Reconstructing Quantum States and Expectations via Dynamical Tomography

Abstract

When the dynamics of a quantum system of interest is known, an informationally-complete set of observables is not needed for state reconstruction via tomographic techniques: letting the system evolve before performing the measurement allows one to effectively extend the available ways to probe the system. This idea leads to dynamical quantum tomography, whose feasibility we characterize for general quantum dynamics using Krylov-based methods. Specializing to Markovian ones, we also provide deterministic tests, and randomized ones to effectively assess parametric dynamics. The limits of the methods are explored comparing unitary and open dynamics when a single observable is available, and the set of observables whose expectation can be reconstructed from the available ones characterized. The framework is illustrated with applications to a spin chain (with or without dissipation) and an electron-nuclear system

Figures (6)

• Figure 1: Equivalences between Observability, Feasibility of DQST and the requirement that every state for the system is UDDA. The three conditions are equivalent.
• Figure 2: Example 1: The 4-spin chain considered in Example I. The dotted line indicate the subsystems on which we are allowed to perform arbitrary measurements, contained in $\mathcal{X}$. The dashed lines group the subsystems that are directly interacting dynamically.
• Figure 3: Example 1 Scenario 2 Dissipative Dynamics: Pairs of times and measurement operators to perform DQST identified by the AOT described in section \ref{['sec:quality']}data_av.
• Figure 4: Squared error ($\varepsilon^2_\rho$) scaling in function of the number $N$ of instances of the system involved in computing the estimate $\hat{\rho}$ of the state $\rho$ of the system. We considered three states $\rho_S, \rho_{GKS}$ and $\rho_{Gibbs}$ as described in the main text data_av.
• Figure 5: Example 2: coupled electron (E) nuclear (N) system. The dotted line indicates the subsystem on which the measurement operators act nontrivially $\mathcal{X}$.

Theorems & Definitions (41)

• Definition 1: UDDA states
• Definition 2: Non-observable/observable subspaces
• Definition 3
• Proposition 1
• proof
• Lemma 1
• proof
• Proposition 2
• proof
• Remark 1