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Structure, Positivity and Classical Simulability of Kirkwood-Dirac Distributions

Jędrzej Burkat, Sergii Strelchuk

Abstract

The Kirkwood-Dirac (KD) quasiprobability distribution is known for its role in quantum metrology, thermodynamics, as well as quantum foundations. In this work we classify unitary evolutions that preserve KD positivity. We identify conditions under which positivity preservation is equivalent to $l_1$-norm preservation, and exhibit unitaries that preserve positivity on KD-positive distributions while failing to preserve the $l_1$-norm of non-positive ones. We further prove that unitaries inducing stochastic updates of KD quasiprobabilities form a strict subset of the positivity-preserving unitaries. By adapting the classical sampling algorithm of Pashayan et al. [Phys. Rev. Lett. 115, 070501], we obtain efficient simulation methods for all identified classes of positivity-preserving unitaries. Our classification is complete for distributions defined on Fourier-conjugate bases in dimensions $d = p^k$ and $d = pq$, where $p, q$ are distinct primes, as well as for generic randomly chosen bases. As a consequence, no resource theory in the Fourier-conjugate $d=pq$ setting can simultaneously regard KD non-positivity as a monotone and include all efficiently simulable positivity-preserving unitaries among its free operations.

Structure, Positivity and Classical Simulability of Kirkwood-Dirac Distributions

Abstract

The Kirkwood-Dirac (KD) quasiprobability distribution is known for its role in quantum metrology, thermodynamics, as well as quantum foundations. In this work we classify unitary evolutions that preserve KD positivity. We identify conditions under which positivity preservation is equivalent to -norm preservation, and exhibit unitaries that preserve positivity on KD-positive distributions while failing to preserve the -norm of non-positive ones. We further prove that unitaries inducing stochastic updates of KD quasiprobabilities form a strict subset of the positivity-preserving unitaries. By adapting the classical sampling algorithm of Pashayan et al. [Phys. Rev. Lett. 115, 070501], we obtain efficient simulation methods for all identified classes of positivity-preserving unitaries. Our classification is complete for distributions defined on Fourier-conjugate bases in dimensions and , where are distinct primes, as well as for generic randomly chosen bases. As a consequence, no resource theory in the Fourier-conjugate setting can simultaneously regard KD non-positivity as a monotone and include all efficiently simulable positivity-preserving unitaries among its free operations.

Paper Structure

This paper contains 8 sections, 9 theorems, 100 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterising (Non-) Positivity-Preserving and KD-Stochastic Unitaries
  4. Estimating Born Rule Probabilities via Kirkwood–Dirac Distributions
  5. Estimating KD Superoperator Elements
  6. Self-Similarity of KD Distributions
  7. Bounds on Kirkwood–Dirac Quasiprobabilities
  8. Extended Proof of Theorem \ref{['thm:ustar']}

Key Result

Lemma 1

If $\mathcal{C} = \emptyset$, i.e. $\mathcal{E}^\mathrm{pure}_{\mathrm{KD}+} = \mathcal{A} \cup \mathcal{B}$, then any positivity-preserving unitary $U$ either satisfies $U(\mathcal{A}) = \mathcal{A}$ and $U(\mathcal{B}) = \mathcal{B}$ (type I), or $U(\mathcal{A}) = \mathcal{B}$ and $U(\mathcal{B}) $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Left: The cycle test for estimating $\text{Re}[Q_{ij}(\rho)]$ ($s = 0$), and $\text{Im}[Q_{ij}(\rho)]$ ($s = 1$). $H$ is the Hadamard gate, and $P = \text{diag}(1, i)$. After many measurements, $Q_{ij}(\rho)$ is estimated from the outcome frequencies via $p_{s=0}(0) = (1 + \text{Re}[Q_{ij}]) / 2$ and $p_{s=1}(0) = (1 + \text{Im}[Q_{ij}]) / 2$. Right: The cycle gate may be implemented by a cascade of qudit controlled-SWAP gates.
  • Figure 2: Cycle Test for Estimating KD Superoperator elements. The circuit estimates the quantity $|\langle b_l | a_k \rangle|^2 \times (\hat{\mathcal{E}}_U)_{ij, kl}$, where $p(0)_{s=1} = (1 + |\langle b_l | a_k \rangle|^2 \times \mathrm{Re}[(\hat{\mathcal{E}}_U)_{ij, kl}])/2$, and $p(0)_{s=0} = (1 + |\langle b_l | a_k \rangle|^2 \times \mathrm{Im}[(\hat{\mathcal{E}}_U)_{ij, kl}])/2$.

Theorems & Definitions (31)

  • Definition 1: Kirkwood--Dirac Distributions arvidssonshukur2024properties
  • Definition 2: Total Non-Positivity
  • Definition 3: Positivity Preservation
  • Definition 4: Total Non-Positivity Preservation
  • Definition 5: Kirkwood--Dirac Stochasticity
  • Definition 6: Generalised Permutations
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 21 more