Quantum tomography from the evolution of a single expectation

TL;DR

This work investigates whether full quantum tomography can be achieved from the homogeneous evolution of a single expectation value in finite-dimensional systems. It shows a no-go for unitary, noise-free dynamics beyond the qubit and establishes Takens-inspired embedding results indicating when a finite time-series suffices to reconstruct the full dynamics, with bounds tied to spectral data and Minkowski dimension. The authors prove time-series extension theorems for both discrete and continuous time, derive no-go results under limited information, and provide conditions under which noisy evolutions generically enable full tomography for states or observables, including detailed qubit-specific phenomena. They also analyze statistical stability under finite data, deriving explicit error bounds, and discuss the special role of qubits where tomography can be optimal or become feasible with minimal noise. The work highlights a spectrum-driven pathway to tomography and opens questions about optimal data sampling, channel choice, and noise design for practical tomography protocols.

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.

Figures (3)

• Figure 1: Time-series extension. Suppose a sufficient number of consecutive data points of a homogeneous discrete-time evolution of a $d$-dimensional quantum system w.r.t. a known quantum channel is given. Then the series can be extended to its infinite counterpart by a linear map that does neither depend on the state, nor on the observable (Thm.\ref{['thm:seriesextension']}). Similar applies to continuous time evolution: generically, $d^2$ arbitrarily spaced data points suffice to complete the graph without knowing state or observable (Thm.\ref{['thm:Lextension']}).
• Figure 2: Simplified graphical depiction of why noise added to unitary evolution can be beneficial for full quantum tomography from time evolution. Left: Unitary/coherent evolution of an expectation value leads to a periodic signal (1a) whose information is carried in the frequency components (1b). Right: If noise is added, here by a superimposed decay leading to the signal (2a), additional information is carried by the decay components (dashed line in 2b).
• Figure 3: The least singular value of the map $\beta$ from Eq.(\ref{['eq:qubitbeta']}) for the qubit-channel of Eq.(\ref{['eq:qubitchannel']}) quantifies how stably the map can be inverted, and (according to Sec.\ref{['sec:stability']}) how much statistics is required. Here, $p$ quantifies the amount of depolarizing noise, and $\theta$ the degree of unitary rotation. While a unitary evolution ($p=0$) prohibits invertibility, additional noise ($p>0$) enables invertibility and thus full tomography of qubit observables.

Theorems & Definitions (29)

• Lemma 1
• proof
• Lemma 2
• proof
• Definition 1: Covering number & Minkowski dimension
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3