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:1506.01218 · Reviews in Mathematical Physics

Covariant KSGNS construction and quantum instruments

Erkka Haapasalo, Juha-Pekka Pellonpää

Published Jun 13, 2017• 6 citations
arXivHTMLPDF

Abstract

We study completely positive (CP) [Formula: see text]-sesquilinear-form-valued maps on a unital [Formula: see text]-algebra [Formula: see text], where the sesquilinear forms operate on a module over a [Formula: see text]-algebra [Formula: see text]. We also study the cases when either one or both of the algebras are von Neumann algebras. Moreover, we assume that the CP maps are covariant with respect to actions of a symmetry group. This allows us to view these maps as generalizations of covariant quantum instruments. We determine minimal covariant dilations (KSGNS constructions) for covariant CP maps to find necessary and sufficient conditions for a CP map to be extreme in convex subsets of normalized covariant CP maps. As a special case, we study covariant quantum observables and instruments whose value space is a transitive space of a unimodular type-I group. Finally, we discuss the case of instruments that are covariant with respect to a square-integrable representation.

DOI
10.1142/s0129055x17500209

Related papers

On a dilation of a some class of completely positive maps

Эльсаев Якуб Витальевич

In this article we investigate sesquilinear forms defined on the Cartesian product of Hilbert C^*-module M over C^*-algebra B and taking values in B. The set of all such defined sesquilinear forms is denoted by S_B (M). We consider completely positive maps from locally C^*-algebra A to S_B (M). Moreover we assume that these completely positive maps are covariant with respect to actions of a group symmetry. This allow us to view these maps as generalizations covariant quantum instruments which are very important for the modern quantum mechanic and the quantum field theory. We analyze the dilation problem for these class of maps. In order to solve this problem we construct the minimal Stinespring representation and prove that every two minimal representations are unitarily equivalent.

Tambov University Reports Series Natural and Technical Sciences•Jan 1, 2019•0 citations

Integration Theory for Completely positive Instruments: A Lyapunov-Type Theorem and Applications

Arghya Chongdar

We develop a theory of integration with respect to quantum instruments through two complementary approaches: a vector measure formulation based on Bartle's integration theory and a tensor product construction. As a principal application, we establish a CP map-valued Lyapunov theorem by characterizing the convexity of the range of non-atomic completely positive instruments via the associated integration map. This extends the work of Plosker and Ramsey~\cite{Plosker_Ramsey} from POVMs to completely positive instruments and gives a general characterization of the phenomenon exhibited in their setting. We also establish a correspondence between completely positive instruments and completely positive maps, prove a Krein--Milman type theorem for the $C^*$-convex set of unital completely positive instruments, and show that the tensor product construction connects the theory of CP instruments with the integration theory for POVMs due to Farenick \emph{et al}(\cite{douglus_plosker_ramsey_povmintegration_1}).

arXiv:2608.10191•Aug 10, 2026•0 citations

The Choi-Jamiolkowski isomorphism and covariant quantum channels

Erkka Haapasalo

A generalization of the Choi-Jamiolkowski isomorphism for completely positive maps between operator algebras is introduced. Particular emphasis is placed on the case of normal unital completely positive maps defined between von Neumann algebras. This generalization is applied especially to the study of maps which are covariant under actions of a symmetry group. We highlight with the example of, e.g., phase-shift-covariant quantum channels the ease of this method in particular in the case of a compact symmetry group. We also discuss the case of channels which are covariant under actions of the Euclidean group of rigid motions in 3 dimensions.

arXiv:1906.11442•Jun 27, 2019•1 citation

Optimal covariant quantum measurements

Erkka Haapasalo, Juha-Pekka Pellonpää

Abstract We discuss symmetric quantum measurements and the associated covariant observables modelled, respectively, as instruments and positive-operator-valued measures. The emphasis of this work are the optimality properties of the measurements, namely, extremality, informational completeness, and the rank-1 property which contrast the complementary class of (rank-1) projection-valued measures. The first half of this work concentrates solely on finite-outcome measurements symmetric w.r.t. finite groups where we derive exhaustive characterizations for the pointwise Kraus-operators of covariant instruments and necessary and sufficient extremality conditions using these Kraus-operators. We motivate the use of covariance methods by showing that observables covariant with respect to symmetric groups contain a family of representatives from both of the complementary optimality classes of observables and show that even a slight deviation from a rank-1 projection-valued measure can yield an extreme informationally complete rank-1 observable. The latter half of this work derives similar results for continuous measurements in (possibly) infinite dimensions. As an example we study covariant phase space instruments, their structure, and extremality properties.

arXiv:2009.14080•Journal of Physics A Mathematical and Theoretical•Feb 17, 2021•5 citations

Extreme covariant quantum observables in the case of an Abelian symmetry group and a transitive value space

Erkka Haapasalo, Juha-Pekka Pellonpää

We represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group G. The value space of such observables is a transitive G-space. We characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables. The results are applied to position, position difference, and time observables.

arXiv:1105.1084•Journal of Mathematical Physics•Dec 1, 2011•6 citations

1 Normal Completely Positive Maps on the Space of Quantum Operations

Giulio Chiribella, Alessandro Toigo

Quantum supermaps are higher-order maps transforming quantum operations into quantum operations. Here we extend the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps transforming quantum operations with input in a separable von Neumann algebra and output in the algebra of the bounded operators on a given separable Hilbert space. In this setting we prove two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. Finally, we consider the case of quantum superinstruments, namely measures with values in the set of quantum supermaps, and derive a dilation theorem for them that is analogue to Ozawa's theorem for quantum instruments. The three dilation theorems presented here show that all the supermaps defined in this paper can be implemented by connecting devices in quantum circuits.

