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LIMITING OPERATIONS FOR SEQUENCES OF QUANTUM RANDOM VARIABLES AND A CONVERGENCE THEOREM FOR QUANTUM MARTINGALES

We study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of Lebesgue’s dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits non-classical behaviour; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.


Introduction
In 2011, while studying classical and non-classical convexity properties of the space of positive operator valued measures on the Borel sets of a locally compact Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, Farenick, Plosker, and Smith [4] introduced a transform 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 was achieved by using an operator valued integral in which 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.
Farenick and Kozdron [2] helped provide a better mathematical understanding of quantum probability by introducing a quantum analogue for the expected value E ν [ψ] of a quantum random variable ψ relative to a quantum probability measure ν using the operator valued integral of [4]. This led to theorems for a change of quantum measure and a change of quantum variables. They also introduced a quantum conditional expectation which resulted in quantum versions of some standard identities for Radon-Nikodým derivatives, and allowed them to formulate and prove a quantum analogue of Bayes' rule.
It is a basic fact of functional analysis that if ψ : X → C is an essentially bounded function on a probability space (X, F(X), µ), then the essential range of ψ is precisely the spectrum of ψ, where one considers ψ as an element of the von Neumann algebra L ∞ (X, µ). Recently, Farenick, Kozdron, and Plosker [3] arrived at a similar result for essentially bounded quantum random variables on quantum probability spaces using higher dimensional spectra. Their investigation of quantum variance also involved notions from spectral theory, and they discovered that the quantum moment problem admits a characterisation entirely within spectral terms.
In the present work, we build on these earlier results by considering for the first time limiting operations for sequences of quantum random variables and quantum probability measures including a quantum analogue of Lebesgue's dominated convergence theorem and a discrete Fubini-type theorem. As in those earlier investigations, the noncommutativity of operator algebra leads to some structure that simply does not appear in the classical setting. Using the quantum conditional expectation of Farenick and Kozdron [2], we also establish a quantum martingale convergence theorem for quantum martingales obtained by conditioning on a fixed quantum random variable. This theorem is of particular interest since it strongly exhibits non-classical behaviour; even though the limit of the martingale exists and is unique, it is not identifiable. However, we provide a partial classification of the limit through a study of the space of quantum random variables having quantum expectation zero.
The outline of the paper is as follows. In Section 2 we introduce our notation and summarize the relevant results of [2], [3], and [4]. We provide our first limiting results in Section 3 and then study quantum random variables having quantum expectation zero in Section 4. Finally, in Section 5 we develop our quantum martingale convergence theorem.

