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:2307.16327

Properties of Sequential Products

Stanley Gudder

Published Jul 30, 2023• 0 citations
arXivHTMLPDF

Abstract

Our basic concept is the set $\mathcal{E}(H)$ of effects on a finite dimensional complex Hilbert space $H$. If $a,b\in\mathcal{E}(H)$, we define the sequential product $a[\mathcal{I}]b$ of $a$ then $b$. The sequential product depends on the operation $\mathcal{I}$ used to measure $a$. We begin by studying the properties of this sequential product. It is observed that $b\mapsto a[\mathcal{I}]b$ is an additive, convex morphism and we show by examples that $a\mapsto a[\mathcal{I}]b$ enjoys very few conditions. This is because a measurement of $a$ can interfere with a later measurement of $b$. We study sequential products relative to Kraus, Lüders and Holevo operations and find properties that characterize these operations. We consider repeatable effects and conditions on $a[\mathcal{I}]b$ that imply commutativity. We introduce the concept of an effect $b$ given an effect $a$ and study its properties. We next extend the sequential product to observables and instruments and develop statistical properties of real-valued observables. This is accomplished by employing corresponding stochastic operators. Finally, we introduce an uncertainty principle for conditioned observables.

DOI
10.48550/arxiv.2307.16327

Related papers

Dual Instruments and Sequential Products of Observables

Stanley Gudder

We first show that every operation possesses an unique dual operation and measures an unique effect. If a and b are effects and J is an operation that measures a, we define the sequential product of a then b relative to J. Properties of the sequential product are derived and are illustrated in terms of Lüders and Holevo operations. We next extend this work to the theory of instruments and observables. We also define the concept of an instrument (observable) conditioned by another instrument (observable). Identity, state-constant and repeatable instruments are considered. Sequential products of finite observables relative to Lüders and Holevo instruments are studied. Quanta 2022; 11: 15–27.

arXiv:2208.07923•Quanta•Aug 30, 2022•3 citations

Conditional Effects, Observables and Instruments

Stanley Gudder

We begin with a study of operations and the effects they measure. We define the probability that an effect $a$ occurs when the system is in a state $\rho$ by $P_{\rho}(a)=\textrm{Tr}(\rho a)$. If $P_{\rho}(a)\ne0$ and $\mathcal{I}$ is an operation that measures $a$, we define the conditional probability of an effect $b$ given $a$ relative to $\mathcal{I}$ by $P_{\rho}(b\mid a)=\textrm{Tr}[\mathcal{I}(\rho)b]/P_{\rho}(a)$. We characterize when Bayes' quantum second rule $P_{\rho}(b\mid a)=\frac{P_{\rho}(b)}{P_{\rho}(a)}\,P_{\rho}(a\mid b)$ holds. We then consider Lüders and Holevo operations. We next discuss instruments and the observables they measure. If $A$ and $B$ are observables and an instrument $\mathcal{I}$ measures $A$, we define the observable $B$ conditioned on $A$ relative to $\mathcal{I}$ and denote it by $(B\mid A)$. Using these concepts, we introduce Bayes' quantum first rule. We observe that this is the same as the classical Bayes' first rule, except it depends on the instrument used to measure $A$. We then extend this to Bayes' quantum first rule for expectations. We show that two observables $B$ and $C$ are jointly commuting if and only if there exists an atomic observable $A$ such that $B=(B\mid A)$ and $C=(C\mid A)$. We next obtain a general uncertainty principle for conditioned observables. Finally, we discuss observable conditioned quantum entropies. The theory is illustrated with many examples. Quanta 2024; 13: 1–10.

arXiv:2303.15640•Quanta•Feb 6, 2024•1 citation

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

Interval Effect Algebras and Holevo Instruments

Stanley Gudder

This article begins with a study of convex effect-state spaces. We point out that such spaces are equivalent to interval effect algebras that generate an ordered linear space and possess an order-determining set of states. We then discuss operations and instruments on interval effect algebras under the assumption of an unrestrictive condition. Effects measured by operations and sequential products of effects relative to operations are considered. Observables are introduced and coexistence of effects are discussed. We also present properties of sequential products of observables and conditioning of observables related to instruments. The final section is devoted to Holevo instruments. Pure and mixed Holevo operations are defined and extended to instruments. The Holevo sequential product of two observables is defined and the marginals of these products are computed. We define the commutant of two effects and derive its properties. Examples are given that illustrate the properties of previously presented concepts.

arXiv:2403.18925•Mar 27, 2024•0 citations

Convex and Sequential Effect Algebras

Stanley Gudder

We present a mathematical framework for quantum mechanics in which the basic entities and operations have physical significance. In this framework the primitive concepts are states and effects and the resulting mathematical structure is a convex effect algebra. We characterize the convex effect algebras that are classical and those that are quantum mechanical. The quantum mechanical ones are those that can be represented on a complex Hilbert space. We next introduce the sequential product of effects to form a convex sequential effect algebra. This product makes it possible to study conditional probabilities and expectations.

arXiv:1802.01265•Feb 5, 2018•7 citations

Dual Quantum Instruments and Sub-observables

Stanley Gudder

We introduce the concepts of dual instruments and sub-observables. We show that although a dual instruments measures a unique observable, it determines many sub-observables. We define a unique minimal extension of a sub-observable to an observable and consider sequential products and conditioning of sub-observables. Sub-observable effect algebras are characterized and studied. Moreover, the convexity of these effect algebras is considered. The sequential product of instruments is discussed. These concepts are illustrated with many examples of instruments. In particular, we discuss Lüders, Holero and constant state instruments. Various conjectures for future research are presented.

