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Mathematica Slovaca

Remarks on the order for quantum observables

Sylvia Pulmannová, Elena Vinceková

Published Nov 30, 2007• 28 citations
Publisher

Abstract

Abstract Relations between generalized effect algebras and the sets of classical and quantum observables endowed with an ordering recently introduced in [GUDDER, S.: An order for quantum observables, Math. Slovaca 56 (2006), 573–589] are studied. In the classical case, a generalized OMP, while in the quantum case a weak generalized OMP is obtained. Existence of infima for arbitrary sets and suprema for above bounded sets in the quantum case is shown. Compatibility in the sense of Mackey is characterized.

DOI
10.2478/s12175-007-0048-x

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Abstract Relations between generalized effect algebras and the sets of classical and quantum observables endowed with an ordering recently introduced in [GUDDER, S.: An order for quantum observables, Math. Slovaca 56 (2006), 573–589] are studied. In the classical case, a generalized OMP, while in the quantum case a weak generalized OMP is obtained. Existence of infima for arbitrary sets and suprema for above bounded sets in the quantum case is shown. Compatibility in the sense of Mackey is characterized.
The set of effects is not a lattice with respect to its natural order. Projection operators do have the greatest lower bounds (and the least upper bounds) in that set, but there are also other (incomparable) effects which share this property. However, the coexistence, the commutativity, and the regularity of a pair of effects are not sufficient for the existence of their infima and suprema. The structure of the range of an observable (as a normalized POV measure) can vary from that of a commutative Boolean to a noncommutative non-Boolean subset of effects.
Continuing the development [4] of an aspect of the approach to the axiomatization of quantum mechanics of G. W. Mackey [3], we consider here the real linear space X of signed measures on the set P of events generated by the states, and the set F0 of linear functionals on X which are induced in a natural way by the bounded observables. A necessary and sufficient condition for two events to be simultaneously measurable is found in terms of the order structure of F0, with the following consequence: if F0 is a lattice, P is deterministic. At the opposite extreme, F0 is said to be an anti-lattice1 if the greatest lower bound exists only for comparable pairs of its elements and we show in this case that the center of P is trivial. Our results extend those of R. V. Kadison [l], in which F0 and P are the self-adjoint operators and projections respectively in a uniformly closed self-adjoint operator algebra. While the framework and plan of the proofs were inspired by Kadison's work, almost none of the apparatus used by him is available here with the result that, in detail, our techniques are quite different from his. Let P be a weakly modular partially ordered set (see [4]). A function x from P to the non-negative real numbers and + oo is said to be a measure if x(0) =0 and x is countably additive in the sense that whenever {a,} is a pairwise orthogonal sequence of elements of P, then x(Ua?) = 2Zxiad- If x is a measure and {bt} EP is an increasing (decreasing) sequence with supremum (infimum) b, then x(&?) —*x(6). A countably additive function x from P to the extended real numbers is a signed measure if x(0) =0 and x takes on at most one of the values + =o and — oo ; x is finite if x(l) is finite. Define the functions 5 and i on the signed measures on P by six) = sup(x(a) : aEP\, fx) =inf {x(a): aEP} and set ||x|| =s(x)—?(x). Clearly ||x|| < oo if and only if x is finite. It iseasy toseethat||x|| = sup {x(a)— xia'):aE P} ■ Lemma 1. Let X be a real linear space of finite signed measures on P. Then the function || || defined above is a norm for X and, under this norm and its natural partial ordering, X is a partially ordered normed linear space. That is,2
In 2006, Gudder [Math. Slovaca 56, 573 (2006)] introduced a logic order ≼ on bounded quantum observable set S(H). In 2007, Pulmannova and Vincekova [Math Slovaca 57, 589 (2007)] proved that for each subset D of S(H), the infimum of D exists with respect to the logic order ≼. In 2008, Liu and Wu [J. Math. Phys. 49, 073521 (2008)] found a representation of the infimum A∧B for A,B∊S(H), and by using the limit methods, they gave out a representation for the infimum of D. But, that representation is complicated. In this paper, we present a simpler spectral representation for the infimum of D with respect to the logic order ≼.
The quantum effects for a physical system can be described by the set E(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. For A, B∊E(H), the operation of sequential product A∘B=A1∕2BA1∕2 was proposed as a model for sequential quantum measurements. A nice investigation of properties of the sequential product has been carried over [Gudder, S. and Nagy, G., “Sequential quantum measurements,” J. Math. Phys. 42, 5212 (2001)]. In this note, we extend some results of this reference. In particular, a gap in the proof of Theorem 3.2 in this reference is overcome. In addition, some properties of generalized infimum A⨅B are studied.