Quantum-Bayesian Coherence

Christopher A. Fuchs; Ruediger Schack

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Quantum-Bayesian Coherence

Christopher A. Fuchs, Ruediger Schack

arXiv:0906.2187v1Jun 11, 2009quant-ph

TL;DR

The paper asks what role the Born Rule has if quantum states do not objectively set measurement probabilities. It recasts the rule as a Bayesian constraint relating probabilities for performable measurements to those for an imagined reference measurement, and explores how much quantum-state structure follows.

Problem
If quantum states are subjective rather than independent objective entities, what role does the Born Rule play in setting measurement probabilities?
Source: §1 · p. 6
Method
The paper represents states with probabilities for a fiducial SIC measurement and treats the Born Rule as a constraint relating these to probabilities for other measurements.
Source: §1 · p. 5, §3 · p. 12
Results
The coherence-plus assumptions yield state-space features including basis distributions, probability bounds, and extreme points on a sphere, while not every flat zeros-bound vector is valid.
Source: §5.6 · pp. 39–40
Takeaways & Limitations
The Born Rule is presented as an empirical addition to Bayesian probability that relates measurement probabilities rather than setting them from an objective quantum state.
Source: §1 · p. 5
The quantum-Bayesian account has not yet derived Hilbert space, only indicated it.
Source: §4.2 · p. 27

Abstract

from arXiv · show

In a quantum-Bayesian take on quantum mechanics, the Born Rule cannot be interpreted as a rule for setting measurement-outcome probabilities from an objective quantum state. But if not, what is the role of the rule? In this paper, we argue that it should be seen as an empirical addition to Bayesian reasoning itself. Particularly, we show how to view the Born Rule as a normative rule in addition to usual Dutch-book coherence. It is a rule that takes into account how one should assign probabilities to the consequences of various intended measurements on a physical system, but explicitly in terms of prior probabilities for and conditional probabilities consequent upon the imagined outcomes of a special counterfactual reference measurement. This interpretation is seen particularly clearly by representing quantum states in terms of probabilities for the outcomes of a fixed, fiducial symmetric informationally complete (SIC) measurement. We further explore the extent to which the general form of the new normative rule implies the full state-space structure of quantum mechanics. It seems to get quite far.

1 Introduction: Unperformed Measurements Have No Outcomes

The paper recasts the Born Rule as an empirical addition to Bayesian probability: it relates probabilities for performable measurements to those for an imagined counterfactual measurement. It also previews how this rule can recover features of quantum-state space.

1 Introduction: Unperformed Measurements Have No Outcomes

Feynman’s double-slit argument relates probabilities for experiments that will be done to probabilities from experiments that will not be done.The authors recast interference as a relation among probabilities, rather than interference in a material field.Source: §1 · p. 4
The authors argue that probability remains personalistic Bayesian probability, while quantum mechanics supplies a new recipe for calculating probabilities from old ones.They present the Born Rule as a normative addition to ordinary Bayesian reasoning, with empirical content beyond Dutch-book coherence.Source: §1 · p. 5
Quantum mechanics adds a rule for deriving probabilities for performable experiments from probabilities assigned to a designated counterfactual experiment.The authors describe the Born Rule as an empirical addition to Bayesian probability, not a radical change to the concept of probability.Source: §1 · p. 5
The paper asks how the Born Rule can remain necessary when quantum states are not independent objective entities that set probabilities.From the Bayesian perspective, quantum states are as subjective as Bayesian probabilities.Source: §1 · p. 6
A single designated observable for counterfactual reasoning frames the paper’s urgleichung, whose consistency is explored as a source of quantum-state-space structure.The authors say this approach appears to recover a significant part of that structure.Source: §1 · p. 6

