Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Temporal Kirkwood-Dirac Quasiprobability Distribution and Unification of Temporal State Formalisms through Temporal Bloch Tomography

Zhian Jia, Kavan Modi, Dagomir Kaszlikowski

TL;DR

The paper extends KD/MH quasiprobabilities to temporal and spatiotemporal quantum processes by defining left, right, and doubled KD distributions and their MH real parts, with CPTP marginals obeying Kolmogorov consistency. It introduces a temporal characteristic function and an interferometric measurement scheme to reconstruct these distributions, and clarifies the link between KD and weak values in temporal settings. Crucially, temporal Bloch tomography unifies diverse temporal state formalisms (including PDOs and process tensors) under a single KD-based framework, enabling information-complete temporal and spatiotemporal tomography. The approach provides operational tools to quantify temporal nonclassicality, analyze causal structure in quantum dynamics, and explore metrological advantages in temporal processes.

Abstract

Temporal quantum states generalize the multipartite density operator formalism to the time domain, enabling a unified treatment of quantum systems with both timelike and spacelike correlations. Despite a growing body of temporal state formalisms, their precise operational relationships and conceptual distinctions remain unclear. In this work, we resolve this issue by extending the Kirkwood-Dirac (KD) quasiprobability distribution to arbitrary multi-time quantum processes and, more broadly, to general spatiotemporal settings. We define left, right, and doubled temporal KD quasiprobabilities, together with their real components, which we identify as temporal Margenau-Hill (MH) quasiprobabilities. All of these quantities are experimentally accessible through interferometric measurement schemes. By characterizing their nonclassical features, we show that the generalized KD framework provides a unified operational foundation for a wide class of temporal state approaches and can be directly implemented via temporal or spatiotemporal Bloch tomography.

Temporal Kirkwood-Dirac Quasiprobability Distribution and Unification of Temporal State Formalisms through Temporal Bloch Tomography

TL;DR

The paper extends KD/MH quasiprobabilities to temporal and spatiotemporal quantum processes by defining left, right, and doubled KD distributions and their MH real parts, with CPTP marginals obeying Kolmogorov consistency. It introduces a temporal characteristic function and an interferometric measurement scheme to reconstruct these distributions, and clarifies the link between KD and weak values in temporal settings. Crucially, temporal Bloch tomography unifies diverse temporal state formalisms (including PDOs and process tensors) under a single KD-based framework, enabling information-complete temporal and spatiotemporal tomography. The approach provides operational tools to quantify temporal nonclassicality, analyze causal structure in quantum dynamics, and explore metrological advantages in temporal processes.

Abstract

Temporal quantum states generalize the multipartite density operator formalism to the time domain, enabling a unified treatment of quantum systems with both timelike and spacelike correlations. Despite a growing body of temporal state formalisms, their precise operational relationships and conceptual distinctions remain unclear. In this work, we resolve this issue by extending the Kirkwood-Dirac (KD) quasiprobability distribution to arbitrary multi-time quantum processes and, more broadly, to general spatiotemporal settings. We define left, right, and doubled temporal KD quasiprobabilities, together with their real components, which we identify as temporal Margenau-Hill (MH) quasiprobabilities. All of these quantities are experimentally accessible through interferometric measurement schemes. By characterizing their nonclassical features, we show that the generalized KD framework provides a unified operational foundation for a wide class of temporal state approaches and can be directly implemented via temporal or spatiotemporal Bloch tomography.
Paper Structure (8 sections, 10 theorems, 127 equations, 4 figures, 2 tables)

This paper contains 8 sections, 10 theorems, 127 equations, 4 figures, 2 tables.

Table of Contents

  1. Spatiotemporal generalization of Kirkwood–Dirac and Margenau-Hill quasiprobability distributions
  2. Two-time temporal Kirkwood–Dirac and Margenau-Hill quasiprobability distributions
  3. Multi-time temporal Kirkwood–Dirac and Margenau-Hill quasiprobability distributions
  4. Spatiotemporal Kirkwood–Dirac and Margenau-Hill quasiprobability distributions
  5. Temporal characteristic function and interferometric measurement scheme for temporal Kirkwood-Dirac quasiprobability distributions
  6. Weak values and temporal Kirkwood-Dirac quasiprobability distribution
  7. Proof of Theorem \ref{['thm:nogo']}
  8. Unification of spatiotemporal states through spatiotemporal Bloch tomography

Key Result

Lemma 1

Let $\mathfrak{P}_{t_0,\ldots, t_n} = (\rho_{t_0}, \mathcal{E}_{t_1 \leftarrow t_0}, \ldots, \mathcal{E}_{t_n \leftarrow t_{n-1}})$ be a quantum process over $n+1$ time steps. Consider a sub-process $\mathfrak{P}'_{t_0,t_{i_1},\ldots,t_{i_k}}$ over $k+1$ time steps, where the initial state is the sa

Figures (4)

  • Figure 1: Illustration of a temporal state at two time steps, from $t_A$ to $t_B$. Pauli measurements are performed at these times to carry out temporal state tomography. In the PDO framework, one obtains the LvN distribution, from which the corresponding PDO can be reconstructed via tomography. In the right temporal KD case, the corresponding right temporal KD quasiprobability distribution is obtained, from which the joint expectation value $\langle \{ \sigma_{\mu}(t_B), \sigma_{\nu}(t_A) \} \rangle$ is computed. This procedure yields the temporal KD state $\overrightarrow{\Upsilon}_{t_B t_A}$. The left and doubled KD cases, as well as the left/right and doubled MH cases, proceed analogously.
  • Figure 2: The relationships between different temporal states arising from temporal Bloch tomography. Starting from the doubled temporal KD quasiprobability distributions, all temporal states can be obtained.
  • Figure S1: Illustration of a spatiotemporal quantum process, where a multipartite initial state $\rho_{t_0}$ evolves over several time steps. The gray dots represent spacetime points where quantum operations, such as measurements, can be implemented. Since the initial multipartite state may be entangled, tracing out some local degrees of freedom results in a process that goes beyond the simple picture of a mixed state evolving under a quantum channel.
  • Figure S2: Quantum circuits illustrating the interferometric measurement scheme. (a) The scheme for unitary evolution. (b) The Stinespring dilation of a CPTP evolution. (c) The scheme for CPTP evolution.

Theorems & Definitions (21)

  • Lemma 1: Kolmogorov consistency condition
  • Theorem 1: Classicality criterion
  • Example 1: Replacement channel
  • Example 2: Measure-and-replace channel
  • Definition 1: Temporal state
  • Theorem 2
  • Definition 2: Temporal quasiprobability distribution
  • Lemma 2
  • proof
  • Remark 1: Remark on spatiotemporal Kirkwood–Dirac quasiprobability from process matrix
  • ...and 11 more