Quantum Belief Characterization: A Multi-Shot Notion of Equivalence
Abstract
A recent formulation of quantum belief showed that quantum retrodiction depends not only on the reduced density operator of the system of interest, but also on correlations with a hidden reference system, which leads to the notion of quantum belief equivalence. In this paper we formulate a multi-shot problem, where the extended posterior obtained after one update is used again as the prior for the next update. We first define a formal notion of n-shot equivalence and derive several propositions for arbitrary extended beliefs. We then argue that the agent must also specify why the data obtained in different rounds concern the same underlying object. Through the quantum de Finetti representation, this narrows the one-copy belief to a special class: candidate-state ensemble belief. In that sector, we show that two beliefs are n-shot equivalent if and only if their (n+1)-th moments are equal. We also give a physical interpretation of this moment condition and illustrate it through the equivalence between Haar priors and projective-design priors at bounded update depth.
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BibTeX
@misc{liu2026quantum,
title = {{Quantum Belief Characterization: A Multi-Shot Notion of Equivalence}},
author = {Mingxuan Liu and Pavel Sekatski and Valerio Scarani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29450},
eprint = {2610.29450},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29450},
}
RIS
TY - UNPB
TI - Quantum Belief Characterization: A Multi-Shot Notion of Equivalence
AU - Mingxuan Liu
AU - Pavel Sekatski
AU - Valerio Scarani
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29450
UR - https://arxiv.org/abs/2610.29450
AB - A recent formulation of quantum belief showed that quantum retrodiction depends not only on the reduced density operator of the system of interest, but also on correlations with a hidden reference system, which leads to the notion of quantum belief equivalence. In this paper we formulate a multi-shot problem, where the extended posterior obtained after one update is used again as the prior for the next update. We first define a formal notion of n-shot equivalence and derive several propositions for arbitrary extended beliefs. We then argue that the agent must also specify why the data obtained in different rounds concern the same underlying object. Through the quantum de Finetti representation, this narrows the one-copy belief to a special class: candidate-state ensemble belief. In that sector, we show that two beliefs are n-shot equivalent if and only if their (n+1)-th moments are equal. We also give a physical interpretation of this moment condition and illustrate it through the equivalence between Haar priors and projective-design priors at bounded update depth.
ER -