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International audience; Fortement investis par les politiques publiques dans le cadre du plan Neige (1964-1977), les territoires de montagne sont aujourd’hui confrontés à des enjeux de diversification : enjeu de renouvellement du parc immobilier, évolution des pratiques touristiques, enneigement incertain et de plus en plus rare en moyenne montagne, etc.Les deux départements de Savoie et de Haute-Savoie, dont le secteur touristique est capital pour l’économie, n’échappent pas à ces mutations. Ainsi, le modèle de masse hérité du plan Neige est aujourd’hui remis en question et certains organismes (notamment Savoie- Mont-Blanc Tourisme) cherchent des solutions pour pérenniser un tourisme savoyard qui risque, à plus ou moins long terme, de s’essouffler. C’est dans ce contexte que la destination lacustre se voit mise en valeur, avec la nécessité de définir au mieux le phénomène touristique lié à ces lacs pour développer ces aménités paysagères en accord avec les attentes nouvelles des visiteurs et les projections des acteurs territoriaux. Dans le cadre de cette étude, nous cherchons à définir de manière plus fine le tourisme lié aux 4 grands lacs périalpins que sont Aiguebelette, Annecy, Le Bourget et le Léman. Pour ce faire, nous proposons de partir des notions théoriques avancées par C.MICHAËL HALL et TUIJA HÄRKÖNEN (2006) sur la définition de l’espace touristique lacustre, lequel se définit comme : « un ensemble incluant le lac, ses rives et les équipements qui y sont implantés, mais également les services et les infrastructures à proximité qui contribuent à faire du lac une attraction touristique». L’objectif de notre étude est de dépasser cette première définition centrée sur les équipements pour interroger l’espace lacustre au niveau des images touristiques, élément essentiel dans la constitution et le fonctionnement de destination touristique. Aussi, il s’agit de définir l’espace touristique de ces quatre lacs à partir de la place qu’ils occupent dans les discours et les images produites et véhiculées par les offices de tourisme des deux départements. D’un point de vue méthodologique, on cherche à définir, via l’analyse quantitative des outils de communication des OT (sites web et présentoirs de brochures), le lien entre chacun de ces lacs périalpins et les territoires administrés par les OT. Parallèlement, des entretiens ont été menés afin de distiller la vision qu’ont les agents d’accueil du territoire dont ils font la promotion auprès des visiteurs. Le but étant ensuite de croiser les résultats des deux méthodes afin de déterminer de manière précise la nature du phénomène touristique en place. Cette mise en perspective permet d’appréhender s’il y a congruence ou divergence entre la place occupée par les lacs dans le matériel de promotion touristique et les discours tenus par les personnes en charge de renseigner les visiteurs.In fine, cette analyse croisée permet de discuter dans quelle mesure les OT, prescripteurs d’activités touristiques, considèrent les lacs comme des ressources touristiques à mettre en avant sur leur territoire de renseignement, et d’avancer dans la définition des contours des espaces touristiques de chacun de ces quatre lacs.

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.

We study very generally towhat extent the uncertainty with which a phase shift can be estimated in quantum metrology can be reduced by extending the Hamiltonian that generates the phase shift to an ancilla system with a Hilbert space of arbitrary dimension, and allowing arbitrary interactions between the original system and the ancilla. Such Hamiltonian extensions provide a general framework for open quantum systems, as well as for “non-linear metrology schemes” that have been investigated over the last few years. We prove that such Hamiltonian extensions cannot improve the sensitivity of the phase shift measurement when considering the quantum Fisher information optimized over input states.

A well-studied scenario in quantum parameter estimation theory arises when the parameter to be estimated is imprinted on the initial state by a Hamiltonian of the form $\theta G$. For such "phase shift Hamiltonians" it has been shown that one cannot improve the channel quantum Fisher information by adding ancillas and letting the system interact with them. Here we investigate the general case, where the Hamiltonian is not necessarily a phase shift, and show that in this case in general it \emph{is} possible to increase the quantum channel information and to reach an upper bound. This can be done by adding a term proportional to the derivative of the Hamiltonian, or by subtracting a term to the original Hamiltonian. Both methods do not make use of any ancillas and show therefore that for quantum channel estimation with arbitrary parameter-dependent Hamiltonian, entanglement with an ancillary system is not necessary to reach the best possible sensitivity. By adding an operator to the Hamiltonian we can also modify the time scaling of the channel quantum Fisher information. We illustrate our techniques with NV-center magnetometry and the estimation of the direction of a magnetic field in a given plane using a single spin-1 as probe., Comment: New references included

International audience

We consider quantum channel-estimation for depolarizing channels and phase-flip channels extended by ancilla qubits and fed with a GHZ orWstate. After application of the channel one or several qubits can be lost, and we calculate the impact of the loss on the quantum Fisher information that determines the smallest uncertainty with which the parameters of these channels can be estimated.