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arXiv:1309.0559 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Non-abelian Weyl commutation relations and the series product of quantum stochastic evolutions

D. Gwion Evans, John Edward Gough, M. R. James

Published Oct 22, 2012• 2 citations
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Abstract

We show that the series product, which serves as an algebraic rule for connecting state-based input-output systems, is intimately related to the Heisenberg group and the canonical commutation relations. The series product for quantum stochastic models then corresponds to a non-abelian generalization of the Weyl commutation relation. We show that the series product gives the general rule for combining the generators of quantum stochastic evolutions using a Lie-Trotter product formula.

DOI
10.1098/rsta.2011.0525

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We show that the series product, which serves as an algebraic rule for connecting state-based input-output systems, is intimately related to the Heisenberg group and the canonical commutation relations. The series product for quantum stochastic models then corresponds to a non-abelian generalization of the Weyl commutation relation. We show that the series product gives the general rule for combining the generators of quantum stochastic evolutions using a Lie-Trotter product formula.
If the state space of a quantum system is finite dimensional it may be considered as the carrier space of an irreducible unitary representation of some compact Lie group G. In this case the set of operators may be put in one-to-one correspondence with a class of functions defined on the group parameters (Q representation). For these functions a binary relation is derived that corresponds to the product of two operators. The general formalism is applied to G=SU(2) (spin coherent states) and this result is used to derive a product formula for the non-compact Heisenberg-Weyl group (coherent oscillator states.).
A theory is developed of product integrals of the form ∏a<s<b ∏c<t<d(1+g[h] (ds,dt)). Here [a,b[ and [c,d[ are disjoint finite subintervals of R+, and g[h] is a formal power series in the indeterminate h whose constant term is zero and whose coefficients are elements of L⊗L, where ℒ is the space of basic differentials of a multidimensional quantum stochastic calculus. The product integrals are themselves formal power series in h whose coefficients are finite sums of iterated stochastic integrals against the elements of ℒ. They are symmetrized in such a way that ∏a<s<b ∏c<t<d(1+g[h](ds,dt)) is the image, obtained by applying the representation J[a,b[⊗J[c,d[ to the coefficients, where J[a,b[ is the representation canonically associated with the interval [a,b[, of a formal power series ∏ ∏(1+dg[h]) whose coefficients lie in U⊗U, where 𝒰 is the universal enveloping algebra of the Lie algebra ℒ. It is shown that the naturally conjectured multiplication rule, analogous to the multiplication rule for simple product integrals, holds in the commutative case.
Motivated by the search for solutions of the quantum Yang–Baxter equation, an algebraic theory of quantum stochastic product integrals is developed. The product integrators are formal power series in an indeterminate h whose coefficients are elements of the Lie algebra ℒ labelling the usual integrators of a many-dimensional quantum stochastic calculus. The product integrals are also formal power series in h, whose coefficients are finite iterated additive stochastic integrals which act on the exponential domain in the Fock space of the calculus and which represent elements of the universal enveloping algebra 𝒰 of ℒ. They obey a multiplication rule suggested by the quantum Itô product formula, and are characterized among all such formal power series by a grouplike property.
Abstract We introduce the notion of stochastic product as a binary operation on the convex set of quantum states (the density operators) that preserves the convex structure, and we investigate its main consequences. We consider, in particular, stochastic products that are covariant wrt a symmetry action of a locally compact group. We then construct an interesting class of group-covariant, associative stochastic products, the so-called twirled stochastic products . Every binary operation in this class is generated by a triple formed by a square integrable projective representation of a locally compact group, by a probability measure on that group and by a fiducial density operator acting in the carrier Hilbert space of the representation. The salient properties of such a product are studied. It is argued, in particular, that, extending this binary operation from the density operators to the whole Banach space of trace class operators, this space becomes a Banach algebra, a so-called twirled stochastic algebra . This algebra is shown to be commutative in the case where the relevant group is abelian. In particular, the commutative stochastic products generated by the Weyl system are treated in detail. Finally, the physical interpretation of twirled stochastic products and various interesting connections with the literature are discussed.