Short-time-interaction quantum measurement through an incoherent mediator

We propose a method of indirect measurements where a probe is able to read, in short interaction times, the quantum state of a remote system through an incoherent third party, hereafter called a mediator. The probe and system can interact briefly with the mediator in an incoherent state but not directly among themselves and, nevertheless, the transfer of quantum information can be achieved with robustness. We exemplify our measurement scheme with a paradigmatic example of this tripartite problem—a qubit-oscillator-qubit setup—and discuss different physical scenarios, pointing out the associated advantages and limitations.

Figure 1

Sketch of the model. P, A, S, and R are the probe, mediator, system, and reservoir, respectively.

Figure 2

Realization of a qubit in a single trapped ${}^{40}\mathrm{Ca}$${}^{+}$ ion. The qubit states $|S\rangle$, $|D\rangle$ are encoded in a pair of Zeeman states interacting with a laser pulse of frequency $\omega$ and coupling strength $\Omega$.

Figure 3

Plots of $P_{\mathrm{e}}({\tau }_{\mathrm{eff}})$ and its derivatives as a function of the dimensionless effective time, ${\tau }_{\mathrm{eff}}=g_p^{2}t/\delta$, for a dispersive case where the system state is $|{\psi }_s\rangle =0.1|1\rangle +e^{i\pi /3}\sqrt{1-0.1^2}|2\rangle$, the probe state is $|{\psi }_p\rangle =(|\mathrm{g}\rangle +|\mathrm{e}\rangle )/\sqrt{2}$, the detuning is $\delta =30g_p$, and the mediator is in a thermal state with ${\mathrm{n̄}}_a=1$. The fast oscillating blue lines correspond to a calculation done with the ab initio Hamiltonian of Eq. (1) and the green line to the dispersive Hamiltonian of Eq. (13).

Figure 4

Plots of the relative errors in the measurement of the system coherence and population, ${\epsilon }_{{\rho }_{12}}$ and ${\epsilon }_{{\rho }_{22}}$, respectively, and the infidelity, $1-F$, of the measured density matrix as a function of the population inversion $\Delta \mathrm{P}$. The state of the system is $|{\psi }_s\rangle =0.3|1\rangle +\sqrt{1-0.3^2}|2\rangle$ and that of the probe is $|{\psi }_p\rangle =(|\mathrm{g}\rangle +|\mathrm{e}\rangle )/\sqrt{2}$.