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Advantages of the Kirkwood-Dirac distribution among general quasi-probabilities for finite-state quantum systems

Shun Umekawa, Jaeha Lee, Naomichi Hatano

Abstract

We investigate features of the quasi-joint-probability distribution for finite-state quantum systems, especially the two-state and three-state quantum systems, comparing different types of quasi-joint-probability distributions based on the general framework of quasi-classicalization. We show from two perspectives that the Kirkwood-Dirac distribution is the quasi-joint-probability distribution that behaves nicely for the finite-state quantum systems. One is the similarity to the genuine probability and the other is the information that we can obtain from the quasi-probability. By introducing the concept of the possible values of observables, we show for the finite-state quantum systems that the Kirkwood-Dirac distribution behaves more similarly to the genuine probability distribution in contrast to most of the other quasi-probabilities including the Wigner function. We also prove that the states of the two-state and three-state quantum systems can be completely distinguished by the Kirkwood-Dirac distribution of only two directions of the spin and point out for the two-state system that the imaginary part of the quasi-probability is essential for the distinguishability of the state.

Advantages of the Kirkwood-Dirac distribution among general quasi-probabilities for finite-state quantum systems

Abstract

We investigate features of the quasi-joint-probability distribution for finite-state quantum systems, especially the two-state and three-state quantum systems, comparing different types of quasi-joint-probability distributions based on the general framework of quasi-classicalization. We show from two perspectives that the Kirkwood-Dirac distribution is the quasi-joint-probability distribution that behaves nicely for the finite-state quantum systems. One is the similarity to the genuine probability and the other is the information that we can obtain from the quasi-probability. By introducing the concept of the possible values of observables, we show for the finite-state quantum systems that the Kirkwood-Dirac distribution behaves more similarly to the genuine probability distribution in contrast to most of the other quasi-probabilities including the Wigner function. We also prove that the states of the two-state and three-state quantum systems can be completely distinguished by the Kirkwood-Dirac distribution of only two directions of the spin and point out for the two-state system that the imaginary part of the quasi-probability is essential for the distinguishability of the state.
Paper Structure (19 sections, 17 theorems, 67 equations, 5 figures)

This paper contains 19 sections, 17 theorems, 67 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Quasi-Joint-Probability Distributions
  3. The general framework of quasi-probabilities and quantizations
  4. Quasi-classicalizations generating real quasi-probabilities
  5. Finite-state quantum systems and possible values of observables
  6. Quasi-Probabilities for Two-State Quantum System
  7. Spin $1/2$
  8. Examples
  9. General features of Kirkwood-Dirac distributions of spin $1/2$
  10. Other quasi-classicalizations
  11. Other observables
  12. Quasi-Probabilities for Three-State Quantum System
  13. Independ components of a state
  14. General feature of the Kirkwood-Dirac distributions of spin 1
  15. Other observables
  16. ...and 4 more sections

Key Result

Proposition 3.1

The Kirkwood-Dirac distribution of the $x$ and $y$ components of the spin $1/2$ with respect to an arbitrary state of the two-state quantum systems takes non-zero values only at $(J_1,J_2)=(\pm1/2,\pm1/2)$:

Figures (5)

  • Figure 1: The duality of quasi-classicalization and quantization.
  • Figure 2: Bloch sphere. Any states of spin $1/2$ is specified as a point $(\langle J_1\rangle,\,\langle J_2\rangle,\,\langle J_3\rangle)$.
  • Figure 3: The real and imaginary parts of the Kirkwood-Dirac distribution for the $\ket{z+}$ and $\ket{z-}$ states. The blue square indicates the value $+1/4$, while the red square indicates the value $-1/4$.
  • Figure 4: Quasi-joint-probability distributions generated from $\hat{\#}^{S_{1/2}}_{J_1,J_2}(s,t)$ with respect to (a) the $\ket{z+}$ state and (b) the $\ket{z-}$ state. Each blue square indicates the value $+1/4$
  • Figure 5: (a) the Kirkwood-Dirac distribution and (b) the quasi-joint-probability distribution generated from $\hat{\#}^{S_{1/2}}_{J_1,J_2}(s,t)$ with respect to the $\ket{y+}$ state. The blue square indicates the value $+1/4$, while the light blue circle indicates the value $+1/2$ and the orange circle indicates the value $-1/2$.

Theorems & Definitions (17)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 3.1
  • Proposition 3.4
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Proposition 3.5
  • Proposition 4.1
  • ...and 7 more