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:quant-ph/0506252 · Physical Review A

Clauser-Horne-Shimony-Holt violation and the entropy-concurrence plane

Łukasz Derkacz, Lech Jakóbczyk

Published Oct 21, 2005• 27 citations
arXivHTMLPDF

Abstract

We characterize violation of Clauser-Horne-Shimony-Holt (CHSH) inequalities for mixed two-qubit states by their mixedness and entanglement. The class of states that have maximum degree of CHSH violation for a given linear entropy is also constructed.
DOI
10.1103/physreva.72.042321

Related papers

Nonlinear Inequalities and Entropy-Concurrence Plane

Bovino, F A

Nonlinear inequalities based on the quadratic Renyi entropy for mixed two-qubit states are characterized on the Entropy-Concurrence plane. This class of inequalities is stronger than Clauser-Horne-Shimony-Holt (CHSH) inequalities and, in particular, are violated "in toto" by the set of Type I Maximally-Entangled-Mixture States (MEMS I).

•Dec 20, 2006•1 citation

Entanglement estimation from Bell inequality violation

Karol Bartkiewicz, Bohdan Horst, Karel Lemr +1 more

It is well known that the violation of Bell's inequality in the form given by Clauser, Horne, Shimony, and Holt (CHSH) in two-qubit systems requires entanglement, but not vice versa, i.e., there are entangled states which do not violate the CHSH inequality. Here we compare some standard entanglement measures with violations of the CHSH inequality (as given by the Horodecki measure) for two-qubit states generated by Monte Carlo simulations. We describe states that have extremal entanglement according to the negativity, concurrence, and relative entropy of entanglement for a given value of the CHSH violation. We explicitly find these extremal states by applying the generalized method of Lagrange multipliers based on the Karush-Kuhn-Tucker conditions. The found minimal and maximal states define the range of entanglement accessible for any two-qubit states that violate the CHSH inequality by the same amount. We also find extremal states for the concurrence versus negativity by considering only such states which do not violate the CHSH inequality. Furthermore, we describe an experimentally efficient linear-optical method to determine the highest Horodecki degree of the CHSH violation for arbitrary mixed states of two polarization qubits. By assuming to have access simultaneously to two copies of the states, our method requires only six discrete measurement settings instead of nine settings, which are usually considered.

arXiv:1306.6504•Physical Review A•Nov 7, 2013•96 citations

Variant of the Clauser-Horne-Shimony-Holt inequality

Zeqian Chen

We construct a Bell inequality from the Clauser-Horne-Shimony-Holt inequality for two qubits that provides a stronger bound on the correlations of entangled states than allowed by the CHSH inequality. The argument involved here can be generalized to $n$ qubits. The inequalities obtained are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits.

arXiv:0711.0810•Nov 6, 2007•0 citations

Theoretical and Experimental Characterization of Entropic Inequalities

Fabio Antonio Bovino, Concita Sibilia, Diederik Sybolt Wiersma

Nonlinear inequalities based on the quadratic Renyi entropy for mixed two‐qubit states are characterized on the Entropy‐Concurrence plane. This class of inequalities is stronger than Clauser‐Horne‐Shimony‐Holt (CHSH) inequalities and, in particular, are violated “in toto” by the set of Type I Maximally‐Entangled‐Mixture States (MEMS I). Renyi entropy is experimentally obtained by local measurements on two pairs of polarization‐entangled photons. The novel “phase marking” technique allows the selection of uncorrupted outcomes even with nondeterministic sources of entangled photons. Experimental data demonstrate the violation of entropic inequalities which are an example of nonlinear entanglement witnesses.

AIP conference proceedings•Jan 1, 2007•1 citation

ENTANGLEMENT, PURITY AND VIOLATION OF BELL INEQUALITY

Dong-Ling Deng, Jing‐Ling Chen

We use the Clauser–Horne–Shimony-Holt (CHSH) inequality to investigate the relationship among entanglement, purity and violation of the Bell inequality. On the one hand, we show numerically that all two-dimensional (qubit) states, whose entanglement of formation (EOF) is larger than [Formula: see text], violate the CHSH inequality. On the other hand, any state with purity smaller than 0.5562 may not violate it.

