Draft:Computable Cross Norm Criterion

• Comment: This is effectively written in the first person. Devonian Wombat (talk) 12:34, 18 May 2026 (UTC)

• Comment: The topic may be notable, but it requires a complete rewrite:
  1. Please read WP:Lead. It should be a brief summary for a technical audience.
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  Please use the AfC process. WP pages are different from other science texts, it takes time to understand the differences and many academics get it wrong at first. Ldm1954 (talk) 13:41, 16 February 2026 (UTC)

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The Computable Cross Norm / Realignment (CCNR) Criterion is a necessary condition, for the joint density matrix $\rho$ of two quantum mechanical systems $A$ and $B$, to be separable. It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply. The criterion can detect some entangled states that are not detected by the Peres-Horodecki criterion. The Computable Cross Norm Criterion has been formulated by Oliver Rudolph. The Realignment criterion has been found by Kai Chen and Ling-An Wu. The two methods turned out to be equivalent.

Mathematical description

The bipartite quantum state is defined with its density matrix as

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

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

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

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

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

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

For separable states

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

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

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

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

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

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

Maximal violation of the criterion

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 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 $4\times 4$ case is also found analytically.

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

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

See also

• Bound entanglement