Basic package usage—optimization of standard strategies

In this section we present the basic ways to use the package that utilize high-level functions from protocols and bounds subpackages. These functions allow the user to compute and bound the QFI within three standard scenarios: single-channel, parallel and adaptive strategies, by simply specifying the parameter-encoding channel and the number of uses.

In the following subsections we will apply these methods to study the problem of phase estimation in the presence of two paradigmatic decoherence models: dephasing (d) or amplitude damping (a).

The parameter-encoding channels \(\Lambda_\theta\) that we will consider are given by Kraus operators of the form

\[K_{\theta,k} = U_{\theta} K_k, \quad U_\theta= e^{-\frac{i}{2}\theta\sigma_z},\]

where \(\sigma_z\) is a Pauli \(z\)-matrix, \(U_\theta\) represents unitary parameter encoding, whereas \(K_k\) are Kraus operators corresponding to one of the two decoherence models. For simplicity, we will always assume that estimation is performed around the parameter value \(\theta \approx 0\), so after differentiating the Kraus operators we set \(\theta=0\)—this does not affect the values of the QFI as finite unitary rotations could always be incorporated in control operations, measurements or the state preparation stage and hence will not affect the optimized value of QFI.

In case of the dephasing model, the two Kraus operators may be parametrized with a single parameter \(p\in[0, 1]\):

(16)\[K^{(d)}_0=\sqrt{p}\mathbb{1}, \quad K^{(d)}_1 = \sqrt{1-p}\sigma_z,\]

where \(p=1\) corresponds to no dephasing, \(p=1/2\) to the strongest dephasing case, where all equatorial qubit states are mapped to the maximally mixed state, while \(p=0\) represents a phase flip channel. Equivalently, we may use a different Kraus representation where the two Kraus operators are given by:

(17)\[K^{(d)}_+=\frac{1}{\sqrt{2}} U_\epsilon, \quad K^{(d)}_-=\frac{1}{\sqrt{2}} U_{-\epsilon},\]

which may be interpreted as random rotations by angles \(\pm \epsilon\) with probability \(1/2\) each—in order for the two representations to match we need to fix \(p=\cos^2(\epsilon/2)\).

In case of amplitude damping, the Kraus operators take the form

(18)\[K^{(a)}_0=\ket{0}\bra{0} + \sqrt{p} \ket{1}\bra{1},\quad K^{(a)}_1 = \sqrt{1-p} \ket{0}\bra{1},\]

where \(p=1\) represents the no-decoherence case, while \(p=0\) the full relaxation case, where all the states of the qubit are mapped to the ground \(\ket{0}\) state.

Alternatively, dephasing and amplitude damping models can be considered in a continuous time regime. In this case, the state undergoes a process for some time \(t\) which imprints on it information about the parameter \(\omega\) related to \(\theta\) by \(\theta=\omega t\). The derivative of the state with respect to time is defined using the Lindbladian operator \(\mathcal{L}_\omega\):

\[\frac{d\rho_{\omega, t}}{dt} = \mathcal{L}_{\omega}[\rho_{\omega, t}] = \frac{1}{i\hbar}[H_\omega, \rho_{\omega, t}] + \mathcal{D}[\rho_{\omega, t}],\]

where \(H_\omega = \hbar\omega\sigma_z/2\) is a Hamiltonian generating the evolution \(U_\theta\) and \(\mathcal{D}\) is a dissipation term. The dissipation term has a general form:

\[\mathcal{D}[\rho] = \sum_i \gamma_i \left( L_i\rho L_i^\dagger - \frac{1}{2}\{L_i^\dagger L_i, \rho\} \right),\]

where \(\{\cdot,\cdot\}\) is the anticommutator, \(L_i\) are jump operators that determine the type of dissipation and \(\gamma_i\geq0\) are damping rates. In case of dephasing and amplitude damping only one pair \(L_i, \gamma_i\) is required and it is respectively:

\[ \begin{align}\begin{aligned}L^{(d)} = \sigma_z/\sqrt{2}, \quad 2p-1=e^{-\gamma^{(d)}t},\\L^{(a)} = \ket{0}\bra{1}, \quad p=e^{-\gamma^{(a)}t}.\end{aligned}\end{align} \]