Continuous decomposition of quantum measurements via Hamiltonian feedback

TL;DR

This paper characterizes which quantum measurements can be implemented continuously through a feedback-controlled process involving weak probe interactions and Hamiltonian tuning, revealing a specific algebraic structure of allowable Hamiltonians.

Contribution

It introduces a framework for decomposing generalized quantum measurements into continuous processes using feedback and characterizes the necessary algebraic structure of the Hamiltonians involved.

Findings

The allowed Hamiltonians form a finite dimensional Jordan algebra.

The scheme enables a large class of generalized measurements to be performed continuously.

The process results in a stochastic evolution with a one-dimensional random walk structure.

Abstract

We characterize the set of generalized quantum measurements that can be decomposed into a continuous measurement process using a stream of probe qubits and a tunable interaction Hamilto- nian. Each probe in the stream interacts weakly with the target quantum system, then is measured projectively in a standard basis. This measurement result is used in a closed feedback loop to tune the interaction Hamiltonian for the next probe. The resulting evolution is a stochastic process with the structure of a one-dimensional random walk. To maintain this structure, and require that at long times the measurement outcomes be independent of the path, the allowed interaction Hamil- tonians must lie in a restricted set, such that the Hamiltonian terms on the target system form a finite dimensional Jordan algebra. This algebraic structure of the interaction Hamiltonians yields a large class of generalized measurements that can be continuously performed by our scheme, and we fully describe this set.