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Journal of Physics A Mathematical and General

Characterisations of compatibility, comparability and orthogonality of quantum propositions in terms of chains of filters

Fedor Herbut

Published Oct 21, 1985• 2 citations
Publisher

Abstract

Let p(F.E, rho ) denote the probability of a yes result of the yes-no effect obtained by putting the filter of the quantum proposition F (a projection operator) after that of the proposition E in the quantum state rho (a statistical operator). It is pointed out that EF=FE implies/implied by For all rho :p(E.F, rho )=p(F.E, rho ), EF=E implies/implied by For all rho : p(E.F, rho )=p(E, rho ), and EF=0 implies/implied by For all rho :p(E.F, rho )=0. Proof is given for generalisations of these simple characterisations to arbitrary strings (products of operators) and chains (of filters). Other cases of parallelism between strings and chains and departure from it are also studied.

DOI
10.1088/0305-4470/18/15/018

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Let p(F.E, rho ) denote the probability of a yes result of the yes-no effect obtained by putting the filter of the quantum proposition F (a projection operator) after that of the proposition E in the quantum state rho (a statistical operator). It is pointed out that EF=FE implies/implied by For all rho :p(E.F, rho )=p(F.E, rho ), EF=E implies/implied by For all rho : p(E.F, rho )=p(E, rho ), and EF=0 implies/implied by For all rho :p(E.F, rho )=0. Proof is given for generalisations of these simple characterisations to arbitrary strings (products of operators) and chains (of filters). Other cases of parallelism between strings and chains and departure from it are also studied.
A quantum effect is a yes-no measurement that may be unsharp. An effect is represented by an operator E E on a Hilbert space that satisfies 0 ≤ E ≤ I 0\le E\le I . We define effects E 1 , E 2 , … , E n E_1,E_2,\ldots ,E_n to be sequentially independent if the result of any sequential measurement of E 1 , E 2 , … , E n E_1,E_2,\ldots ,E_n does not depend on the order in which they are measured. We show that two effects are sequentially independent if and only if they are compatible. That is, their corresponding operators commute. We also show that three effects are sequentially independent if and only if all permutations of the product of their corresponding operators coincide. It is noted that this last condition does not imply that the three effects are mutually compatible.
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.
We consider the standard quantum logic L(H){\mathcal L}(H)L(H) associated to a complex Hilbert space HHH, i.e. the lattice of closed subspaces of HHH together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic L(H){\mathcal L}(H)L(H)) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of L(H){\mathcal L}(H)L(H) and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.
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As has already been pointed out by Birkhoff and von Neumann, quantum logic can be formulated in terms of projective geometry. In three-dimensional Hilbert space, elementary logical propositions are associated with one-dimensional subspaces, corresponding to points of the projective plane. It is shown that, starting with three such propositions corresponding to some basis {u,v,w}, successive application of the binary logical operation (x,y)⟼(x∨y)⊥ generates a set of elementary propositions which is countable infinite and dense in the projective plane if and only if no vector of the basis {u,v,w} is orthogonal to the other ones.
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].