Teiko Heinosaari

Nondisturbing quantum measurements

We consider pairs of discrete quantum observables (POVMs) and analyze the relation between the notions of nondisturbance, joint measurability, and commutativity. We specify conditions under which these properties coincide or differ—depending, for instance, on the interplay between the number of outcomes and the Hilbert space dimension or on algebraic properties of the effect operators. We also show that (non-)disturbance is, in general, not a symmetric relation and that it can be decided and quantified by means of a semidefinite program.

Canonical phase measurement is pure

We show that the canonical phase measurement is pure in the sense that the corresponding positive operator valued measure (POVM) is extremal in the convex set of all POVMs. This means that the canonical phase measurement cannot be interpreted as a noisy measurement even if it is not a projection valued measure.

Unambiguous comparison of quantum measurements

The goal of comparison is to reveal the difference of comparednobjects as fast and reliably as possible. In this paper wenformulate and investigate the unambiguous comparison of unknownnquantum measurements represented by non-degenerate sharp POVMs.nWe distinguish between measurement devices with apriori labelednand unlabeled outcomes. In both cases we can unambiguouslynconclude only that the measurements are different. For thenlabeled case it is sufficient to use each unknown measurementnonly once and the average conditional success probabilityndecreases with the Hilbert space dimension as 1/d. If thenoutcomes of the apparatuses are not labeled, then the problemnis more complicated. We analyze the case of two-dimensionalnHilbert space. In this case single shot comparison isnimpossible and each measurement device must be used (at least)ntwice. The optimal test state in the two-shots scenario givesnthe average conditional success probability 3/4. Interestingly,nthe optimal experiment detects unambiguously the differencenwith nonvanishing probability for any pair of observables.

COEXISTENCE OF QUANTUM OPERATIONS

Quantum operations are used to describe the observed probability distributions and conditional states of the measured system. In this paper, we address the problem of their joint mea- surability (coexistence). We derive two equivalent coexistence cri- teria. The two most common classes of operations — Luders opera- tions and conditional state preparators — are analyzed. It is shown that Luders operations are coexistent only under very restrictive conditions, when the associated effects are either proportional to each other, or disjoint.

Operations and channels

In Chapters 2 and 3 we assumed a model according to which an experiment is split into two parts – preparation and measurement. This led us to the associated concepts of states and observables, respectively. On the basis of this picture we can think about two types of apparatus, which can be characterized abstractly as follows. Preparators are devices producing quantum states. No input is required, but a quantum output is produced. Measurements are input–output devices that accept a quantum system as their input and produce a classical output in the form of measurement outcome distributions. This setting is summarized in Figure 2.2. Furthermore, in subsection 3.5.2 we discussed coarse-graining relations; coarse-graining can be depicted as a box with a classical input and a classical output. This line of thought indicates that at least one type of input–output box is so far missing from our discussion – a device taking a quantum input and producing a quantum output. Hence, we say that • quantum channels are input–output devices transforming quantum states into quantum states. Each quantum channel can be placed between arbitrary preparation and measurement devices (Figure 4.1). Channels, and a slightly more general concept, that of operations , are the topics of this chapter. Transforming quantum systems In this section we discuss the mathematical definition of operations and channels. The notion of an operation was introduced by Haag and Kastler [70] within the framework of algebraic quantum field theory. Later Kraus [88], Lindblad [93] and Davies [51] discussed the requirement for operations to be completely positive.