Hidden Variable Models

A hidden-variable model is a specified probability construction, not a single premise shared by every no-go theorem. Preparations determine probability measures on an underlying space, and measurements determine response kernels. Additional requirements such as noncontextuality, Bell factorization, or preparation independence constrain different parts of that construction. The canonical contextuality article explains the operational distinctions; this page supplies the measure-level tools needed to compare such models without informal support arguments.

Required background. Measure Theory Minimum supplies probability measures and almost-everywhere statements; Contextuality explains preparation and measurement equivalence.

Ontological models: preparation measures and measurement kernels

Let $(\Lambda,\Sigma)$ be a measurable space. A preparation procedure $P$ has a probability measure $\mu_P$. For a finite-outcome measurement $M$, the response functions satisfy

$$\xi_M(k\mid\lambda)\ge0, \qquad \sum_k\xi_M(k\mid\lambda)=1,$$

and are measurable in $\lambda$. The prediction is

$$p(k\mid P,M)=\int_\Lambda \xi_M(k\mid\lambda)\,\mu_P(d\lambda).$$

For a general measurable outcome space, replace the finite list by a Markov kernel: for fixed $\lambda$ it is a probability measure on outcomes, and for each outcome event it is measurable in $\lambda$. An underlying Euclidean phase space or a density relative to Lebesgue measure is not required.

The label $P$ matters. Two procedures can prepare the same density operator while having different underlying measures in a preparation-contextual model. A classical random choice of procedures gives the corresponding mixture of their measures when that randomization is modeled explicitly. Identifying all procedures with the same operational density operator is the stronger preparation-noncontextuality premise, not a consequence of the integral formula alone.

Overlap without choosing coordinates

Take two probability measures $\mu_0,\mu_1$. The finite measure $\nu=\mu_0+\mu_1$ dominates both. By the Radon–Nikodym theorem write $f_i=d\mu_i/d\nu$ and define the common submeasure

$$d\chi=\min(f_0,f_1)\,d\nu, \qquad \omega=\chi(\Lambda).$$

The number $\omega$ is independent of the choice of dominating measure. It measures common probability mass, rather than a geometric intersection of topological supports. Directly,

$$\begin{aligned} D_{\rm TV}(\mu_0,\mu_1) &=\tfrac12\int|f_0-f_1|\,d\nu\\ &=1-\omega. \end{aligned}$$

Here total variation ranges from zero to one. The measures are mutually singular exactly when $\omega=0$. Two singular measures may have overlapping topological supports, so phrases such as “disjoint supports” need a measure-theoretic interpretation in a theorem proof.

Operational distinguishability cannot exceed ontic distinguishability

For an outcome event, a response kernel gives a measurable function $0\le h\le1$. Therefore

$$\left|\int h\,d\mu_0-\int h\,d\mu_1\right| \le D_{\rm TV}(\mu_0,\mu_1).$$

This is the variational characterization of total variation, or classical data processing. If the model reproduces all quantum POVMs for two density operators, optimize over the quantum effects to obtain

$$D(\rho_0,\rho_1)\le1-\omega.$$

In particular, for pure states with overlap magnitude $s$, $\omega\le1-\sqrt{1-s^2}$. Orthogonal states force $\omega=0$ already from one perfect-discrimination measurement. Distinct nonorthogonal states do not: this inequality permits a positive overlap. Removing that remaining possibility requires further hypotheses, as in the PBR argument.

Pairwise singularity of preparation measures is the relevant $\psi$-ontic condition. It does not, by itself, furnish an experimentally accessible measurement of $\lambda$, a unique interpretation, or a globally measurable reconstruction algorithm for an arbitrary uncountable state family. Those are additional questions.

Composition is a separate mathematical choice

For two independently operated preparation devices, the product preparation-independence assumption takes their ontic space to be $\Lambda\times\Lambda$ and their joint measure to be $\mu_i\otimes\mu_j$. The existence of two correct single-system marginals is weaker than this factorization. Correlated couplings can have the same marginals and very different overlap properties.

This independence concerns preparation composition. Bell locality instead factorizes the response to separated measurements conditional on a shared variable, and measurement independence constrains the source law relative to the setting choices. See Local Hidden Variables. Writing the probability objects explicitly prevents these distinct premises from collapsing into the word “independence.”

Exercise: identical marginals, disjoint joint laws

On two bits, let one joint distribution assign probability $1/2$ to each of $00,11$, and another assign probability $1/2$ to each of $01,10$. Compute their single-bit marginals, joint overlap, and distance from the product of their marginals.

Solution

Every single-bit marginal is uniform. The joint supports are disjoint, so their overlap is zero and their mutual total-variation distance is one. The product of the marginals is uniform on all four bit pairs. Each stated joint law has total-variation distance $\tfrac12(4\times\tfrac14)=1/2$ from that product. Thus identical marginals do not establish a product law or determine joint overlap.

References

• Harrigan, Nicholas, and Robert W. Spekkens, “Einstein, Incompleteness, and the Epistemic View of Quantum States,” Foundations of Physics 40, 125–157 (2010), doi:10.1007/s10701-009-9347-0 — ontological models and the ontic/epistemic distinction.
• Pusey, Matthew F., Jonathan Barrett, and Terry Rudolph, “On the reality of the quantum state,” Nature Physics 8, 475–478 (2012), doi:10.1038/nphys2309 — preparation composition and overlap constraints.
• Spekkens, Robert W., “Contextuality for preparations, transformations, and unsharp measurements,” Physical Review A 71, 052108 (2005), doi:10.1103/PhysRevA.71.052108 — operational equivalence and generalized noncontextuality.
• Watrous, John, The Theory of Quantum Information, Cambridge University Press, 2018, doi:10.1017/9781316848142 — the variational trace-distance formula.