End-point measurement approach to assess quantum coherence in energy fluctuations

We discuss the role of quantum coherence in the energy fluctuations of open quantum systems. To this aim, we introduce a protocol to which we refer as the end-point measurement scheme, allowing us to define the statistics of energy changes as a function of energy measurements performed only after the evolution of the initial state. At the price of an additional uncertainty on the initial energies, this approach prevents the loss of initial quantum coherences and enables the estimation of their effects on energy fluctuations. We demonstrate our findings by running an experiment on the IBM Quantum Experience superconducting qubit platform.

Figure 1

Protocol for the quantification of energy fluctuations and the extraction of information about coherence. An ensemble of identical systems, prepared in the initial state ${\rho }_{\mathrm{i}}$, is divided in three subgroups. One is used to obtain $p_{\mathrm{i}}^{\ell }=\mathrm{Tr}\left({\rho }_i{\mathrm{\Pi }}_{\mathrm{i}}^{\ell }\right)$ via an initial energy measurement. The second goes through a dephasing channel, returning a state $\mathcal{P}$ diagonal in the energy basis. This then undergoes map ${\mathrm{\Phi }}_t$ and is used to determine $p_{\mathcal{P}}^k=\mathrm{Tr}\left({\mathrm{\Phi }}_{t_{\mathrm{f}}}\left[\mathcal{P}\right]{\mathrm{\Pi }}_{\mathrm{f}}^k\right)$. The systems in the third subgroup are not initially measured but subjected to the dynamics and used to obtain $p_{\mathrm{f}}^k=\mathrm{Tr}\left({\mathrm{\Phi }}_{t_{\mathrm{f}}}\left[{\rho }_i\right]{\mathrm{\Pi }}_{\mathrm{f}}^k\right)$.

Figure 2

Top: Circuits implemented in IBMQ. The initial state is prepared by applying two identical single-qubit gates $U\left({\theta }_0\right)$ onto $|00\rangle$ (we use ${\theta }_0=2$ [36]). In TPM, two initial projective measurements destroy any coherence in the computational basis, while in EPM such measurements (enclosed in the dashed red box) are absent. We then implement the controlled gate $U\left({\theta }_n\right)$, with ${\theta }_n\equiv n\pi /10$ and $n=0,...,20$, followed by two projective measurements in the computational basis. The results are stored in four classical registers to allow the analysis of the energy-change statistics. Bottom: Comparison of the characteristic functions for EPM and TPM. The lines show the theoretical predictions, while the points (with their error bars) the experimental results. Each data point has been obtained from 2048 experimental runs. The solid red line and circles are related to the results obtained by applying TPM. The dashed blue line and squared refer to the EPM characteristic function. Finally, the dotted magenta line and rhombuses (dot-dashed black line and triangles) show the contribution of the diagonal (off-diagonal) parts of the initial state ${\rho }_{\mathrm{i}}$ in the computational basis. The inverse (physical) temperature of the diagonal part of the initial state is $\beta =0.443/\epsilon$, where $\epsilon \sim 5$ MHz is the energy gap for the superconducting qubits, as provided by the IBMQ documentation.