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Comparing quantum incompatibility of device sets from an operational perspective

Kensei Torii, Ryo Takakura, Ryotaro Imamura

TL;DR

The paper introduces an operational preorder $\preceq_{\mathrm{inc}}$ to compare quantum device incompatibility based on how readily incompatibility is detected with restricted state sets $\mathcal{S}_0$. This ordering preserves fundamental monotonicity under convex mixing with compatible devices, observable post-processing, and channel concatenation, and it yields a fine-grained hierarchy beyond traditional measures. In the prominent case of unbiased qubit observables, the induced equivalence $\sim_{\mathrm{inc}}$ uniquely identifies mutually unbiased qubit pairs, contrasting them with all other unbiased pairs, and links the ordering to the distributed sampling task as a QC certification tool. Numerically, the authors map regions where certain Mub configurations are ordered, revealing new classifications and suggesting that a restricted family of state sets $\mathcal{S}_0^{(R)}$ suffices to characterize the ordering for these observables. The work paves the way for applying the ordering to broader quantum-device scenarios and GPT frameworks, with potential extensions and computational approaches (e.g., SDP) for higher-dimensional systems.

Abstract

To effectively utilize quantum incompatibility as a resource in quantum information processing, it is crucial to evaluate how incompatible a set of devices is. In this study, we propose an ordering to compare incompatibility and reveal its various properties based on the operational intuition that larger incompatibility can be detected with fewer states. We especially focus on typical class of incompatibility exhibited by mutually unbiased qubit observables and numerically demonstrate that the ordering yields new classifications among sets of devices. Moreover, the equivalence relation induced by this ordering is proved to uniquely characterize mutually unbiased qubit observables among all pairs of unbiased qubit observables. The operational ordering also has a direct implication for a specific protocol called distributed sampling.

Comparing quantum incompatibility of device sets from an operational perspective

TL;DR

The paper introduces an operational preorder to compare quantum device incompatibility based on how readily incompatibility is detected with restricted state sets . This ordering preserves fundamental monotonicity under convex mixing with compatible devices, observable post-processing, and channel concatenation, and it yields a fine-grained hierarchy beyond traditional measures. In the prominent case of unbiased qubit observables, the induced equivalence uniquely identifies mutually unbiased qubit pairs, contrasting them with all other unbiased pairs, and links the ordering to the distributed sampling task as a QC certification tool. Numerically, the authors map regions where certain Mub configurations are ordered, revealing new classifications and suggesting that a restricted family of state sets suffices to characterize the ordering for these observables. The work paves the way for applying the ordering to broader quantum-device scenarios and GPT frameworks, with potential extensions and computational approaches (e.g., SDP) for higher-dimensional systems.

Abstract

To effectively utilize quantum incompatibility as a resource in quantum information processing, it is crucial to evaluate how incompatible a set of devices is. In this study, we propose an ordering to compare incompatibility and reveal its various properties based on the operational intuition that larger incompatibility can be detected with fewer states. We especially focus on typical class of incompatibility exhibited by mutually unbiased qubit observables and numerically demonstrate that the ordering yields new classifications among sets of devices. Moreover, the equivalence relation induced by this ordering is proved to uniquely characterize mutually unbiased qubit observables among all pairs of unbiased qubit observables. The operational ordering also has a direct implication for a specific protocol called distributed sampling.

Paper Structure

This paper contains 9 sections, 11 theorems, 84 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operational comparison of incompatibility
  4. Operational ordering of incompatibility
  5. Relation between operational ordering and distributed sampling
  6. Comparing incompatibility of mutually unbiased qubit observables
  7. Operational equivalence of incompatibility
  8. Numerical comparison of incompatibility
  9. Conclusion

Key Result

Proposition 1

Let $\mathbb{D}=\{\mathsf{D}_1,\ldots,\mathsf{D}_n\},~\mathbb{E}=\{\mathsf{E}_1,\ldots,\mathsf{E}_n\}$ be device sets and let $\mathbb{N}=\{\mathsf{N}_1,\ldots,\mathsf{N}_n\}$ be a compatible device set. Suppose that $\mathbb{E}$ is a convex combination of $\mathbb{D}$ and $\mathbb{N}$, that is, the for every $i=1,\ldots,n$. Then $\mathbb{E} \preceq_\mathrm{inc} \mathbb{D}$ holds.

Figures (3)

  • Figure 1: Parametrization of a set $\mathcal{S}_0$ consisting of three linearly independent states. Such $\mathcal{S}_0$ is identified with an intersection of a 2D plane and the Bloch ball, characterized by $r$ and $\mathbf{n}$. The real parameter $r\in [0,1)$ denotes the distance from the origin to the intersection, and the vector $\mathbf{n} \in \mathbb{R}^3$ is a normal vector.
  • Figure 3: The vectors $\mathbf{a}_1,\mathbf{a}_2,\pm\mathbf{b}_1,\mathbf{b}_2,\mathbf{c}_1$ and $\mathbf{c}_2$ in the $xy$ plane of the Bloch ball under the condition of \ref{['psi_plus_omega']}.
  • Figure 4: The region $(t,\theta)$ where the ordering $\mathbb{B}_\mathrm{mub}^t(\theta) \preceq_\mathrm{inc} \mathbb{A}_\mathrm{mub}^{t=1}$ holds for mutually unbiased qubit observables $\mathbb{A}_\mathrm{mub}^{t=1} = \{\mathsf{A}^\mathbf{x},\mathsf{A}^\mathbf{y}\}$ and $\mathbb{B}_\mathrm{mub}^t(\theta) = \{\mathsf{A}^{t\mathbf{x}},\mathsf{A}^{t(\mathbf{y} \cos\theta + \mathbf{z} \sin\theta) }\}$. Blue region: The ordering $\mathbb{B}_\mathrm{mub}^t(\theta) \preceq_\mathrm{inc} \mathbb{A}_\mathrm{mub}^{t=1}$ is explained by Proposition \ref{['fund_features']} and \ref{['prop:post-processing']}. Gray region: We confirm $\mathbb{B}_\mathrm{mub}^t(\theta) \npreceq_\mathrm{inc} \mathbb{A}_\mathrm{mub}^{t=1}$ through only analyzing $\mathcal{S}_0^{(R)}$. White region: We need to investigate general $\mathcal{S}_0$ to judge whether the ordering holds. Solid lines: For $\theta=\pi/12,\pi/6,\pi/4,\pi/3$, we newly found that $\mathbb{B}_\mathrm{mub}^t(\theta) \preceq_\mathrm{inc} \mathbb{A}_\mathrm{mub}^{t=1}$ holds. The maximum value of $t$ that realizes $\mathbb{B}_\mathrm{mub}^t(\theta) \preceq_\mathrm{inc} \mathbb{A}_\mathrm{mub}^{t=1}$ for each $\theta$ lies in the boundary of the white and the gray regions.

Theorems & Definitions (34)

  • Definition 1
  • Example 1: Observable set
  • Example 2: Channel set
  • Example 3: Observable-channel pair
  • Remark 1
  • Definition 2
  • Example 4: $\mathcal{S}_0$-incompatibility of qubit observables
  • Remark 2
  • Definition 3
  • Definition 4
  • ...and 24 more