bibkey: fullwoodmawu2026spatiotemporalborn
authors: James Fullwood; Zhihao Ma; Zhen Wu
year: 2026
title: "Quasiprobabilistic nature of the spatiotemporal Born rule"
doi: 10.1103/cw25-pqx1
url: https://arxiv.org/abs/2507.16919v2
claim: "The paper compares the Lüders-von Neumann distribution of two-point sequential measurements with the Terletsky-Margenau-Hill quasiprobability distribution, shows that a spatiotemporal Born rule exists for the Lüders-von Neumann distribution when the initial state is maximally mixed or the channel is discard-and-prepare, and asks whether one of these two conditions is also necessary in arbitrary dimension."
strata_touched:
  - D5/S3/Quantum/Measurements/SpatiotemporalBornRuleDichotomy
license: citation-only
triage: anchor

Quasiprobabilistic nature of the spatiotemporal Born rule

J. Fullwood, Z. Ma and Z. Wu, arXiv:2507.16919v2 (10 January 2026; v1 22 July 2025), titled “The spatiotemporal Born rule is quasiprobabilistic”; Phys. Rev. A 113(3) (2026), 032204. Quotations are from the v2 TeX source, with the macros \E, \A, \B, \H written as $\mathcal{E}$, $\mathcal{A}$, $\mathcal{B}$, $\mathcal{H}$ and \define as plain text.

Channels and projective measurements:

A linear map $\mathcal{E}:\mathcal{A}\to \mathcal{B}$ which is completely positive and trace-preserving (CPTP) will be referred to as a quantum channel. … A collection of operators $\{P_i\}$ on a finite-dimensional Hilbert space $\mathcal{H}$ is said to be a projective measurement if and only if $P_i{P}_j={\delta }_{ij}{P}_i$ and ${\sum }_i{P}_i=1$.

The Lüders-von Neumann distribution of a two-point sequential measurement $(\rho ,\{P_i\},\mathcal{E},\{Q_j\})$ is $\mathbf{P}(i,j)=\mathrm{Tr}[\mathcal{E}(P_i\rho P_i)Q_j]$.

The spatiotemporal Born rule:

A spatiotemporal Born rule is then said to exist for the LvN distribution with respect to the pair $(\mathcal{E},\rho )$ if and only if there exists an operator ${\varrho }_{AB}$ such that for all TPSM scenarios $(\rho ,\{P_i\},\mathcal{E},\{Q_j\})$, $\mathrm{Tr}[{\varrho }_{AB}(P_i\otimes Q_j)]=\mathbf{P}(i,j)\equiv \mathrm{Tr}[\mathcal{E}(P_i\rho P_i)Q_j]$.

The open question:

It is then natural to ask whether $\rho$ being maximally mixed or $\mathcal{E}$ being a discard-and-prepare channel is also a necessary condition for the existence of the spatiotemporal Born rule for the LvN distribution. While it is straightforward to show that this is indeed the case when $A$ consists of a single qubit, for systems of arbitrary dimension we leave it as an open question.

A discard-and-prepare channel is $\mathcal{E}(a)=\mathrm{Tr}[a]\sigma$ for a state $\sigma$ (Proposition 1).

Verified locator

Source: https://arxiv.org/abs/2507.16919v2

• URL: https://arxiv.org/abs/2507.16919v2 (the e-print is a tar archive; member PRA_rev.tex, 623 lines, md5 ccc824023e2958b612ebb8c2be819e2f):
  • channels and projective measurements, l. 231;
  • the Lüders-von Neumann distribution, l. 247–250;
  • discard-and-prepare channels, Proposition 1, l. 271;
  • the spatiotemporal Born rule, l. 534–537;
  • the open question, l. 577.
• DOI: 10.1103/cw25-pqx1 (https://doi.org/10.1103/cw25-pqx1; Crossref lists Physical Review A, volume 113, issue 3, article 032204, published 5 March 2026).