Hilbert Space Structure Induced by Quantum Probes

Abstract

It is unrealistic to control all of the degrees of freedom of a high-dimensional quantum system. Here, we consider a scenario where our direct access is restricted to a small subsystem S that is constantly interacting with the rest of the system E. What we investigate is the fundamental structures of the Hilbert space and the algebra of hamiltonians that are caused solely by the restrictedness of the direct control. One key finding is that hamiltonians form a Jordan algebra, and this leads to a significant observation that there is a sharp distinction between the cases of dimHS ≥ 3 and dimHS = 2 in terms of the nature of possible operations in E. Since our analysis is totally free from specific properties of any physical systems, it would form a solid basis for obtaining deeper insights into quantum control related issues, such as controllability and observability.

Keywords

• quantum control
• Jordan algebra
• dynamical Lie algebra