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Compatibility of Binary Qubit Measurements.

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This study introduces a new geometric method to determine if quantum measurements can be performed simultaneously. This approach provides a complete characterization for unbiased qubit measurements and offers insights into quantum correlations and steering.

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Area of Science:

  • Quantum Information Theory
  • Quantum Measurement Theory
  • Quantum Correlations

Background:

  • Simultaneous readout of quantum measurements is a key challenge in quantum measurement theory.
  • Existing analytical criteria for measurement incompatibility are limited.
  • Incompatible measurements are crucial for exploring quantum correlations and quantum steering.

Purpose of the Study:

  • To develop a complete geometric characterization for joint measurability of quantum measurements.
  • To provide a necessary condition for biased measurements.
  • To translate these findings into steering inequalities and explore their tightness.

Main Methods:

  • Reformulation of the joint measurability problem using functions on the Boolean hypercube and Fourier transformations.
  • Geometric characterization of joint measurability for unbiased binary qubit measurements.
  • Analysis of measurement incompatibility using second-order cone programming.

Main Results:

  • A complete geometric characterization of joint measurability for finite sets of unbiased binary qubit measurements.
  • A necessary condition for the joint measurability of biased measurements.
  • A family of steering inequalities derived from the geometric criteria, which are tight under specific unbiasedness conditions.

Conclusions:

  • The developed geometric approach offers a powerful tool for analyzing measurement incompatibility.
  • The results fully characterize quantum steering problems involving a qubit and multiple binary measurements.
  • The study introduces a novel second-order cone programming approach and falsifies a conjecture on measurement incompatibility.