Quantum-Bayesian coherence

In the quantum-Bayesian interpretation of quantum theory (or QBism), the Born rule cannot be interpreted as a rule for setting measurement-outcome probabilities from an objective quantum state. But if not, what is the role of the rule? In this paper, the argument is given that it should be seen as an empirical addition to Bayesian reasoning itself. Particularly, it is shown how to view the Born rule as a normative rule in addition to usual Dutch-book coherence. It is a rule that takes into account how one should assign probabilities to the consequences of various intended measurements on a physical system, but explicitly in terms of prior probabilities for and conditional probabilities consequent upon the imagined outcomes of a special counterfactual reference measurement. This interpretation is exemplified by representing quantum states in terms of probabilities for the outcomes of a fixed, fiducial symmetric informationally complete measurement. The extent to which the general form of the new normative rule implies the full state-space structure of quantum mechanics is explored.

Figure 1

The planar surface represents the convex set of all probability distributions over $d^2$ outcomes—the probability simplex ${\Delta }_{d^2}$. With respect to any fiducial POVM, the probability distributions valid for representing the set of quantum states, however, form a smaller convex set within the simplex, here depicted as an ellipsoid. In actual fact, however, the convex shape is quite complex. The choice of a SIC for defining the mapping makes the shape as simple as it can be with respect to the natural geometry of the simplex.

Figure 2

The basic conceptual apparatus of this review. The measurement on the ground, with outcomes $j=1,\dots ,m$, is some potential measurement that could be performed in the laboratory, i.e., one that could be factualized. The measurement in the sky, on the other hand, with outcomes $i=1,\dots ,n$, is a fixed measurement one can contemplate independently. The probability distributions $p(i)$ and $r(j|i)$ represent how an agent would gamble if a conditional lottery based on the measurement in the sky were operative. The probability distribution $q(j)$ represents instead how the agent would gamble on outcomes of the ground measurement if the measurement in the sky and the associated conditional lottery were nullified, i.e., they were to never take place at all. In the quantum case, the measurement in the sky is a SIC with $n=d^2$ outcomes; the measurement on the ground is any POVM. In pure Bayesian reasoning, there is no necessity that $q(j)$ be related to $p(i)$ and $r(j|i)$ at all. In quantum mechanics, however, there is the specific relation $q(j)=(d+1)\sum _{i=1}^{d^2}p(i)r(j|i)-(1/d)\sum _{i=1}^{d^2}r(j|i)$. This equation contains the sum content of the Born rule to which it is equivalent.

Figure 3

The Paulian idea (57)—in the form of a figure inspired by John Archibald Wheeler (101). In contemplating a quantum measurement, one makes a conceptual split in the world: one part is treated as an agent, and the other as a kind of reagent or catalyst (one that brings about change in the agent). In older terms, the former is an observer and the latter a quantum system of some finite dimension $d$. [For arguments in support of finite fundamental theories see, e.g., 99, 135, and 25.] A quantum measurement consists first of the agent taking an action on the quantum system. The action is formally captured by some POVM $\{E_i\}$. The action leads generally to an incompletely predictable consequence, a particular personal experience $E_i$ for the agent (56). The quantum state $|\psi ⟩$ makes no appearance but in the agent’s head; for it only captures his degrees of belief concerning the consequences of his actions, and (in contrast to the quantum system itself) has no existence in the external world. Measurement devices are depicted as prosthetic hands to make it clear that they should be considered an integral part of the agent. (This contrasts with Bohr’s view where the measurement device is always treated as a classically describable system external to the observer.) The sparks between the measurement-device hand and the quantum system represent the idea that the consequence of each quantum measurement is a unique creation within the previously existing universe (56). Wolfgang Pauli characterized this picture as a “wider form of the reality concept” than that of Einstein’s, which he labeled “the ideal of the detached observer” (84; 102; 67). The particular character of the catalysts (James’s “materials of your universe’s composition”) leaves its trace in the formal rules that allow us to conceptualize factualizable measurements in terms of a standard counterfactual one.