Quantum Tomography and Schwinger's Picture of Quantum Mechanics

TL;DR

This paper explores quantum state reconstruction using Schwinger's groupoid framework, focusing on spin tomography and providing a formula for state reconstruction based on groupoid structures, especially for finite sets of outcomes.

Contribution

It introduces a novel reconstruction formula within Schwinger's groupoid picture, linking groupoid bisections to quantum state tomography, particularly for finite outcome sets.

Findings

Derived a reconstruction formula for states on the groupoid-algebra.

Established a frame using groupoid bisections for tomographic reconstruction.

Analyzed the case of outcomes as integers modulo n, with n prime, and identified a quorum.

Abstract

In this paper the problem of tomographic reconstruction of states is investigated within the so-called Schwinger's picture of Quantum Mechanics in which a groupoid is associated with every quantum system. The attention is focused on spin tomography: In this context the groupoid of interest is the groupoid of pairs over a finite set. In a nutshell, this groupoid is made up of transitions between all possible pairs of outcomes belonging to a finite set. In addition, these transitions possess a partial composition rule, generalizing the notion of groups. The main goal of the paper consists in providing a reconstruction formula for states on the groupoid-algebra associated with the observables of the system. Using the group of bisections of this groupoid, which are special subsets in one-to-one correspondence with the outcomes, a frame is defined and it is used to prove the validity of the tomographic reconstruction. The special case of the set of outcomes being the set of integers modulo n, with n odd prime, is considered in detail. In this case the subgroup of discrete affine linear transformations, whose graphs are linear subspaces of the groupoid, provides a \textit{quorum} in close analogy with the continuos case.