Notation and background results
We will always write H for a d-dimensional Hilbert space, B(H) for the C * -algebra of linear operators acting on H, and B(H) + for the cone of positive operators. The predual of B(H) is denoted by T (H), the space of trace-class operators. Since H is finite dimensional, B(H) and T (H) coincide as sets. Finally, X shall denote a locally compact Hausdorff space and F(X) a σ-algebra of subsets of X. A particular σ-algebra of interest is O(X), the Borel sets of X. A density operator, or state, on H is a positive trace-class operator ρ such that Tr(ρ) = 1; the set of all density operators is denoted by S(H). By a quantum effect we mean a positive operator h ∈ B(H) + with the property that every eigenvalue λ of h satisfies 0 ≤ λ ≤ 1, and we let Eff(H) denote the set of quantum effects. Note that every state ρ ∈ S(H) is also a quantum effect. A set function ν : If, in addition, ν(X) = 1 ∈ B(H), then ν is called a quantum probability measure. The convergence in (1) above is normally assumed to be with respect to the ultraweak topology; however, because B(H) has finite dimension, the convergence in (1) may be taken with respect to any of the usual operator topologies on B(H). The POVM ν : F(X) → B(H) induces the classical (i.e., scalar valued) measure µ via µ = 1 d Tr •ν, where Tr is the canonical trace on B(H). Note that if ν is a quantum probability measure, then µ is a classical probability measure. We call the triple (X, F(X), ν) a quantum probability space.
A function ψ : X → B(H) is said to be measurable (i.e., a quantum random variable) if, for every pair ξ, η ∈ H, the complex valued function x → ψ(x)ξ, η is measurable (i.e., a random variable) in the classical sense. In fact, it is known [3]  The predual of the von Neumann algebra L ∞ (X, µ)⊗B(H) is given by L 1 T (H) (X, µ); see Theorem IV.7.17 of [9]. In particular, if Ψ ∈ L ∞ (X, µ)⊗B(H), then there is a bounded quantum random variable ψ : X → B(H) such that, for each f ∈ L 1 T (H) (X, µ), the complex number Ψ(f ) is given by Although ψ is not unique, it is unique up to a set of µ-measure zero. We therefore identify Ψ and ψ and consider the elements of L ∞ (X, µ)⊗B(H) to be bounded quantum random variables ψ : We end this section by stating a number of theorems and definitions from [2], [3], and [4] relevant for our purposes. Recall that if ν 1 and ν 2 are both positive operator valued measures on (X, F(X)), then ν 2 is absolutely continuous with respect to ν 1 , written ν 2 ac ν 1 , if ν 2 (E) = 0 for every E ∈ F(X) with ν 1 (E) = 0. Furthermore, if µ is a classical measure, then we can always view µ as the scalar valued POVM µ · 1.
Theorem 2.1. If ν is a POVM on (X, F(X)), then ν is absolutely continuous with respect to the induced classical measure µ, and there exists an F(X)-measurable function dν dµ such that for all E ∈ F(X) and all ξ ∈ H. The function dν dµ is called the principal Radon-Nikodým derivative of ν and is a positive operator for µ-almost all x ∈ X.
is µ-integrable. The integral of a ν-integrable function ψ : X → B(H) is defined to be the unique operator acting on H having the property that for every density operator ρ. for every E ∈ F(X). Moreover, and is called the non-principal Radon-Nikodým derivative of ν 2 with respect to ν 1 .
Recall from [6] and [8] that if a, b ∈ B(H) + are both invertible, then the geometric mean of a and b is the positive operator a#b defined by a#b = a 1/2 (a −1/2 ba −1/2 ) 1/2 a 1/2 . If a, b ∈ B(H) + are non-invertible, then a#b is defined by with convergence in the strong operator topology. If ν 1 and ν 2 are both quantum probability measures with ν 2 ac ν 1 and if ψ : X → B(H) is a quantum random variable, then we define In particular, is a quantum probability measure, then the quantum expectation of ψ with respect to ν is the map E ν : The following example shows that one can view E ν [ψ] as a quantum averaging of ψ. A version of this first appeared in [4]; see also Theorem 2.3(4) of [2].

Continuity of quantum expectation
In this section we establish a natural quantum analogue of the classical dominated convergence theorem, namely Theorem 3.4, continuity of quantum expectation, along with some related results.
It is an easy fact that the ultraweak µ-almost sure limit ψ of the previous definition is itself a quantum random variable.
Lemma 3.2. Let ψ : X → B(H) and suppose that {ψ n } ∞ n=1 is a sequence of quantum random variables. If ψ n converges ultraweakly µ-almost surely to ψ, then ψ is a quantum random variable.
Proof. Since ψ n converges ultraweakly µ-almost surely to ψ, it follows that Tr(ρψ n (x)) → Tr(ρψ(x)) for all ρ ∈ S(H) and µ-almost all x ∈ X. But since each Tr(ρψ n (x)) is a complex valued random variable, the limit of the sequence {Tr(ρψ n (x))} ∞ n=0 converges to a complex valued random variable, namely Tr(ρψ(x)) for each x ∈ X, and therefore ψ is a quantum random variable.
Proof. For ρ ∈ S(H) and x ∈ X, letρ x = Tr ρ dν dµ , and notice thatρ x ∈ S(H). Therefore, using the assumption that ψ n converges ultraweakly µ-almost surely to ψ along with properties of the trace functional, as required.
We now prove the main result of this section, namely continuity of quantum expectation.
is a sequence of ν-integrable quantum random variables that converges ultraweakly µ-almost surely to ψ, and if there exists a µ-integrable random variable Z : X → C such that Proof. Begin by defining the sequence of complex valued random variables {ψ As a first application of the continuity of quantum expectation we prove that, under certain conditions, quantum expectation is linear over infinite sums. In fact, this could even be considered as a special case of a quantum Fubini-type theorem.
By Lemma 3.2, ϕ is a quantum random variable, and by Theorem 3.4, ϕ is ν-integrable and

