Math 2.0

Quantum theory Foundations and measurement Theorem

PBR theorem

Builds on Hidden-variable theoryProbability measureTensor product of Hilbert spacesQuantum state

Statement

Consider a hidden-variable theory for a finite-dimensional quantum system and for all its finite collections of copies. Let Λ\Lambda be the space of hidden variables of one copy, and let Λn\Lambda^n be that of nn copies. Assume preparation independence: for pure states ψ1,…,ψn\psi_1, \dots, \psi_n, the measure of the product state ψ1⊗⋯⊗ψn\psi_1 \otimes \dots \otimes \psi_n is μψ1×⋯×μψn\mu_{\psi_1} \times \dots \times \mu_{\psi_n}. Let ψ\psi and φ\varphi be unit vectors in the same space that define different rays. Then the probability measures μψ\mu_\psi and μφ\mu_\varphi are mutually singular: some measurable set has μψ\mu_\psi-measure 11 and μφ\mu_\varphi-measure 00.

Remarks

Mutually singular measures mean that the hidden variable determines the quantum state, so the state is ontic rather than a mere description of knowledge. The key case is ψ=e0\psi = e_0 and φ=(e0+e1)/2\varphi = (e_0 + e_1)/\sqrt2 in ℂ2\C^2. Prepare two independent copies, so that one of the four product states e0⊗e0e_0 \otimes e_0, e0⊗φe_0 \otimes \varphi, φ⊗e0\varphi \otimes e_0, φ⊗φ\varphi \otimes \varphi is present. There is an entangled measurement on ℂ2⊗ℂ2\C^2 \otimes \C^2 whose kk-th outcome is orthogonal to the kk-th of these product states. Suppose μψ\mu_\psi and μφ\mu_\varphi overlap. By preparation independence, with positive probability the hidden variable of the pair is compatible with all four states. Then some outcome has positive probability for a state orthogonal to it, which is impossible. Non-orthogonal states that are closer together need more copies and a similar argument. Without preparation independence the conclusion fails: there are models in which distinct quantum states share hidden variables.

Going deeper

For a student

The theorem was published by Pusey, Barrett and Rudolph in 2012, in the framework of ontological models. It distinguishes "psi-ontic" models, such as de Broglie–Bohm theory, from "psi-epistemic" ones, in which distributions for different states overlap. Preparation independence is essential, and it plays a role similar to that of measurement independence in Bell's theorem. Unlike Bell's theorem, the PBR theorem concerns preparations of independent systems rather than correlations of entangled ones.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Hidden-variable theory, concept, no evidence levelHidden-variable theoryconceptProbability measure, concept, no evidence levelProbability measureconceptQuantum state, concept, no evidence levelQuantum stateconceptTensor product of Hilbert spaces, concept, no evidence levelTensor product of Hil…conceptPBR theorem, statement, no evidence levelPBR theoremstatement
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  1. Hidden-variable theory concept
  2. Probability measure concept
  3. Quantum state concept
  4. Tensor product of Hilbert spaces concept
  5. PBR theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Pusey%E2%80%93Barrett%E2%80%93Rudolph_theorem
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AboutHidden-variable theory, Quantum state
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Changed3 November 2026

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