Abstract
Gleason-type theorems for quantum theory allow one to recover the quantum state space by assuming that (i) states consistently assign probabilities to measurement outcomes and that (ii) there is a unique state for every such assignment. We identify the class of general probabilistic theories which also admit Gleason-type theorems. It contains theories satisfying the no-restriction hypothesis as well as others which can simulate such an unrestricted theory arbitrarily well when allowing for post-selection on measurement outcomes. Our result also implies that the standard no-restriction hypothesis applied to effects is not equivalent to the dual no-restriction hypothesis applied to states which is found to be less restrictive.
Highlights
More than sixty years ago, Mackey [1] asked whether the density operator represents the most general notion of a quantum state that is consistent with the standard description of observables as self-adjoint operators
We will prove this result in two steps: (i) in Lemma 6, a frame function on a general probabilistic theories (GPTs) effect space E is found to correspond to a vector in the set W (E) defined in Section 3.1; (ii) the set W (E) is found to correspond to the state space of a GPT system if and only if the GPT is in the class of almost noisy unrestricted (aNU) GPTs, in Lemmata 7 and 8
From a conceptual point of view, the results of this paper imply that each general probabilistic theory belongs to one of two distinct classes: either it admits, like quantum theory, a Gleason-type theorem which allows us to construct the set of the possible states of the theory, or it does not admit a GTT
Summary
More than sixty years ago, Mackey [1] asked whether the density operator represents the most general notion of a quantum state that is consistent with the standard description of observables as self-adjoint operators. Gleason and Busch establish a bijection between frame functions and density operators in quantum theory. The rationale behind a frame function is that the probabilities of all measurement outcomes for all observables should define a unique state. If this were not the case, two “different” states would be indistinguishable, both practically and theoretically. The existence of a GTT for GPTs such as quantum theory or real-vector-space quantum theory [14,15,16,17] has a number of consequences It becomes possible, for example, to modify the axiomatic structure of the theories as it is no longer necessary to—separately and independently—stipulate both the state space and the observables of the theory.
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