Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2506.23727

Realignment Criterion: A necessary and sufficient condition for two-qubit $X$-states

Amish Gogia, Bagga, Aakhyat, Satyabrata Adhikari, Mukhtiyar Singh

Published Jun 30, 2025• 0 citations
arXivHTMLPDF

Abstract

The Computable Cross Norm (CCN), or realignment criterion, is a widely used method for entanglement detection in quantum systems; however, it typically provides only a necessary condition. In this work, we advance the applicability of the realignment criterion by deriving a condition that is both necessary and sufficient for detecting entanglement in two-qubit. $X$-states derive their name from the characteristic 'X' shape of their density matrix, which contains seven independent matrix parameters. Notably, they incorporate several important subclasses of entangled states, including Bell states, Werner states, and maximally entangled mixed states. $X$-states have proven highly useful in entanglement studies due to their sparse structure and the ease with which entanglement-related quantities can be computed. This refined criterion improves the identification of entangled states that the standard CCN approach fails to detect, thereby extending the utility of the method.

DOI
10.48550/arxiv.2506.23727

Related papers

A few entanglement criterion for two-qubit and two-qudit system based on realignment operation

Shweta Kalson, Anchal Singh, Satyabrata Adhikari

It is known that realignment crierion is necessary but not a sufficient criterion for lower as well as higher dimensional system. In this work, we first consider a two-qubit system and derived the necessary and sufficient condition based on realignment operation for a particular class of two-qubit system. Thus we solved the problem of if and only if condition partially for a particular class of two-qubit state. We have shown that the derived necessary and sufficient condition detects two-qubit entangled states, which are not detected by the realignment criterion. Next, we discuss the higher dimensional system and obtained the necessary condition on the minimum singular value of the realigned matrix of $d\otimes d$ dimensional separable states. Moreover, we provide the geometrical interpretation of the derived separability criterion for $d\otimes d$ dimensional system. Furthermore, we show that our criterion may also detect bound entangled state. The entanglement detection criterion studied here is beneficial in the sense that it requires to calculate only minimum singular value of the realigned matrix while on the other hand realignment criterion requires all singular values of the realigned matrix. Thus, our criterion has computational advantage over the realignment criterion.

arXiv:2204.14014•Apr 29, 2022•0 citations

Some properties of the computable cross-norm criterion for separability

Oliver Rudolph

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.

arXiv:quant-ph/0212047•Physical Review A•Mar 27, 2003•239 citations

Bipartite quantum systems: on the realignment criterion and beyond

Cosmo Lupo, Paolo Aniello, Antonello Scardicchio

Inspired by the `computable cross norm' or `realignment' criterion, we propose a new point of view about the characterization of the states of bipartite quantum systems. We consider a Schmidt decomposition of a bipartite density operator. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are regarded as quantities that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states is derived. We conjecture that these conditions, which are weaker than the parent criterion, can be strengthened in such a way to obtain a new family of criteria that are independent of the original one. This conjecture is supported by numerical examples for the low dimensional cases. These ideas can be applied to the study of quantum channels, leading to a relation between the rate of contraction of a map and its ability to preserve entanglement.

arXiv:0802.2019•Journal of Physics A Mathematical and Theoretical•Sep 15, 2008•32 citations

Further results on entanglement detection and quantification from the correlation matrix criterion

Julio I de Vicente

The correlation matrix (CM) criterion is a recently derived powerful sufficient condition for the presence of entanglement in bipartite quantum states of arbitrary dimensions. It has been shown that it can be stronger than the positive partial transpose (PPT) criterion, as well as the computable cross norm or realignment (CCNR) criterion in different situations. However, it remained as an open question whether there existed sets of states for which the CM criterion could be stronger than both criteria simultaneously. Here, we give an affirmative answer to this question by providing examples of entangled states that scape detection by both the PPT and CCNR criteria whose entanglement is revealed by the CM condition. We also show that the CM can be used to measure the entanglement of pure states and obtain lower bounds for the entanglement measure known as tangle for general (mixed) states.

arXiv:0705.2583•Journal of Physics A Mathematical and Theoretical•Jan 29, 2008•76 citations

Entanglement criterion of computable cross norm and realignment for continuous-variable bipartite symmetric states

Li-zhen Jiang, Xiaoyu Chen, Ping Yu +1 more

We provide the entanglement criterion of computable cross norm and realignment for any two-mode continuous-variable state with permutational symmetry. The entanglement criterion for a non-Gaussian state is obtained from its Gaussian kernel by employing the technique of functional analysis. The entanglement conditions turn out to be necessary and sufficient for the non-Gaussian states prepared by one-photon subtracting from or adding to symmetric Gaussian states. The entanglement criterion of non-Gaussian states evolved in thermal noise and amplitude damping environment also is obtained.

arXiv:1211.5725•Physical Review A•Jan 27, 2014•8 citations

Correlation Minor Norm as a Detector and Quantifier of Entanglement

Bar Y. Peled, Amit Te’eni, Avishy Y. Carmi +1 more

In this paper we develop an approach for detecting entanglement, which is based on measuring quantum correlations and constructing a correlation matrix. The correlation matrix is then used for defining a family of parameters, named Correlation Minor Norms, which allow one to detect entanglement. This approach generalizes the computable cross-norm or realignment (CCNR) criterion, and moreover requires measuring a state-independent set of operators. Furthermore, we illustrate a scheme which yields for each Correlation Minor Norm a separable state that maximizes it. The proposed entanglement detection scheme is believed to be advantageous in comparison to other methods because correlations have a simple, intuitive meaning and in addition they can be directly measured in experiment. Moreover, it is demonstrated to be stronger than the CCNR criterion. We also illustrate the relation between the Correlation Minor Norm and entanglement entropy for pure states. Finally, we discuss the relation between the Correlation Minor Norm and quantum discord, and briefly discuss possible generalizations for multipartite scenarios.

