This theorem concerns the interpretational status of pure quantum states. Under the classification of hidden variable models of Nicholas Harrigan and
Robert Spekkens, the interpretation of the quantum wavefunction |\psi\rangle can be categorized as either
ψ-ontic if "every complete physical state or ontic state in the theory is consistent with only one pure quantum state" or
ψ-epistemic if "there exist ontic states that are consistent with more than one pure quantum state." The PBR theorem proves that either the quantum state |\psi\rangle is
ψ-ontic, or else non-
entangled quantum states violate the assumption of preparation independence, which would entail
action at a distance. More specifically, the theorem applies to models that treat quantum states as probability distributions over hidden variables, or ontic states. In such a model, writing the space of ontic states as \Lambda, a quantum state \psi is a probability distribution p_\psi(\lambda) defined on the set \Lambda. An observable A is represented as a set of response functions, or conditional probability densities: A(S,\lambda) is the probability that the measurement A has the outcome S if the ontic state of the system being measured is \lambda. In order to reproduce the predictions of quantum mechanics, the probability of obtaining an outcome S given a state \psi as calculated by the
Born rule must satisfy P(S,\psi) = \int_\Lambda A(S,\lambda) p_\psi(\lambda) \, d\lambda. The theorem concludes that if two quantum states are distinct, they must correspond to probability distributions that do not overlap. The PBR theorem employs the concept of an "antidistinguishable" set of quantum states. A finite set of quantum states \{\rho_i:i=1,\ldots,N\}, written as
density matrices to include the possibility of mixed states, is antidistinguishable if there exists a generalized measurement (a
POVM) such that, for each value of i, some outcome of the POVM is assigned probability zero by the state \rho_i. In other words, for an antidistinguishable set of density matrices, there exists a POVM \{E_i\} such that \mathrm{tr}(E_i \rho_i) = 0 for all i = 1,\ldots,N. This concept was introduced by
Carlton M. Caves,
Christopher A. Fuchs and Rüdiger Schack under the name "post-Peierls incompatibility", as it generalizes a condition proposed by
Rudolf Peierls. An antidistinguishable, or post-Peierls incompatible, set is also sometimes termed a set that allows "conclusive exclusion". ==See also==