Hypothesis Testing with Open Quantum Systems

Using a quantum circuit model we derive the maximal ability to distinguish which of several candidate Hamiltonians describe an open quantum system. This theory, in particular, provides the maximum information retrievable from continuous quantum measurement records, available when a quantum system is perturbatively coupled to a broadband quantized environment.

Figure 1

Panel (a) illustrates the evolution of a quantum system S and a broadband environment E governed by a Hamiltonian $H_{\theta }(t)$. In (b), the state of an ancilla qubit controls the application of candidate Hamiltonians $H_0(t)$, $H_1(t)$. The reduced system and ancilla density matrix ${\rho }^{AS}$ follows by elimination of the environment.

Figure 2

Time-dependent probability for erroneously assigning whether an atom with decay rate $\kappa$ is excited resonantly with a Rabi frequency of ${\mathrm{\Omega }}_0=0$ or ${\mathrm{\Omega }}_1=4\kappa$. The error probability is shown for a purely atomic measurement (green, dash-dotted curve), photon counting (blue, thin solid curve), and photon counting and a measurement of the final atomic excitation (red, dashed curve). The black, fat solid curve shows the minimal error achievable by any measurement on the system and the radiation field.

Figure 3

The numerically smallest real part of the eigenvalues of Eq. (7) governing an exponentially converging distinction of different discrete values of the detuning $\delta$ from zero. The values of the convergence rate are shown as a function of the Rabi frequency $\mathrm{\Omega }$ for $\delta =0.5\kappa$, $\kappa$, $1.5\kappa$, $2\kappa$, $2.5\kappa$ (from above).