•Aug 17, 2016•7 citations
We study completely positive (CP) [Formula: see text]-sesquilinear-form-valued maps on a unital [Formula: see text]-algebra [Formula: see text], where the sesquilinear forms operate on a module over a [Formula: see text]-algebra [Formula: see text]. We also study the cases when either one or both of the algebras are von Neumann algebras. Moreover, we assume that the CP maps are covariant with respect to actions of a symmetry group. This allows us to view these maps as generalizations of covariant quantum instruments. We determine minimal covariant dilations (KSGNS constructions) for covariant CP maps to find necessary and sufficient conditions for a CP map to be extreme in convex subsets of normalized covariant CP maps. As a special case, we study covariant quantum observables and instruments whose value space is a transitive space of a unimodular type-I group. Finally, we discuss the case of instruments that are covariant with respect to a square-integrable representation.
In this article we investigate sesquilinear forms defined on the Cartesian product of Hilbert C^*-module M over C^*-algebra B and taking values in B. The set of all such defined sesquilinear forms is denoted by S_B (M). We consider completely positive maps from locally C^*-algebra A to S_B (M). Moreover we assume that these completely positive maps are covariant with respect to actions of a group symmetry. This allow us to view these maps as generalizations covariant quantum instruments which are very important for the modern quantum mechanic and the quantum field theory. We analyze the dilation problem for these class of maps. In order to solve this problem we construct the minimal Stinespring representation and prove that every two minimal representations are unitarily equivalent.
We develop a theory of integration with respect to quantum instruments through two complementary approaches: a vector measure formulation based on Bartle's integration theory and a tensor product construction. As a principal application, we establish a CP map-valued Lyapunov theorem by characterizing the convexity of the range of non-atomic completely positive instruments via the associated integration map. This extends the work of Plosker and Ramsey~\cite{Plosker_Ramsey} from POVMs to completely positive instruments and gives a general characterization of the phenomenon exhibited in their setting. We also establish a correspondence between completely positive instruments and completely positive maps, prove a Krein--Milman type theorem for the C∗C^*C∗-convex set of unital completely positive instruments, and show that the tensor product construction connects the theory of CP instruments with the integration theory for POVMs due to Farenick \emph{et al}(\cite{douglus_plosker_ramsey_povmintegration_1}).
A generalization of the Choi-Jamiolkowski isomorphism for completely positive maps between operator algebras is introduced. Particular emphasis is placed on the case of normal unital completely positive maps defined between von Neumann algebras. This generalization is applied especially to the study of maps which are covariant under actions of a symmetry group. We highlight with the example of, e.g., phase-shift-covariant quantum channels the ease of this method in particular in the case of a compact symmetry group. We also discuss the case of channels which are covariant under actions of the Euclidean group of rigid motions in 3 dimensions.
Abstract We discuss symmetric quantum measurements and the associated covariant observables modelled, respectively, as instruments and positive-operator-valued measures. The emphasis of this work are the optimality properties of the measurements, namely, extremality, informational completeness, and the rank-1 property which contrast the complementary class of (rank-1) projection-valued measures. The first half of this work concentrates solely on finite-outcome measurements symmetric w.r.t. finite groups where we derive exhaustive characterizations for the pointwise Kraus-operators of covariant instruments and necessary and sufficient extremality conditions using these Kraus-operators. We motivate the use of covariance methods by showing that observables covariant with respect to symmetric groups contain a family of representatives from both of the complementary optimality classes of observables and show that even a slight deviation from a rank-1 projection-valued measure can yield an extreme informationally complete rank-1 observable. The latter half of this work derives similar results for continuous measurements in (possibly) infinite dimensions. As an example we study covariant phase space instruments, their structure, and extremality properties.
We represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group G. The value space of such observables is a transitive G-space. We characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables. The results are applied to position, position difference, and time observables.
Quantum supermaps are higher-order maps transforming quantum operations into quantum operations. Here we extend the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps transforming quantum operations with input in a separable von Neumann algebra and output in the algebra of the bounded operators on a given separable Hilbert space. In this setting we prove two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. Finally, we consider the case of quantum superinstruments, namely measures with values in the set of quantum supermaps, and derive a dilation theorem for them that is analogue to Ozawa's theorem for quantum instruments. The three dilation theorems presented here show that all the supermaps defined in this paper can be implemented by connecting devices in quantum circuits.