Gleason's theorem

Gleason's theorem (named after Andrew M. Gleason) is a mathematical result which shows that the rule one uses to calculate probabilities in quantum physics follows logically from particular assumptions about how measurements are represented mathematically. More specifically, it proves that the Born rule for the probability of obtaining specific results for a given measurement follows naturally from the structure formed by the lattice of events in a real or complex Hilbert space. This result is of particular importance for the field of quantum logic. Furthermore, it was historically significant for the role it played in showing that local hidden variable theories are inconsistent with quantum physics. The theorem states:

Theorem. Suppose H is a separable Hilbert space. A measure on H is a function f that assigns a nonnegative real number to each closed subspace of H in such a way that, if

{\textstyle \{A_{i}\}}

is a countable collection of mutually orthogonal subspaces of H, and the closed linear span of this collection is B, then

{\textstyle f(B)=\sum _{i}f(A_{i})}

. If the Hilbert space H has dimension at least three, then every measure f can be written in the form

{\textstyle f(A)=\mathrm {Tr} (WP_{A})}

, where W is a positive semidefinite trace class operator and

{\textstyle P_{A}}

is the orthogonal projection onto A.

The trace-class operator W can be interpreted as the density matrix of a quantum state. Effectively, the theorem says that any legitimate probability measure on the space of measurement outcomes is generated by some quantum state.

Overview

Consider a quantum system with a Hilbert space of dimension 3 or larger, and suppose that there exists some function that assigns a probability to each outcome of any possible measurement upon that system. The probability of any such outcome must be a real number between 0 and 1 inclusive, and in order to be consistent, for any individual measurement the probabilities of the different possible outcomes must add up to 1. Gleason's theorem shows that any such function—that is, any consistent assignment of probabilities to measurement outcomes—must be expressible in terms of a quantum-mechanical density operator and the Born rule. In other words, given that each quantum system is associated with a Hilbert space, and given that measurements are described by particular mathematical entities defined on that Hilbert space, both the structure of quantum state space and the rule for calculating probabilities from a quantum state then follow.

For simplicity, we can assume that the dimension of the Hilbert space is finite. A quantum-mechanical observable is a self-adjoint operator on that Hilbert space. Equivalently, we can say that a measurement is defined by an orthonormal basis, with each possible outcome of that measurement corresponding to one of the vectors comprising the basis. A density operator is a positive-semidefinite operator whose trace is equal to 1. In the language of von Weizsäcker, a density operator is a "catalogue of probabilities": for each measurement that can be defined, we can compute the probability distribution over the outcomes of that measurement from the density operator.^[1] We do so by applying the Born rule, which states that

P(x_{i})=\mathrm {Tr} (\Pi _{i}W),

where ${\textstyle W}$ is the density operator and ${\textstyle \Pi _{i}}$ is the projection operator onto the basis vector associated with the measurement outcome ${\textstyle x_{i}}$ .

Let ${\textstyle f}$ be a function from projection operators to the unit interval with the property that, if a set ${\textstyle \{\Pi _{i}\}}$ of projection operators sum to the identity matrix—that is, if they correspond to an orthonormal basis—then

\sum _{i}f(\Pi _{i})=1.

Such a function expresses an assignment of probability values to the outcomes of measurements, an assignment that is "noncontextual" in the sense that the probability for an outcome does not depend upon which measurement that outcome is embedded within, but only upon the mathematical representation of that specific outcome, i.e., its projection operator.^[2] Gleason's theorem states that for any such function ${\textstyle f}$ , there exists a positive semidefinite operator with unit trace ${\textstyle W}$ such that

f(\Pi _{i})=\mathrm {Tr} (\Pi _{i}W).

Both the Born rule and the fact that "catalogues of probability" are positive semidefinite operators of unit trace follow from the assumptions that measurements are represented by orthonormal bases, and that probability assignments are "noncontextual". The theorem presumes that the underlying set of numbers that the Hilbert space is defined over are real numbers or complex numbers.

