nLab wave function collapse

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).

Related concepts

• interpretation of quantum mechanics

• Bayesian interpretation of quantum mechanics

• propositions as projections

• deferred measurement principle

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

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

• John von Neumann, §III.3 and §VI of:

  Mathematische Grundlagen der Quantenmechanik, Springer (1932, 1971) [doi:10.1007/978-3-642-96048-2]

  Mathematical Foundations of Quantum Mechanics Princeton University Press (1955) [doi:10.1515/9781400889921, Wikipedia entry]

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:

• Gerhart Lüders:

  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:

• Michael A. Nielsen, Isaac L. Chuang, §2.2.3 of: Quantum computation and quantum information, Cambridge University Press (2000) [doi:10.1017/CBO9780511976667, pdf, pdf]

• Joseph M. Renes, eq. (3.2) in: Quantum Information Theory (2015) [pdf], (A.2) in: De Gruyter (2022) [doi:10.1515/9783110570250]

Discussion from the point of view of quantum probability:

• Greg Kuperberg, A concise introduction to quantum probability, quantum mechanics, and quantum computation (2005) [pdf, pdf]

• Miklos Redei, Stephen Summers, Quantum Probability Theory (arXiv:quant-ph/0601158)

• Qiaochu Yuan, Finite noncommutative probability, the Born rule, and wave function collapse, 2012

• Jürg Fröhlich, B. Schubnel, Quantum Probability Theory and the Foundations of Quantum Mechanics. In: Blanchard P., Fröhlich J. (eds.) The Message of Quantum Science. Lecture Notes in Physics, vol 899. Springer 2015 (arXiv:1310.1484, doi:10.1007/978-3-662-46422-9_7)

• Jürg Fröhlich, The structure of quantum theory, Chapter 6 in The quest for laws and structure, EMS 2016 (doi, doi:10.4171/164-1/8)

See also

• Wikipedia, Wave function collapse

• Wikipedia, Born rule

More speculative discussion of state collapse as caused by stochastic terms added to the Schrödinger equation:

from the point of view of the symplectic formulation of quantum mechanics:

• Lane P. Hughston, Geometry of stochastic state vector reduction, Proceedings of the Royal Society A 452 1947 (1996) [doi:10.1098/rspa.1996.0048, jstor:52944]