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

On the sharpness of an effect

Paul Busch

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

Abstract. I present a measure of the sharpness of an effect A as a function A↦ → S(A) that assumes values between 0 and 1 such that S(A) = 0 exactly when A is trivial and S(A) = 1 exactly when A is a nontrivial projection. This is followed by a list of open questions arising from this definition. 1. General Let H be a complex separable Hilbert space with inner product 〈 ϕ | ψ 〉, ϕ, ψ ∈ H. An effect is a selfadjoint operator A acting on H with the property that ≤ A ≤. Here and are the null and identity operators, respectively, and the partial order A ≤ B is defined as 〈ϕ|Aϕ 〉 ≤ 〈ϕ|Bϕ 〉, stipulated for all ϕ ∈ H. States are represented by positive operators T of trace equal to one, the convex set of all states being denoted S(H). The extreme elements of S(H) are the vector states, that is the rank-one projections T = |ϕ 〉〈ϕ|, where ϕ is any unit vector of H. An effect A together with a state T gives the number tr[TA] ∈ [0, 1], which represents the probability for an outcome of a measurement performed on the state T. The set of effects E(H) is thus the operator interval [ , ] with respect to ≤,

Related papers

Sequential quantum measurements

Stanley Gudder, Gabriel Nagy

A quantum effect is an operator A on a complex Hilbert space H that satisfies 0⩽A⩽I. We denote the set of quantum effects by E(H). The set of self-adjoint projection operators on H corresponds to sharp effects and is denoted by P(H). We define the sequential product of A,B∈E(H) by A∘B=A1/2BA1/2. The main purpose of this article is to study some of the algebraic properties of the sequential product. Many of our results show that algebraic conditions on A∘B imply that A and B commute for the usual operator product. For example, if A∘B satisfies certain distributive or associative laws, then AB=BA. Moreover, if A∘B∈P(H), then AB=BA and A∘B=B or B∘A=B if and only if AB=BA=B. A natural definition of stochastic independence is introduced and briefly studied.

Journal of Mathematical Physics•Nov 1, 2001•110 citations

On the infimum of quantum effects

Aurelian Gheondea, Stanley Gudder, Peter Jonas

The quantum effects for a physical system can be described by the set E(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator. While a general effect may be unsharp, the collection of sharp effects is described by the set of orthogonal projections P(H)⊆E(H). Under the natural order, E(H) becomes a partially ordered set that is not a lattice if dimH⩾2. A physically significant and useful characterization of the pairs A,B∊E(H) such that the infimum A∧B exists is called the infimum problem. We show that A∧P exists for all A∊E(H), P∊P(H) and give an explicit expression for A∧P. We also give a characterization of when A∧(I−A) exists in terms of the location of the spectrum of A. We present a counterexample which shows that a recent conjecture concerning the infimum problem is false. Finally, we compare our results with the work of Ando on the infimum problem.

Journal of Mathematical Physics•May 10, 2005•22 citations

Probability Theory on the Lattice of Projections in a Hilbert Space

Kalyanapuram Rangachari Parthasarathy

In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra \( \mathcal{B} \) X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on \( \mathcal{B} \) X constitute a convex set whose extreme points are degenerate probability measures.

Texts and readings in mathematics•Jan 1, 2005•0 citations

Properties of Sequential Products

Stanley Gudder

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.

arXiv:2307.16327•Jul 30, 2023•0 citations

Joint Distributions and Commutability of Observables

Kazimierz Urbanik

In the quantum probability theory the 6-field of random events is replaoed by the lattice of orthogonal projectors in a separable infinite dimensional Hilbert space H.A countably additive function from this lattices to the unit interval constitutes a state, the non-commutative analogue of a probability measure.The Theorem of Gleason [4] asserts that every state is of the form n--tr jtT, where jt runs over all projectors and T is a probability operator on H, i.e. a positive linear operator of unit trace.Conversely, every probability operator determines a state by the Gleason formula.From now onwards let S stand for the set of all states, i.e. all probability operators on H.We shall dienote by the space of-? ell nuclear linear operators acting in H with the norm jjïj|.j= tr{TT*) 1 ^2.Of coarse S is e closed and convex subset Of TyIn quantum theory to every physical quantity or observable th^re corresponds e sslf»adjoint not necessarily bounded linear operator on Hs By 0 ws shall denote the set of all obt ;ry£bles-Given he-0 f tac pi'cb^ivility distribution of A at the etato T is defined for ail Eoi'oi subsets E of the real

