Quantum Probability

Quantum mechanics assigns probabilities, but not on a Boolean $\sigma$-algebra of events. Its events are the projections of a Hilbert space, which form an orthomodular but non-distributive lattice, and the Kolmogorov axioms do not apply to it. This is not a defect of the physical theory nor a failure of rigour: it is a mathematically precise generalization, and Gleason's theorem shows that once the event structure is changed, the form of the probability assignment is forced.

This page is the promised generalization of kolmogorov-axioms.md §6 — what happens when distributivity, the assumption that all events have simultaneous truth values, is dropped. It builds on the spectral theory of functional analysis and serves the postulate discussions in QM and QFT.

References: Gleason (1957); Varadarajan, Geometry of Quantum Theory; Busch, "Quantum states and generalized observables: a simple proof of Gleason's theorem" (2003); Redei–Summers, "Quantum probability theory" (2007).

1. The Event Structure

In the classical setting, events form a Boolean $\sigma$-algebra $\mathcal{F}$; a random variable is a measurable map, and its spectral resolution is the family of events $\{X\in B\}$, which is a $\sigma$-homomorphism $\mathcal{B}(\mathbb{R})\to \mathcal{F}$.

In the quantum setting, replace $\mathcal{F}$ by

$$\mathcal{P}(\mathcal{H})=\{\Pi \in \mathcal{B}(\mathcal{H}):\Pi ={\Pi }^{\ast }={\Pi }^2\},$$

the orthogonal projections on a complex Hilbert space, ordered by $\Pi \le {\Pi }^{\prime }⟺\mathrm{ran}\Pi \subseteq \mathrm{ran}{\Pi }^{\prime }$, with complement ${\Pi }^{\perp }=I-\Pi$. Observables are self-adjoint operators, and their spectral measures (spectral-theorem.md) are the exact analogue of $B\mapsto \{X\in B\}$: projection-valued measures $E:\mathcal{B}(\mathbb{R})\to \mathcal{P}(\mathcal{H})$ with $E(\mathbb{R})=I$ and countable additivity on disjoint sets.

The lattice is not distributive. Take $\mathcal{H}={\mathbb{C}}^2$ and rays ${\Pi }_a,{\Pi }_b,{\Pi }_c$ spanned by three distinct vectors. Then ${\Pi }_a\wedge >({\Pi }_b\vee {\Pi }_c)={\Pi }_a\wedge I={\Pi }_a$, while $({\Pi }_a\wedge {\Pi }_b)\vee ({\Pi }_a\wedge {\Pi }_c)=0\vee 0=0$. Distributivity fails at the first opportunity. What survives is orthomodularity: $\Pi \le >{\Pi }^{\prime }⟹{\Pi }^{\prime }=\Pi \vee ({\Pi }^{\prime }\wedge {\Pi }^{\perp })$ — distributivity restricted to commuting (compatible) projections.

Two projections commute iff the corresponding events are jointly decidable, and the sublattice generated by a commuting family is Boolean. So classical probability is recovered exactly on each maximal commuting family, and quantum probability is the theory of how those Boolean patches fail to glue.

2. States as Probability Measures

Definition. A state on $\mathcal{P}(\mathcal{H})$ is a map $\mu :\mathcal{P}(\mathcal{H})\to [0,1]$ with $\mu (I)=1$ and

$$\mu \left(\sum _i{\Pi }_i\right)=\sum _i\mu ({\Pi }_i)\text{for countable families of mutually orthogonal }{\Pi }_i.$$

This is precisely Kolmogorov's third axiom with "disjoint" replaced by "orthogonal" — additivity is required only where the events are compatible. Nothing else is assumed: no continuity, no linearity, no relation between non-commuting projections.

Gleason's theorem (1957). Let $\dim \mathcal{H}\ge 3$ (real, complex, or quaternionic, separable or not). Every state $\mu$ on $\mathcal{P}(\mathcal{H})$ is of the form

$$\mu (\Pi )=\mathrm{Tr}(\rho \Pi )$$

for a unique density operator $\rho$: positive, self-adjoint, trace one.

