Gleason-Type Theorem for Projective Measurements, Including Qubits: The Born Rule Beyond Quantum Physics

Abstract

Born's quantum probability rule is traditionally included among the quantum postulates as being given by the squared amplitude projection of a measured state over a prepared state, or else as a trace formula for density operators. Both Gleason's theorem and Busch's theorem derive the quantum probability rule starting from very general assumptions about probability measures. Remarkably, Gleason's theorem holds only under the physically unsound restriction that the dimension of the underlying Hilbert space H must be larger than two. Busch's theorem lifted this restriction, thereby including qubits in its domain of validity. However, while Gleason assumed that observables are given by complete sets of orthogonal projectors, Busch made the mathematically stronger assumption that observables are given by positive operator-valued measures. The theorem we present here applies, similarly to the quantum postulate, without restricting the dimension of H and for observables given by complete sets of orthogonal projectors. We also show that the Born rule applies beyond the quantum domain, thereby exhibiting the common root shared by some quantum and classical phenomena.

Beyond Bell’s theorem: realism and locality without Bell-type correlations

“…The 2 D case was afterwards covered by Busch’s approach 31 , but at the cost of conflicting Kolmogorov’s axiomatic. A third attempt was presented recently 32 – 34 , in which the 2 D case is covered without conflicting Kolmogorov’s axiomatic. Moreover, this axiomatic appears as a special case of what can be seen as the most basic physical concept, namely the concept of a signed measure.…”

Gudder’s Theorem and the Born Rule

“…The reason can be stated very simply and in advance: the assumptions underlying our approach imply that any function we deal with is a linear one. This was not explicitly shown in [ 11 ], but only implicitly, by deriving Born’s linear expression. We present here an explicit demonstration of linearity, and moreover, go beyond the goals of our previous work.…”

“…Recently, we have presented an alternative derivation of the Born rule [ 11 ], starting from Gudder’s theorem [ 12 ]—a theorem which is in a sense the reciprocal of Pythagoras’s theorem. Such a derivation begins with two-dimensional systems and then extends to higher-dimensional ones, including both pure and mixed states.…”

“…Indeed, Hall’s criticisms represent a welcome opportunity to expand the scope of Ref. [ 11 ], as well as to clear up the physical content of the proposed extension of Gleason’s theorem.…”

“…In Section 2 we recall Gleason’s theorem and in Section 3 we reproduce—for the sake of completeness—the essential points of Ref. [ 11 ]. At the same time, we extend somewhat the results presented in Ref.…”