This PhD responds to a two fold problem with the philosophy of design and the practice of design. The philosophical problem is stated as the discrepancy between a dominant philosophical framework that orders the world according to eternal essences and the actual conditions of the world in which we exist: the conditions of becoming and of flux. Commencing with a critique of the western metaphysical tradition of statics this research project proposes that we need to find a way of describing an evolutionary model of practice, and by so doing to provide a revitalised narration of process. Consequently, the PhD explores the meanings of process through a critical examination of an ensemble of projects created by the author. Within this framework, a number of questions are posed in order to explore the proposition of a process practice. These questions are: What is process? How does one think process? Indeed, how do we get to grasp change? What are the consequences of process thinking on the practices of design, their fields of operation, and their productions? And, how can the thematising of process contribute to the design of the constructed environment, as well as reconfigure the practices of design? This thematising of process is argued to involve a necessary address to the constitutive and interrelated characteristics of process: space/time, movement, change, form and matter. Such an address is also seen to problematise the status of the object, the paradigms of representation, the modes of creation, the economies of exchange, and the structures of community, and to offer a modality of practice which would re-imagine the forms of social exchange to offer an ethical alternative to the tyranny of supply and demand, and thereby reconfigure the potential for dwelling. Making an overview of the discourses and practices engaging with theories of becoming, this thesis argues that almost all of these re-inscribe statics and that consequently the practice of design seems to drag behind our understanding of the world. Through a meditation on dis/appearance, in which the dynamics of being and becoming and the restless ambiguity of the gap are examined, the work establishes a process vocabulary, and makes clear through a material practice, the domains of process thinking, its inclinations, and the kinds of operations and procedures that flourish there. Foregrounding the fertile character of process practice, the PhD then proceeds to introduce notions of the movement-form, the duration-form, the transformational-form, the geometry of encounter, and to argue for physical form as an in-movement poise. Advocating new modes of approach and of attentiveness, and demonstrating new generative methods, this PhD argues that process thinking is not simply an operational stance, but an ethical position that identifies a field of care, and that consequently the design practices be expanded by taking seriously the relationship between process thinking and place making. Thus, this thesis concludes by advocating a mode of place making which, rather than reproduce planned environments as systems of control, configures place as the discursive contested place of encounter and exchange.

At Cape Gantheaume, Kangaroo Island (South Australia), adult male, lactating female and juvenile New Zealand (NZ) and Australian fur seals regularly return to the same colony, creating the potential for intra- and inter-specific foraging competition in nearby waters. I hypothesised that these demographic groups would exhibit distinct foraging strategies, which reduce competition and facilitate their coexistence. I analysed the diet of adult male, adult female and juvenile NZ fur seals and adult male Australian fur seals and studied the diving behaviour of adult male and lactating female NZ fur seals and the at-sea movements of juvenile, adult male and lactating female NZ fur seals. Female diet reflected that of a generalist predator, influenced by prey availability and their dependant pups� fasting abilities. In contrast, adult male NZ and Australian fur seals used larger and more energy-rich prey, most likely because they could more efficiently access and handle such prey. Juvenile fur seals primarily utilised small lantern fish, which occur south of the shelf break, in pelagic waters. Juveniles undertook the longest foraging trips and adult males conducted more lengthy trips than lactating females, which perform relatively brief trips in order to regularly nurse their pups. Unlike lactating females, some adult males appeared to rest underwater by performing dives that were characterised by a period of passive drifting through the water column. The large body sizes of adult males and lactating females facilitated the use of both benthic and pelagic habitats, but adult males dived deeper and for longer than lactating females, facilitating vertical separation of their foraging habitats. Spatial overlap in foraging habitats among the age/sex groups was minimal, because lactating females typically utilised continental shelf waters and males used deeper water over the shelf break, beyond female foraging grounds. Furthermore, juveniles used pelagic waters, up to 1000 km south of the regions used by lactating females and adult males. The age and sex groups in this study employed dramatically different strategies to maximise their survival and reproductive success. Their prey and foraging habitats are likely to be shaped by body size differences, which determine their different physiological constraints and metabolic requirements. I suggest that these physiological constraints and the lactation constraints on females are the primary factors that reduce competition, thereby facilitating niche partitioning.

