Dilation Theory

Eugene P. Gross

Partha Sarathi Chakraborty, Debashish Goswami, Kalyan B. Sinha

A covariant version of the dilation theory for quantum dynamical semigrops is established, including both Bhat-Parthasarathy type and Evans-Hudson type dilation. It is shown in particular that every uniformly continuous quantum dynamical semigroup on a separable C*-algebra or a von Neumann algebra, which is covariant under a suitable group-action, admits a covariant Evans-Hudson dilation in some appropriate Fock space. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

K. T. Hecht

Nicholas Gauguin Houghton-Larsen

The motivation for this thesis was to recast quantum self-testing [MY98,MY04] in operational terms. The result is a category-theoretic framework for discussing the following general question: How do different implementations of the same input-output process compare to each other? In the proposed framework, an input-output process is modelled by a causally structured channel in some fixed theory, and its implementations are modelled by causally structured dilations formalising hidden side-computations. These dilations compare through a pre-order formalising relative strength of side-computations. Chapter 1 reviews a mathematical model for physical theories as semicartesian symmetric monoidal categories. Many concrete examples are discussed, in particular quantum and classical information theory. The key feature is that the model facilitates the notion of dilations. Chapter 2 is devoted to the study of dilations. It introduces a handful of simple yet potent axioms about dilations, one of which (resembling the Purification Postulate [CDP10]) entails a duality theorem encompassing a large number of classic no-go results for quantum theory. Chapter 3 considers metric structure on physical theories, introducing in particular a new metric for quantum channels, the purified diamond distance, which generalises the purified distance [TCR10,Tom12] and relates to the Bures distance [KSW08a]. Chapter 4 presents a category-theoretic formalism for causality in terms of '(constructible) causal channels' and 'contractions'. It simplifies aspects of the formalisms [CDP09,KU17] and relates to traces in monoidal categories [JSV96]. The formalism allows for the definition of 'causal dilations' and the establishment of a non-trivial theory of such dilations. Chapter 5 realises quantum self-testing from the perspective of chapter 4, thus pointing towards the first known operational foundation for self-testing.

Ralph E. Christoffersen

Kalyan B. Sinha, Debashish Goswami

In Chapter 6 we built a theory of quantum stochastic dilation ‘naturally’ associated with an arbitrary Q.D.S. on a von Neumann or C * -algebra with bounded generator. There the computations involved C * or von Neumann Hilbert modules, using the results of [24], map-valued quantum stochastic processes on modules and quantum stochastic integration with respect to them, developed in Chapter 5. It is now natural to consider the case of a Q.D.S. with unbounded generator and ask the same questions about the possibility of dilation. As one would expect, the problem is too intractable in this generality and we need to impose some further structures on it. In this chapter we shall consider a few classes of such Q.D.S. and try to construct H–P and E–H dilation for them. At first, we shall work under the framework of a Lie group action on the underlying algebra, and consider covariant Q.D.S. For H–P dilation, symmetry with respect to a trace is also assumed, whereas a general theory for E–H dilation has been built under the assumption of covariance under the action of a compact group, but without it being symmetric. Then, in the last section, we deal with a class of Q.D.S. on the U.H.F. algebra, described in Chapter 3. In this case, E–H dilation is constructed by a direct iteration using some natural estimates. However, what is common to the methods used in constructing dilation for the different kinds of Q.D.S. mentioned above is the use of a natural locally convex topology, in which the generator (unbounded in the norm topology) is continuous.