Another way of phrasing the theorem uses the terminology of quantum logic, which makes heavy use of lattice theory. Quantum logic treats quantum events (or measurement outcomes) as logical propositions, and studies the relationships and structures formed by these events, with specific emphasis on quantum measurement. In quantum logic, the logical propositions that describe events are organized into a lattice in which the distributive law, valid in classical logic, is weakened, to reflect the fact that in quantum physics, not all pairs of quantities can be measured simultaneously. The representation theorem in quantum logic shows that such a lattice is isomorphic to the lattice of subspaces of a vector space with a scalar product.^[3] It remains an open problem in quantum logic to constrain the field K over which the vector space is defined. Solèr's theorem implies that, granting certain hypotheses, the field K must be either the real numbers, complex numbers, or the quaternions.^[4] In order for Gleason's theorem to be applicable, the vector space must be a real or complex Hilbert space. (Gleason's argument is inapplicable if, for example, one tries to construct an analogue of quantum mechanics using p-adic numbers.)

We let A represent an observable with finitely many potential outcomes: the eigenvalues of the Hermitian operator A, i.e. $\alpha _{1},\alpha _{2},\alpha _{3},\ldots ,\alpha _{n}$ . An "event", then, is a proposition $x_{i}$ , which in natural language can be rendered "the outcome of measuring A on the system is $\alpha _{i}$ ". Let H denote the Hilbert space associated with the physical system, and let L denote the lattice of subspaces of H. The events $x_{i}$ generate a sublattice of L which is a finite Boolean algebra, and if n is the dimension of the Hilbert space, then each event is an atom of the lattice L.

A quantum probability function over H is a real function P on the atoms in L that has the following properties:

$P(0)=0$ , and $P(y)\geq 0$ for all $y \in L$
$\sum _{{j=1}}^{n}P(x_{j})=1$ , if $x_{1},x_{2},x_{3},\ldots ,x_{n}$ are orthogonal atoms

This means for every lattice element y, the probability of obtaining y as a measurement outcome is known, since it may be expressed as the union of the atoms under y:

P(y)=\sum \{P(x_{j})\mid x_{j}\leq y\}.

In this context, Gleason's theorem states:

Given a quantum probability function P over a space of dimension $\geq 3$ , there is an Hermitian, non-negative operator W on H, whose trace is unity, such that $P(x)=\langle {\mathbf {x}},W{\mathbf {x}}\rangle$ for all atoms $x\in L$ , where $\langle \,,\,\rangle$ is the inner product, and $\mathbf {x}$ is a unit vector along $x$ .

As one consequence: if some $x_{0}$ satisfies $P(x_{0})=1$ , then W is the projection onto the complex line spanned by $x_{0}$ and $P(x)=\left|\langle {\mathbf {x_{0}}},{\mathbf {x}}\rangle \right|^{2}$ for all $x\in L$ .

Implications

Gleason's theorem highlights a number of fundamental issues in quantum measurement theory. Fuchs argues that the theorem "is an extremely powerful result," because "it indicates the extent to which the Born probability rule and even the state-space structure of density operators are dependent upon the theory's other postulates." As a consequence, quantum theory is "a tighter package than one might have first thought."^[5] The fact that the logical structure of quantum events dictates the probability measure of the formalism is taken by some to demonstrate an inherent stochasticity in the very fabric of the world. To some researchers, such as Pitowsky, the result is convincing enough to conclude that quantum mechanics represents a new theory of probability. Alternatively, such approaches as relational quantum mechanics make use of Gleason's theorem as an essential step in deriving the quantum formalism from information-theoretic postulates.^[6]

The theorem is often taken to rule out the possibility of hidden variables in quantum mechanics. This is because the theorem implies that there can be no bivalent probability measures, i.e. probability measures having only the values 1 and 0. To see this, note that the mapping $u\rightarrow \langle Wu,u\rangle$ is continuous on the unit sphere of the Hilbert space for any density operator W. Since this unit sphere is connected, no continuous function on it can take only the values of 0 and 1.^[7] But, a hidden variables theory which is deterministic implies that the probability of a given outcome is always either 0 or 1: either the electron's spin is up, or it isn't (which accords with classical intuitions). Gleason's theorem therefore seems to hint that quantum theory represents a deep and fundamental departure from the classical way of looking at the world. (This has been argued to support a variety of philosophical perspectivism.^[8])

