Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2308.09785

Measurement Models with Separable Interaction Channels

Stanley Gudder

Published Aug 18, 2023• 0 citations
arXivHTMLPDF

Abstract

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 $ν$ between the system being measured and a probe system. For a general $ν$, the measured observable $A$ need not have an explicit useful form. In this work we introduce a condition for $ν$ called separability and in this case $A$ has an explicit form. Under the assumption that $ν$ 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.

DOI
10.48550/arxiv.2308.09785

Related papers

Quantum Channel Conditioning and Measurement Models

Stanley Gudder

If $H_1$ and $H_2$ are finite-dimensional Hilbert spaces, a channel from $H_1$ to $H_2$ is a completely positive, linear map $\mathcal{I}$ that takes the set of states $\mathcal{S}(H_1)$ for $H_1$ to the set of states $\mathcal{S}(H_2)$ for $H_2$. Corresponding to $\mathcal{I}$ there is a unique dual map $\mathcal{I}^*$ from the set of effects $\mathcal{E}(H_2)$ for $H_2$ to the set of effects $\mathcal{E}(H_1)$ for $H_1$. We call $\mathcal{I}^*(b)$ the effect $b$ conditioned by $\mathcal{I}$ and the set $\mathcal{I}^c = \mathcal{I}^*(\mathcal{E}(H_2))$ the conditioned set of $\mathcal{I}$. We point out that $\mathcal{I}^c$ is a convex subeffect algebra of the effect algebra $\mathcal{E}(H_1)$. We extend this definition to the conditioning $\mathcal{I}^*(B)$ for an observable $B$ on $H_2$ and say that an observable $A$ is in $\mathcal{I}^c$ if $A=\mathcal{I}^*(B)$ for some observable $B$. We show that $\mathcal{I}^c$ 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.

arXiv:2403.08126•Mar 12, 2024•0 citations

Nondisturbing Quantum Measurement Models

Stanley Gudder

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.

arXiv:2009.12655•Sep 26, 2020•0 citations

Parts and Composites of Quantum Systems

Stanley Gudder

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.

arXiv:2009.07371•Symmetry•Jun 8, 2021•6 citations

A Theory of Quantum Instruments

Stanley Gudder

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.

arXiv:2305.17584•Quanta•Jun 6, 2023•2 citations

Quantum Instruments and Conditioned Observables

Stanley Gudder

Observables and instruments have played significant roles in recent studies on the foundations of quantum mechanics. Sequential products of effects and conditioned observables have also been introduced. After an introduction in Section~1, we review these concepts in Section~2. Moreover, it is shown how these ideas can be unified within the framework of measurement models. In Section~3, we illustrate these concepts and their relationships for the simple example of a qubit Hilbert space. Conditioned observables and their distributions are studied in Section~4. Section~5 considers joint probabilities of observables. We introduce a definition for joint probabilities and discuss why we consider this to be superior to the standard definition.

arXiv:2005.08117•May 16, 2020•6 citations

Resource theory of interactive quantum instruments

Chung-Yun Hsieh, Armin Tavakoli, Huan-Yu Ku +1 more

Quantum instruments describe both the classical outcome and the updated quantum state in a measurement process. To do this in a non-trivial way, instruments must have the capability to interact coherently with the state that they measure. Here, we develop a resource theory for instruments. We consider a relevant quantifier of the separation between interactive and non-interactive instruments and show that it admits three distinct operational interpretations in terms of quantum information tasks. These concern (i) the preservation of maximally entangled states after a local measurement, (ii) the average ability to preserve random states after measurement, and (iii) the ability to recover the classical information generated from measuring half of a maximally entangled state. We also introduce a natural set of allowed operations and show that the third task fully characterises the resource content of instruments. Our general framework reproduces as special cases established resource theories for channels and measurements.

quant-pharXiv:2603.27676•Physical Review A•Mar 31, 2026•1 citation
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.
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.
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.
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.
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.
Observables and instruments have played significant roles in recent studies on the foundations of quantum mechanics. Sequential products of effects and conditioned observables have also been introduced. After an introduction in Section~1, we review these concepts in Section~2. Moreover, it is shown how these ideas can be unified within the framework of measurement models. In Section~3, we illustrate these concepts and their relationships for the simple example of a qubit Hilbert space. Conditioned observables and their distributions are studied in Section~4. Section~5 considers joint probabilities of observables. We introduce a definition for joint probabilities and discuss why we consider this to be superior to the standard definition.
Quantum instruments describe both the classical outcome and the updated quantum state in a measurement process. To do this in a non-trivial way, instruments must have the capability to interact coherently with the state that they measure. Here, we develop a resource theory for instruments. We consider a relevant quantifier of the separation between interactive and non-interactive instruments and show that it admits three distinct operational interpretations in terms of quantum information tasks. These concern (i) the preservation of maximally entangled states after a local measurement, (ii) the average ability to preserve random states after measurement, and (iii) the ability to recover the classical information generated from measuring half of a maximally entangled state. We also introduce a natural set of allowed operations and show that the third task fully characterises the resource content of instruments. Our general framework reproduces as special cases established resource theories for channels and measurements.