A universal property for sequential measurement

Abstract

We study the sequential product, the operation $p * q = \sqrt{p} q \sqrt{p}$ on the set of effects of a von Neumann algebra that represents sequential measurement of first $p$ and then $q$. We give four axioms which completely determine the sequential product

Sequential product spaces are Jordan algebras

“…It should be noted that the standard sequential product a & b = √ ab √ a on B(H) sa is not fully characterised by these axioms, as there are multiple binary operations that satisfy these axioms [27]. It is possible however to characterise the standard sequential product using related sets of axioms [11,28,25]. It has been established in the authors previous work [25] that Euclidean Jordan algebras allow a binary operation satisfying these properties and hence are examples of sequential product spaces.…”

Three characterisations of the sequential product

“…In particular, all the proofs rely on showing that some order isomorphism must square to the identity and then showing that these isomorphisms must already have been a square. This same proof step also occurs in the characterisations of [9] and [20] and is the reason the authors of these papers include the axiom that a • (a • b) = (a • a) • b. It might be interesting to see whether a characterisation of the sequential product can be found that does not include a variation on this axiom.…”

“…It was shown in [18] that the properties required of an abstract sequential product as defined by Gudder and Greechie are not enough to characterise the standard sequential product a • b = √ ab √ a. There do however exist characterisations based on a related set of axioms [9,20]. We will add to these efforts by giving characterisations based on a variety of different ideas, including an order-theoretic property, symmetry with respect to an inner product, and an algebraic property.…”

Pure Maps between Euclidean Jordan Algebras

“…We completely characterize the pure -positive maps and show that they exactly correspond to a generalization of the sequential product maps b → √ ab √ a in von Neumann algebras, (and we'll deduce from this that pure -positivity and †-positivity coincide.) This result can be seen as a characterization of the sequential product like the ones given in [15,32,25].…”

“…With the preliminaries out of the way we will start to look at additional structure that is present in the category EJA psu . The proofs in this section are heavily inspired by [25,27] where the existence of this structure was shown for the category of von Neumann algebras. As stated in the introduction, our notion of purity is based on filters and corners.…”