Operational Theories

An operational theory specifies the probabilities of experimental outcomes given preparations, transformations, and measurements. Its states summarize what an experiment can distinguish. This definition makes no claim that the operational state is a complete description of reality; that additional question belongs to ontological models.

Experiments and probability assignments

Write a preparation as $P$, a transformation as $T$, and a measurement choice as $M$. The theory supplies

$$p(k\mid P,T,M)\geq0,\qquad \sum_k p(k\mid P,T,M)=1.$$

The labels describe procedures, including relevant conditioning and laboratory boundaries. A detector that sometimes fails still has a no-click outcome. Dropping it and renormalizing changes the experiment into a conditional one.

Two preparations are operationally equivalent when every allowed subsequent test gives the same probabilities. The qualification “allowed” matters: equivalence established by an incomplete collection of measurements may disappear when the measurement set is enlarged. A state is an equivalence class of preparations relative to the theory’s full experimental repertoire.

Measurements also admit equivalence classes. An effect represents an outcome probability functional, while an instrument additionally describes the postmeasurement state. Two detectors can implement the same effect and disturb the system differently.

Randomization produces affine states

Suppose a classical randomizer chooses $P_1$ with probability $q$ and $P_2$ otherwise, and its record is discarded. The law of total probability gives

$$p(k\mid P_{\mathrm{mix}},M) =q\,p(k\mid P_1,M)+(1-q)p(k\mid P_2,M).$$

Consequently states form a convex set and effects act affinely on it. This is the starting point of generalized probabilistic theories, not a specifically quantum postulate.

For a qubit, equiprobable preparations of the two $Z$ eigenstates and equiprobable preparations of the two $X$ eigenstates both give $\rho=I/2$. Every POVM returns the same statistics for these ensembles. Their laboratory preparation histories differ, but those histories cannot be recovered by measuring the qubit alone. Retaining a correlated classical register changes the accessible system and therefore changes this conclusion.

Transformations and ancillary systems

A deterministic transformation maps normalized states to normalized states and respects randomization. In quantum theory it is represented by a completely positive trace-preserving map. Positivity on isolated states is insufficient: the transformation must remain valid when an untouched ancillary system is present.

For example, transposition preserves the eigenvalues of an isolated density matrix. Partial transposition of an entangled pair can have a negative eigenvalue, so it cannot represent an unrestricted local physical operation. The compositional requirement contains physical information beyond the one-system probability table.

An operational description therefore needs states, effects, transformations, and a composition rule. A list of allowed correlations by itself is not yet a complete dynamical theory. Barrett develops this distinction; Spekkens uses operational equivalences to formulate generalized noncontextuality.

Exercises

Explain why equal outcome frequencies for one measurement do not establish operational equivalence.

Solution

The states $\lvert0\rangle$ and $\lvert+\rangle$ give different $Z$ statistics but both can agree on another test, such as $Y$. Equality must hold for every allowed test. In practice, tomography uses an informationally complete family to infer that stronger equality within a specified model.

References

• Barrett, Jonathan, “Information processing in generalized probabilistic theories,” Physical Review A 75, 032304 (2007). Paper.
• Spekkens, Robert W., “Contextuality for preparations, transformations, and unsharp measurements,” Physical Review A 71, 052108 (2005). Paper.