Single-channel QFI optimization

Fig. 7 (A) Diagram of a strategy with a single parameter-dependent channel \(\Lambda_\theta\) and ancilla \(\mathcal{A}\). Values of QFI for (B) dephasing and (C) amplitude damping, for various values of \(p\) and for different methods: MOP – black ×, ISS with \(d_\mathcal{A}=1\) – blue dashed line, ISS with \(d_\mathcal{A}=2\) – red dotted line.

After defining the parameter-dependent channel using the ParamChannel class, we may now easily compute the single-channel QFI (4), including the possible use of an ancilla, see Fig. 7 (A). The channel QFI can be computed using one of the two functions:

• iss_channel_qfi for the ISS method,

• mop_channel_qfi for the MOP method.

The function iss_channel_qfi takes two arguments: the channel whose QFI is computed and the dimension of the ancillary system. It returns five items: the optimized QFI, a list of pre-QFI values per algorithm iteration, the density matrix of the optimal input state, the SLD matrix, and the information whether the optimization converged successfully.

On the other hand, mop_channel_qfi takes only the channel and returns only the QFI—here, by the nature of the method, ancillary system dimension is unspecified, see Sec. Minimization over purifications. The following code shows both of these functions in practice:

from qmetro import *

p = 0.75
channel = par_dephasing(p)

ancilla_dim = 2
iss_qfi, qfis, rho0, sld, status = iss_channel_qfi(channel, ancilla_dim)

mop_qfi = mop_channel_qfi(channel)

Fig. 7 (B) and Fig. 7 (C) present plots of the results obtained using iss_channel_qfi and mop_channel_qfi for the two decoherence models respectively. Since the MOP method gives the optimal QFI implicitly assuming arbitrary ancilla dimension \(d_\mathcal{A}\), these plots indicate that for the dephasing case (B) the ancilla is in fact unnecessary, while in the amplitude damping case (C) the ancillary system of dimension \(d_\mathcal{A}=2\) is required for an optimal precision.

This example clearly shows the benefits of having two independent optimization procedures. One method (MOP) provides the true optimal QFI, but does not allow us to gain insight into the required ancilla size, while the other one (ISS) allows us to study the impact of the ancilla size.