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arXiv:quant-ph/0212047 · Physical Review A

Some properties of the computable cross-norm criterion for separability

Oliver Rudolph

Published Mar 27, 2003• 239 citations
arXivHTMLPDF

Abstract

The computable cross-norm (CCN) criterion is a powerful analytical and computable separability criterion for bipartite quantum states, which is also known to systematically detect bound entanglement. In certain aspects this criterion complements the well-known Peres positive partial transpose (PPT) criterion. In the present paper we study important analytical properties of the CCN criterion. We show that in contrast to the PPT criterion it is not sufficient in dimension $2\ifmmode\times\else\texttimes\fi{}2.$ In higher dimensions, theorems connecting the fidelity of a quantum state with the CCN criterion are proved. We also analyze the behavior of the CCN criterion under local operations and identify the operations that leave it invariant. It turns out that the CCN criterion is in general not invariant under local operations.

DOI
10.1103/physreva.67.032312

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Realignment Criterion: A necessary and sufficient condition for two-qubit XXX-states
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Quantum entanglement is an important form of quantum non-locality, and it is a very important physical resource in quantum computing, quantum cryptography and other applications. It can be said that quantum entanglement is the core of the whole quantum information technology. Therefore, determining whether a quantum state is entangled or not is the core of the whole problem. This thesis introduces some basic quantum entanglement criteria, including the Positive Partial Transpose Criterion (PPT Criterion), Computable Cross Norm Realignment Criterion (CCNR Criterion). This thesis focuses on the PPT criterion given by Peres and Horodecki, which is recognized as the strongest operable criterion at present and has quite strong judgment ability. This thesis elaborates the implementation process of PPT criteria and gives the specific process of PPT criteria MATLAB programming. In addition, the programmed PPT criterion is applied to the entanglement judgment of some quantum states, and the difference of the entanglement judgment ability between the PPT criterion and the CCNR criterion and their respective limitations are reflected in the entanglement judgment problem.