Kirkwood-Dirac Quasiprobability as a Universal Framework for Quantum Measurements Across All Regimes

TL;DR

This work argues that the Kirkwood-Dirac quasiprobability provides a universal framework for quantum measurements across all strength regimes. By introducing a pointer-induced decoherence mechanism governed by a time-dependent factor $F(t)$, the authors show the KD distribution deforms continuously from its complex form (governing weak values) to the real, non-negative Wigner form (for projective measurements) while preserving informational completeness. The non-classicality of the KD distribution decays linearly with $F(t)$, establishing a quantitative link between decoherence and the suppression of quantum features, and yielding a dynamical KD distribution $Q_{ij}(\rho(t),F)=F Q_{ij}(\rho)+(1-F)Q_{wigner}$. This framework unifies normal, conditional, and weak values within a single, physically transparent description and provides experimentally testable predictions for the transition between measurement regimes via the decoherence time $\tau_D$.

Abstract

The question of when the Kirkwood-Dirac quasiprobability serves as the most appropriate description for quantum measurements has remained unresolved, particularly across different measurement strengths. While known to generate anomalous weak values in the weak measurement regime and to reduce to classical probabilities under projective measurement, the physical mechanism governing its continuous transformation has been lacking. Here we demonstrate that the KD quasiprobability provides a general framework for all measurement regimes by identifying pointer-induced decoherence as the universal mechanism controlling this transition. We show that the decoherence factor F(t) simultaneously quantifies the loss of quantum coherence and interpolates the measurement strength from weak to strong. Within this framework, the KD quasiprobability naturally deforms from its full complex form-governing weak values-to the real, non-negative Wigner formula describing projective measurements, while maintaining informational completeness throughout the transition. Our work resolves the fundamental question of the KD distribution's applicability by establishing it as the universal framework that seamlessly connects all quantum measurement regimes through a physically transparent decoherence pathway.

Figures (2)

• Figure 1: Schematics of conditional expectation value and weak value . According to the TSVF formalism of the quantum theory the system state prepared in $\rho$ at $t_{0}$ and postselected to $\vert f_{j}\rangle$ at $t_{f}$. The expectation value of the system observable $A=\sum_{i}a_{i}\Pi_{a_{i}}$ give by $A_{c}$ or weak value $\langle A\rangle_{w}$ for different measurement strengths.
• Figure 2: The initial state $\rho=\sum_{ik}\rho_{ik}\vert a_{i}\rangle\langle a_{k}\vert$ of the system interacts with the initial state�$\vert\phi\rangle\langle\phi\vert$ of the pointer, and the interaction operator is $U(t)=\sum_{i}\vert a_{i}\rangle\langle a_{i}\vert\otimes U_{i}(t)$. $A$ and $P$ are observables of the system and pointer, respectively. After performing an operation on the system, the pointer state becomes $\vert\phi_{i}(t)\rangle\langle\phi_{k}(t)\vert=\vert\phi\left(x-ga_{i}t\right)\rangle\langle\phi\left(x-ga_{k}t\right)\vert$.