Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2107.05861 · Journal of Physics A Mathematical and Theoretical

Quantum conditional probabilities

Ignacio Pérez, Alfredo Luis

Published Jul 21, 2022• 1 citation
arXivHTMLPDF

Abstract

Abstract We investigate the consistency of conditional quantum probabilities. This is whether there is compatibility between the Kolmogorov–Bayes conditional probabilities and the Born rule. We show that they are not compatible in the sense that there are situations where there is no legitimate density matrix that may reproduce the conditional statistics of the other observable via the Born rule. This is to say that the Gleason theorem does not apply to conditional probabilities. Moreover, we show that when this occurs the joint statistics is nonclassical. We show that conditional probabilities are not equivalent to state reduction, so these results do not affect the validity of the Lüders expression.

DOI
10.1088/1751-8121/ac8321

Related papers

Conditional quantum state reduction in joint positive operator-valued measures.

Ignacio Pérez, Alfredo Luis

We investigate conditional quantum state reduction associated with positive operator-valued measures describing joint measurements of incompatible observables. To this end we invoke consistency with very basic rules of conditional probabilities, as formulated in the standard Kolmogorovian-Bayesian formulation of statistics. We find an interesting relation with the nonclassical character of the statistics.

•Jul 13, 2021•0 citations

Extending Kolmogorov’s Axioms for a Generalized Probability Theory on Collections of Contexts

Karl Svozil

Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason's theorem.

arXiv:1903.10424•Entropy•Sep 12, 2022•5 citations

Possibilities for probabilities

Don Nelson Page

Abstract In ordinary situations involving a small part of the universe, Born's rule seems to work well for calculating probabilities of observations in quantum theory. However, there are a number of reasons for believing that it is not adequate for many cosmological purposes. Here a number of possible generalizations of Born's rule are discussed, explaining why they are consistent with the present statistical support for Born's rule in ordinary situations but can help solve various cosmological problems.

arXiv:1711.05740•Journal of Cosmology and Astroparticle Physics•Oct 1, 2022•6 citations

Derivation of the Born rule from operational assumptions

Simon W. Saunders

The Born rule is derived from operational assumptions, together with assumptions of quantum mechanics that concern only the deterministic development of the state. Unlike Gleason's theorem, the argument applies even if probabilities are defined for only a single resolution of the identity, so it applies to a variety of foundational approaches to quantum mechanics. It also provides a probability rule for state spaces that are not Hilbert spaces.

arXiv:quant-ph/0211138•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Apr 5, 2004•138 citations

Born Again

Don Nelson Page

A simple proof is given that the probabilities of observations in a large universe are not given directly by Born's rule as the expectation values of projection operators in a global quantum state of the entire universe. An alternative procedure is proposed for constructing an averaged density matrix for a random small region of the universe and then calculating observational probabilities indirectly by Born's rule as conditional probabilities, conditioned upon the existence of an observation.

arXiv:0907.4152•Jul 23, 2009•8 citations

A Gleason-type theorem for qubits based on mixtures of projective measurements

Victoria J Wright, Stefan Weigert

Abstract We derive Born’s rule and the density-operator formalism for quantum systems with Hilbert spaces of dimension two or larger. Our extension of Gleason’s theorem only relies upon the consistent assignment of probabilities to the outcomes of projective measurements and their classical mixtures. This assumption is significantly weaker than those required for existing Gleason-type theorems valid in dimension two.

arXiv:1808.08091•Journal of Physics A Mathematical and Theoretical•Dec 18, 2018•18 citations
Abstract We investigate the consistency of conditional quantum probabilities. This is whether there is compatibility between the Kolmogorov–Bayes conditional probabilities and the Born rule. We show that they are not compatible in the sense that there are situations where there is no legitimate density matrix that may reproduce the conditional statistics of the other observable via the Born rule. This is to say that the Gleason theorem does not apply to conditional probabilities. Moreover, we show that when this occurs the joint statistics is nonclassical. We show that conditional probabilities are not equivalent to state reduction, so these results do not affect the validity of the Lüders expression.
Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason's theorem.
Abstract In ordinary situations involving a small part of the universe, Born's rule seems to work well for calculating probabilities of observations in quantum theory. However, there are a number of reasons for believing that it is not adequate for many cosmological purposes. Here a number of possible generalizations of Born's rule are discussed, explaining why they are consistent with the present statistical support for Born's rule in ordinary situations but can help solve various cosmological problems.
A simple proof is given that the probabilities of observations in a large universe are not given directly by Born's rule as the expectation values of projection operators in a global quantum state of the entire universe. An alternative procedure is proposed for constructing an averaged density matrix for a random small region of the universe and then calculating observational probabilities indirectly by Born's rule as conditional probabilities, conditioned upon the existence of an observation.