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

Physics Letters A

Optimized contextuality for two qubits

Asma Al-Qasimi, Daniel F. V. James

Published Dec 9, 2024• 2 citations
Publisher

Abstract

Generalizing the ideas of Peres [1] , we investigate a measure for the contextuality of a two-qubit state and derive an analytic expression applicable to the most general mixed state. Further, we investigate the relationships contextuality has with quantum entanglement and quantum discord .
DOI
10.1016/j.physleta.2024.130148

Related papers

Comment on "Contextuality and quantum discord"

Chellasamy Jebarathinam

In a paper, Al-Qasimi [Physics Letters A 449 (2022) 128347] proposed a criterion for contextuality of two-qubit systems extending Peres' proof of contextuality [Phys. Lett. A 151 (1990) 107]. Using this criterion, Al-Qasimi argued that certain discordant states can be contextual even if they are not entangled. Here I demonstrate that Al-Qasimi's criterion does not, in general, imply contextuality of two-qubit system. However, I point out that Al-Qasimi's criterion is related to a different notion of operational nonclassicality of discordant feature, which is inequivalent to contextuality. This nonclassicality beyond contextuality has been formalized in C. Jebarathinam and R. Srikanth [Int. J. Quantum Inf. 23 (2025) 2550037]. Identifying Al-Qasimi's criterion as demonstration of quantum correlation beyond contextuality, it then follows that Al-Qasimi's main conclusion that one cannot equate quantum correlation and discordant feature is made precise.

quant-pharXiv:2512.11450•Dec 15, 2025•0 citations

Quantum discord for general two-qubit states: Analytical progress

Davide Girolami, Gerardo Adesso

We present a reliable algorithm to evaluate quantum discord for general two-qubit states, amending and extending an approach recently put forward for the subclass of $X$ states. A closed expression for the discord of arbitrary states of two qubits cannot be obtained, as the optimization problem for the conditional entropy requires the solution to a pair of transcendental equations in the state parameters. We apply our algorithm to run a numerical comparison between quantum discord and an alternative, computable measure of nonclassical correlations, namely, the geometric discord. We identify the extremally nonclassically correlated two-qubit states according to the (normalized) geometric discord, at a fixed value of the conventional quantum discord. The latter cannot exceed the square root of the former for systems of two qubits.

arXiv:1103.3189•Physical Review A•May 9, 2011•317 citations

Contextuality and quantum discord

Asma Al-Qasimi

One of the defining differences between classical and quantum systems is how measurements affect them. Here, we compare the approaches of contextuality and quantum discord in capturing quantum correlations in special classes of two-qubit states, demonstrating that although non-discordant states are non-contextual, discordant states are not always contextual.

arXiv:2208.01698•Physics Letters A•Jul 30, 2022•3 citations

Quantum discord for two-qubit systems

Shunlong Luo

Quantum discord, as introduced by Olliver and Zurek [Phys. Rev. Lett. 88, 017901 (2001)], is a measure of the discrepancy between two natural yet different quantum analogs of the classical mutual information. This notion characterizes and quantifies quantumness of correlations in bipartite states from a measurement perspective, and is fundamentally different from the various entanglement measures in the entanglement vs separability paradigm. The phenomenon of nonzero quantum discord is a manifestation of quantum correlations due to noncommutativity rather than due to entanglement, and has interesting and significant applications in revealing the advantage of certain quantum tasks. We will evaluate analytically the quantum discord for a large family of two-qubit states, and make a comparative study of the relationships between classical and quantum correlations in terms of the quantum discord. We furthermore compare the quantum discord with the entanglement of formation, and illustrate that the latter may be larger than the former, although for separable states, the entanglement of formation always vanishes and thus is less than the quantum discord.

Physical Review A•Apr 3, 2008•1,060 citations

Quantum discord for two-qubit X states: Analytical formula with very small worst-case error

Yichen Huang

Quantum discord is a measure of quantum correlation beyond entanglement. Computing quantum discord for simple quantum states is a basic problem. An analytical formula of quantum discord for two-qubit $X$ states is first claimed in [Ali, Rau, and Alber, Phys. Rev. A 81, 042105 (2010)], but later found to be not always correct. I observe numerically that the formula is valid with worst-case absolute error 0.0021. For symmetric two-qubit $X$ states, I give a counterexample to the analytical formula derived in [F. F. Fanchini et al., Phys. Rev. A 81, 052107 (2010)], but observe that the formula is valid with worst-case absolute error 0.0006. The formula has been used in many research papers. The results in all these works are approximately correct, even if they may not be exactly correct.

arXiv:1306.0228•Physical Review A•Jul 18, 2013•135 citations

Faithful nonclassicality indicators and extremal quantum correlations in two-qubit states

