Abstract
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.
Highlights
Entanglement is a ubiquitous feature of quantum system and key element in quantum information processing, whereas yet far from fully understood1
We present an applicable criterion for the separability of bipartite mixed state through exploring the multiplicative Horn’s problem21
Example II: The relation with PPT5 scheme The partial transposition of a bipartite density matrix corresponds to the sign flips of columns or rows of, whose indices are that of skew symmetric generators, i.e., λμT = −λμ
Summary
Quantum entanglement has been regarded as one of the key physical resources in quantum information sciences. 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. We find that the solution to the multiplicative Horn’s problem yields a complete and finite set of inequalities, a new criterion, which in practice provides a systematic procedure for the decomposition of separable mixed states. Relations between this new criterion and other related ones are investigated through concrete examples, including the separable decomposition of arbitrary dimensional Werner and isotropic states. Results manifest that the criterion raised in this work is to our knowledge the most applicable one at present in determining the separability of entangled quantum states
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