Quantum Tomography from the Evolution of a Single Expectation

Conventional quantum tomography assumes a high degree of control over the system in question in order to measure many observables. On the other hand, a classical dynamical system may be reconstructed from a time series of a single observable by means of delay embeddings. Inspired by this, we investigate the possibility of tomography of a quantum system undergoing a known homogenous time evolution when we have control over only the time of measurement. Remarkably, full quantum state tomography is possible for every non-trivial binary measurement evolved by any quantum channel, bar a null set. 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 need for non-trivial noise. We explore various further aspects of the problem, including prior information, expectation bounds for finite statistics, and recovery of an infinite time series of expectation values from a finite one given only spectral properties of the evolution.