Contents

# Contents

## Idea

In the context of quantum mechanics, the collapse of the wave function, also known as the reduction of the wave packet, is said to occur after observation or measurement, when a wave function expressed as the sum of eigenfunctions of the observable is projected randomly onto one of them. Different interpretations of quantum mechanics understand this process differently.

The perspective associated with the Bayesian interpretation of quantum mechanics observes (see below) that the apparent collapse is just the mathematical reflection of the formula for conditional expectation values in quantum probability theory.

## Relation to conditional expectation values

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, see also Fröhlich-Schubnel 15, (5.49), Fröhlich 19 (45))

(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 a 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”

$\vert \psi \rangle \mapsto P \vert \psi \rangle \,.$

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

Early explicit discussion of wavefunction collapse in quantum measurements, including discussion relating to historical experiments:

But the formulation in von Neumann 1932 postulated that a pure state would collapse to a mixed state when the quantum observable‘s eigenvalues are degenerate.

This problem was pointed out and the modern formulation of the collapse postulate was formulated by:

• Über die Zustandsänderung durch den Meßprozeß, Ann. Phys. 8 (1951) 322–328 [doi:10.1002/andp.19504430510]

Concerning the state-change due to the measurement process, Ann. Phys. 15 9 (2006) 663-670 [pdf, pdf]

Early review:

• Erhard Scheibe, pp 137 in: The logical analysis of quantum mechanics&, Pergamon Press Oxford (1973)

Textbook accounts in quantum information theory:

Discussion from the point of view of quantum probability: