This paper i a continuation of the first part under the same title. The author studies a joint distribution in
σ-finite measures for noncompatible observables of a quantum logic defined on some system of
σ-independent Boolean sub-
σ-algebras of a Boolean
σ-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.
The notion of a joint distribution in
σ-finite measures of observables of a quantum logic defined on some system of
σ-independent Boolean sub-
σ-algebras of a Boolean
σ-algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.
This paper deals with the topics of modeling joint distributions on a generalized probability space. An algebraic structure known as quantum logic is taken as the basic model. There is a brief summary of some earlier published findings concerning a function s-map, which is a mathematical tool suitable for constructing virtual joint probabilities of even non-compatible propositions. The paper completes conclusions published in 2020 and extends the results for three or more random variables if the marginal distributions are known. The existence of a (n+1)-variate joint distribution is shown in special cases when the quantum logic consists of at most n blocks of Boolean algebras.
We analyze finitely additive orthogonal states whose values lie in a real Hilbert space. We call them
h-states. We first consider the important case of
h-states on a standard Hilbert logic
L(H) of projectors in
H-we describe the
h-states
s:
L(H1)→H2, where
dim H2≤ \text {dim} H_1 < \infty$. In particular, we show that, up to a unitary mapping, every
h-state
s:
L(H)→H(3≤dim H<∞) has to be concentrated on a one-dimensional projection. We also study the
h-states
s:
L(H1)→H2 for the case of
dim H1=∞. The results of the first part complement the papers [10] and [13]. In the second part we investigate
h-states on general logics. Being motivated by the quantum axiomatics, the main question we ask here is as follow: Given a Hilbert space
H with
dim H<∞, what is the class of such logics
L that, for any Boolean subalgebra
B of
L, every
h-states
s:
B→H extends over
L? We answer this question by finding a simple condition characterizing the class (Theorem 3.4]. It turns out that the class is considerably large-it contains e.g. all concrete logics-but, on the other hand, it does not contain all finite logics (we construct a counterexample in the appendix).
In this note, we analyze joint probability distributions that arise from outcomes of sequences of quantum measurements performed on sets of quantum states. First, we identify some properties of these distributions that need to be fulfilled to get a classical behavior. Secondly, we prove that a joint distribution exists iff measurement operators "on-state" permute (permutability is the commutativity of more than two operators). By "on-state" we mean properties of operators that hold only on a subset of states in the Hilbert space. Then, we disprove a conjecture proposed by Carstens, Ebrahimi, Tabia, and Unruh (eprint 2018), which states that the property of partial on-state permutation implies full on-state permutation. We disprove this conjecture with a counterexample where pairwise "on-state" commutativity does not imply on-state permutability, unlike in the case of commutativity for all states in the Hilbert space. Finally, we explore the new concept of on-state commutativity by showing a simple proof that if two projections almost on-state commute, then there is a commuting pair of operators that are on-state close to the originals. This result was originally proven by Hastings (Communications in Mathematical Physics, 2019) for general operators.