The consequence is remarkable and is what the QFT remarks page records: the Born rule is not an independent postulate about the form of quantum probabilities. Assume only that probabilities exist and are additive over orthogonal alternatives, and the trace formula — hence $|⟨\psi |\varphi ⟩{|}^2$ for pure states $\rho =|\psi ⟩⟨\psi |$ — is forced. What remains postulated is that the projections carry probabilities at all, and how those probabilities connect to measurement outcomes.

Why $\dim \ge 3$. In dimension 2 the projection lattice has no non-trivial compatibility structure — any two distinct rays generate the whole lattice with no constraint linking them — and non-trace states exist in abundance. The theorem is a statement about how three-dimensional orthogonality constrains a frame function, and its proof (originally hard analysis on the sphere; later elementary treatments by Cooke–Keane–Moran and Busch) is where the geometry enters.

3. No Hidden Boolean Model

A natural hope is that the quantum structure is an artefact of ignorance: that underneath there is a classical probability space assigning definite values to all observables, with quantum probabilities arising by marginalization. Two theorems say this cannot be done under mild assumptions.

Kochen–Specker (1967). For $\dim \mathcal{H}\ge 3$ there is no map $v:\mathcal{P}(\mathcal{H})\to \{0,1\}$ that is additive on orthogonal resolutions of the identity. Equivalently, no dispersion-free state exists.

This is an immediate corollary of Gleason (a $\{0,1\}$-valued state would need $\mathrm{Tr}(\rho \Pi )\in \{0,1\}$ for all $\Pi$, impossible), and also admits finite combinatorial proofs using explicit sets of vectors. Its content: noncontextual value assignments do not exist — an observable cannot be thought of as possessing a value independent of which compatible set it is measured alongside.

Bell (1964). No local hidden-variable model reproduces the quantum correlations for entangled systems; the CHSH inequality $|S|\le 2$, valid for any classical joint distribution, is violated up to $2\sqrt{2}$ by quantum states.

Read probabilistically, Bell's theorem is a statement about the non-existence of a joint distribution: the four pairwise correlations of a CHSH experiment are individually well defined (each pair is compatible) but no single Kolmogorov probability space carries all four random variables with those marginals. This is Fine's theorem, and it identifies the exact classical assumption that fails — not determinism, not locality alone, but the existence of a global Boolean model. The philosophical reading is discussed in philosophy of Bell's theorem.

4. POVMs and the General Framework

Projection-valued measures are the special case of a wider notion.

Definition. A POVM (positive operator-valued measure) is a countably additive map $E:\mathcal{B}(\mathbb{R})\to \mathcal{B}(\mathcal{H})$ with $E(B)\ge 0$ and $E(\mathbb{R})=I$, the values being positive operators, not necessarily projections. Probabilities are again $\mu (B)=\mathrm{Tr}(\rho >E(B))$.

POVMs model measurements with noise, unsharp observables, and measurements on subsystems (via Naimark dilation, every POVM is a PVM on a larger space). Gleason's theorem has a much easier POVM version, provable in a page and valid in dimension 2 as well, which is one reason the POVM framework is now standard.

The full generalization is algebraic: states are positive normalized functionals on a $C^{\ast }$-algebra $\mathfrak{A}$ of observables (operator-algebras.md), and the GNS construction turns each state into a Hilbert-space representation. Classical probability is the commutative case — by Gelfand duality, a commutative $C^{\ast }$-algebra is $C(X)$ and states are Radon probability measures on $X$ (Riesz–Markov). So:

$$\text{classical probability}=\text{commutative quantum probability}.$$

Kolmogorov's theory is not superseded; it is the abelian corner of a larger theory, which is exactly why it continues to apply within any single compatible family of observables.

5. What Is and Is Not Settled

• Settled. Given the projection lattice and additivity over orthogonal families, the form of the probability measure is unique (Gleason). Given the quantum correlations, no global Boolean model exists (Bell/Kochen–Specker/Fine).
• Not settled by any of this. Why the state space is a Hilbert-space projection lattice; what a probability means here; and how deterministic unitary evolution issues in probabilistic outcomes — the measurement problem. Those are interpretive questions, and the mathematics constrains without resolving them; see QM postulates and QFT remarks.

6. Where This Is Used

• The spectral theorem — PVMs, the object states are evaluated on.
• C*- and von Neumann algebras — the algebraic framework of §4.
• QM postulates and QFT remarks — the physics consumers.
• Kolmogorov's axiomatization §6 — the distributivity assumption dropped here.