Active optical media leading to interaction Hamiltonians of the form $ H = tilde{lambda}, (a + a^{dagger})^{zeta}$ represent a crucial resource for quantum optical technology. In this paper, we address the characterization of those nonlinear media using quantum probes, as opposed to semiclassical ones. In particular, we investigate how squeezed probes may improve individual and joint estimation of the nonlinear coupling $tilde{lambda}$ and of the nonlinearity order $zeta$. Upon using tools from quantum estimation, we show that: i) the two parameters are compatible, i.e. the may be jointly estimated without additional quantum noise; ii) the use of squeezed probes improves precision at fixed overall energy of the probe; iii) for low energy probes, squeezed vacuum represent the most convenient choice, whereas for increasing energy an optimal squeezing fraction may be determined; iv) using optimized quantum probes, the scaling of the corresponding precision with energy improves, both for individual and joint estimation of the two parameters, compared to semiclassical coherent probes. We conclude that quantum probes represent a resource to enhance precision in the characterization of nonlinear media, and foresee potential applications with current technology.
We address the use of asymptotic incompatibility (AI) to assess the quantumness of a multiparameter quantum statistical model. AI is a recently introduced measure which quantifies the difference between the Holevo and the SLD scalar bounds, and can be evaluated using only the symmetric logarithmic derivative (SLD) operators of the model. At first, we evaluate analytically the AI of the most general quantum statistical models involving two-level (qubit) and single-mode Gaussian continuous-variable quantum systems, and prove that AI is a simple monotonous function of the state purity. Then, we numerically investigate the same problem for qudits ($d$-dimensional quantum systems, with $2 < d leq 4$), showing that, while in general AI is not in general a function of purity, we have enough numerical evidence to conclude that the maximum amount of AI is achievable only for quantum statistical models characterized by a purity larger than $mu_{sf min} = 1/(d-1)$. In addition, by parametrizing qudit states as thermal (Gibbs) states, numerical results suggest that, once the spectrum of the Hamiltonian is fixed, the AI measure is in one-to-one correspondence with the fictitious temperature parameter $beta$ characterizing the family of density operators. Finally, by studying in detail the definition and properties of the AI measure we find that: i) given a quantum statistical model, one can readily identify the maximum number of asymptotically compatibile parameters; ii) the AI of a quantum statistical model bounds from above the AI of any sub-model that can be defined by fixing one or more of the original unknown parameters (or functions thereof), leading to possibly useful bounds on the AI of models involving noisy quantum dynamics.
We address the discrimination of structured baths at different temperatures by dephasing quantum probes. We derive the exact reduced dynamics and evaluate the minimum error probability achievable by three different kinds of quantum probes, namely a qubit, a qutrit and a quantum register made of two qubits. Our results indicate that dephasing quantum probes are useful in discriminating low values of temperature, and that lower probabilities of error are achieved for intermediate values of the interaction time. A qutrit probe outperforms a qubit one in the discrimination task, whereas a register made of two qubits does not offer any advantage compared to two single qubits used sequentially.
