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Connection between the contextuality breaking and incompatibility breaking qubit channels

Swati Kumari, Sumit Mukherjee, R. Prabhu

Abstract

Contextuality and measurement incompatibility are two fundamental aspects of nonclassicality, and their manifestations in observed quantum correlations are often deeply interconnected. Recently, measurement incompatibility has been studied in connection with nonlocality, particularly in terms of their robustness under various quantum channels. This line of investigation helps establish a connection between the channels that break nonlocality and those that break incompatibility. In this study, we focus on an asymmetric bipartite Bell scenario involving three and four inputs on Alice and Bob sides, respectively, with each of these inputs having dichotomous outcomes. Under the assumption of locality, the observed statistics in this asymmetric scenario obeys the Elegant Bell inequality (EBI). Here, we use a different version of the EBI that relies on the assumption of the preparation noncontextuality. By taking the violation of this noncontextual version of EBI as a witness of preparation contextuality we establish a connection between the channels that break contextuality and the channels that break triple-wise measurement incompatibility. Our results suggest that any channel which breaks EBI contextuality will also break Clauser-Horne-Shimony-Holt (CHSH) nonlocality; however, the reverse does not hold. We also show that a depolarising channel that breaks N-wise incompatibility can also break a certain form of contextuality, witnessed by a generalised inequality involving N measurements on one wing of a bipartite Bell scenario.

Connection between the contextuality breaking and incompatibility breaking qubit channels

Abstract

Contextuality and measurement incompatibility are two fundamental aspects of nonclassicality, and their manifestations in observed quantum correlations are often deeply interconnected. Recently, measurement incompatibility has been studied in connection with nonlocality, particularly in terms of their robustness under various quantum channels. This line of investigation helps establish a connection between the channels that break nonlocality and those that break incompatibility. In this study, we focus on an asymmetric bipartite Bell scenario involving three and four inputs on Alice and Bob sides, respectively, with each of these inputs having dichotomous outcomes. Under the assumption of locality, the observed statistics in this asymmetric scenario obeys the Elegant Bell inequality (EBI). Here, we use a different version of the EBI that relies on the assumption of the preparation noncontextuality. By taking the violation of this noncontextual version of EBI as a witness of preparation contextuality we establish a connection between the channels that break contextuality and the channels that break triple-wise measurement incompatibility. Our results suggest that any channel which breaks EBI contextuality will also break Clauser-Horne-Shimony-Holt (CHSH) nonlocality; however, the reverse does not hold. We also show that a depolarising channel that breaks N-wise incompatibility can also break a certain form of contextuality, witnessed by a generalised inequality involving N measurements on one wing of a bipartite Bell scenario.

Paper Structure

This paper contains 16 sections, 4 theorems, 48 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Contextuality in Bipartite Bell-type Scenario
  4. Measurement incompatibility
  5. Quantum Channels
  6. Results
  7. Contextuality Breaking Conditions
  8. White-noise robustness of contextuality under channels
  9. Depolarizing channel (DPoC).
  10. Amplitude-damping channel (ADC).
  11. Loss channel (LC).
  12. Dephasing channel (DPC).
  13. Generalization to $N$-incompatibility breaking Channels
  14. Conclusive remarks and future directions
  15. Acknowledgements
  16. ...and 1 more sections

Key Result

Lemma 1

Consider the POVM with Bloch vector $\mathbf{n} \in \mathbb{R}^3$ (satisfying $|\mathbf{n}|\le 1$). If $\Lambda$ is a unital qubit channel, then its Heisenberg image $E' = \Lambda^*(E)$, has the same form for some real vector $\mathbf{n}'$, with $|\mathbf{n}'| \le 1$. In other word, under the action of an unital channel the transformed POVM remain unbiased if the initial POVM is unbiased. $\blac

Figures (3)

  • Figure 1: Figure depicts the action of contextuality breaking channels ($\mathcal{E}\otimes \mathbb{I}$) and ($\mathbb{I} \otimes \mathcal{E}$) on first and second qubit of the shared state $\rho_{AB}$ between Alice and Bob respectively.
  • Figure 2: Plot of quantum value of Bell functional $\mathfrak{B}_{Q}$ as a function of the channel parameter $p$ with the channel applied to (a) a single qubit, (b) both the qubits.
  • Figure 3: Plots of the white-noise robustness $R_{WN}(p)$ as a function of the channel parameter $p$ for Different Channels acting on the subsystem of Alice's wing.

Theorems & Definitions (8)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof