2. The objectivity of quantum probabilities

Any attempt to understand quantum theory must address the significance of probabilities derived from the Born Rule, which I write as followsprobρ
(A
εΔ
)=Tr
(ρP
A[Δ]) (Born Rule)

whereA
is a dynamical variable (an “observable”) pertaining to a systems
, ρ represents a quantum state of that system by a density operator on a Hilbert space Hs , Δ is aBorel
set of real numbers (soA
εΔ
states that the value ofA
lies in Δ), and P A[Δ] is the value for Δ of the projection-valued measure defined by the unique self-adjoint
operator on Hs corresponding toA
. Born probabilities yielded by systems’ quantum states are the key to successful applications of quantum theory to explain and predict natural phenomena involving them. If one denies that the quantum state describes or represents the physical properties or relations of any system or ensemble of systems, then its main job is simply to yield these probabilities. But what kind of probabilities are these, and what, exactly, are they probabilities of?

If one clear conclusion has been established by foundational work, it is that not every probability derivable by applying theBorn
rule to a system with quantum state ρ can be taken as a quantitative measure of ignorance or uncertainty of the real-numbered value of a dynamical variable on that system. Born probabilities are not analogous to probabilities in classical statistical mechanics in that they cannot be jointly represented on any classical phase space: quantum observables are not random variables on a common probability space.
However, as I expressed it theBorn
rule specifies, for any state ρ, a probability for each sentence of the formS
: The value ofA
lies in Δ. The traditional way to resolve the resulting tension is to take each instantiation of the Born rule to observableA
to be (perhaps implicitly) conditional onmeasurement
ofA
, and to assume or postulate that only observables represented by commuting operators can be measured together.
Whether this resolution is satisfactory has been the topic of a heated debate.
I will address aspects of this in the next section, which offers an account of what theBorn
probabilities are probabilities of. But whatever they concern, what kind of probabilities are these?

It is common to classify an interpretation of probability as either objective or subjective. Accounts of probability in terms of frequency, propensity or single-case chance count as objective, while thepersonalist
Bayesian interpretation counts as subjective. But then how should one classify classical (Laplacean
), logical and “objective” Bayesian notions of probability? It will be best to leave the tricky issue of objectivity aside for a while, so I begin instead by classifying accounts of probability on the basis of their answers to the question ‘Does a probability judgment function as a description of anything in the natural world?’ vonMises’s
([1922]; [2003], p.194) answer to this question was clear:

Probability calculus is part of theoretical physics in the same way as classical mechanics or optics, it is an entirely self-contained theory of certain phenomenaSo
was Popper’s ([1967], pp.32-3)

In proposing the propensity interpretation I propose to look upon probability statements as statements about some measure of a property (a physical property, comparable to symmetry or asymmetry) of the whole experimental arrangement; a measure, more precisely, of a virtual frequency

These are expressions of what I will call anatural property
account of probability.

Accounts of probability as a natural property typically take this to be a property of something in the physical world independent of the epistemic state of anyone making judgments about it. When deFinetti
wrote in the preface to his ([1974]) ‘PROBABILITY DOES NOT EXIST’, this was what he meant to deny. But he wrote elsewhere ([1968], p.48) that probability

means degree of belief (as actually held by someone, on the ground of his whole knowledge, experience, information) regarding the truth of a sentence, or eventE
(a fully specified ‘single’ event or sentence, whose truth or falsity is, for whatever reason, unknown to the person).

and
it is at least plausible to suppose that an actual degree of belief is a natural property of the person holding it. If so, even the arch subjectivist deFinetti
here adopts a natural property account of probability! Of course, he would insist that different persons may, and often do, hold different beliefs, which makes probabilitypersonalist
- varying from person to person - and to that extent subjective.

On other “subjectivist” views, an agent’s degrees of belief count as probabilities only in so far as its overall epistemic state meets a normative constraint ofcoherence
,
since otherwise these partial beliefs will not satisfy an analog ofKolmogorov’s
([1933]) axioms defining probability mathematically as a finitely additive, unit-normed
, non-negative function on a field of sets. Ramsey ([1926]), for one, took probability theory as a branch of logic, the logic of partial belief and inconclusive argument.
So viewed, probability theory offers an agent prescriptions for adjusting the corpus of its beliefs so that its total epistemic state meets minimal internal standards of rationality - standards that are nevertheless met by the total epistemic state of few if any actual agents. A probability judgment made by an agent then counts as an expression of its partial degree of belief and a commitment to hold its epistemic state to this minimal standard of rationality. Such a probability judgment does not function as a description of the agent’s own belief state, and is certainly not a description of a natural property of anything else in the physical world.

On the present approach, quantum probabilities given by theBorn
rule do not describe any natural property of the system or systems to which they pertain, or of any other physical system or situation: nor is it their function to describe any actual agent’s state of belief, knowledge or information. Their function is to offer advice to any actual or hypothetical agent on the extent of its commitment to claims expressible by sentences of the formS
: The value ofA
ons
lies in Δ - roughly, what degree of belief or credence to attach to such a claim.