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Journal of Mathematical Physics

A note on the infimum problem of Hilbert space effects

Yuan Li, Hong-Ke Du

Published Oct 1, 2006• 8 citations
Publisher

Abstract

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.

DOI
10.1063/1.2358392

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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.
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.
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. While a general effect may be unsharp, the collection of sharp effects is described by the set of orthogonal projections P(H)⊆E(H). Under the natural order, E(H) becomes a partially ordered set that is not a lattice if dimH⩾2. A physically significant and useful characterization of the pairs A,B∊E(H) such that the infimum A∧B exists is called the infimum problem. We show that A∧P exists for all A∊E(H), P∊P(H) and give an explicit expression for A∧P. We also give a characterization of when A∧(I−A) exists in terms of the location of the spectrum of A. We present a counterexample which shows that a recent conjecture concerning the infimum problem is false. Finally, we compare our results with the work of Ando on the infimum problem.
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).
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 ≼.