Dilation of states and processes in operational-probabilistic theories

Abstract

This paper provides a concise summary of the framework of operational-probabilistic theories, aimed at emphasizing the interaction between category-theoretic and probabilistic structures. Within this framework, we review an operational version of the GNS construction, expressed by the so-called purification principle [12], which under mild hypotheses leads to an operational version of Stinespring's theorem.

Microcanonical thermodynamics in general physical theories

“…This means that all the natural candidates for the sets of free operations induce the same notion of resource. This resource is generally called purity, and plays a fundamental role in many quantum protocols [53].In this paper we extend the paradigm of microcanonical thermodynamics from quantum theory to arbitrary physical theories [54][55][56][57][58][59][60][61]. We propose two minimal requirements a probabilistic theory must satisfy in order to support a microcanonical description, and, when these requirements are satisfied, we provide a general operational definition of random reversible, noisy, and unital operations.…”

“…where  = { } x x n 1 are reversible transformations on A, chosen so that n 1 of the channels are equal to  1 , n 2 are equal to  2 , and so on. Since the theory satisfies Purification, the channel  has a reversible extension [57,58] Applying a ¢ † i to both sides, we obtain…”

Necessary and sufficient conditions on measurements of quantum channels

“…From the proof of proposition 5 we have The theory of quantum supermaps, where generic evolutions of quantum maps are described by supermaps, can be analyzed using the framework of operational-probabilistic theories (OPTs) [57,[60][61][62][65][66][67], which is a formalism to describe arbitrary physical theories admitting probabilistic processes. OPTs differ from the convex set approach to general probabilistic theories [68][69][70] in that they take the composition of physical processes and systems as a primitive.…”

Purity through Factorisation

“…Given any collection A of morphisms, we can generate a pair (L , R) by letting R = A and then L = R. This pair then has the property that L = R and R = L . 3 Thus for any collection of morphisms A we can generate a pair (L , R) and then ask if this pair happens to form a factorisation system. Definition 3.…”

“…The goal of this article is to give an operational definition of purity. In contrast to other approaches [4,6,3,5,10,11], our definition merely needs the structure of symmetric monoidal categories. That is, we define what it means for a morphism to be pure using only its relationships with the other morphisms in the category (so without daggers or dual objects), and without any reference to the interpretation of morphisms as being processes between systems.…”