Probability in Operational Theories

An operational theory starts with preparations, transformations, and outcomes whose statistics can be compared. A state represents an equivalence class of preparations that give the same probabilities for every allowed test. Probability is primitive in this description; the theory organizes its structure rather than deriving its meaning from an underlying ontology.

Required background. Born rule: Translate a specified state and measurement into outcome probabilities.

States and effects

Let $\omega$ denote a state and $e$ an effect. Their pairing is a probability $e(\omega)\in[0,1]$. A measurement is a set of effects summing to the unit effect $u$:

$$\sum_a e_a=u,\qquad u(\omega)=1.$$

Classical random mixing requires affine dependence,

$$e(t\omega_1+(1-t)\omega_2) =te(\omega_1)+(1-t)e(\omega_2).$$

This follows operationally when a classical randomizer selects the preparation and its selection is ignored. It is an assumption about how randomized procedures are represented.

Quantum theory realizes the pairing as $e_E(\rho)=\operatorname{Tr}(\rho E)$. A classical finite system instead has a probability vector $\mathbf p$ and effects represented by vectors with entries between zero and one.

Geometry constrains decompositions

A classical bit has a line-segment state space. A qubit has a Bloch ball. The center of the ball has many decompositions:

$$\frac I2 =\frac12|0\rangle\langle0|+\frac12|1\rangle\langle1| =\frac12|+\rangle\langle+|+\frac12|-\rangle\langle-|.$$

All single-system effects give the same probabilities on these preparations once their classical labels are forgotten. Operational equivalence does not prove their underlying ontic distributions are identical; imposing that identification is a preparation-noncontextuality assumption.

The operational theories and generalized probabilistic theories pages develop these structures further.

Composition is an extra ingredient

A single-system state space does not determine which bipartite states and measurements are allowed. Composite-system rules govern entanglement, local tomography, and nonlocal correlations. Even theories with well-behaved local probabilities can differ substantially in their joint correlations.

No-signaling requires that summing over one party’s outcomes makes the other party’s marginal independent of the distant measurement choice. This is weaker than specifying the quantum composite. PR-box correlations show why merely imposing normalized local probabilities and no-signaling does not recover the quantum CHSH bound.

What a reconstruction explains

A reconstruction identifies axioms under which operational state spaces and transformations take the quantum form. It may thereby recover the Born pairing as part of a representation theorem. Probability remains part of the input language; the reconstruction does not decide whether its numbers should be interpreted as propensities or personal expectations.

This limitation is useful. Operational predictions can be compared while ontological questions remain open, and competing nonquantum theories can be expressed in a common mathematical framework.

Exercises

Show that two different ensemble decompositions of the same density operator are indistinguishable by every POVM on that system alone.

Solution

If $\rho=\sum_i p_i\rho_i=\sum_jq_j\sigma_j$, affine dependence gives $\sum_ip_i\operatorname{Tr}(E_a\rho_i)=\operatorname{Tr}(E_a\rho)=\sum_jq_j\operatorname{Tr}(E_a\sigma_j)$ for every effect. An accessible record of the ensemble label changes the experiment by enlarging the system; it is excluded by the phrase “system alone.”

References

• J. Barrett, “Information processing in generalized probabilistic theories,” Physical Review A 75, 032304 (2007), doi:10.1103/PhysRevA.75.032304.
• G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Informational derivation of quantum theory,” Physical Review A 84, 012311 (2011), doi:10.1103/PhysRevA.84.012311.