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Journal of Mathematical Physics

Lattice properties of quantum effects

Stanley Gudder

Published Jun 1, 1996• 36 citations
Publisher

Abstract

Sufficient conditions for the existence of the infimum AΛB of two quantum effects A and B are given. The existence of AΛB is characterized for commuting A and B with pure point spectrum. Properties of a generalized infimum and supremum are studied. Some previous finite dimensional, commutative results are extended to the infinite dimensional and noncommutative case.

DOI
10.1063/1.531533

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Sufficient conditions for the existence of the infimum AΛB of two quantum effects A and B are given. The existence of AΛB is characterized for commuting A and B with pure point spectrum. Properties of a generalized infimum and supremum are studied. Some previous finite dimensional, commutative results are extended to the infinite dimensional and noncommutative case.
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 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 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.
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.
A parametrized family of normal states on a von Neumann algebra is called a statistical experiment, which generalizes the corresponding concepts in classical statistics and finite-dimensional quantum systems. We introduce randomization preorder and equivalence relations for statistical experiments with a fixed parameter set and for normal channels with a fixed input space by post-processing completely positive channels. In this paper, we prove that the set of equivalence classes of statistical experiments or those of normal channels is an upper and lower directed-complete partially ordered set with respect to the randomization order, i.e. any increasing or decreasing net of statistical experiments or channels has its supremum or infimum in the randomization order. We also show that if the outcome space of each statistical experiment or channel of a randomization-monotone net is commutative, the outcome space of the supremum or infimum can also be taken to be commutative. We consider two examples of homogeneous Markov processes of channels on infinite-dimensional separable Hilbert spaces, namely block-diagonalization with irrational translation and ideal quantum linear amplifier channels, and explicitly derive their infima. Throughout the paper, the concept of channel conjugation is used to obtain results for decreasing channels from those for increasing channels.
We introduce a theory of noncommutative majorization that extends the classical majorization theory introduced by Hardy, Littlewood and Pólya to tuples of self-adjoint matrices that do not necessarily commute. We define and characterize a noncommutative majorization order that extends the classical majorization order. The definition is in terms of convex noncommutative functions, and we utilize the noncommutative convexity theory and noncommutative Choquet theory recently introduced by Davidson and the first author. As an application, we obtain a new necessary and sufficient condition for the existence of a trace-preserving completely positive map, i.e. a quantum channel, that interpolates between two finite sets of matrices. We give examples demonstrating that it is not always possible for this map to be chosen mixed unitary, even locally. We also address the computational difficulty of verifying the noncommutative majorization order. Our results further apply beyond the tracial case, and we obtain more general results characterizing the existence of unital completely positive maps that preserve an arbitrary faithful state or even unital completely positive map.