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Journal of Mathematical Physics

On the infimum of bounded quantum observables

Xiaoming Xu, Hong-Ke Du, Xiaochun Fang

Published Sep 1, 2010• 4 citations
Publisher

Abstract

The set of all bounded observables corresponding to a quantum system is usually represented by S(H), the set of all bounded linear self-adjoint operators on a Hilbert space H. In 2006, Gudder introduced a logic order on S(H). Given A, B∊S(H), let A∧B be the greatest lower bound (infimum) of A and B with respect to the Gudder order. In this paper, we characterize A∧B using the invariant subspaces of A and B.

DOI
10.1063/1.3493645

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The set of all bounded observables corresponding to a quantum system is usually represented by S(H), the set of all bounded linear self-adjoint operators on a Hilbert space H. In 2006, Gudder introduced a logic order on S(H). Given A, B∊S(H), let A∧B be the greatest lower bound (infimum) of A and B with respect to the Gudder order. In this paper, we characterize A∧B using the invariant subspaces of A and B.
Bounded observables corresponding to a quantum system are usually represented by S(H), the set of all bounded linear self-adjoint operators on a Hilbert space H. In 2006, Gudder introduced a logic order, ≼, on S(H). For A,B∊S(H), A≼B if and only if there exists C∊S(H) such that AC=0 and A+C=B. Given A,B∊S(H), let A∨B be the least upper bound (supremum) for A and B with respect to the Gudder order. In 2007, Pulmannová and Vincenková proved that A∨B exists if and only if A and B have an upper bound for the Gudder order. In this paper, we present some new necessary and sufficient conditions for which A∨B exists and give an explicit representation of A∨B (if A∨B exists).
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 ≼.
In quantum mechanics, many observables for a quantum system are represented as self-adjoint linear operators (possibly unbounded) on a complex Hilbert space H\mathcal{H}H. In this paper, we introduce a binary relation ⊥\perp⊥ on S(H)S(\mathcal{H})S(H) of all self-adjoint linear operators (including unbounded) on H\mathcal{H}H. We obtain that A⊥BA\perp BA⊥B implies that A,BA,BA,B are affiliated with some abelian von Neumann algebra at the same time. Depending on ⊥\perp⊥,S(H), S(\mathcal{H}),S(H) can be constructed to be a generalized orthoalgebra. Then we present the orthoalgebra order ⪯\preceq⪯ and the traditional order ≤\leq≤ on S(H)S(\mathcal{H})S(H) and give some properties about these two orders. We prove that for A,B∈S(H)A,B\in S(\mathcal{H})A,B∈S(H), A⪯BA\preceq BA⪯B if and only if AAA has a value in Δ\DeltaΔ implies that BBB has a value in Δ\DeltaΔ for every Borel set Δ\DeltaΔ not containing 000. Moreover, we study the existence of the infimum A∧BA\wedge BA∧B and supremum A∨BA\vee BA∨B for A,B∈S(H)A,B\in S(\mathcal{H})A,B∈S(H) with respect to ⪯\preceq⪯. Finally, we prove that Q∧P=0Q\wedge P=0Q∧P=0 where QQQ and PPP are the significant quantum observables corresponding to the position and momentum.}
The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators S(H) on a complex Hilbert space H, and the quantum effects for a physical system can be described by the set E(H) of positive contractive operators on a complex Hilbert space H. In this note, by the techniques of operator block and spectral, we give the simpler representation of A∧P and obtained the new necessary and sufficient conditions for A∨P, for A∊S(H) and P∊P(H), where P(H) is the set of all orthogonal projection operators on H. In particular, we get that if A∨P exists, then A∨P∊E(H) for A∊E(H) and P∊P(H). In addition, we consider the relations between the existence of A∨B, A−∨B−, and A+∨B+, where A+, B+, A−, and B− are the positive and negative parts of A,B∊S(H).
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. The infimum problem of Hilbert space effects is to find under what condition the infimum A∧B exists for two quantum effects A and B∊E(H). The problem has been studied in different contexts by Kadison, Gudder, Moreland, and Ando. In this Note, using the method of the spectral theory of operators, we give a affirmative answer of a conjecture of [S. Gudder, J. Math. Phys. 37, 2637–2642 (1996)]. In addition, some properties of generalized infimum A⨅B were considered.