1.1 Outline of the Paper

Section 2 reviews personalist Bayesian probability and argues that Dutch-book coherence alone does not compel Bayes Rule or the Law of Total Probability without a conditional lottery.The paper distinguishes an actually gambled-upon conditional lottery from an imagined one.Source: §1.1 · p. 7
Section 3 presents SIC-based quantum-state representations, while Section 4 recasts the Born Rule using an imagined conditional lottery tied to a counterfactual SIC.The authors say the resulting rule resembles the Law of Total Probability; they also discuss the complex number field.Source: §1.1 · p. 7
Section 5 derives some quantum-state-space features from the modified probability rule, while Section 6’s preliminary account of its constants remains more mathematical than insightful.The paper concludes that Hilbert space has been indicated from a quantum-Bayesian view, not yet derived.Source: §1.1 · pp. 7–8

2 Personalist Bayesian Probability

Personalist Bayesian probability treats probabilities as agents’ degrees of belief, constrained by Dutch-book coherence against sure loss. The Law of Total Probability follows when a conditional lottery is available, but removing that lottery does not require retaining its original unconditional probability.

2 Personalist Bayesian Probability

Dutch-book coherence requires agents to avoid betting combinations that guarantee a loss and yields the usual probability calculus.The account treats probability as personal in value, while its function is constrained by consistency.Source: §2 · p. 8
When a conditional lottery is available, coherence requires the Law of Total Probability relating unconditional and conditional probabilities.The authors state that Bayes’ rule and the Law of Total Probability have force when an actual conditional lottery can be gambled upon.Source: §2 · p. 10, §2.1 · p. 11

2.1 When a Conditional Lottery Is Not Without Consequence

Without a conditional lottery, p(B|A) has no good meaning within this account, so Bayes’ rule is not a coherence requirement.The authors argue that without such a lottery there is no means to define the conditional-probability terms.Source: §2.1 · p. 10, §2 · p. 10
Nullifying the imagined A-lottery does not normatively require the agent to keep the original price p(B) for betting on B.The situation may change how the agent would gamble on B; keeping the original valuation is a judgment, not a necessity of coherence.Source: §2.1 · p. 11
A proposed relation between original and revised probabilities can indirectly restrict the probabilities assigned to the counterfactual lottery.The example function F must yield nonnegative values summing to one, and its domain may restrict allowable p(B) assignments.Source: §2.1 · p. 11

3 Expressing Quantum-State Space in Terms of SICs

The section represents quantum states as probability vectors for an informationally complete measurement, with SICs furnishing a particularly simple representation. It describes restrictions on valid vectors and expresses unitary evolution in SIC coordinates as a doubly stochastic mapping.

3 Expressing Quantum-State Space in Terms of SICs

A minimal informationally complete measurement maps each quantum state injectively to d2 outcome probabilities that completely specify it.The measurement operators are linearly independent and form a basis for operators on the Hilbert space.Source: §3 · pp. 12–13
Only a restricted subset of probability vectors in the simplex represents quantum states, and the section gives algebraic and parameterized descriptions of that state space.The pure-state boundary is characterized using quadratic and cubic constraints, while the full set is formed by convex combinations and also parameterized explicitly.Source: §3 · pp. 13, 15, 16, 17, 18
SIC measurements are informationally complete and make the quantum-state representation especially simple, but their existence is unknown in every finite dimension.The paper proceeds on the assumption that SICs exist in all dimensions, citing analytic proofs for dimensions 2–15 and 19 and numerical observations for dimensions 2–67.Source: §3 · p. 15

3.1 Aside on Unitarity

SIC coordinates make unitary time evolution formally resemble stochastic evolution: the transition matrix is doubly stochastic.The authors describe the resemblance as formal rather than claiming that quantum evolution is ordinary stochastic evolution.Source: §3.1 · p. 19

4 Expressing the Born Rule in Terms of SICs

The section expresses the Born Rule as a relation between probabilities for a hypothetical SIC measurement and a possible laboratory measurement, treating quantum mechanics’ formal structure as an addition to Dutch-book coherence. It explores the rule’s implications, including Eq. (56) without assuming Hilbert-space structure, while deferring questions of underlying ontology.

