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On compatibility of binary qubit measurements

Dmitry Grinko, Roope Uola

TL;DR

This work develops a Fourier-analytic framework on the Boolean hypercube to characterize joint measurability of finite sets of binary qubit POVMs. It yields a general necessary condition for compatibility that subsumes known two- and three-measurement results (recovering Busch and Fermat–Torricelli criteria) and extends to arbitrary $N$, with tight results for unbiased measurements and a practical SOCP-based numerical approach. The authors connect joint measurability to quantum steering, deriving steering inequalities that become complete characterizations under unbiased steering-equivalent POVMs, and demonstrate the method's utility by falsifying a conjecture on measurement incompatibility. The approach offers a computationally competitive alternative to SDP-based methods and points toward higher-dimensional extensions and geometric interpretations of Fermat–Torricelli-type conditions.

Abstract

Deciding which sets of quantum measurements allow a simultaneous readout is a central problem in quantum measurement theory. The problem is relevant not only from the foundational perspective but also has direct applications in quantum correlation problems fueled by incompatible measurements. Although central, only a few analytical criteria exist for deciding the incompatibility of general sets of measurements. This work approaches the problem through functions defined on the Boolean hypercube and their Fourier transformations. We show that this reformulation of the problem leads to a complete geometric characterisation of joint measurability of any finite set of unbiased binary qubit measurements and gives a necessary condition for the biased case. We discuss our results in the realm of quantum steering, where they translate into a family of steering inequalities. When certain unbiasedness conditions are fulfilled, these criteria are tight, hence fully characterizing the steering problem when the trusted party holds a qubit, and the untrusted party performs any finite number of binary measurements. We further discuss how our results point towards a second-order cone programming approach to measurement incompatibility and compare this to the predominantly used semi-definite programming-based techniques. We use our approach to falsify an existing conjecture on measurement incompatibility of special sets of measurements.

On compatibility of binary qubit measurements

TL;DR

This work develops a Fourier-analytic framework on the Boolean hypercube to characterize joint measurability of finite sets of binary qubit POVMs. It yields a general necessary condition for compatibility that subsumes known two- and three-measurement results (recovering Busch and Fermat–Torricelli criteria) and extends to arbitrary , with tight results for unbiased measurements and a practical SOCP-based numerical approach. The authors connect joint measurability to quantum steering, deriving steering inequalities that become complete characterizations under unbiased steering-equivalent POVMs, and demonstrate the method's utility by falsifying a conjecture on measurement incompatibility. The approach offers a computationally competitive alternative to SDP-based methods and points toward higher-dimensional extensions and geometric interpretations of Fermat–Torricelli-type conditions.

Abstract

Deciding which sets of quantum measurements allow a simultaneous readout is a central problem in quantum measurement theory. The problem is relevant not only from the foundational perspective but also has direct applications in quantum correlation problems fueled by incompatible measurements. Although central, only a few analytical criteria exist for deciding the incompatibility of general sets of measurements. This work approaches the problem through functions defined on the Boolean hypercube and their Fourier transformations. We show that this reformulation of the problem leads to a complete geometric characterisation of joint measurability of any finite set of unbiased binary qubit measurements and gives a necessary condition for the biased case. We discuss our results in the realm of quantum steering, where they translate into a family of steering inequalities. When certain unbiasedness conditions are fulfilled, these criteria are tight, hence fully characterizing the steering problem when the trusted party holds a qubit, and the untrusted party performs any finite number of binary measurements. We further discuss how our results point towards a second-order cone programming approach to measurement incompatibility and compare this to the predominantly used semi-definite programming-based techniques. We use our approach to falsify an existing conjecture on measurement incompatibility of special sets of measurements.
Paper Structure (10 sections, 4 theorems, 48 equations)

This paper contains 10 sections, 4 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Joint measurability
  3. A brief review on known geometric results in the qubit case
  4. Main Results
  5. Quantum steering
  6. Numerical method
  7. Conclusions
  8. Acknowledgements
  9. Appendix
  10. Proof of Theorems \ref{['thm:necessary_condition']}-\ref{['thm:sufficient_condition']}

Key Result

Theorem 1

If $N$ binary qubit observables, specified by the Bloch vectors $\bm{z}_1,\dotsc,\bm{z}_N$, are jointly measurable, then where the configuration set $\mathcal{D}^{N}$ is defined as The optimisation is done over the set $\mathcal{F}_{N}$ of $\mathbbm{R}^3$-vectors defined as The vectors $\bm{z}(S)$ with $\abs{S} = 1$ correspond to the Bloch vectors of the given $N$ qubit POVMs. We use notation $

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Conjecture 1
  • Proposition 1
  • proof
  • proof : Proof of Theorem \ref{['thm:necessary_condition']}
  • Remark 1
  • proof : Proof of Theorem \ref{['thm:sufficient_condition']}
  • Remark 2
  • ...and 1 more