“It from Bit” and the Quantum Probability Rule

Abstract

I argue that, on the subjective Bayesian interpretation of probability, “it from bit” requires a generalization of probability theory. This does not get us all the way to the quantum probability rule because an extra constraint, known as noncontextuality, is required. I outline the prospects for a derivation of noncontextuality within this approach and argue that it requires a realist approach to physics, or “bit from it”. I then explain why this does not conflict with “it from bit”. This version of the essay includes an addendum responding to the open discussion that occurred on the FQXi website. It is otherwise identical to the version submitted to the contest.

Technical Endnotes

In general, a betting context $\mathcal {B}$ is a Boolean algebra, which we take to be finite for simplicity. All such algebras are isomorphic to the algebra generated by the subsets of some finite set $\Omega _{\mathcal {B}}$, where AND is represented by set intersection, OR by union, and NOT by complement.

In quantum theory, a betting context corresponds to a set of measurements that can be performed together that is as large as possible. A measurement is represented by a self-adjoint operator $M$ and all such operators have a spectral decomposition of the form

$$\begin{aligned} M = \sum _j \lambda _j \Pi _j, \end{aligned}$$

(2.3)

with eigenvalues $\lambda _j$ and orthogonal projection operators $\Pi _j$ that sum to the identity $\sum _j \Pi _j = I$. The eigenvalues are the possible measurement outcomes and, when the system is assigned the density operator $\rho$, the Born rule states that the outcome $\lambda _j$ is obtained with probability

$$\begin{aligned} p(\lambda _j) = \text {Tr} \left( \Pi _j \rho \right) . \end{aligned}$$

(2.4)

The eigenvalues just represent an arbitrary labelling of the measurement outcomes, so a measurement can alternatively be represented by a set of orthogonal projection operators $\{\Pi _j\}$ that sum to the identity $\sum _j \Pi _j = I$, which is sometimes known as a Projection Valued Measure (PVM).¹²

Two PVMs $A = \{\Pi _j\}$ and $B = \{\Pi '_j\}$ can be measured together if and only if each of the projectors commute, i.e. $\Pi _j \Pi '_k = \Pi '_k \Pi _j$ for all $j$ and $k$. If this is the case then $\Pi _j\Pi '_k$ is also a projector and $\sum _{jk}\Pi _j\Pi _k = I$. Therefore, one way of performing the joint measurement is to measure the PVM $C = \{\Pi ''_{jk}\}$ with projectors $\Pi ''_{jk} = \Pi _j\Pi '_k$ and, upon obtaining the outcome $(jk)$, report the outcome $j$ for $A$ and $k$ for $B$. This fine graining procedure can be iterated by adding further commuting PVMs and forming the product of their elements with those of $C$. The procedure terminates when the resulting PVM is as fine grained as possible and this will happen when it consists of rank-$1$ projectors onto the elements of an orthonormal basis. The outcome of any other commuting PVM is determined by coarse graining the projectors onto the orthonormal basis elements.

Therefore, in quantum theory, we can take the sets $\Omega _{\mathcal {B}}$ that generate the betting contexts $\mathcal {B}$ to consist of the elements of orthonormal bases. An event $E \in \mathcal {B}$ is then a subset of the basis elements and corresponds to a projection operator $\Pi _E = \sum _{\vert \psi \rangle \in E} \vert \psi \rangle \langle \psi \vert$. The Boolean operations on $\mathcal {B}$ can be represented in terms of these projectors as

• Conjunction: $G = E \, {\text {AND}} \, F \,\, \Rightarrow \,\, \Pi _G = \Pi _E\Pi _F$.

• Disjunction: $G = E \, {\text {OR}} \, F \,\, \Rightarrow \,\, \Pi _G = \Pi _G + \Pi _F - \Pi _G\Pi _F$, which reduces to $\Pi _G = \Pi _G + \Pi _F$ when $E \cap F = \emptyset$.

• Negation: $G = {\text {NOT}} \, E \,\, \Rightarrow \,\, \Pi _G = I - \Pi _E$.

From the Dutch book argument applied within a betting context, we have that our degrees of belief should be represented by a set of probability measures $p(E|\mathcal {B})$ satisfying

• For any event $E \subseteq \Omega _{\mathcal {B}}$, $p(E|\mathcal {B}) \ge 0$.

• For the certain events $\Omega _{\mathcal {B}}$, $p(\Omega _{\mathcal {B}}|\mathcal {B}) = 1$.

• For disjoint events within the same betting context $E,F \subseteq \Omega _{\mathcal {B}}$, $E \cap F = \emptyset$, $p(E \cup F|\mathcal {B}) = p(E|\mathcal {B}) + p(F|\mathcal {B})$.

The Born rule is an example of such an assignment, and in this language it takes the form

$$\begin{aligned} p(E|\mathcal {B}) = \text {Tr} \left( \Pi _E \rho \right) . \end{aligned}$$

(2.5)

The Born rule also has the property that the probability only depends on the projector associated with an event, and not on the betting context that it occurs in. For example, in a three dimensional Hilbert space, consider the betting contexts $\Omega _{\mathcal {B}} = \{\vert 0 \rangle ,\vert 1 \rangle ,\vert 2 \rangle \}$ and $\Omega _{\mathcal {B}'} = \{\vert + \rangle ,\vert - \rangle ,\vert 2 \rangle \}$, where $\vert \pm \rangle = \frac{1}{\sqrt{2}} \left( \vert 0 \rangle \pm \vert 1 \rangle \right)$. The Born rule implies that $p(\{\vert 2 \rangle \}|\mathcal {B}) = p(\{\vert 2 \rangle \}|\mathcal {B}')$ and also that $p(\{\vert 0 \rangle ,\vert 1 \rangle \}|\mathcal {B}) = p(\{\vert + \rangle ,\vert - \rangle \}|\mathcal {B}')$ because, in each case, the events correspond to the same projectors. The Dutch book argument alone does not imply this because it does not impose any constraints across different betting contexts.

A probability assignment is called noncontextual if $p(E|\mathcal {B}) = p(F|\mathcal {B}')$ whenever $\Pi _E = \Pi _F$. Gleason’s theorem [34] says that, in Hilbert spaces of dimension $3$ or larger, noncontextual probability assignments are exactly those for which there exists a density operator $\rho$ such that $p(E|\mathcal {B}) = \text {Tr} \left( \Pi _E \rho \right)$, i.e. they must take the form of the Born rule. Therefore, the Born rule follows from the conjunction of the Dutch book constraints and noncontextuality, at least in Hilbert spaces of dimension $3$ or greater.