4 Expressing the Born Rule in Terms of SICs

The Born Rule relates ground-measurement probabilities q(j) to sky-measurement probabilities p(i) and conditional probabilities r(j|i), adding a quantum constraint to the Law of Total Probability.The authors call this relation the sum content of the Born Rule.Source: §4 · p. 20, Figure 2 · p. 20
Because the unperformed SIC has no outcomes, the direct-measurement probabilities need not equal the probabilities from Dutch-book coherence alone, and the rule constrains how sharp p(i) and r(j|i) can be.If the direct probabilities q(j) fall outside 0 to 1, they violate Dutch-book coherence; the authors suggest these constraints may reflect quantum state-space structure.Source: §4 · p. 21

4.1 Why Complex Hilbert Space, Rather Than Real?

Quantum two-designs beyond minimal SICs can yield the general Quantum Law of Total Probability, but nonminimal designs use a redundant probability representation.Complete mutually unbiased bases form such a design when they exist, while continuous designs occupy only a finite-dimensional subspace of an infinite-dimensional simplex.Source: §4.1 · pp. 22–24
Real-vector-space quantum mechanics has no minimal informationally complete sky measurement that yields this form of the Quantum Law of Total Probability.The required measurement would need to form a real-vector-space version of a SIC-POVM, which generally does not exist.Source: §4.1 · p. 25

4.2 Why “Coherence-Plus” Instead of Objective Quantum States?

The coherence-plus rule restricts agents’ probability assignments but preserves their subjectivity, giving no guidance on which quantum state to choose or which probabilities to adjust.The authors argue that quantum-state assignments depend on agents’ priors; violating the rule risks undesirable consequences by failing to account for the context of a potential SIC measurement.Source: §4.2 · pp. 25–27
Bayesian statistics is prescriptive rather than descriptive: it specifies how people should act to avoid undesirable consequences, not how they actually behave.Its aim is disciplined accounting for uncertainty.Source: §4.2 · p. 27
The authors defer specifying the ontology behind the proposed coherence-plus rules and explore Eq. (56) without assuming the Hilbert-space structure that gave rise to it.They suggest the full formal structure of quantum mechanics, including dynamics and tensor-product structure, may be all or nearly all an addition to Dutch-book coherence.Source: §4.2 · p. 27

5 Deriving Quantum-State Space from “Coherence-Plus”

The section explores how the urgleichung and added assumptions about measurement constrain quantum-state space, deriving restrictions on its structure and geometry. It treats these results as suggestive rather than a full derivation, identifying assumptions and mathematical properties still needed to complete that derivation.

5 Deriving Quantum-State Space from “Coherence-Plus”

For consistent maximal sets of sky priors and stochastic response matrices, both sets must be convex under the urungleichung.The section notes that further assumptions are needed and does not guarantee uniqueness of maximal consistent sets.Source: §5 · p. 28
Reciprocity requires every posterior to be a valid prior and every valid prior to arise as a posterior; in quantum mechanics, a suitable POVM can generate any state.The authors call reciprocity a nontrivial assumption, particularly when unperformed measurements are taken to have no outcomes.Source: §5 · p. 29

5.1 Basis Distributions

Under the Principle of Reciprocity and the full-simplex spanning assumption, P includes the uniform distribution and at least d² other distributions, called basis distributions, which are SIC states in SIC language.Source: §5.1 · pp. 30–31

5.2 A Bloch Sphere

If each basis distribution arises from an in-step-unpredictable measurement, the urgleichung’s bound requires m ≥ d and is tight for basis states only when m = d.Assumption 3 further requires every extreme state to arise from a d-outcome in-step-unpredictable measurement and achieve equality in the bound.Source: §5.2 · p. 32
Under Assumption 3, every extreme state lies on a sphere; in d=2, the smaller sphere is isomorphic to the usual Bloch sphere.The authors also argue that if P contains the actual quantum-state space, it cannot contain additional states.Source: §5.2 · pp. 32–33
The authors distinguish this epistemic constraint from an information-only restriction: it arises from coherence between factual and counterfactual gambles and accommodates the idea that unperformed measurements have no outcomes.They note that Spekkens’ hidden-variable toy model has maximal information that is not complete, so that condition alone cannot rule out hidden variables.Source: §5.2 · pp. 34–35