•May 25, 2020•0 citations
Realignment Criterion: A necessary and sufficient condition for two-qubit XXX-states
The Computable Cross Norm (CCN), or realignment criterion, is a widely used method for entanglement detection in quantum systems; however, it typically provides only a necessary condition. In this work, we advance the applicability of the realignment criterion by deriving a condition that is both necessary and sufficient for detecting entanglement in two-qubit. XXX-states derive their name from the characteristic 'X' shape of their density matrix, which contains seven independent matrix parameters. Notably, they incorporate several important subclasses of entangled states, including Bell states, Werner states, and maximally entangled mixed states. XXX-states have proven highly useful in entanglement studies due to their sparse structure and the ease with which entanglement-related quantities can be computed. This refined criterion improves the identification of entangled states that the standard CCN approach fails to detect, thereby extending the utility of the method.
It is known that realignment crierion is necessary but not a sufficient criterion for lower as well as higher dimensional system. In this work, we first consider a two-qubit system and derived the necessary and sufficient condition based on realignment operation for a particular class of two-qubit system. Thus we solved the problem of if and only if condition partially for a particular class of two-qubit state. We have shown that the derived necessary and sufficient condition detects two-qubit entangled states, which are not detected by the realignment criterion. Next, we discuss the higher dimensional system and obtained the necessary condition on the minimum singular value of the realigned matrix of d⊗dd\otimes dd⊗d dimensional separable states. Moreover, we provide the geometrical interpretation of the derived separability criterion for d⊗dd\otimes dd⊗d dimensional system. Furthermore, we show that our criterion may also detect bound entangled state. The entanglement detection criterion studied here is beneficial in the sense that it requires to calculate only minimum singular value of the realigned matrix while on the other hand realignment criterion requires all singular values of the realigned matrix. Thus, our criterion has computational advantage over the realignment criterion.
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.
Inspired by the `computable cross norm' or `realignment' criterion, we propose a new point of view about the characterization of the states of bipartite quantum systems. We consider a Schmidt decomposition of a bipartite density operator. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are regarded as quantities that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states is derived. We conjecture that these conditions, which are weaker than the parent criterion, can be strengthened in such a way to obtain a new family of criteria that are independent of the original one. This conjecture is supported by numerical examples for the low dimensional cases. These ideas can be applied to the study of quantum channels, leading to a relation between the rate of contraction of a map and its ability to preserve entanglement.
The correlation matrix (CM) criterion is a recently derived powerful sufficient condition for the presence of entanglement in bipartite quantum states of arbitrary dimensions. It has been shown that it can be stronger than the positive partial transpose (PPT) criterion, as well as the computable cross norm or realignment (CCNR) criterion in different situations. However, it remained as an open question whether there existed sets of states for which the CM criterion could be stronger than both criteria simultaneously. Here, we give an affirmative answer to this question by providing examples of entangled states that scape detection by both the PPT and CCNR criteria whose entanglement is revealed by the CM condition. We also show that the CM can be used to measure the entanglement of pure states and obtain lower bounds for the entanglement measure known as tangle for general (mixed) states.
We provide the entanglement criterion of computable cross norm and realignment for any two-mode continuous-variable state with permutational symmetry. The entanglement criterion for a non-Gaussian state is obtained from its Gaussian kernel by employing the technique of functional analysis. The entanglement conditions turn out to be necessary and sufficient for the non-Gaussian states prepared by one-photon subtracting from or adding to symmetric Gaussian states. The entanglement criterion of non-Gaussian states evolved in thermal noise and amplitude damping environment also is obtained.
In this paper we develop an approach for detecting entanglement, which is based on measuring quantum correlations and constructing a correlation matrix. The correlation matrix is then used for defining a family of parameters, named Correlation Minor Norms, which allow one to detect entanglement. This approach generalizes the computable cross-norm or realignment (CCNR) criterion, and moreover requires measuring a state-independent set of operators. Furthermore, we illustrate a scheme which yields for each Correlation Minor Norm a separable state that maximizes it. The proposed entanglement detection scheme is believed to be advantageous in comparison to other methods because correlations have a simple, intuitive meaning and in addition they can be directly measured in experiment. Moreover, it is demonstrated to be stronger than the CCNR criterion. We also illustrate the relation between the Correlation Minor Norm and entanglement entropy for pure states. Finally, we discuss the relation between the Correlation Minor Norm and quantum discord, and briefly discuss possible generalizations for multipartite scenarios.