International Journal of Quantum Information•Oct 1, 2009•3 citations

Maximal Clauser-Horne-Shimony-Holt violation for qubit-qudit states

Alexander Bernal, J.A. Casas, Jesús M. Moreno

We evaluate the maximal Clauser-Horne-Shimony-Holt (CHSH) violation for a generic (typically mixed) qubit-qudit state, obtaining easily computable expressions in arbitrary qudit dimension. This represents the optimal (2-2-2) Bell nonlocality for this kind of systems. The work generalizes the well-known Horodeckis result for a qubit-qubit setup. We also give simple lower and upper bounds on that violation. We apply our general results to address a number of issues, namely, we obtain a bound on the degree of purity required in a system to exhibit nonlocality and study the statistics of nonlocality in random density matrices. In addition, we show the impossibility of improving the amount of CHSH violation by embedding the qudit in a Hilbert space of larger dimension. Finally, the general result is illustrated with a family of density matrices in the context of a qubit-qutrit system.

arXiv:2404.02092•Physical Review A•Jul 22, 2025•4 citations
It is well known that the violation of Bell's inequality in the form given by Clauser, Horne, Shimony, and Holt (CHSH) in two-qubit systems requires entanglement, but not vice versa, i.e., there are entangled states which do not violate the CHSH inequality. Here we compare some standard entanglement measures with violations of the CHSH inequality (as given by the Horodecki measure) for two-qubit states generated by Monte Carlo simulations. We describe states that have extremal entanglement according to the negativity, concurrence, and relative entropy of entanglement for a given value of the CHSH violation. We explicitly find these extremal states by applying the generalized method of Lagrange multipliers based on the Karush-Kuhn-Tucker conditions. The found minimal and maximal states define the range of entanglement accessible for any two-qubit states that violate the CHSH inequality by the same amount. We also find extremal states for the concurrence versus negativity by considering only such states which do not violate the CHSH inequality. Furthermore, we describe an experimentally efficient linear-optical method to determine the highest Horodecki degree of the CHSH violation for arbitrary mixed states of two polarization qubits. By assuming to have access simultaneously to two copies of the states, our method requires only six discrete measurement settings instead of nine settings, which are usually considered.
We construct a Bell inequality from the Clauser-Horne-Shimony-Holt inequality for two qubits that provides a stronger bound on the correlations of entangled states than allowed by the CHSH inequality. The argument involved here can be generalized to nnn qubits. The inequalities obtained are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits.
Nonlinear inequalities based on the quadratic Renyi entropy for mixed two‐qubit states are characterized on the Entropy‐Concurrence plane. This class of inequalities is stronger than Clauser‐Horne‐Shimony‐Holt (CHSH) inequalities and, in particular, are violated “in toto” by the set of Type I Maximally‐Entangled‐Mixture States (MEMS I). Renyi entropy is experimentally obtained by local measurements on two pairs of polarization‐entangled photons. The novel “phase marking” technique allows the selection of uncorrupted outcomes even with nondeterministic sources of entangled photons. Experimental data demonstrate the violation of entropic inequalities which are an example of nonlinear entanglement witnesses.
We evaluate the maximal Clauser-Horne-Shimony-Holt (CHSH) violation for a generic (typically mixed) qubit-qudit state, obtaining easily computable expressions in arbitrary qudit dimension. This represents the optimal (2-2-2) Bell nonlocality for this kind of systems. The work generalizes the well-known Horodeckis result for a qubit-qubit setup. We also give simple lower and upper bounds on that violation. We apply our general results to address a number of issues, namely, we obtain a bound on the degree of purity required in a system to exhibit nonlocality and study the statistics of nonlocality in random density matrices. In addition, we show the impossibility of improving the amount of CHSH violation by embedding the qudit in a Hilbert space of larger dimension. Finally, the general result is illustrated with a family of density matrices in the context of a qubit-qutrit system.