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:0808.0722 · Physics Letters B

Insufficiency of the quantum state for deducing observational probabilities

Don Nelson Page

Published Jun 10, 2009• 17 citations
arXivHTMLPDF

Abstract

It is usually assumed that the quantum state is sufficient for deducing all probabilities for a system. This may be true when there is a single observer, but it is not true in a universe large enough that there are many copies of an observer. Then the probability of an observation cannot be deduced simply from the quantum state (say as the expectation value of the projection operator for the observation, as in traditional quantum theory). One needs additional rules to get the probabilities. What these rules are is not logically deducible from the quantum state, so the quantum state itself is insufficient for deducing observational probabilities. This is the measure problem of cosmology.

DOI
10.1016/j.physletb.2009.05.064

Related papers

Born's Rule Is Insufficient in a Large Universe

Don Nelson Page

Probabilities in quantum theory are traditionally given by Born's rule as the expectation values of projection operators. Here it is shown that Born's rule is insufficient in universes so large that they contain identical multiple copies of observers, because one does not have definite projection operators to apply. Possible replacements for Born's rule include using the expectation value of various operators that are not projection operators, or using various options for the average density matrix of a region with an observation. The question of what replacement to use is part of the measure problem in cosmology.

arXiv:1003.2419•Mar 11, 2010•12 citations

Quantum Cosmology Lectures

Don Nelson Page

Quantum cosmology is the quantum theory of the entire universe. Although strange at first sight, it is appropriate because (1) our world appears to be fundamentally quantum, (2) the classical description of gravity breaks down at singularities it would give at the beginning of the universe, and (3) our universe has many properties that cannot be explained by a classical description of them. A quantum state of the universe should obey certain constraints, such as the Wheeler-DeWitt equations. One approach to interpreting such a state is {\it Sensible Quantum Mechanics} (SQM), in which nothing is probabilistic, except, in a certain frequency sense, conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued {\it awareness operators}. These may be defined even when the quantum state itself is not normalizable, though there still seem to be problems of divergences when one has an infinite amount of inflation, producing infinitely large spatial volumes with presumably infinite measures of perceptions. Ratios of the measures for sets of perceptions (if finite) can be interpreted as frequency-type probabilities for many actually existing sets rather than as propensities for potentialities to be actualized, so there is nothing indeterministic in a specific SQM. One can do a Bayesian analysis to test between different specific SQMs. One can also make statistical predictions of what one might perceive within a specific SQM by invoking the {\it Conditional Aesthemic Principle}: among the set of all conscious perceptions, our perceptions are likely to be typical.

arXiv:gr-qc/9507028•To be published in the proceedings of•Jul 14, 1995•4 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

1 Rescuing the Born Rule for Quantum Cosmology

Joshua H. Cooperman

Page has recently argued that the Born rule does not suffice for computing all probabilities in quantum cosmology. He further asserts that the Born rule's failure gives rise to the cosmological measure problem. Here I contend that Page's result stems from his use of an overly restrictive definition of the Born rule. In particular, I demonstrate that all of the probabilities he wishes to compute follow from the Born rule when generalized measurements are permitted. I also register two comments on Page's theoretical setting, relating respectively to Hilbert space dimensionality and permutation symmetry. These considerations lead me to conclude that the claimed insufficiency of the Born rule is by no means specific to the cosmological context.

•Jan 1, 2011•2 citations

In Search of a Viable Epistemic Account

Simon Friederich

A quantum state, via the Born Rule, assigns probabilities to the different possible values of observables of a quantum system. It is only in virtue of these probabilities — or, alternatively, expectation values — that quantum mechanics is empirically testable. Since assigning a state to a system means assigning probabilities to the values of observables, it may seem natural to read Peierls’ claim that the state ‘represents our knowledge of the system’ (Peierls [1991], p. 19) as a shorthand for saying that the state represents our knowledge of these probabilities. The latter view is sometimes even straightforwardly identified with the epistemic conception of states, for example by Marchildon, who claims that ‘[i]n the epistemic view [of quantum states], the state vector (or wave function or density matrix) does not represent the objective state of a microscopic system […], but rather our knowledge of the probabilities of outcomes of future measurements’ (Marchildon [2004], p. 1454). However, according to an argument due to Fuchs and endorsed by Timpson, 31 the idea that quantum states represent our knowledge about quantum probabilities should not be regarded as the core claim of the epistemic conception of quantum states, simply because, as they argue, this idea is in fact incompatible with the latter. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Palgrave Macmillan UK eBooks•Jan 1, 2015•0 citations

Do Our Observations Depend upon the Quantum State of the Universe?

