# nLab quantum probability

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

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 $\mathcal{A}$, typically regarded as a star algebra, and encode the probability measure as a state on this star algebra

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

Hence this primarily axiomatizes the concept of expectation values $\langle A\rangle$ (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 $\mathcal{A}$ is assumed/required to be a von Neumann algebra (e.g. Kuperberg 05, section 1.8). Often $\mathcal{A}$ is taken to be the full algebra of bounded operators on some Hilbert space (e.g. Attal, def. 7.1).

In quantum physics, $\mathcal{A}$ 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”).

## Properties

### Conditional expectation and Wave function collapse

There is a close relation between wave function collapse and conditional expectation values in quantum probability (e.g. Kuperberg 05, section 1.2, Yuan 12):

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

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

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

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

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

thought of as an event, then for any observable $A \in \mathcal{A}$ the conditional expectation value of $A$, conditioned on the observation of $P$, is (e.g. Redei-Summers 06, section 7.3)

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

Now assume a star-representation $\rho \;\colon\; \mathcal{A} \to End(\mathcal{H})$ 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 $\psi \in \mathcal{H}$ (“wave function”) via the Hilbert space inner product $\langle (-), (-)\rangle \;\colon\; \mathcal{H} \otimes \mathcal{H} \to \mathbb{C}$ as

\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 $A$ conditioned on an idempotent observable $P$ becomes (notationally suppressing the representation $\rho$)

\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 (2).

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

This is the statement of “wave function collapse”: The original wave function is $\psi \in \mathcal{H}$, and after observing $P$ it “collapses” to $P \psi \in \mathcal{H}$ (up to normalization).

## References

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

• Irving Segal, Algebraic integration theory, Bull. Amer. Math. Soc. Volume 71, Number 3, Part 1 (1965), 419-489 (Euclid)

• Peter Whittle, Probability via expectation, Springer 1992

A good exposition of quantum physics from this perspective is in

• Jonathan Gleason, The $C^*$-algebraic formalism of quantum mechanics, 2009 (pdf, pdf)

Further introduction to quantum probability theory includes

A textbook account is

• Klaas Landsman, Foundations of quantum theory – From classical concepts to Operator algebras, Springer Open 2017 (pdf)

Last revised on September 19, 2018 at 11:08:21. See the history of this page for a list of all contributions to it.