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arXiv:2009.12655

Nondisturbing Quantum Measurement Models

Stanley Gudder

Published Sep 26, 2020• 0 citations
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Abstract

A measurement model is a framework that describes a quantum measurement process. In this article we restrict attention to $MM$s on finite-dimensional Hilbert spaces. Suppose we want to measure an observable $A$ whose outcomes $A_x$ are represented by positive operators (effects) on a Hilbert Space $H$. We call $H$ the base or object system. We interact $H$ with a probe system on another Hilbert space $K$ by means of a quantum channel. The probe system contains a probe (or meter or pointer) observable $F$ whose outcomes $F_x$ are measured by an apparatus that is frequently (but need not be) classical in practice. The $MM$ protocol gives a method for determining the probability of an outcome $A_x$ for any state of $H$ in terms of the outcome $F_x$. The interaction channel usually entangles this state with an initial probe state of $K$ that can be quite complicated. However, if the channel is nondisturbing in a sense that we describe, then the entanglement is considerably simplified. In this article, we give formulas for observables and instruments measured by nondisturbing $MM$s. We begin with a general discussion of nondisturbing operators relative to a quantum context. We present two examples that illustrate this theory in terms of unitary nondisturbing channels.

DOI
10.48550/arxiv.2009.12655

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A measurement model is a framework that describes a quantum measurement process. In this article we restrict attention to MMMMMMs on finite-dimensional Hilbert spaces. Suppose we want to measure an observable AAA whose outcomes AxA_xAx​ are represented by positive operators (effects) on a Hilbert Space HHH. We call HHH the base or object system. We interact HHH with a probe system on another Hilbert space KKK by means of a quantum channel. The probe system contains a probe (or meter or pointer) observable FFF whose outcomes FxF_xFx​ are measured by an apparatus that is frequently (but need not be) classical in practice. The MMMMMM protocol gives a method for determining the probability of an outcome AxA_xAx​ for any state of HHH in terms of the outcome FxF_xFx​. The interaction channel usually entangles this state with an initial probe state of KKK that can be quite complicated. However, if the channel is nondisturbing in a sense that we describe, then the entanglement is considerably simplified. In this article, we give formulas for observables and instruments measured by nondisturbing MMMMMMs. We begin with a general discussion of nondisturbing operators relative to a quantum context. We present two examples that illustrate this theory in terms of unitary nondisturbing channels.
If H1H_1H1​ and H2H_2H2​ are finite-dimensional Hilbert spaces, a channel from H1H_1H1​ to H2H_2H2​ is a completely positive, linear map I\mathcal{I}I that takes the set of states S(H1)\mathcal{S}(H_1)S(H1​) for H1H_1H1​ to the set of states S(H2)\mathcal{S}(H_2)S(H2​) for H2H_2H2​. Corresponding to I\mathcal{I}I there is a unique dual map I∗\mathcal{I}^*I∗ from the set of effects E(H2)\mathcal{E}(H_2)E(H2​) for H2H_2H2​ to the set of effects E(H1)\mathcal{E}(H_1)E(H1​) for H1H_1H1​. We call I∗(b)\mathcal{I}^*(b)I∗(b) the effect bbb conditioned by I\mathcal{I}I and the set Ic=I∗(E(H2))\mathcal{I}^c = \mathcal{I}^*(\mathcal{E}(H_2))Ic=I∗(E(H2​)) the conditioned set of I\mathcal{I}I. We point out that Ic\mathcal{I}^cIc is a convex subeffect algebra of the effect algebra E(H1)\mathcal{E}(H_1)E(H1​). We extend this definition to the conditioning I∗(B)\mathcal{I}^*(B)I∗(B) for an observable BBB on H2H_2H2​ and say that an observable AAA is in Ic\mathcal{I}^cIc if A=I∗(B)A=\mathcal{I}^*(B)A=I∗(B) for some observable BBB. We show that Ic\mathcal{I}^cIc is closed under post-processing and taking parts. We also define the conditioning of instruments by channels. These concepts are illustrated using examples of Holevo instruments and channels. We next discuss measurement models and their corresponding observables and instruments. We show that calculations can be simplified by employing Kraus and Holevo separable channels. Such channels allow one to separate the components of a tensor product.
Measurement models (MMs) stand at the highest structural level of quantum measurement theory. MMs can be employed to construct instruments which stand at the next level. An instrument is thought of as an apparatus that is used to measure observables and update states. Observables, which are still at the next level, are used to determine probabilities of quantum events. The main ingredient of an MM is an interaction channel ννν between the system being measured and a probe system. For a general ννν, the measured observable AAA need not have an explicit useful form. In this work we introduce a condition for ννν called separability and in this case AAA has an explicit form. Under the assumption that ννν is separable, we study product MMs and conditioned MMs. We also consider the statistics of MMs and their uncertainty principle. Various concepts are illustrated using examples of Lüders and Holevo instruments.
This article considers quantum systems described by a finite-dimensional complex Hilbert space HHH. We first define the concept of a finite observable on HHH. We then discuss ways of combining observables in terms of convex combinations, post-processing and sequential products. We also define complementary and coexistent observables. We then introduce finite instruments and their related compatible observables. The previous combinations and relations for observables are extended to instruments and their properties are compared. We present four types of instruments; namely, identity, trivial, Lüders and Kraus instruments. These types are used to illustrate different ways that instruments can act. We next consider joint probabilities for observables and instruments. The article concludes with a discussion of measurement models and the instruments they measure.
We consider three types of entities for quantum measurements. In order of generality, these types are observables, instruments and measurement models. If α and β are entities, we define what it means for α to be a part of β. This relationship is essentially equivalent to α being a function of β and in this case β can be employed to measure α. We then use the concept to define the coexistence of entities and study its properties. A crucial role is played by a map α^ which takes an entity of a certain type to one of a lower type. For example, if I is an instrument, then I^ is the unique observable measured by I. Composite systems are discussed next. These are constructed by taking the tensor product of the Hilbert spaces of the systems being combined. Composites of the three types of measurements and their parts are studied. Reductions in types to their local components are discussed. We also consider sequential products of measurements. Specific examples of Lüders, Kraus and trivial instruments are used to illustrate various concepts. We only consider finite-dimensional systems in this article. Finally, we mention the role of symmetry representations for groups using quantum channels.
In the general theory of quantum measurement, one associates a positive-semidefinite operator on a ddd-dimensional Hilbert space to each of the nnn possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert-Schmidt orthogonal, but when n>dn>dn>d, orthogonality is restricted by positivity. This restriction allows us to more precisely state the quantum adage: information gain of a system is always accompanied by unavoidable disturbance. Specifically, we investigate three properties of a measurement with L\"uders rule updating: its disturbance, a measure of how the expected postmeasurement state deviates from the input; its measurement strength, a measure of the intrinsic information producing capacity of the measurement; and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. These quantities satisfy an information-disturbance tradeoff relation that highlights the additional role played by orthogonality. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.
Until recently, a quantum instrument was defined to be a completely positive operation-valued measure from the set of states on a Hilbert space to itself. In the last few years, this definition has been generalized to such measures between sets of states from different Hilbert spaces called the input and output Hilbert spaces. This article presents a theory of such instruments. Ways that instruments can be combined such as convex combinations, post-processing, sequential products, tensor products and conditioning are studied. We also consider marginal, reduced instruments and how these are used to define coexistence (compatibility) of instruments. Finally, we present a brief introduction to quantum measurement models where the generalization of instruments is essential. Many of the concepts of the theory are illustrated by examples. In particular, we discuss Holevo and Kraus instruments. Quanta 2023; 12: 27–40.