Quantum Bayesian Probability

Quantum Bayesian probability treats a quantum state as encoding an agent’s expectations about outcomes of contemplated interventions. The probabilities are personal assignments, but the Born rule and the geometry of valid quantum states constrain how those assignments fit together. This is not the claim that an observer may choose experimental results.

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

Coherence beyond one probability table

For a qubit, write

$$\rho=\frac12(I+\mathbf r\cdot\boldsymbol\sigma), \qquad |\mathbf r|\le1.$$

The assigned probabilities for the positive outcomes of Pauli measurements are $p_x=(1+r_x)/2$ and similarly for $y,z$. Positivity implies

$$(2p_x-1)^2+(2p_y-1)^2+(2p_z-1)^2\le1.$$

Each probability separately lies between zero and one, but that is insufficient. Assigning $p_x=p_y=p_z=1$ violates the inequality. A coherent classical probability for each separate binary question need not form a physically admissible quantum assignment across incompatible questions.

This example displays the distinctive quantum constraint without choosing a particular interpretation of the Bloch vector. Quantum Bayesianism gives that constraint a normative reading for an agent using the theory.

Updating is not always ordinary conditioning

If a recorded intervention is represented by an instrument $\mathcal I_a$, the posterior state used for subsequent predictions is

$$\rho_a=\frac{\mathcal I_a(\rho)} {\operatorname{Tr}\mathcal I_a(\rho)}.$$

The instrument contains physical information about the intervention; a POVM specifies only its outcome probabilities. Learning an outcome and changing the physical system are therefore distinguishable aspects of a state update. Ordinary Bayes conditioning applies to a fixed joint distribution, whereas incompatible sequential experiments require a model of the implemented transformations.

For example, a projective $Z$ measurement on $\lvert+\rangle$ followed by an $X$ measurement yields equal $X$ probabilities after either $Z$ outcome. Ignoring the physical disturbance and treating the process as merely learning an old $Z$ value would miss that change.

Quantum Bayesianism and QBism

QBism is a particular interpretive program that treats measurement outcomes as an agent’s experiences and the Born rule as a normative relation among personal probabilities. Bayesian statistical methods themselves do not commit one to QBism. One may use Bayesian tomography while regarding quantum states as objective physical objects.

Likewise, a personal probability distribution over unknown preparations is not automatically a $\psi$-epistemic hidden-variable model. The latter introduces an underlying ontic state $\lambda$ with a specific representation of preparations and measurements. QBism need not make that move.

Agreement and empirical accountability

Agents with different information may assign different states. Shared data and shared modeling assumptions can bring their predictions closer, but convergence is conditional: different likelihood models or unsupported priors can prevent agreement.

A probability-one assignment is a maximal commitment within this interpretation; it is not by itself a proof that a value existed before measurement. That distinction blocks a common inference from certainty to an observer-independent property.

Exercises

Take $p_x=p_z=0.9$ and $p_y=0.5$. Can these probabilities arise from any qubit density operator?

Solution

The Bloch vector would be $(0.8,0,0.8)$, whose squared length is $1.28>1$. The associated operator has a negative eigenvalue and is not a density operator. Each binary distribution is normalized, but the three assignments violate quantum compatibility.

References

• C. A. Fuchs and R. Schack, “Quantum-Bayesian coherence,” Reviews of Modern Physics 85, 1693–1715 (2013), doi:10.1103/RevModPhys.85.1693.
• C. A. Fuchs, N. D. Mermin, and R. Schack, “An introduction to QBism with an application to the locality of quantum mechanics,” American Journal of Physics 82, 749–754 (2014), author manuscript.