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.
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.
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 I. The infimum problem of Hilbert space effects is to find under what condition the infimum A∧B exists for two quantum effects A and B∊E(H). The problem has been studied in different contexts by Kadison, Gudder, Moreland, and Ando. In this Note, using the method of the spectral theory of operators, we give a affirmative answer of a conjecture of [S. Gudder, J. Math. Phys. 37, 2637–2642 (1996)]. In addition, some properties of generalized infimum A⨅B were considered.
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 I. For A, B∊E(H), the operation of sequential product A∘B=A1∕2BA1∕2 was proposed as a model for sequential quantum measurements. A nice investigation of properties of the sequential product has been carried over [Gudder, S. and Nagy, G., “Sequential quantum measurements,” J. Math. Phys. 42, 5212 (2001)]. In this note, we extend some results of this reference. In particular, a gap in the proof of Theorem 3.2 in this reference is overcome. In addition, some properties of generalized infimum A⨅B are studied.
The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators S(H) on a complex Hilbert space H, and the quantum effects for a physical system can be described by the set E(H) of positive contractive operators on a complex Hilbert space H. In this note, by the techniques of operator block and spectral, we give the simpler representation of A∧P and obtained the new necessary and sufficient conditions for A∨P, for A∊S(H) and P∊P(H), where P(H) is the set of all orthogonal projection operators on H. In particular, we get that if A∨P exists, then A∨P∊E(H) for A∊E(H) and P∊P(H). In addition, we consider the relations between the existence of A∨B, A−∨B−, and A+∨B+, where A+, B+, A−, and B− are the positive and negative parts of A,B∊S(H).
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.
A quantum effect is an operator A on a complex Hilbert space H that satisfies 0≤A≤I, E(H) is the set of all quantum effects on H. In 2001, Professor Gudder and Nagy studied the sequential product A∘B=A1/2BA1/2 of A,B∈E(H). In 2005, Professor Gudder asked: Is A∘B=A1/2BA1/2 the only sequential product on E(H)? Recently, Liu and Wu presented an example to show that the answer is negative. In this paper, firstly, we characterize some algebraic properties of the abstract sequential product on E(H); secondly, we present a general method for constructing sequential products on E(H); finally, we study some properties of the sequential products constructed by the method