Don Nelson Page

Generically the probabilities of observational results depend upon both the quantum state and the rules for extracting the probabilities from it. It is often argued that inflation may make our observations independent of the quantum state. In a framework in which one considers the state and the rules as logically separate, it is shown how it is possible that the probabilities are indeed independent of the state, but the rules for achieving this seem somewhat implausible.

arXiv:0907.4751•Jul 27, 2009•3 citations
It is usually assumed that the quantum state is sufficient for deducing all probabilities for a system. This may be true when there is a single observer, but it is not true in a universe large enough that there are many copies of an observer. Then the probability of an observation cannot be deduced simply from the quantum state (say as the expectation value of the projection operator for the observation, as in traditional quantum theory). One needs additional rules to get the probabilities. What these rules are is not logically deducible from the quantum state, so the quantum state itself is insufficient for deducing observational probabilities. This is the measure problem of cosmology.
Probabilities in quantum theory are traditionally given by Born's rule as the expectation values of projection operators. Here it is shown that Born's rule is insufficient in universes so large that they contain identical multiple copies of observers, because one does not have definite projection operators to apply. Possible replacements for Born's rule include using the expectation value of various operators that are not projection operators, or using various options for the average density matrix of a region with an observation. The question of what replacement to use is part of the measure problem in cosmology.
Quantum cosmology is the quantum theory of the entire universe. Although strange at first sight, it is appropriate because (1) our world appears to be fundamentally quantum, (2) the classical description of gravity breaks down at singularities it would give at the beginning of the universe, and (3) our universe has many properties that cannot be explained by a classical description of them. A quantum state of the universe should obey certain constraints, such as the Wheeler-DeWitt equations. One approach to interpreting such a state is {\it Sensible Quantum Mechanics} (SQM), in which nothing is probabilistic, except, in a certain frequency sense, conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued {\it awareness operators}. These may be defined even when the quantum state itself is not normalizable, though there still seem to be problems of divergences when one has an infinite amount of inflation, producing infinitely large spatial volumes with presumably infinite measures of perceptions. Ratios of the measures for sets of perceptions (if finite) can be interpreted as frequency-type probabilities for many actually existing sets rather than as propensities for potentialities to be actualized, so there is nothing indeterministic in a specific SQM. One can do a Bayesian analysis to test between different specific SQMs. One can also make statistical predictions of what one might perceive within a specific SQM by invoking the {\it Conditional Aesthemic Principle}: among the set of all conscious perceptions, our perceptions are likely to be typical.
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.
Page has recently argued that the Born rule does not suffice for computing all probabilities in quantum cosmology. He further asserts that the Born rule's failure gives rise to the cosmological measure problem. Here I contend that Page's result stems from his use of an overly restrictive definition of the Born rule. In particular, I demonstrate that all of the probabilities he wishes to compute follow from the Born rule when generalized measurements are permitted. I also register two comments on Page's theoretical setting, relating respectively to Hilbert space dimensionality and permutation symmetry. These considerations lead me to conclude that the claimed insufficiency of the Born rule is by no means specific to the cosmological context.
A quantum state, via the Born Rule, assigns probabilities to the different possible values of observables of a quantum system. It is only in virtue of these probabilities — or, alternatively, expectation values — that quantum mechanics is empirically testable. Since assigning a state to a system means assigning probabilities to the values of observables, it may seem natural to read Peierls’ claim that the state ‘represents our knowledge of the system’ (Peierls [1991], p. 19) as a shorthand for saying that the state represents our knowledge of these probabilities. The latter view is sometimes even straightforwardly identified with the epistemic conception of states, for example by Marchildon, who claims that ‘[i]n the epistemic view [of quantum states], the state vector (or wave function or density matrix) does not represent the objective state of a microscopic system […], but rather our knowledge of the probabilities of outcomes of future measurements’ (Marchildon [2004], p. 1454). However, according to an argument due to Fuchs and endorsed by Timpson, 31 the idea that quantum states represent our knowledge about quantum probabilities should not be regarded as the core claim of the epistemic conception of quantum states, simply because, as they argue, this idea is in fact incompatible with the latter. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.