Process tomography for unitary quantum channels

TL;DR

This paper demonstrates that quantum process tomography for unitary channels can be significantly optimized using interactive observables, reducing the measurement complexity from quartic to quadratic in the system dimension.

Contribution

The authors introduce the concept of interactive observables and prove that unitary channels can be uniquely identified with only O(d^2) measurements, improving efficiency over previous methods.

Findings

Unitary channels can be identified with O(d^2) interactive observables.

Channels with up to q Kraus operators require O(qd^2) measurements.

Explicit constructions of large subspaces of Hermitian matrices underpin the results.

Abstract

We study the number of measurements required for quantum process tomography under prior information, such as a promise that the unknown channel is unitary. We introduce the notion of an interactive observable and we show that any unitary channel acting on a $d$-level quantum system can be uniquely identified among all other channels (unitary or otherwise) with only $O(d^2)$ interactive observables, as opposed to the $O(d^4)$ required for tomography of arbitrary channels. This result generalizes, so that channels with at most $q$ Kraus operators can be identified with only $O(qd^2)$ interactive observables. Slight improvements can be obtained if we wish to identify such a channel only among unital channels or among other channels with $q$ Kraus operators. These results are proven via explicit construction of large subspaces of Hermitian matrices with various conditions on rank, eigenvalues, and partial trace. Our constructions are built upon various forms of totally nonsingular matrices.