College of Liberal Arts & Sciences

# Math Physics Seminar

**Abstract:**

The mathematical framework for infinite dimensional quantum systems suffers from a fundamental chicken-and-egg problem: Do states come first, and observables are functionals on the state space, or does the observable algebra come first, and states are just expectation value functionals? This corresponds to the choice of a $C^*$-algebraic vs $W^*$-algebraic approach. In the abelian case it is the decision to emphasize either the topology or the measure theory of the underlying parameter space.

A quantum classical hybrid must surely be described in some sort of tensor product of operator algebras, but which? I describe here a structure designed as a general but useful framework for quasi-free operations, including preparation, measurement (returning a partially classical output), continual measurement, dynamical semigroups, feedback of classical information by linear controls and many more. One would like to include pure states on the classical side (speaking for a $C^*$-approach), but no Stone-Cech-type non-constructive states at infinity. On the quantum side one would like to use standard ($W^*$) quantum mechanics, including Weyl operators. For every channel in the category there should be a well-defined Schr"odinger picture (action on states) and a Heisenberg picture (action on observables). This is achieved by choosing a third option in the chicken-and-egg problem. Namely, one starts from a subalgebra $A$ of observables without unit: $C0(X)$ in the classical case with a locally compact $X$, and the compacts on the quantum side. At this level the $C^*$-tensor product works well. This gives us pure states in $A^*$, but the state space is not closed, because $X$ is not compact. Observables proper are now in the bidual $A^{**}$, with some manageable subspaces singled out by monotone limits. This includes the Weyl operators and, more generally, elements which are continuous under phase space translations. The resulting framework for quasifree operations is quite powerful, easy to use, and has some strong general results, like a factorization of any operation into tensoring with a noisy state and a noiseless part.