Quantum probability addresses the challenge of merging the apparently distinct domains of algebra and probability, in view of physical applications. Those fields typically involve radically different types of thinking, which are not often mastered simultaneously. Indeed, the framework of algebra is often abstract and noncommutative while probability addresses the “fluctuating” but classical notion of a random variable, and requires a good amount of statistical intuition. On the other hand, those two fields combined yield natural applications, for e.g., quantum mechanics. On a more general level, the noncommutativity of operations is a common real life phenomenon which can be connected to classical probability via quantum mechanics. Algebraic approaches to probability also have applications in theoretical computer science, cf. e.g., [39].

In the framework of this noncommutative (or algebraic) approach to probability, often referred to as quantum probability, real-valued random variables on a classical probability space become special examples of noncommutative (or quantum) random variables. For this, a real-valued random variable on a probability space (Ω, F, P) is viewed as (unbounded) self-adjoint multiplication operators acting on the Hilbert space L2(Ω, P). This has led to the suggestion by several authors to develop a theory of quantum probability within the framework of operators and group representations in a Hilbert space, cf. [87] and references therein.

In this monograph, our approach is to focus on the links between the commutation relations within a given noncommutative algebra A on the one hand, and the combinatorics of the moments of a given probability distribution on the other hand. This approach is exemplified in Chapters 1–3. In this respect our point of view is consistent with the description of quantum probability by P.A. Meyer in [80] as a set of prescriptions to extract probability from algebra, based on various choices for the algebra A.

For example, it is a well-known fact that the Gaussian distribution arises from the Heisenberg–Weyl algebra which is generated by three elements {P, Q, I} linked by the commutation relation

It turns out similarly that other infinitely divisible distributions such as the gamma and continuous binomial distributions can be constructed via noncommutative random variables using representations of the special linear algebra sl2(R), …

This chapter introduces the notion of joint (Wigner) density of random variables, for future use in quantum Malliavin calculus. For this we will rely on functional calculus on general Lie algebras, starting with the Heisenberg– Weyl algebra. We also consider some applications to quantum optics and timefrequency analysis.

Joint moments of noncommuting random variables

The notion of moment ⟨ϕ, Pnψ⟩ of order n of P in the state ⟨ϕ, ·ψ⟩ is well understood as the noncommutative analogue of E[Xn]. Indeed, in the single variable case, various functional calculi are available for elements of an algebra. In particular,

1. • polynomials can be applied to any element in an algebra;

2. • holomorphic functions can be applied to elements of a Banach algebra provided the function is holomorphic on a neighborhood of the spectrum of the element;

3. • continuous functions can be applied to normal elements of C*-algebras provided they are continuous on the spectrum of the element;

4. • measurable (i.e., Borel) functions can be applied to normal elements in a von Neumann algebra (e.g., B(h)), provided they are measurable on the spectrum of the operator.

In addition, the Borel functional calculus can be extended to unbounded normal operators (where normal means that XX∗ and X∗X have the same domain and coincide on this domain).

All the above functional calculi work due to a restriction to a commutative subalgebra. In that sense, Weyl calculus is fundamentally different because it applies to functions of noncommuting operators.

We now need to focus on the meaning of joint moments. For example, which is the noncommutative analogue of E[XY]? A possibility is to choose ⟨ϕ,PQψ⟩ as the the noncommutative analogue of E[XY], however this would lead to ⟨ϕ,QPψ⟩ as the the noncommutative analogue of E[YX] and we have E[XY] = E[YX] while

due to noncommutativity. A way to solve this contradiction can be to define the first joint moment as

Next, how can we define the analogue of E[XY2]? Proceeding similarly, a natural extension could be

Then what is the correct way to construct E[X2Y3]? Clearly there is a combinatorial pattern which should be made explicit.