Quantum tomography from the evolution of a single expectation

TL;DR

This paper explores how full quantum state tomography can be achieved through the evolution of a single expectation value over time, emphasizing the role of non-trivial noise and spectral analysis.

Contribution

It introduces a method for quantum tomography based on the evolution of a single expectation value, extending Takens' embedding theorem to quantum channels.

Findings

Full tomography possible with evolved binary measurements and quantum channels.

Unitary evolution with simple noise is insufficient for tomography beyond qubits.

Spectral properties enable recovery of expectation values from finite data.

Abstract

We investigate the possibility of performing full quantum tomography based on the homogeneous time evolution of a single expectation value. Remarkably, every non-trivial binary measurement evolved by any quantum channel, except for a null set, in principle enables full quantum state tomography. We show that this remains true when restricted to Lindblad semigroups, although unitary evolution -- even with added simply depolarizing noise -- is insufficient beyond the qubit case, highlighting the necessity of non-trivial noise. We establish an analog of Takens' embedding theorem for quantum channels, which incorporates prior information into the framework. We also provide estimation bounds for finite statistics and analyze the feasibility of recovering an infinite time series of expectation values from a finite one using only spectral properties of the evolution.