5.3 But Only Part of It

Except for qubits, valid states occupy only the sphere-simplex intersection; their zero components are bounded by 1/2 d(d−1), and the urungleichung trims the sphere further.For d = 2, the sphere lies wholly within the simplex and gives the Bloch sphere.Source: §5.3 · pp. 35–36

5.4 Consequently an Underlying ‘Dimensionality’

At most d sphere states can be pairwise maximally distant, suggesting—but not establishing—the quantum-state-space dimension: Hausdorff dimension 2d−2 remains unproved.When n = d, the corresponding Gram matrix has rank d−1; the paper cautions that this does not establish the manifold’s Hausdorff dimension.Source: §5.4 · pp. 37–38

5.5 Example: Excising Some Flat Zeros-Bound Vectors

For d ≥ 4, the urungleichung excludes at least one of each too-distant flat zero-bound pair; in d = 3 some such vectors are valid, yet other constraints still exclude some.In the d = 3 example, three linearly independent states span the space, ruling out a vector supported only on the remaining components.Source: §5.5 · pp. 38–39

5.6 Summary of the Argument So Far

The reconstruction supplements the Urgleichung with reciprocity, basis states spanning the simplex, and an extreme-point preparation assumption.The reciprocity principle treats ground-measurement posteriors as valid sky-measurement priors, while the extreme-point assumption specifies how extreme states may be generated.Source: §5.6 · p. 39
Under the stated assumptions, basis distributions belong to P, probabilities satisfy p(k) ≤ 1/d, and at most d extreme states can be mutually maximally distant.The argument also bounds the number of zero components and excludes some flat zero-bound vectors.Source: §5.6 · pp. 39–40
The results are only hints of isomorphism: deriving the required algebraic variety without further assumptions remains unresolved, and substantial technical and conceptual work remains.The authors also question whether assumptions 1–3 are the best additional assumptions for reaching full quantum mechanics.Source: §5.6 · p. 40

6 Relaxing the Constants and Regaining Them

The section starts from a generalized urgleichung with arbitrary constants and outcome count, then asks what assumptions recover the standard quantum form. A specialized measurement and consistency requirements recover that form and constrain the relation between the outcome counts.

6 Relaxing the Constants and Regaining Them

The generalized urgleichung allows arbitrary parameters α and β and an outcome count n not initially assumed to be a perfect square.The target is to recover α = d + 1, β = 1/d, and n = d^2 for some d.Source: §6 · pp. 40–41
The consistency of the generalized urgleichung also imposes constraints on n, α, and β, including a condition obtained by requiring nonnegative outcome probabilities.The section combines these consistency constraints with those from the certainty assumptions.Source: §6 · p. 41
Under in-step unpredictability and availability of certainty, the number m of measurement outcomes is bounded by the relation derived from the Gram-matrix constraint.The derivation rules out m + 1 probability distributions satisfying the required inner products.Source: §6 · p. 44

6.1 Putting the Ingredients Together

The standard urgleichung is regained by combining consistency requirements with the assumptions of availability of certainty, reciprocity, identification, and the specified basis states.The authors treat the generalized urgleichung as universal and use a specialized measurement to fix its constants.Source: §6.1 · p. 45
The specialized measurement can exist only if n is a perfect square, distinguishing the sky outcome count n from the measurement’s maximal count m.The authors use this distinction to recover the quantum-mechanical target parameters.Source: §6.1 · p. 45

7 Summary: From Quantum Interference to Bayesian Coherence

The paper recasts quantum interference and the Born Rule in probabilistic terms, using a counterfactual measurement without invoking probability amplitudes. It argues that the urgleichung captures substantial quantum state structure, while acknowledging that the full pure-state structure remains unrecovered.

