Information Gain in Tomography–A Quantum Signature of Chaos

We find quantum signatures of chaos in various metrics of information gain in quantum tomography. We employ a quantum state estimator based on weak collective measurements of an ensemble of identically prepared systems. The tomographic measurement record consists of a sequence of expectation values of a Hermitian operator that evolves under repeated application of the Floquet map of the quantum kicked top. We find an increase in information gain and, hence, higher fidelities in the reconstruction algorithm when the chaoticity parameter map increases. The results are well predicted by random matrix theory.

Figure 1

Information gain in tomography as quantified by various metrics, given a measurement record generated by iterations of a quantum-kicked-top Floquet map $U_{\tau }=\exp \{-i\lambda j_z^2/(2j)\}\exp \{-i\alpha j_x\}$, for a spin $j=10$. The value of $\alpha$ is fixed and $\lambda$ serves as the chaoticity parameter, varying from regular dynamics, $\lambda =0.5$, to fully chaotic, $\lambda =7.0$. All results are shown as the average of 100 Haar-random pure states. (a) Fidelity of state reconstruction. (b) The Fisher information of estimating the parameters that define the state. (c) The Shannon entropy of the normalized eigenvalues of the inverse of covariance matrix of the likelihood function. In all cases, both the rate of growth and the final value of the information metric are increased with higher values of the chaoticity parameter, $\lambda$. In the fully chaotic regime, $\lambda =7.0$, the results are well predicted by tomography performed with a measurement record generated by a Haar-random matrix picked from the COE. The COE results, averaged over 100 random pure states, are plotted as the dashed line in (a)–(c).