Demonstratio Mathematica•Jan 1, 1985•6 citations

Classical and nonclassical randomness in quantum measurements

Douglas Farenick, Sarah Plosker, Jerrod Smith

The space POVM H(X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra C(X)⊗B(H) into B(H). This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ:C(X)⊗B(H)→B(H). The extremal and C*-extremal points of POVM H(X) are determined.

arXiv:1110.1645•Journal of Mathematical Physics•Dec 1, 2011•19 citations
Abstract. I present a measure of the sharpness of an effect A as a function A↦ → S(A) that assumes values between 0 and 1 such that S(A) = 0 exactly when A is trivial and S(A) = 1 exactly when A is a nontrivial projection. This is followed by a list of open questions arising from this definition. 1. General Let H be a complex separable Hilbert space with inner product 〈 ϕ | ψ 〉, ϕ, ψ ∈ H. An effect is a selfadjoint operator A acting on H with the property that ≤ A ≤. Here and are the null and identity operators, respectively, and the partial order A ≤ B is defined as 〈ϕ|Aϕ 〉 ≤ 〈ϕ|Bϕ 〉, stipulated for all ϕ ∈ H. States are represented by positive operators T of trace equal to one, the convex set of all states being denoted S(H). The extreme elements of S(H) are the vector states, that is the rank-one projections T = |ϕ 〉〈ϕ|, where ϕ is any unit vector of H. An effect A together with a state T gives the number tr[TA] ∈ [0, 1], which represents the probability for an outcome of a measurement performed on the state T. The set of effects E(H) is thus the operator interval [ , ] with respect to ≤,
A quantum effect is an operator A on a complex Hilbert space H that satisfies 0⩽A⩽I. We denote the set of quantum effects by E(H). The set of self-adjoint projection operators on H corresponds to sharp effects and is denoted by P(H). We define the sequential product of A,B∈E(H) by A∘B=A1/2BA1/2. The main purpose of this article is to study some of the algebraic properties of the sequential product. Many of our results show that algebraic conditions on A∘B imply that A and B commute for the usual operator product. For example, if A∘B satisfies certain distributive or associative laws, then AB=BA. Moreover, if A∘B∈P(H), then AB=BA and A∘B=B or B∘A=B if and only if AB=BA=B. A natural definition of stochastic independence is introduced and briefly studied.
The quantum effects for a physical system can be described by the set E(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator. While a general effect may be unsharp, the collection of sharp effects is described by the set of orthogonal projections P(H)⊆E(H). Under the natural order, E(H) becomes a partially ordered set that is not a lattice if dimH⩾2. A physically significant and useful characterization of the pairs A,B∊E(H) such that the infimum A∧B exists is called the infimum problem. We show that A∧P exists for all A∊E(H), P∊P(H) and give an explicit expression for A∧P. We also give a characterization of when A∧(I−A) exists in terms of the location of the spectrum of A. We present a counterexample which shows that a recent conjecture concerning the infimum problem is false. Finally, we compare our results with the work of Ando on the infimum problem.
In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra B \mathcal{B} B X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on B \mathcal{B} B X constitute a convex set whose extreme points are degenerate probability measures.
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.
In the quantum probability theory the 6-field of random events is replaoed by the lattice of orthogonal projectors in a separable infinite dimensional Hilbert space H.A countably additive function from this lattices to the unit interval constitutes a state, the non-commutative analogue of a probability measure.The Theorem of Gleason [4] asserts that every state is of the form n--tr jtT, where jt runs over all projectors and T is a probability operator on H, i.e. a positive linear operator of unit trace.Conversely, every probability operator determines a state by the Gleason formula.From now onwards let S stand for the set of all states, i.e. all probability operators on H.We shall dienote by the space of-? ell nuclear linear operators acting in H with the norm jjïj|.j= tr{TT*) 1 ^2.Of coarse S is e closed and convex subset Of TyIn quantum theory to every physical quantity or observable th^re corresponds e sslf»adjoint not necessarily bounded linear operator on Hs By 0 ws shall denote the set of all obt ;ry£bles-Given he-0 f tac pi'cb^ivility distribution of A at the etato T is defined for ail Eoi'oi subsets E of the real
The space POVM H(X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra C(X)⊗B(H) into B(H). This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ:C(X)⊗B(H)→B(H). The extremal and C*-extremal points of POVM H(X) are determined.