7 Summary: From Quantum Interference to Bayesian Coherence

The paper formulates quantum interference as calculating probabilities for a factualizable experiment using an explicitly assumed counterfactual, without probability amplitudes.The authors use SIC measurements as mathematical machinery for this probabilistic account.Source: §7 · pp. 45–46
The Born Rule can be viewed as a relation between probabilities rather than as a rule that sets probabilities from an objective quantum state.In the quantum-Bayesian view, quantum states depend on agents’ prior beliefs.Source: §7 · p. 46
The urgleichung and related considerations specify a significant fraction of quantum-state structure, but the paper does not recover the manifold of pure quantum states.The authors leave open whether more careful reasoning can recover it without question-begging assumptions.Source: §7 · p. 46
The paper presents its framework as a new setting for quantifying the idea that unperformed measurements have no outcomes.Source: §7 · p. 46

8 Forward to an Eventual Foreword!

The section frames quantum theory as a guide for agents organizing expectations about an external world, with measurement understood as an agent’s action whose consequences are personal rather than pre-existing. It presents the urgleichung as a possible postulate connecting personal decision theory to a partly undetermined world in which interactions generate unforeseeable consequences and agency and physicality are reciprocal aspects.

8 Forward to an Eventual Foreword!

Quantum theory is a theory for agents immersed in a world, not a direct description of that world; its limits reflect the world’s character.The authors argue that treating quantum mechanics as a theory of the world has fueled discomfort in quantum foundations.Source: §8 · p. 48
The authors urge replacing “measurement” with action because its ordinary sense suggests passive comparison and obscures the agent-set context of quantum experiments.In their account, actions produce consequences in an agent’s experience; an unperformed measurement therefore has no outcome to find.Source: §8 · pp. 47, 49, §8.1 · p. 52

8.1 The Paulian Idea and the Jamesian Multiverse

Under the Paulian Idea [14], quantum states and measurement outcomes belong to a single user’s expectations and personal experiences arising from actions on external systems.The formal theory guides how an agent organizes Bayesian probabilities for the consequences of potential actions.Source: §8.1 · p. 51
Unitary evolution and other quantum operations represent changes in an agent’s beliefs over time or after actions, rather than objective underlying dynamics.The framework uses a density operator to organize an agent’s beliefs across potential actions.Source: §8.1 · p. 51
QBism is not solipsistic: measurement presupposes an autonomous external system, and the outcome remains beyond the agent’s control once an action is chosen.Others must elicit outcomes through their own actions; the external world is real, while quantum theory serves as a handbook for agents within it.Source: §8.1 · pp. 52–53
Quantum theory remains empirically informative as a theory of acts, decisions, and consequences, portraying consequences as generated through interactions rather than waiting in advance.The authors say this interpretation follows if their hunch about taking the urgleichung as quantum mechanics’ main postulate is on the right track.Source: §8.1 · p. 53
Agents facing uncertainty should use Bayesian probability, and quantum theory is already complete as a decision theory rather than a hidden-variable world theory.The passage cautions that a theory of decision does not transfer into a theory or vision of the world in that way.Source: §8.1 · pp. 53, 55
The proposed world picture allows partial cohesion among partially independent powers, leaving possible complements indeterminate until particular facts actually occur.The passage compares this to a tune whose first notes allow many possible endings, while no one part controls the whole view.Source: §8.1 · p. 54
This is objective indeterminism without objective chance in Lewis’s Principal Principle sense; the future is undetermined within parts and, since no actual whole exists, overall.The footnote rejects the universal wavefunction as an all-encompassing whole that would make everything available in advance.Source: §8.1 · p. 54
Agents and physical systems are reciprocal categories: their interactions catalyze unforeseen experiences, and the paper presents its formalism as an entrance toward an ontology joining these aspects.The authors leave whether to understand or reject this dual-aspect idea as a long road ahead.Source: §8.1 · p. 55