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Teiko Heinosaari

Publications and source records attributed to Teiko Heinosaari.

At least 91 records · Page 5Linked to original sources
arXiv2011

Generalized Coherent States and Extremal Positive Operator Valued Measures

We present a correspondence between positive operator valued measures (POVMs) and sets of generalized coherent states. Positive operator valued measures describe quantum observables and, similarly to quantum states, also quantum observables can be mixed. We show how the formalism of generalized coherent states leads to a useful characterization of extremal POVMs. We prove that covariant phase space observables related to squeezed states are extremal, while the ones related to number states are not extremal.

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arXiv2011

Informationally complete joint measurements on finite quantum systems

We show that there are informationally complete joint measurements of two conjugated observables on a finite quantum system, meaning that they enable to identify all quantum states from their measurement outcome statistics. We further demonstrate that it is possible to implement a joint observable as a sequential measurement. If we require minimal noise in the joint measurement, then the joint observable is unique. If the dimension d is odd, then this observable is informationally complete. But if d is even, then the joint observable is not informationally complete and one has to allow more noise in order to obtain informational completeness.

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arXiv2011

Quantum measurements on finite dimensional systems: relabeling and mixing

Concentrating on finite dimensional systems, we show that one can limit to extremal rank-1 POVMs if two simple procedures of mixing and relabeling are permitted. We demonstrate that any finite outcome POVM can be obtained from extremal rank-1 POVMs with these two procedures. In particular, extremal POVMs with higher rank are just relabelings of extremal rank-1 POVMs and their structure is therefore clarified.

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arXiv2011

Extreme commutative quantum observables are sharp

It is well known that, in the description of quantum observables, positive operator valued measures (POVMs) generalize projection valued measures (PVMs) and they also turn out be more optimal in many tasks. We show that a commutative POVM is an extreme point in the convex set of all POVMs if and only if it is a PVM. This results implies that non-commutativity is a necessary ingredient to overcome the limitations of PVMs.

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arXiv2011

Sequential measurements of conjugate observables

We present a unified treatment of sequential measurements of two conjugate observables. Our approach is to derive a mathematical structure theorem for all the relevant covariant instruments. As a consequence of this result, we show that every Weyl-Heisenberg covariant observable can be implemented as a sequential measurement of two conjugate observables. This method is applicable both in finite and infinite dimensional Hilbert spaces, therefore covering sequential spin component measurements as well as position-momentum sequential measurements.

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arXiv2010

Approximating incompatible von Neumann measurements simultaneously

We study the problem of performing orthogonal qubit measurements simultaneously. Since these measurements are incompatible, one has to accept additional imprecision. An optimal joint measurement is the one with the least possible imprecision. All earlier considerations of this problem have concerned only joint measurability of observables, while in this work we also take into account conditional state transformations (i.e., instruments). We characterize the optimal joint instrument for two orthogonal von Neumann instruments as being the Luders instrument of the optimal joint observable.

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arXiv2010

Non-disturbing quantum measurements

We consider pairs of quantum observables (POVMs) and analyze the relation between the notions of non-disturbance, 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.

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arXiv2009

The 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.

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arXiv2009

Covariant quantum instruments

The structure of covariant instruments is studied and a general structure theorem is derived. A detailed characterization is given to covariant instruments in the case of an irreducible representation of a locally compact group.

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arXiv2009

The semigroup structure of Gaussian channels

We investigate the semigroup structure of bosonic Gaussian quantum channels. Particular focus lies on the sets of channels which are divisible, idempotent or Markovian (in the sense of either belonging to one-parameter semigroups or being infinitesimal divisible). We show that the non-compactness of the set of Gaussian channels allows for remarkable differences when comparing the semigroup structure with that of finite dimensional quantum channels. For instance, every irreversible Gaussian channel is shown to be divisible in spite of the existence of Gaussian channels which are not infinitesimal divisible. A simpler and known consequence of non-compactness is the lack of generators for certain reversible channels. Along the way we provide new representations for classes of Gaussian channels: as matrix semigroup, complex valued positive matrices or in terms of a simple form describing almost all one-parameter semigroups.

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arXiv2009

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 measurability (coexistence). We derive two equivalent coexistence criteria. The two most common classes of operations - Luders operations 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.

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arXiv2009

Unambiguous comparison of quantum measurements

The goal of comparison is to reveal the difference of compared objects as fast and reliably as possible. In this paper we formulate and investigate the unambiguous comparison of unknown quantum measurements represented by non-degenerate sharp POVMs. We distinguish between measurement devices with apriori labeled and unlabeled outcomes. In both cases we can unambiguously conclude only that the measurements are different. For the labeled case it is sufficient to use each unknown measurement only once and the average conditional success probability decreases with the Hilbert space dimension as 1/d. If the outcomes of the apparatuses are not labeled, then the problem is more complicated. We analyze the case of two-dimensional Hilbert space. In this case single shot comparison is impossible and each measurement device must be used (at least) twice. The optimal test state in the two-shots scenario gives the average conditional success probability 3/4. Interestingly, the optimal experiment detects unambiguously the difference with nonvanishing probability for any pair of observables.

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arXiv2009

Commutative POVMs and Fuzzy Observables

In this paper we review some properties of fuzzy observables, mainly as realized by commutative positive operator valued measures. In this context we discuss two representation theorems for commutative positive operator valued measures in terms of projection valued measures and describe, in some detail, the general notion of fuzzification. We also make some related observations on joint measurements.

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arXiv2008

Notes on Joint Measurability of Quantum Observables

For sharp quantum observables the following facts hold: (i) if we have a collection of sharp observables and each pair of them is jointly measurable, then they are jointly measurable all together; (ii) if two sharp observables are jointly measurable, then their joint observable is unique and it gives the greatest lower bound for the effects corresponding to the observables; (iii) if we have two sharp observables and their every possible two outcome partitionings are jointly measurable, then the observables themselves are jointly measurable. We show that, in general, these properties do not hold. Also some possible candidates which would accompany joint measurability and generalize these apparently useful properties are discussed.

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arXiv2008

Guide to Mathematical Concepts of Quantum Theory

Quantum Theory is one of the pillars of modern science developed over the last hundred years. In this review paper we introduce, step by step, the quantum theory understood as a mathematical model describing quantum experiments. We start with splitting the experiment into two parts: a preparation process and a measurement process leading to a registration of a particular outcome. These two ingredients of the experiment are represented by states and effects, respectively. Further, the whole picture of quantum measurement will be developed and concepts of observables, instruments and measurement models representing the three different descriptions on experiments will be introduced. In the second stage, we enrich the model of the experiment by introducing the concept of quantum channel describing the system changes between preparations and measurements. At the very end we review the elementary properties of quantum entanglement. The text contains many examples and exercise covering also many topics from quantum information theory and quantum measurement theory. The goal is to give a mathematically clear and self-containing explanation of the main concepts of the modern language of quantum theory.

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arXiv2008

Coexistence of qubit effects

We characterize all coexistent pairs of qubit effects. This gives an exhaustive description of all pairs of events allowed, in principle, to occur in a single qubit measurement. The characterization consists of three disjoint conditions which are easy to check for a given pair of effects. Known special cases are shown to follow from our general characterization theorem.

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