PBR Theorem: Two-Copy Proof

The Pusey–Barrett–Rudolph theorem excludes overlapping ontic preparation measures for distinct pure states when a model reproduces quantum predictions and obeys a specified product preparation-independence assumption. The assumption is essential. This page proves the two-copy case for $|0\rangle$ and $|+\rangle$ and derives an error-tolerant bound; the general many-copy theorem is cited rather than proved. The foundations treatment develops the broader interpretive discussion and the distinction between preparation independence and other foundational assumptions.

Required background. Hidden-Variable Models defines preparation measures, response kernels, and overlap as common probability mass; Hilbert Space Tensor Products supplies the two-system state space.

PBR preparation independence and common probability mass

Let $\mu_0,\mu_1$ describe two preparation procedures for $\psi_0=|0\rangle$ and $\psi_1=|+\rangle$, where $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt2$. Let $\chi$ be their maximal common submeasure and $\omega=\chi(\Lambda)$. Thus $\chi\le\mu_i$ and $\omega>0$ means overlap.

Assume that independent choices $i,j\in\{0,1\}$ yield joint ontic law $\mu_i\otimes\mu_j$ on $\Lambda^2$, and that the response to a fixed joint measurement depends on the joint ontic state, not additionally on which preparation label was used. Every joint preparation then dominates $\chi\otimes\chi$, whose mass is $\omega^2$.

This product law is stronger than the claim that the two observed quantum states form a tensor product or that each device has the right marginal statistics. It is an assumption about the underlying model’s composition. No spatial separation of the later joint measurement is required.

A measurement excludes each possible preparation once

Define four unit vectors in the two-qubit space:

$$\begin{aligned} \xi_{00}&=(|01\rangle+|10\rangle)/\sqrt2,\\ \xi_{01}&=(|0-\rangle+|1+\rangle)/\sqrt2,\\ \xi_{10}&=(|+1\rangle+|-0\rangle)/\sqrt2,\\ \xi_{11}&=(|+-\rangle+|-+\rangle)/\sqrt2. \end{aligned}$$

They form an orthonormal basis. In the computational basis their real coordinate vectors are respectively

$$\begin{gathered} (0,1,1,0)/\sqrt2,\qquad (1,-1,1,1)/2,\\ (1,1,-1,1)/2,\qquad (1,0,0,-1)/\sqrt2. \end{gathered}$$

Their scalar products verify both orthogonality and $\langle\xi_{ij},\psi_i\otimes\psi_j\rangle=0$ for each label pair. Thus the PVM with projectors $|\xi_{ij}\rangle\langle\xi_{ij}|$ has four outcomes, and quantum theory assigns probability zero to outcome $ij$ on preparation $ij$. An outcome excludes one candidate preparation; it does not identify the actual preparation uniquely.

Zero probabilities contradict common mass

Let $r_{ij}(\lambda_1,\lambda_2)$ be the response probability for outcome $ij$ of that one PVM. These four nonnegative functions sum to one. Reproduction of the four quantum zeroes requires

$$\int r_{ij}\,d(\mu_i\otimes\mu_j)=0.$$

Since $\chi\otimes\chi\le\mu_i\otimes\mu_j$, each integral against the common submeasure is also zero. Summing the four equations gives

$$\begin{aligned} 0&=\int\sum_{i,j}r_{ij}\,d(\chi\otimes\chi)\\ &=(\chi\otimes\chi)(\Lambda^2)=\omega^2. \end{aligned}$$

Therefore $\omega=0$: the two preparation measures are mutually singular. This argument avoids selecting a particular point from an informal overlap region and keeps all almost-everywhere qualifications under one common measure. A finite sum of null integrals suffices.

Noise gives an overlap bound

Suppose the four nominally excluded outcome probabilities are instead $\varepsilon_{ij}\ge0$. The same domination argument gives

$$\omega^2\le\sum_{i,j}\varepsilon_{ij}.$$

If every excluded probability is at most $\varepsilon$, then $\omega\le2\sqrt\varepsilon$, with the trivial upper bound one also understood. A statistically estimated frequency needs its own confidence bound before being inserted as a bound on a probability. Imperfect preparation independence is another change of premise and is not absorbed automatically into these four measurement errors.

For arbitrary distinct pure states, Pusey, Barrett, and Rudolph construct a suitable exclusion measurement on sufficiently many independently prepared copies. If a common single-copy submeasure has mass $\omega$, its $n$-fold product has mass $\omega^n$, and the same proof applies to the $2^n$ preparation strings. Establishing the required measurement for the chosen overlap is the additional quantum construction; the two-copy basis above is not universal.

Interpretive limits

The conclusion is pairwise singularity in a declared ontological-model class. It does not prove that the wavefunction is a field in physical three-space, solve the measurement problem, exclude all hidden variables, or select a unique interpretation. Models that reject the product composition assumption are not ruled out by this proof. Neither Bell locality nor measurement noncontextuality was substituted for preparation independence.

Exercise: confirm one excluded outcome

Use the coordinate vectors to verify $\langle\xi_{01},0+\rangle=0$ and $\langle\xi_{01},\xi_{10}\rangle=0$. Explain why four such exclusions are insufficient without response normalization.

Solution

The vector $|0+\rangle$ is $(1,1,0,0)/\sqrt2$, giving the first inner product $(1-1)/(2\sqrt2)=0$. The second is $(1-1-1+1)/4=0$. If all response functions were allowed to vanish on the common component, every excluded integral could vanish without a contradiction. Normalization forbids this: some outcome must occur there, and each of the four outcomes is excluded by one of the possible preparations that dominates the same common measure.

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 — the measure-overlap meaning of 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, arXiv:1111.3328 — the exclusion measurement, general many-copy construction, and robust theorem.