Dimensional reduction without symmetries: an algorithm for finding irreducible representations of operators and its applications

Presenting Author: Oleg Kabernik, University of British Columbia
Contributing Author(s): Jason Pollack, Ashmeet Singh

We propose a general, not inherently numeric, algorithm for finding irreducible representations of operator algebras in applications where symmetry considerations may fall short. This algorithm is motivated by applications where, given a set of operators that is not necessarily characterized by any known symmetries, it is essential to identify the invariant subspaces on which all these operators act irreducibly. One such application is the problem of identifying how the quantum state reduces given a restriction on observables that can be measured. Such state reductions can account for limited resolution measurements, inability to individually address the particles, inaccessible external environments, lack of reference frames or any other scenario where operational constraints can be specified as a restriction of the allowed observables. Another potential application is reduction of dynamics that does not rely on symmetries to identify the simplifying structure. With this approach we can reduce the dynamics by considering the algebras generated by the operators from which the Hamiltonian or the quantum channel is constructed. In particular, we will show how the action of parameterized Hamiltonians with “tunable” terms can be reduced to lower dimensional (virtual) subsystems if the algebra generated by the terms is not trivial.

Read this article online: https://arxiv.org/abs/1909.12851

(Session 5 : Saturday from 5:00pm - 7:00pm)


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