglement Criterion for Rank of Block Matrix Vector Group Based on Density Matrix

Guangrong Liu

Published Oct 1, 2019• 2 citations
Publisher

Abstract

Abstract The determination of quantum state separability is the basic problem of quantum entanglement theory. By judging the rank of the matrix vector group formed by the density matrix block, the necessary and sufficient conditions for the separation of quantum pure States are given, and the necessary conditions for the separation of quantum mixed States are given, too. In the composite system 2 × 2, the rank of the vector group corresponding to the divisible pure state is 1, and the rank corresponding to the entangled state is 4.

DOI
10.1088/1742-6596/1325/1/012137

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Abstract The determination of quantum state separability is the basic problem of quantum entanglement theory. By judging the rank of the matrix vector group formed by the density matrix block, the necessary and sufficient conditions for the separation of quantum pure States are given, and the necessary conditions for the separation of quantum mixed States are given, too. In the composite system 2 × 2, the rank of the vector group corresponding to the divisible pure state is 1, and the rank corresponding to the entangled state is 4.
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In previous paper ( 2019 Int. J. Mod. Phys. B 33 1950197 ; 2020 Int. J. Mod. Phys. B 34 2050022 ), we presented a method to judge the entanglement of 2-qubit system. The necessary and sufficient conditions for the 2qubit system being separable are that if the relevant coefficients is positive and the principal density matrix is separable, then the system is separable, otherwise it is entangled. Now in this paper, we try to generalize this criterion to a 3-qubit system, and then, we further generalize the criterion of 3-qubit system to an N-qubit system. This is a complicated and interesting issue.
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Quantum entanglement has been regarded as one of the key physical resources in quantum information sciences. However, the determination of whether a mixed state is entangled or not is generally a hard issue, even for the bipartite system. In this work we propose an operational necessary and sufficient criterion for the separability of an arbitrary bipartite mixed state, by virtue of the multiplicative Horn's problem. The work follows the work initiated by Horodecki et al. and uses the Bloch vector representation introduced to the separability problem by J. De Vicente. In our criterion, a complete and finite set of inequalities to determine the separability of compound system is obtained, which may be viewed as trade-off relations between the quantumness of subsystems. We apply the obtained result to explicit examples, e.g. the separable decomposition of arbitrary dimension Werner state and isotropic state.
Abstract The state space of quantum system is a Hilbert space with complex inner product. The two states quantum system is called a qubit. Its quantum state gives different geometric representations. The density matrix of a quantum state can describe all the information of a quantum system. In particular, the representation of quantum pure states by means of extended complex plane is given. On this basis, combined with the matrix representation of the quantum states of the composite system, the criteria for quantum entanglement and separability are studied. Finally, the measurement method of quantum pure state entanglement is given.