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Gongcheng shuxue xuebao

Some Properties of Quantum Probability

Zhengli Chen, Huaixin Cao, Hong-Ke Du

Published Feb 15, 2010• 1 citation
OpenAlex

Abstract

The following conclusions are established in this paper. The supremum of the absolute value of the absolute variances of an operator in a rank-one projection is equal to the square of the distance from the operator to the scalar operators; a density operator is faithful if and only if it is injective; the strong convergence on the closure of the range of a density operator ρ implies the ρ-a. s. convergence of the sequence; a sequence of operators is strongly convergent if it is ρ-a. s. convergent for every density operator ρ; if ρ is a faithful density operator, then a sequence operators is strongly convergent to A if and only if it is uniformly bounded and convergent to A a. s. [ρ].

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The following conclusions are established in this paper. The supremum of the absolute value of the absolute variances of an operator in a rank-one projection is equal to the square of the distance from the operator to the scalar operators; a density operator is faithful if and only if it is injective; the strong convergence on the closure of the range of a density operator ρ implies the ρ-a. s. convergence of the sequence; a sequence of operators is strongly convergent if it is ρ-a. s. convergent for every density operator ρ; if ρ is a faithful density operator, then a sequence operators is strongly convergent to A if and only if it is uniformly bounded and convergent to A a. s. [ρ].
Some equalities and inequalities on expectations,variances,covariances,absolute variance and independence of matrices in a given density matrix are established.The following results are proved:(ⅰ) Two matrices A and B are ρ-independent if and only if Covρ(A,B)=0 if and only if Expρ(AB)=Expρ(A)Expρ(B);(ⅱ) If the numerical ranges W(A) and W(B) of A and B are contained in disks with radius R and S,respectively,then |Expρ(AB)-Expρ(A)Expρ(B)|≤4RS and |Covρ(A,B)|≤4ω(A)ω(B),where ω(A),ω(B) are the numerical radius of A,B,respectively.
Up until this point, we have concerned ourselves with a description of quantum mechanics centered upon how a state |ψ〉 evolves in time. Using this we can show that the expectation value of an operator evolves as 〈A(t)〉 = 〈ψ(t)| ˆA|ψ(t)〉 (6.1) However, for many instances, especially if we are interested in describing relaxation processes, it is useful to introduce the density operator ρˆ(t) = |ψ(t)〉〈ψ(t)| (6.2) taken as the outer product of the state vector with itself. From this definition we can write ρˆ = ∑ mn c∗ncm |m〉〈n| (6.3) = ∑ mn ρmn|m〉〈n| (6.4) where the ρmn are the density matrix elements. Expectation values of operator are then given by the trace 〈A(t)〉 = ∑ mn Amnρmn = Tr [ ˆAρ(t)] (6.5) where Tr [ ˆAρ(t)] denotes the trace operation: Tr [ ˆAρ(t)] = ∑ nn′ If ρ is diagonal, then Tr [ ˆAρ(t)] = ∑ Annρnn = ∑ 〈n|A|n〉Pn where Pn = ρnn is the statistical probability of finding the system in state n. These statistical weights must be such that Pn ≤ 1 and∑ Pn = 1 Hence, we conclude that knowing ρ we can compute the statistical average of an operator A.
An overcomplete family of states (OFS) is constructed for a countably infinite linear vector space with an indefinite metric for the case that the metric is diagonal with eigenvalues (−1)n, where n is an integer. A continuous representation is indicated and the properties of a semiclassical description of a quantum mechanical system (the pseudo oscillator whose creation and destruction operators ā and ā+ satisfy [ā, ā+] = −1) defined in this vector space are studied. It is found that a consistent OFS |z> can be constructed if the operator G(z) which generates the state |z> from the vacuum is unitary. Furthermore, with the statistical state of this system specified by a bounded pseudo-Hermitian density matrix ρ̄, the related semiclassical complex function ρA(z) for antinormal ordering of operators in the indefinite metric space is found to be bounded, with ρA(z) and [ρA(z)]2 integrable, continuous, and a boundary value of an entire analytic function of two complex variables. The semiclassical function ρN(z) for normal ordering is associated with a sequence of functions ρN(ν)(z) whose square is integrable and related to a sequence of tempered distributions ρN(ν) such that the corresponding sequence of density matrices ρ̄(ν) converges to ρ̄ in the norm.
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
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.