Compatibility of any pair of 2-outcome measurements characterizes the Choquet simplex
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Compatibility of any pair of 2-outcome measurements characterizes the Choquet simplex Yui Kuramochi1 Received: 1 December 2019 / Accepted: 11 February 2020 © Springer Nature Switzerland AG 2020
Abstract For a compact convex subset K of a locally convex Hausdorff space, a measurement on A(K ) is a finite family of positive elements in A(K ) normalized to the unit constant 1 K , where A(K ) denotes the set of continuous real affine functionals on K . It is proved that a compact convex set K is a Choquet simplex if and only if any pair of 2-outcome measurements are compatible, i.e. the measurements are given as the marginals of a single measurement. This generalizes the finite-dimensional result of Plávala (Phys Rev A 94:042108, 2016) obtained in the context of the foundations of quantum theory. Keywords Choquet simplex · Bauer simplex · General probabilistic theory · Compatibility of measurements Mathematics Subject Classification 46A55 · 46B40 · 81P16 · 81P15
1 Introduction and main results Let K be a compact convex subset of a locally convex Hausforff space over the reals R and let A(K ) denote the set of continuous real affine functionals on K , where A(K ) is a Banach space under the supremum norm f := supx∈K | f (x)| ( f ∈ A(K )). (Throughout the paper all the linear spaces are over R.) A measurement on m A(K ) is a finite sequence ( f k )m k=1 ∈ A(K ) for some integer m such that f k ≥ 0 f = 1 , where 1 (1 ≤ k ≤ m) and m K K ≡ 1 is the unit constant functional. A k=1 k measurement belonging to A(K )m is called m-outcome. Two measurements ( f k )m k=1
This work was supported by Cross-ministerial Strategic Innovation Promotion Program (SIP) (Council for Science, Technology and Innovation (CSTI)).
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Yui Kuramochi [email protected] Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
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Y. Kuramochi
and (g j )nj=1 on A(K ) are called compatible (or jointly measurable) if there exists a family (h k, j )1≤k≤m,1≤ j≤n in A(K ) such that h k, j ≥ 0,
fk =
n j =1
h k, j , g j =
m
h k , j (1 ≤ k ≤ m, 1 ≤ j ≤ n).
k =1
In [5] Plávala showed that for finite-dimensional K , K is a simplex if and only if any pair of 2-outcome measurements are compatible. In the physical context, a compact convex set K corresponds to a state space of a general physical system and Plávala’s result indicates that incompatibility of measurements [3] characterizes non-classicality of a physical system. The proof in [5] depends on the notion of the maximal face and is not straightforwardly applicable to infinite-dimensional compact sets. The purpose of this paper is to generalize this result to an arbitrary compact convex set K (Theorems 1 and 2). The first main result of this paper is the following theorem. Theorem 1 Let K be a compact convex subset of a locally convex Hausdorff space. Then K is a Choquet simplex if and only if any pair ( f k )2k=1 and (g j )2j=1 of 2-outcome measurements on A(K ) are compatible. Here
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