Davide Girolami, Mauro Paternostro, Gerardo Adesso

The state disturbance induced by locally measuring a quantum system yields a signature of nonclassical correlations beyond entanglement. Here, we present a detailed study of such correlations for two-qubit mixed states. To overcome the asymmetry of quantum discord and the unfaithfulness of measurement-induced disturbance (severely overestimating quantum correlations), we propose an ameliorated measurement-induced disturbance as nonclassicality indicator, optimized over joint local measurements, and we derive its closed expression for relevant two-qubit states. We study its analytical relation with discord, and characterize the maximally quantum-correlated mixed states, that simultaneously extremize both quantifiers at given von Neumann entropy: among all two-qubit states, these states possess the most robust quantum correlations against noise.

arXiv:1008.4136•Journal of Physics A Mathematical and Theoretical•Aug 15, 2011•82 citations
In a paper, Al-Qasimi [Physics Letters A 449 (2022) 128347] proposed a criterion for contextuality of two-qubit systems extending Peres' proof of contextuality [Phys. Lett. A 151 (1990) 107]. Using this criterion, Al-Qasimi argued that certain discordant states can be contextual even if they are not entangled. Here I demonstrate that Al-Qasimi's criterion does not, in general, imply contextuality of two-qubit system. However, I point out that Al-Qasimi's criterion is related to a different notion of operational nonclassicality of discordant feature, which is inequivalent to contextuality. This nonclassicality beyond contextuality has been formalized in C. Jebarathinam and R. Srikanth [Int. J. Quantum Inf. 23 (2025) 2550037]. Identifying Al-Qasimi's criterion as demonstration of quantum correlation beyond contextuality, it then follows that Al-Qasimi's main conclusion that one cannot equate quantum correlation and discordant feature is made precise.
We present a reliable algorithm to evaluate quantum discord for general two-qubit states, amending and extending an approach recently put forward for the subclass of XXX states. A closed expression for the discord of arbitrary states of two qubits cannot be obtained, as the optimization problem for the conditional entropy requires the solution to a pair of transcendental equations in the state parameters. We apply our algorithm to run a numerical comparison between quantum discord and an alternative, computable measure of nonclassical correlations, namely, the geometric discord. We identify the extremally nonclassically correlated two-qubit states according to the (normalized) geometric discord, at a fixed value of the conventional quantum discord. The latter cannot exceed the square root of the former for systems of two qubits.
Quantum discord, as introduced by Olliver and Zurek [Phys. Rev. Lett. 88, 017901 (2001)], is a measure of the discrepancy between two natural yet different quantum analogs of the classical mutual information. This notion characterizes and quantifies quantumness of correlations in bipartite states from a measurement perspective, and is fundamentally different from the various entanglement measures in the entanglement vs separability paradigm. The phenomenon of nonzero quantum discord is a manifestation of quantum correlations due to noncommutativity rather than due to entanglement, and has interesting and significant applications in revealing the advantage of certain quantum tasks. We will evaluate analytically the quantum discord for a large family of two-qubit states, and make a comparative study of the relationships between classical and quantum correlations in terms of the quantum discord. We furthermore compare the quantum discord with the entanglement of formation, and illustrate that the latter may be larger than the former, although for separable states, the entanglement of formation always vanishes and thus is less than the quantum discord.
Quantum discord is a measure of quantum correlation beyond entanglement. Computing quantum discord for simple quantum states is a basic problem. An analytical formula of quantum discord for two-qubit XXX states is first claimed in [Ali, Rau, and Alber, Phys. Rev. A 81, 042105 (2010)], but later found to be not always correct. I observe numerically that the formula is valid with worst-case absolute error 0.0021. For symmetric two-qubit XXX states, I give a counterexample to the analytical formula derived in [F. F. Fanchini et al., Phys. Rev. A 81, 052107 (2010)], but observe that the formula is valid with worst-case absolute error 0.0006. The formula has been used in many research papers. The results in all these works are approximately correct, even if they may not be exactly correct.
The state disturbance induced by locally measuring a quantum system yields a signature of nonclassical correlations beyond entanglement. Here, we present a detailed study of such correlations for two-qubit mixed states. To overcome the asymmetry of quantum discord and the unfaithfulness of measurement-induced disturbance (severely overestimating quantum correlations), we propose an ameliorated measurement-induced disturbance as nonclassicality indicator, optimized over joint local measurements, and we derive its closed expression for relevant two-qubit states. We study its analytical relation with discord, and characterize the maximally quantum-correlated mixed states, that simultaneously extremize both quantifiers at given von Neumann entropy: among all two-qubit states, these states possess the most robust quantum correlations against noise.