What Does the Quantum State Represent

A quantum state assigns probabilities to possible measurement outcomes. Whether it also represents a complete observer-independent physical condition is an additional question. Pure versus mixed, complete versus incomplete, and ontic versus epistemic classify different distinctions and should not be used interchangeably.

The operational role is definite

Given a measurement effect $E_a$, the state $\rho$ predicts $p(a)=\operatorname{Tr}(\rho E_a)$. Two density operators that differ can be distinguished statistically with suitable repeated measurements. The density-operator treatment develops this probability calculus; the present question is what licenses the state assignment and what physical meaning it carries.

An assignment can depend on preparation information, a received measurement record, or a subsystem restriction. Updating it after receiving data does not alone tell us whether an underlying physical state changed. To answer that, specify the theory’s ontology and dynamics as well as its probability assignments.

One mixed state, different preparations

For a qubit,

$$I/2=\tfrac12|0\rangle\langle0|+\tfrac12|1\rangle\langle1| =\tfrac12|+\rangle\langle+|+\tfrac12|-\rangle\langle-|.$$

The same operator can describe two deliberately randomized preparations with different retained classical labels. It is also the reduced state of one half of a maximally entangled pair. No measurement on the qubit alone distinguishes these cases. Joint access to a preparation label or the purifying partner can distinguish the larger physical arrangements.

Consequently, writing a mixed state as an ensemble does not identify a unique actual but unknown pure state. Treating it as ordinary ignorance requires an account of the preparation or additional ontology. Calling it “information” is equally incomplete unless one says information about what, available to whom and used for which predictions.

Ontic and epistemic are framework-dependent terms

In an ontological model, a preparation assigns a probability measure over physical states $\lambda$. Distinct pure quantum states are called $\psi$-ontic when their ontic measures do not overlap. They are $\psi$-epistemic when the relevant measures overlap in the specified sense. The model taxonomy defines these notions precisely.

This is not the same distinction as pure versus mixed. An ontic wavefunction can coexist with uncertainty about extra variables. An agent-centered interpretation need not supply the observer-independent ontic model presupposed by the classification. The PBR theorem constrains such models under preparation independence; it does not resolve every ordinary-language question about what a state means.

Completeness and dynamics

A theory that declares the state complete must explain how its physical records are represented. A theory that adds variables must explain how they evolve and reproduce observed statistics. Neither declaration alone solves the measurement problem.

Likewise, a state on a large configuration space need not be interpreted as a field in ordinary three-dimensional space. Ontology proposals must state the relationship between mathematical representation and physical objects. Empirical adequacy constrains that relationship but may not uniquely fix it.

Exercise: local indistinguishability

A source either sends one half of a Bell pair or sends a fair random computational-basis state with a private classical label. Can a receiver distinguish the source from qubit measurements alone? What extra access changes the question?

Solution

Both local states are $I/2$, so every local measurement has identical statistics. Access to the Bell partner permits joint entanglement-sensitive measurements; access to a classical label permits conditional preparation statistics. The identical local density operator does not make the global preparations identical.

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.
• Hughston, Lane P., Richard Jozsa, and William K. Wootters, “A complete classification of quantum ensembles having a given density matrix,” Physics Letters A 183, 14–18 (1993), doi:10.1016/0375-9601(93)90880-9.