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Journal of Mathematical Physics

Spectral properties of observables and convex mappings in quantum mechanics

Gianni Cassinelli, PEKKA J. LAHTI

Published Dec 1, 1993• 12 citations
Publisher

Abstract

Observables of a physical system can be identified with convex mappings sending states into probability measures. The properties of these mappings determine the spectral properties of the observables. The mutually exclusive cases of injective and surjective mappings are characterized, and the convex structure of their ranges are investigated. In particular, the validity of the Krein–Milman property for these convex sets is studied.

DOI
10.1063/1.530316

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Observables of a physical system can be identified with convex mappings sending states into probability measures. The properties of these mappings determine the spectral properties of the observables. The mutually exclusive cases of injective and surjective mappings are characterized, and the convex structure of their ranges are investigated. In particular, the validity of the Krein–Milman property for these convex sets is studied.
Convex sets of quantum states and processes play a central role in quantum theory and quantum information. Many important examples of convex sets in quantum theory are spectrahedra, that is, sets of positive operators satisfying affine constraints. These examples include sets of quantum states with given expectation values of a set of observables, sets of multipartite quantum states with given marginals, sets of quantum measurements, channels and multitime quantum processes, as well as sets of higher-order quantum maps and quantum causal structures. This contribution provides a characterization of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators. The general results are applied to several special cases, and then used to retrieve classic results such as Choi’s characterization of the extreme quantum channels, Parthasarathy’s characterization of the extreme quantum states with given marginals and the quantum version of Birkhoff’s theorem for qubit unital channels. Finally, we propose a notion of positive operator valued measures (POVMs) with general affine constraints for their normalization, and we characterize the extremal POVMs.
We present in this paper a set of postulates for a physical system and deduce from these the main general features of the quantum theory of stationary states. Our theory is strictly operational in the sense that only the observables of the physical system are involved in the postulates. The collection of all bounded self-adj oint operators on a Hilbert space, which has previously been used as a mathematical model for the observables in quantum mechanics, satisfy the postulates, as do a variety of considerably more general mathematical structures. We show that for any two observables there exists a pure state of the system (this term being defined essentially as done by von Neumann and Weyl) in which they have different (expectation) values. Each observable has a spectral resolution, and a state of the system induces, in a natural way, a probability distribution on the range of spectral values of each observable. A number of observables are simultaneously observable if and only if they commute (although the product of two observables is not defined in general, a reasonable notion of commutativity can be introduced). Inasmuch as Hilbert space plays no role in our theory, our proofs are necessarily of a different character from the proofs of these results for the case of the system of all bounded selfadjoint operators. Actually, Hilbert space appears to be somewhat inadequate as a state space even for the latter system, in that there exist pure states of the system which cannot be represented in the usual way by rays of the Hilbert space. The postulates are partly algebraic and partly metric. The algebraic postulates require essentially that an observable can be multiplied by real numbers and raised to integral powers, and that any two observables can be added. It is assumed that the usual algebraic laws are satisfied so that (1) the observables can be treated like the elements of a linear space, and (2) the usual rules for dealing with polynomials in one variable with real coefficients remain valid when the variable is replaced by an observable. It is not assumed that two observables have a product. The metric postulates require that for each observable there be defined a kind of maximum numerical value, which plays the part of a norm, and has various properties in accord with its physical significance. While this norm is quite essential to the development of the theory, an interesting consequence of the theory is that the norm can be (uniquely) defined in a purely algebraic fashion. This shows that the objective features of a physical system,-the spectral values and probability distributions of the observables, and the pure states,-are completely determined by the algebra of observables, i.e., by the rules for addition, scalar multiplication, and powers, of observables.
Entanglement witnesses are observables which when measured, detect entanglement in a measured composed system. It is shown what kind of relations between eigenvectors of an observable should be fulfilled, to allow an observable to be an entanglement witness. Some restrictions on the signature of entaglement witnesses, based on an algebraic-geometrical theorem will be given. The set of entanglement witnesses is linearly isomorphic to the set of maps between matrix algebras which are positive, but not completely positive. A translation of the results to the language of positive maps is also given. The properties of entanglement witnesses and positive maps express as special cases of general theorems for kkk-Schmidt witnesses and kkk-positive maps. The results are therefore presented in a general framework.
A map ϕ between convex sets is said to be convex combination preserving if, for any elements x , y in its domain and any number 0 < t < 1, there is some s with 0 < s < 1, such that ϕ( tx + (1 − t ) y ) = s ϕ( x ) + (1 − s )ϕ( y ). Every quantum measurement described by a measurement operator M introduces a map (i.e. if ) that is convex combination preserving. In this paper, we give a characterization of convex combination preserving bijective maps on the set of all separable states in a bipartite quantum system H 1 ⊗ H 2 and show that, under a mild additional condition, such a map is a composition of an invertible local quantum measurement (i.e. the map of the form with S , T invertible) and some of the following maps: the transpose, the partial transpose and the swap.