A universal property for sequential measurement

The following full text is a preprint version which may differ from the publisher's version. We study the sequential product GN01,GG02,GG05 , the operation p * q = √ pq √ p on the set of effects, [0, 1] A , of a von Neumann algebra A that represents sequential measurement of first p and then q. In GL08 Gudder andLatémolì ere give a list of axioms based on physical grounds that completely determines the sequential product on a von Neumann algebra of type I, that is, a von Neumann algebra B(H) of all bounded operators on some Hilbert space H. In this paper we give a list of axioms that completely determines the sequential product on all von Neumann algebras simultaneously, see Thm 4. These axioms may be formulated in purely categorical terms (although we do not pursue this here, see also Remark 12). In this way this paper contributes to the larger program Jac15,CJWW15b,CJWW15a to identify structure in the category of von Neumann algebras with completely positive normal linear contractions to interpret the constructs in a programming language designed for a quantum computer: with the sequential product one can interpret measurement. Our axioms for the sequential product are based on the following observations. Given a von Neumann algebra A and p ∈ [0, 1] A the expression √ pa √ p makes sense for all a ∈ A (and not only for a ∈ [0, 1] A). The resulting map asrt p : A → A (so asrt p (a) = √ pa √ p) factors as A π : a →→pap / / pA p c : a → √ pa √ p / / A , where p is the least projection above p. (Roughly speaking, the von Neumann algebra pA p represents the subtype of A in which the predicate p holds. The map c is simply the restriction of asrt p to pA p, while π is the map which forgets that p holds. The map c is a more sharply typed version of sequential product than asrt p — much in the same way that the absolute value on the reals is more sharply described as a map R → [0, ∞) than as a map R → R.) The maps c and π have a universal property: c is a compression of p and π is a corner of p (see Definition …