However, finite additivity of quantum expectation gives
ψ n as required.
As an example of the type of calculations possible using the previous result, consider the following.
The following diagram describes the relationships between these statements.
Since X is finite, the principal Radon-Nikodým derivative is easily computed, namely Hence, in this example (A) holds, but neither (C) nor (D) hold. Moreover, one can check that

A quantum martingale convergence theorem
In this section we establish a quantum martingale convergence theorem for quantum martingales obtained by conditioning on a fixed quantum random variable. Recall that a stochastic process {M j } ∞ j=0 defined on a filtered probability space is a martingale with respect to the filtration  When the martingale is obtained by conditioning on a fixed random variable, the martingale convergence theorem takes the following form; see Corollary 3.6.9 of [1].
Corollary 5.2. If X is a random variable on the filtered probability space (Ω, F, {F j } ∞ j=1 , P) and satisfies E[ |X| ] < ∞, then the martingale M j = E[X|F j ] converges both almost surely and in We now turn our attention to quantum conditional expectation. The following result summarizes the relevant facts from [2] that we need; see, in particular, the proof of Theorem III.1.
Theorem 5.3. Suppose that (X, O(X), ν) is a quantum probability space, and that ψ : X → B(H) + is a ν-integrable quantum random variable with E ν [ψ] = 0. If F(X) is a sub-σ-algebra of O(X), then there exists a function ϕ : for every E ∈ F(X).
We call ϕ a version of quantum conditional expectation of ψ given F(X) relative to ν and write ϕ = E ν [ψ|F(X)]. Moreover, ifφ is any other ν-integrable F(X)-measurable function satisfying E ν [ψχ E ] = E ν [φχ E ] for every E ∈ F(X), then ν({x ∈ X : ϕ(x) =φ(x)}) = 0. Thus, instead of saying "ϕ = E ν [ψ|F(X)] ν-almost surely" we identify different versions and say that E ν [ψ|F(X)] is the quantum conditional expectation of ψ given F(X) relative to ν. In fact, if ν = ν| F (X) is the restriction of ν to F(X), andν Since ν -measure zero sets have ν-measure zero, setting implies ϕ : X → B(H) + for all x ∈ X. We are now able to prove the important tower property for quantum conditional expectation. Note that this was not considered in [2].
To prove the theorem, we will verify that The first equality in (7) follows immediately from the fact that ϕ f is G(X)-measurable and F(X) ⊆ G(X). As for the second equality in (7), observe that if F ∈ F(X) and G ∈ G(X), In analogy with the classical definition, we now state the definition of a quantum martingale.
Definition 5.5. Let (X, O(X), ν) be a quantum probability space. A sequence of quantum random variables {ϕ j } ∞ j=0 is called a quantum martingale with respect to the filtration {F j (X)} ∞ j=0 if (a) ϕ j is F j (X)-measurable for all j, (b) ϕ j is ν-integrable for all j, and (c) E ν [ϕ j+1 |F j (X)] = ϕ j for all j.
It is also important to know that a quantum martingale is obtained by conditioning on a fixed quantum random variable.
Our next preliminary result relates the quantum conditional expectation E ν [ψ|F(X)] with the family of classical conditional expectations E µ [ψ ρ |F(X)] for ρ ∈ S(H).
We are now in a position to prove the main result of this paper, namely a quantum martingale convergence theorem for the quantum martingale ϕ j = E ν [ψ|F j (X)]. Although we will prove that the sequence {ϕ j } ∞ j=0 has a unique limit, in contrast to the classical situation, the value of the limiting random variable ϕ ∞ cannot be determined in general. In fact, all that can be said is that ϕ ∞ and E ν [ψ|F ∞ (X)] differ by a quantum random variable Φ satisfying Φ ρ = 0 for all ρ ∈ S(H).
We will now study the set of possible limits from our quantum martingale convergence theorem.
Since this equality holds for all ρ ∈ S(H), it follows that (Ψ 2 − Ψ 1 ) dν dµ = 0 as required. Following the same reasoning in reverse gives the theorem.
We can now use our results from Section 4 to study Γ ν,ψ . We know that if Φ is a quantum random variable then Φ ρ = 0 implies E ν [Φ] = 0 whereas the converse is not necessarily true.