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Measure and probability theory



In probability theory, the concept of noncommutative probability space or quantum probability space is the generalization of that of probability space as the concept of “space” is generalized to non-commutative geometry.

The basic idea is to encode a would-be probability space dually in its algebra of functions 𝒜\mathcal{A}, typically regarded as a star algebra, and encode the probability measure as a state on this star algebra

:𝒜. \langle - \rangle \;\colon\; \mathcal{A} \longrightarrow \mathbb{C} \,.

Hence this primarily axiomatizes the concept of expectation values A\langle A\rangle (Segal 65, Whittle 92) while leaving the nature of the underlying probability measure implicit (in contrast to the classical formalization of probability theory by Andrey Kolmogorov).

Often 𝒜\mathcal{A} is assumed/required to be a von Neumann algebra (e.g. Kuperberg 05, section 1.8). Often 𝒜\mathcal{A} is taken to be the full algebra of bounded operators on some Hilbert space (e.g. Attal, def. 7.1).

In quantum physics, 𝒜\mathcal{A} is an algebra of observables (or a local net thereof) and ()\langle (-)\rangle is a particular quantum state, for instance a vacuum state.

The formulation of non-perturbative quantum field theory from the algebraic perspective of quantum probability is known as algebraic quantum field theory (AQFT).

The formulation of perturbative quantum field theory from the algebraic perspective of quantum probability is known as perturbative algebraic quantum field theory (pAQFT).

The sentiment that quantum physics is quantum probability theory is also referred to as the Bayesian interpretation of quantum mechanics (“QBism”).


As classical probability theory internal to a Bohr topos

The idea that

quantum probability is “just as” classical probability theory but generalized to non-commutative probability spaces, hence, for quantum physics, to quantized phase spaces

may be made precise and fully manifest by understanding quantum probability theory as being classical probability theory internal to the Bohr topos of the given quantum mechanical system.

For details see at Bohr topos the section Kinematics in a Bohr topos.

For going deeper, see at order-theoretic structure in quantum mechanics.

Conditional expectation and Wave function collapse

Quantum probability theory shows that “wave function collapse” is just part of the formula for conditional expectation values in quantum probability theory (e.g. Kuperberg 05, section 1.2, Yuan 12):

Let (𝒜,)(\mathcal{A},\langle -\rangle) be a quantum probability space, hence a complex star algebra 𝒜\mathcal{A} of quantum observables, and a state on a star-algebra :𝒜\langle -\rangle \;\colon\; \mathcal{A} \to \mathbb{C}.

This means that for A𝒜A \in \mathcal{A} any observable, its expectation value in the given state is

𝔼(A)A. \mathbb{E}(A) \;\coloneqq\; \langle A \rangle \in \mathbb{C} \,.

More generally, if P𝒜P \in \mathcal{A} is a real idempotent/projector

(1)P *=P,AAAPP=P P^\ast = P \,, \phantom{AAA} P P = P

thought of as an event, then for any observable A𝒜A \in \mathcal{A} the conditional expectation value of AA, conditioned on the observation of PP, is (e.g. Redei-Summers 06, section 7.3, see also Fröhlich-Schubnel 15, (5.49), Fröhlich 19 (45))

(2)𝔼(A|P)PAPP. \mathbb{E}(A \vert P) \;\coloneqq\; \frac{ \left \langle P A P \right\rangle }{ \left\langle P \right\rangle } \,.

Now assume a star-representation ρ:𝒜End()\rho \;\colon\; \mathcal{A} \to End(\mathcal{H}) of the algebra of observables by linear operators on a Hilbert space \mathcal{H} is given, and that the state \langle -\rangle is a pure state, hence given by an vector ψ\psi \in \mathcal{H} (“wave function”) via the Hilbert space inner product (),():\langle (-), (-)\rangle \;\colon\; \mathcal{H} \otimes \mathcal{H} \to \mathbb{C} as

A ψ|A|ψ ψ,Aψ. \begin{aligned} \langle A \rangle & \coloneqq \left\langle\psi \vert A \vert \psi \right\rangle \\ & \coloneqq \left\langle\psi, A \psi \right\rangle \end{aligned} \,.

In this case the expression for the conditional expectation value (2) of an observable AA conditioned on an idempotent observable PP becomes (notationally suppressing the representation ρ\rho)

𝔼(A|P) =ψ|PAP|ψψ|P|ψ =Pψ|A|PψPψ|Pψ, \begin{aligned} \mathbb{E}(A\vert P) & = \frac{ \left\langle \psi \vert P A P\vert \psi \right\rangle }{ \left\langle \psi \vert P \vert \psi \right\rangle } \\ & = \frac{ \left\langle P \psi \vert A \vert P \psi \right\rangle }{ \left\langle P \psi \vert P \psi \right\rangle } \,, \end{aligned}

where in the last step we used (1).

This says that assuming that PP has been observed in the pure state |ψ\vert \psi\rangle, then the corresponding conditional expectation values are the same as actual expectation values but for the new pure state |Pψ\vert P \psi \rangle.

This is the statement of “wave function collapse”:

|ψP|ψ. \vert \psi \rangle \mapsto P \vert \psi \rangle \,.

The original wave function is ψ\psi \in \mathcal{H}, and after observing PP it “collapses” to PψP \psi \in \mathcal{H} (up to normalization).

quantum probability theoryobservables and states


The understanding of quantum physics as a probabilistic theory originates with the formulation of the Born rule and was made fully explicit in:

see also:

  • Jagdish Mehra, Helmut Rechenberg, The Probability Interpretation and the Statistical Transformation Theory, the Physical Interpretation, and the Empirical and Mathematical Foundations of Quantum Mechanics 1926-1932, Part 1 in: The Historical Development of Quantum Theory. Volume 6: The Completion of Quantum Mechanics, 1926-1941, Springer (2001) [ISBN:978-0-387-98971-6]

with emphasis on POVMs:

The axiomatization of probability theory in terms of the concept of expectation values (instead of probability measures) is amplified in:

Dedicated discussion of quantum probability theory:

Further discussion of quantum probability:

Discussion of density matrices and entropy in quantum probability, via the GNS construction:

Last revised on September 29, 2023 at 09:28:20. See the history of this page for a list of all contributions to it.