arXiv:2302.12243•Feb 23, 2023•0 citations
Our basic concept is the set E(H)\mathcal{E}(H)E(H) of effects on a finite dimensional complex Hilbert space HHH. If a,b∈E(H)a,b\in\mathcal{E}(H)a,b∈E(H), we define the sequential product a[I]ba[\mathcal{I}]ba[I]b of aaa then bbb. The sequential product depends on the operation I\mathcal{I}I used to measure aaa. We begin by studying the properties of this sequential product. It is observed that b↦a[I]bb\mapsto a[\mathcal{I}]bb↦a[I]b is an additive, convex morphism and we show by examples that a↦a[I]ba\mapsto a[\mathcal{I}]ba↦a[I]b enjoys very few conditions. This is because a measurement of aaa can interfere with a later measurement of bbb. We study sequential products relative to Kraus, Lüders and Holevo operations and find properties that characterize these operations. We consider repeatable effects and conditions on a[I]ba[\mathcal{I}]ba[I]b that imply commutativity. We introduce the concept of an effect bbb given an effect aaa and study its properties. We next extend the sequential product to observables and instruments and develop statistical properties of real-valued observables. This is accomplished by employing corresponding stochastic operators. Finally, we introduce an uncertainty principle for conditioned observables.
We first show that every operation possesses an unique dual operation and measures an unique effect. If a and b are effects and J is an operation that measures a, we define the sequential product of a then b relative to J. Properties of the sequential product are derived and are illustrated in terms of Lüders and Holevo operations. We next extend this work to the theory of instruments and observables. We also define the concept of an instrument (observable) conditioned by another instrument (observable). Identity, state-constant and repeatable instruments are considered. Sequential products of finite observables relative to Lüders and Holevo instruments are studied. Quanta 2022; 11: 15–27.
We begin with a study of operations and the effects they measure. We define the probability that an effect aaa occurs when the system is in a state ρ\rhoρ by Pρ(a)=Tr(ρa)P_{\rho}(a)=\textrm{Tr}(\rho a)Pρ​(a)=Tr(ρa). If Pρ(a)≠0P_{\rho}(a)\ne0Pρ​(a)=0 and I\mathcal{I}I is an operation that measures aaa, we define the conditional probability of an effect bbb given aaa relative to I\mathcal{I}I by Pρ(b∣a)=Tr[I(ρ)b]/Pρ(a)P_{\rho}(b\mid a)=\textrm{Tr}[\mathcal{I}(\rho)b]/P_{\rho}(a)Pρ​(b∣a)=Tr[I(ρ)b]/Pρ​(a). We characterize when Bayes' quantum second rule Pρ(b∣a)=Pρ(b)Pρ(a) Pρ(a∣b)P_{\rho}(b\mid a)=\frac{P_{\rho}(b)}{P_{\rho}(a)}\,P_{\rho}(a\mid b)Pρ​(b∣a)=Pρ​(a)Pρ​(b)​Pρ​(a∣b) holds. We then consider Lüders and Holevo operations. We next discuss instruments and the observables they measure. If AAA and BBB are observables and an instrument I\mathcal{I}I measures AAA, we define the observable BBB conditioned on AAA relative to I\mathcal{I}I and denote it by (B∣A)(B\mid A)(B∣A). Using these concepts, we introduce Bayes' quantum first rule. We observe that this is the same as the classical Bayes' first rule, except it depends on the instrument used to measure AAA. We then extend this to Bayes' quantum first rule for expectations. We show that two observables BBB and CCC are jointly commuting if and only if there exists an atomic observable AAA such that B=(B∣A)B=(B\mid A)B=(B∣A) and C=(C∣A)C=(C\mid A)C=(C∣A). We next obtain a general uncertainty principle for conditioned observables. Finally, we discuss observable conditioned quantum entropies. The theory is illustrated with many examples. Quanta 2024; 13: 1–10.
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.
This article begins with a study of convex effect-state spaces. We point out that such spaces are equivalent to interval effect algebras that generate an ordered linear space and possess an order-determining set of states. We then discuss operations and instruments on interval effect algebras under the assumption of an unrestrictive condition. Effects measured by operations and sequential products of effects relative to operations are considered. Observables are introduced and coexistence of effects are discussed. We also present properties of sequential products of observables and conditioning of observables related to instruments. The final section is devoted to Holevo instruments. Pure and mixed Holevo operations are defined and extended to instruments. The Holevo sequential product of two observables is defined and the marginals of these products are computed. We define the commutant of two effects and derive its properties. Examples are given that illustrate the properties of previously presented concepts.
We present a mathematical framework for quantum mechanics in which the basic entities and operations have physical significance. In this framework the primitive concepts are states and effects and the resulting mathematical structure is a convex effect algebra. We characterize the convex effect algebras that are classical and those that are quantum mechanical. The quantum mechanical ones are those that can be represented on a complex Hilbert space. We next introduce the sequential product of effects to form a convex sequential effect algebra. This product makes it possible to study conditional probabilities and expectations.
We introduce the concepts of dual instruments and sub-observables. We show that although a dual instruments measures a unique observable, it determines many sub-observables. We define a unique minimal extension of a sub-observable to an observable and consider sequential products and conditioning of sub-observables. Sub-observable effect algebras are characterized and studied. Moreover, the convexity of these effect algebras is considered. The sequential product of instruments is discussed. These concepts are illustrated with many examples of instruments. In particular, we discuss Lüders, Holero and constant state instruments. Various conjectures for future research are presented.