Gleason's theorem motivated later work by John Stuart Bell, Ernst Specker and Simon Kochen that led to the result often called the Kochen–Specker theorem, which rules out a broad class of hidden-variable models. As noted above, Gleason's theorem shows that there is no bivalent probability measure over the rays of a Hilbert space (as long as the dimension of that space exceeds 2). The Kochen–Specker theorem refines this statement by constructing a specific finite subset of rays on which no bivalent probability measure can be defined.^[9]

Outline of Gleason's proof

Gleason's original proof proceeds in three stages.^[10] In Gleason's terminology, a frame function that is derived in the standard way—i.e., by the Born rule from a quantum state—is regular. Gleason derives a sequence of lemmas concerning when a frame function is necessarily regular, culminating in the final theorem. First, he establishes that every frame function on the Hilbert space $\mathbb {R} ^{3}$ is continuous. Then, he proves the theorem for the special case of $\mathbb {R} ^{3}$ . Finally, he shows that the general problem can be reduced to this special case.

Generalizations

Gleason originally proved the theorem assuming that the measurements applied to the system are of the von Neumann type, i.e., that each possible measurement corresponds to an orthonormal basis of the Hilbert space. Later, Busch, and independently Caves et al., proved an analogous result for a more general class of measurements, known as positive operator valued measures (POVMs). The proof of this result is simpler than that of Gleason's, and unlike the original theorem of Gleason, the generalized version using POVMs also applies to the case of a single qubit, for which the dimension of the Hilbert space equals 2.^[11] This has been interpreted as showing that the probabilities for outcomes of measurements upon a single qubit cannot be explained in terms of hidden variables, provided that the class of allowed measurements is sufficiently broad.^[12]

Gleason's theorem, in its original version, does not hold if the Hilbert space is defined over the rational numbers, i.e., if the components of vectors in the Hilbert space are restricted to be rational numbers, or complex numbers with rational parts. However, when the set of allowed measurements is the set of all POVMs, the theorem holds.^[13]

The original proof by Gleason was not constructive: one of the ideas on which it depends is the fact that every continuous function defined on a compact space obtains its minimum. Because one cannot in all cases explicitly show where the minimum occurs, a proof that relies upon this principle will not be a constructive proof. However, the theorem can be reformulated in such a way that a constructive proof can be found.^[14]

Gleason's theorem can be extended to some cases where the observables of the theory form a von Neumann algebra. Specifically, an analogue of Gleason's result can be shown to hold if the algebra of observables has no direct summand that is representable as the algebra of two-by-two matrices over a commutative von Neumann algebra (i.e., no direct summand of type I₂). In essence, the only barrier to proving the theorem is the fact that Gleason's original result does not hold when the Hilbert space is that of a qubit.^[15]

References

↑ Dreischner, Görnitz and von Weizsäcker (1988)
↑ Barnum et al. (2000); Pitowsky (2003), §1.3; Pitowsky (2006), §2.1; Kunjwal and Spekkens (2015)
↑ Pitowsky (2006), §2
↑ Baez (2010)
↑ Fuchs (2011), pp. 94–95
↑ Wilce (2017), §1.4
↑ Wilce (2017), §1.3
↑ Edwards (1979)
↑ Peres (1991); Mermin (1993)
↑ Hrushovski and Pitowsky (2004), §2
↑ Busch (2003); Caves et al. (2004); Fuchs (2011), p. 116
↑ Spekkens (2005)
↑ Caves et al. (2004), §3.D
↑ Richman and Bridges (1999); Hrushovski and Pitowsky (2004)
↑ Hamhalter (2003)

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