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Probing Kirkwood-Dirac nonpositivity and its operational implications via moments

Sudip Chakrabarty, Bivas Mallick, Saheli Mukherjee, Ananda G. Maity

Abstract

The Kirkwood-Dirac (KD) distribution has recently emerged as a powerful quasiprobability framework with wide-ranging applications in quantum information processing tasks. In this work, we introduce an experimentally motivated criterion for detecting nonclassical signatures of the KD distribution using its statistical moments and demonstrate its effectiveness through explicit examples. We further show that this approach extends naturally to identifying other quantum resources, such as quantum coherence and nonclassical extractable work -- that are intrinsically connected to the KD distribution. Our criteria involves the evaluation of simple functionals, making it well-suited for efficient experimental implementation.

Probing Kirkwood-Dirac nonpositivity and its operational implications via moments

Abstract

The Kirkwood-Dirac (KD) distribution has recently emerged as a powerful quasiprobability framework with wide-ranging applications in quantum information processing tasks. In this work, we introduce an experimentally motivated criterion for detecting nonclassical signatures of the KD distribution using its statistical moments and demonstrate its effectiveness through explicit examples. We further show that this approach extends naturally to identifying other quantum resources, such as quantum coherence and nonclassical extractable work -- that are intrinsically connected to the KD distribution. Our criteria involves the evaluation of simple functionals, making it well-suited for efficient experimental implementation.

Paper Structure

This paper contains 11 sections, 5 theorems, 69 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. KD Distribution
  4. Moment criteria
  5. Detection of KD nonpositivity
  6. Detecting quantum coherence and nonclassical extractable work as a direct consequence of KD nonpositivity detection
  7. Detecting Quantum Coherence
  8. Detecting nonclassical extractable work
  9. Proposal for realization of KD moments using shadow tomography
  10. Conclusions
  11. Acknowledgements

Key Result

Theorem 1

If the KD distribution $(Q (\rho) )$ for a quantum state $\rho$ with respect to bases $\{\ket{a_i}\}$ and $\{\ket{f_j}\}$ is positive, then where $q_2$ and $q_3$ are defined in kdmoments.

Figures (2)

  • Figure 1: The plot compares three quantities as functions of $\theta \in [0, \pi]$: (i) the $\ell_1$ measure of coherence (solid green curve), which is exactly equal to the total nonpositivity of the extended KD distribution , (ii) $-\det(H_1)$ for $\alpha = \beta$, (dashed red curve) showing positivity across the entire range (except $\theta =0,\pi$) and thereby detecting coherence in the entire range of $\theta$ at the first level of the hierarchy, and (iii) $-\det(H_2)$ for $\alpha = \beta + \pi/2$, (dotted blue curve) whose positivity demonstrates detection of coherence at the second level of the hierarchy in Theorem \ref{['theorem4']}.
  • Figure 2: Plot of the nonpositivity measure $\mathcal{N}(Q^{MHQ}_{ij}(\rho))$ as a function of $\Omega$ (solid blue curve). The regions where $\mathcal{N} > 0$ correspond to intervals where nonclassical work extraction is possible. The nonclassical domain is completely identified via Corollary \ref{['theorem5']}, based on the determinant of a second-order Hankel matrix constructed from KD moments (dashed red curve).

Theorems & Definitions (15)

  • Definition 1
  • Theorem 1
  • proof
  • Example 1
  • Theorem 2
  • proof
  • Example 2
  • Definition 2
  • Corollary 1
  • proof
  • ...and 5 more