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arXiv:1309.5248 · Journal of Physics A Mathematical and Theoretical

Coexistence of effects from an algebra of two projections

Teiko Heinosaari, Jukka Kiukas, Daniel Reitzner

Published May 15, 2014• 10 citations
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Abstract

The coexistence relation of quantum effects is a fundamental structure, describing those pairs of experimental events that can be implemented in a single setup. Only in the simplest case of qubit effects is an analytic characterization of coexistent pairs known. We generalize the qubit coexistence characterization to all pairs of effects in arbitrary dimensions that belong to the von Neumann algebra generated by two projections. We demonstrate the presented mathematical machinery by several examples, and show that it covers physically relevant classes of effect pairs.

DOI
10.1088/1751-8113/47/22/225301

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The coexistence relation of quantum effects is a fundamental structure, describing those pairs of experimental events that can be implemented in a single setup. Only in the simplest case of qubit effects is an analytic characterization of coexistent pairs known. We generalize the qubit coexistence characterization to all pairs of effects in arbitrary dimensions that belong to the von Neumann algebra generated by two projections. We demonstrate the presented mathematical machinery by several examples, and show that it covers physically relevant classes of effect pairs.
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Abstract The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig’s formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper’s first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig’s theorem. As a byproduct of our methods we also strengthen a theorem of Molnár.
Quantum effects are represented by operators on a Hilbert space satisfying 0⩽A⩽I, and sharp quantum effects are represented by projection operators. We say that an effect A is almost sharp if A=PQP for projections P and Q. We give simple characterizations of almost sharp effects. We also characterize effects that can be written as longer products of projections. For generality we first work in the formalism of von Neumann algebras. We then specialize to the full operator algebra B(H) and to finite dimensional Hilbert spaces.
This work is a study on the symmetry groups of quantum mechanics accompanied by its applications on the coexistence of qubit effects. We surveyed among some of the mappings known as preservers on the set of quantum effects for their role in conserving specific structures that are important in quantum mechanics. Since the preserves in this work mainly target the quantum effects, we provided principals of its corresponding algebra known as effect algebra. Then we advanced, to study the relations between some of these preservers (here we mean automorphisms) by channels operate on qubit effects. The channels are in convex form and we investigate the role of their convex parameters in conserving or breaking the coexistence of two qubit effects. The preserver that we are interested in is a bijective map which preserves the order and coexistence of effects in both directions. Of course, the coexistence of qubit effects can also be studied in the language of coexistence and compatibility of observables. Then our task is to study the coexistence of observables when there are only two effects in their outcome sets. This task has been studied for observables with arbitrary number of effects in their outcome sets [48]. The bounds they have found for the channels convex parameters are independent of that number. However, when the case narrows for limited number outcomes no analytic solution is available, and the study should be carried numerically. Here, we considered observables each having only two nontrivial effects in their out come sets, and asked for the demanding conditions that break their compatibility partially or completely.
We consider the question of characterizing the incompatibility of sets of high-dimensional quantum measurements. We introduce the concept of measurement incompatibility in subspaces. That is, starting from a set of measurements that is incompatible, one considers the set of measurements obtained by projection onto any strict subspace of fixed dimension. We identify three possible forms of incompatibility in subspaces: (i) incompressible incompatibility---measurements that become compatible in every subspace, (ii) fully compressible incompatibility---measurements that remain incompatible in every subspace, and (iii) partly compressible incompatibility---measurements that are compatible in some subspace and incompatible in another. For each class, we discuss explicit examples. Finally, we present some applications of these ideas. First, we show that joint measurability and coexistence are two inequivalent notions of incompatibility in the simplest case of qubit systems. Second, we highlight the implications of our results for tests of quantum steering.