Draft:Computable Cross Norm Criterion

The bipartite quantum state is defined with its density matrix as^[1]

${\displaystyle \rho =\sum _{ijkl}(\varrho )_{ij,kl}|i\rangle \langle j|\otimes |k\rangle \langle l|,}$

where ${\displaystyle (\varrho )_{ij,kl}}$ are real coefficients. ${\displaystyle |i\rangle }$ and ${\displaystyle |j\rangle }$ are basis vectors for the first subsystem. ${\displaystyle |k\rangle }$ and ${\displaystyle |l\rangle }$ are basis vectors for the second subsystem.

First, present the approach of Ref.^[2]. The density matrix can be given by a matrix Schmidt decomposition as (Corollary 18 of Ref.^[2])

${\displaystyle \varrho =\sum _{k}\lambda _{k}A_{k}\otimes B_{k},}$

where ${\displaystyle \lambda _{k}\geq 0}$ are the Schmidt coefficients. ${\displaystyle A_{k}}$ correspond to the first subsystem, and ${\displaystyle B_{k}}$ correspond to the second subsystem. ${\displaystyle A_{k}}$ and ${\displaystyle B_{k}}$ form pairwise orthogonal bases for operators satisfying

${\displaystyle {\rm {Tr}}(A_{k}A_{l})=\delta _{kl},\quad {\rm {Tr}}(B_{k}B_{l})=\delta _{kl}.}$

For separable states

${\displaystyle \sum _{k}\lambda _{k}\leq 1}$

holds. Any state violating the above inequality is entangled (Corollary 18 of Ref.^[2].)

Next, we present the approach of. Ref.^[3]. The criterion can also be described with the realignment operation defined as

${\displaystyle (\varrho _{R})_{ik,jl}=(\varrho )_{ij,kl}.}$

If ${\displaystyle \varrho }$ is separable then ${\displaystyle ||\varrho _{R}||_{\rm {Tr}}\leq 1}$ holds, where ${\displaystyle ||...||_{\rm {Tr}}}$ is the trace norm. Any quantum state for which ${\displaystyle ||\varrho _{R}||_{\rm {Tr}}>1}$ holds is entangled.

Note that the realignment operation defines a rearrangement of the density matrix elements, similarly as the partial transpoition also rearranges the elements of the density matrix.

The CCNR criterion for separability is neither weaker nor stronger than the Peres-Horodecki criterion criterion.^[5]

For bipatite symmetric states, the Peres-Horodecki criterion and the Computable Cross Norm / Realignment criterion detects the same quantum states.^[6]

It is an important question, how much the CCNR entanglmenet criterion can be violated, and which is the quantum state that violates it the most. The larger the maximal violation, the easer it is to use the quantum state for an experimental test.

There are efficient numerical methods^[7] to find the maximal violation of the CCNR criterion for a given system size. The maximum violation is given in the following table. The quantum state corresponding to the ${\displaystyle 4\times 4}$ case is also found analytically.^[7]

More information The largest values for ...

The largest values for ${\displaystyle \vert \vert R(\varrho )\vert \vert _{\rm {tr}}}$ for ${\displaystyle d_{1}\times d_{2}}$ PPT states for various ${\displaystyle d.}$ The results were obtained by numerical maximization.^[7]

Dimension ${\displaystyle d_{1}\times d_{2}}$	Maximum of ${\displaystyle \vert \vert R(\varrho )\vert \vert _{\rm {tr}}}$
${\displaystyle 2\times 2}$	1
${\displaystyle 2\times 4}$	1
${\displaystyle 3\times 3}$	1.1891
${\displaystyle 3\times 4}$	1.2239
${\displaystyle 4\times 4}$	1.5
${\displaystyle 5\times 5}$	1.5
${\displaystyle 6\times 6}$	1.5881

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• Bound entanglement