Abstract:
This thesis is dedicated to the understanding of the metrology of quantum systems by using
the tools of quantum parameter estimation, in particular the quantum Fisher information (QFI).
Our first project deals with a specific protocol of quantum enhanced measurement known
as coherent averaging [Braun and Martin, 2011]. This protocol is based on a star topology, with
one central object, the so-called quantum bus, connected to N extra subsystems, called probes.
For the estimation of a parameter characteristic of the interaction between the quantum bus
and the probes, coherent averaging leads to a Heisenberg limited (HL) scaling for the QFI (QFI
proportional to N 2 ). Importantly this HL scaling can be obtained while starting with a separable
state. This provides an advantage as generally one needs to use entangled states to achieve
this scaling. Another important aspect in coherent averaging is the possibility to obtain the HL
scaling by performing a measurement on the quantum bus only. These results were obtained using
perturbation theory in the regime of weak interactions.
In this thesis we go one step further in the study of the coherent averaging protocol. We extend
the formalism of perturbation theory to encompass the possibility of estimating any parameter, in
the regimes of strong and weak interactions. To illustrate the validity of our results, we introduce
two models as examples for a coherent averaging scheme. In these models both the quantum bus
and all the probes are qubits. In the ZZXX model, the free Hamiltonians do not commute with
the interaction Hamiltonians and we have to rely on numerics to find non-perturbative solutions
.In the ZZZZ model the free evolution Hamiltonians commute with the interaction Hamiltonians
and we can find the exact solution analytically.
Perturbation theory shows that in the strong interaction regime and starting with a separable
state, we can estimate the parameter of the free evolution of the probes with a HL scaling if the
free Hamiltonians do not commute with the interaction Hamiltonians. This is confirmed by the
non-perturbative numerical results for the ZZXX model. In the weak interaction regime we only
obtain a standard quantum limit (SQL) scaling for the parameter of the free evolution of the
probes (QFI proportional to N ). When one has only access to the quantum bus, we show that the
HL scaling found using the perturbation theory does not necessarily survive outside the regime
of validity of the perturbation. This is especially the case as N becomes large. It is shown by
comparing the exact analytical result to the perturbative result with the ZZZZ model. The same
behaviour is observed with the ZZXX model using the non-perturbative numerical results.
In our second project we investigate the estimation of the depolarizing channel and the
phase-flip channel under non-ideal conditions. It is known that using an ancilla can lead to an
improvement of the channel QFI (QFI maximized over input states feeding the channel) even
if we act with the identity on the ancilla. This method is known as channel extension. In all
generality the maximal channel QFI can be obtained using an ancilla whose Hilbert space has the
same dimension as the dimension of the Hilbert space of the original system. In this ideal scenario
using multiple ancillas — or one ancilla with a larger Hilbert space dimension — is useless.
To go beyond this ideal result we take into account the possibility of loosing either the probe
or a finite number of ancillas. The input states considered are GHZ and W states with n + 1
qubits (the probe plus n ancillas). We show that for any channel, when the probe is lost then
all the information is lost, and the use of ancillas cannot help. For the phase-flip channel the
introduction of ancillas never improves the channel QFI and ancillas are useless.
For the depolarizing channel the maximal channel QFI can be reached using one ancilla and
feeding the extended channel with a Bell state, but if the ancilla is lost then all the advantage
is lost. We show that the GHZ states do not help to fight the loss of ancillas: If one ancilla or
more are lost all the advantage provided by the use of ancillas is lost. More interestingly, we show
that the W states with more than one ancilla are robust against loss. For a given number of lost
ancillas, there always exists an initial number of ancillas for which a W state provides a higher
QFI than the one obtained without ancillas.
Our last project is about Hamiltonian parameter estimation for arbitrary Hamiltonians.
It is known that channel extension does not help for unitary channels. Instead we apply the
idea of extension to the Hamiltonian itself and not to the channel. This is done by adding
to the Hamiltonian an extra term, which is independent of the parameter and which possibly
encompasses interactions with an ancilla. We call this technique Hamiltonian extension. We show
that for arbitrary Hamiltonians there exists an upper bound to the channel QFI that is in general
not saturated. This result is known in the context of non-linear metrology. Here we show explicitly
the conditions to saturate the bound.
We provide two methods for Hamiltonian extensions, called signal flooding and Hamiltonian
subtraction, that allow one to saturate the upper bound for any Hamiltonian. We also introduce a
third method which does not saturate the upper bound but provides the possibility to restore the
quadratic time scaling in the channel QFI when the original Hamiltonian leads only to a periodic
time scaling of the channel QFI.
We finally show how these methods work using two different examples. We study the estimation
of the strength of a magnetic field using a NV center, and show how using signal flooding we
saturate the channel QFI. We also consider the estimation of a direction of a magnetic field using
a spin-1. We show how using signal flooding or Hamiltonian subtraction we saturate the channel
QFI. We also show how by adding an arbitrary magnetic field we restore the quadratic time
scaling in the channel QFI. Eventually we explain how coherent averaging can be scrutinized in
the formalism of Hamiltonian extensions.