Στην συγκεκριμένη εργασία ασχολούμαστε με την μελέτη θερμικά σύμπλεκτων (entangled) καταστάσεων πολλών κβαντικών bit (qubit) σε διάφορα μοντέλα Heisenberg με την μέθοδο Monte Carlo (MC). Αρχικά χρησιμοποιώντας τον μετασχηματισμό Suzuki-Trotter μετατρέπουμε την κβαντική μονοδιάστατη αλυσίδα των spin (μοντέλα Ising, Heisenberg με και χωρίς μαγνητικό πεδίο στις διευθύνσεις x,y,z) σε κλασικό δισδιάστατο πλέγμα. Εξετάζουμε την συμπεριφορά του συγκεκριμένου μετασχηματισμού για το αντισιδηρομαγνητικό Heisenberg ΧΧΧ μοντέλο, για το σιδηρομαγνητικό Heisenberg μοντέλο (ΧΧΧ και ΧΥΖ) χωρίς και με μαγνητικό πεδίο στις διευθύνσεις x,y,z για διάφορα μήκη της αλυσίδας, διαφορετικές διαστάσεις Trotter και διαφορετικό αριθμό Monte Carlo βημάτων (MCΒήματα). Μελετάμε την συμπεριφορά της θερμοχωρητικότητας, της ενέργειας, της μαγνητικής επιδεκτικότητας και της μαγνήτισης στις διευθύνσεις x,y,z. Επιβεβαιώνουμε την σωστή συμπεριφορά τους με βάση τα αναλυτικά αποτελέσματα. Τέλος γνωρίζοντας, ότι η κλασική συσχέτιση είναι το κάτω όριο της ποσότητας Localizable Entanglement, και ότι η ποσότητα Entanglement of Assistance είναι το πάνω όριο, εκτιμούμε για τα ίδια μοντέλα τη συμπεριφορά των ορίων και προσπαθούμε να εκτιμήσουμε το μήκος σύμπλεξης (Entanglement Length) για διάφορες θερμοκρασίες. / In the present Master Thesis we study the thermal entangled states of many qubits in a variety of Heisenberg models with the deployment of the Monte Carlo(MC) method. Initially we are using the Suzuki-Trotter decomposition in order to convert the one dimensional spin chain(models Ising, Heisenberg with and without magnetic field in the x,y,z axis) into a classical two dimensional lattice. We examine the behavior of the latter decomposition for the antiferromagnetic Heisenberg XXX model, the ferromagnetic Heisenberg model (XXX and XYZ) with or without magnetic field in the axis x,y,z for different chain lengths, Trotter dimensions and number of Monte Carlo Steps (MCSteps). We investigate the behavior of the following quantities: specific heat, energy, susceptibility and magnetization in the axis x,y,z. We confirm their proper behavior comparing to analytical and arithmetic results. Finally knowing that the maximum classical correlation function is the lower limit of the quantity Localizable Entanglement (LE) and that the quantity Entanglement of Assistance is the upper limit, we evaluate for the same models the behavior of the limits and we try to evaluate the Entanglement Length for a variety of temperatures.

Semiconductor quantum dots (QDs) are well established as sub-Poissonian sources of entangled photon pairs. To improve the utility of a QD light source, it would be advantageous to extend their emission further into the near infrared, into the low absorption wavelength windows utilised in long-haul optical telecommunication. Initial experiments succeeded in interfering O-band (1260—1360 nm) photons from an InAs/GaAs QD with dissimilar photons from a laser, an important mechanism for quantum teleportation. Interference visibilities as high as 60 ± 6 % were recorded, surpassing the 50 % threshold imposed by classical electrodynamics. Later, polarisation-entanglement of a similar QD was observed, with pairs of telecom-wavelength photons from the radiative cascade of the biexciton state exhibiting fidelities of 92.0 ± 0.2 % to the Bell state. Subsequently, an O-band telecom-wavelength quantum relay was realised. Again using an InAs/GaAs QD device, this represents the first implementation of a sub-Poissonian telecom-wavelength quantum relay, to the best knowledge of the author. The relay proved capable of implementing the famous four-state BB84 protocol, with a mean teleportation fidelity as high as 94.5 ± 2.2 %, which would contribute 0.385 secure bits per teleported qubit. After characterisation by way of quantum process tomography, the performance of the relay was also evaluated to be capable of implementing a six-state QKD protocol. In an effort to further extend the emitted light from a QD into the telecom C-band (1530—1565 nm), alternative material systems were investigated. InAs QDs on a substrate of InP were shown to emit much more readily in the fibre-telecom O- and C-bands than their InAs/GaAs counterparts, largely due to the reduced lattice mismatch between the QD and substrate for InAs/InP (~3 %) compared to InAs/GaAs (~7 %). Additionally, to minimize the fine structure splitting (FSS) of the exciton level, which deteriorates the observed polarisation-entanglement, a new mode of dot growth was investigated. Known as droplet epitaxy (D-E), QDs grown in this mode showed a fourfold reduction in the FSS compared to dots grown in the Stranski-Krastanow mode. This improvement would allow observation of polarisation-entanglement in the telecom C-band. In subsequent work performed by colleagues at the Toshiba Cambridge Research Labs, these D-E QDs were embedded in a p-i-n doped optical cavity, processed with electrical contacts, and found to emit entangled pairs of photons under electrical excitation. The work of this thesis provides considerable technological advances to the field of entangled-light sources, that in the near future may allow for deterministic quantum repeaters operating at megahertz rates, and in the further future could facilitate the distribution of coherent multipartite states across a distributed quantum network.

When I write I, I also write we (say we also write that we’re momentarily in possession of the possibilities, and as for the possibilities: they have us), as in my hyperbolic twin brother and I, and from that point on (as well as before) in company with the gaps between us. The auto-corrections, the imposed circumstances, the complicities, and some conspirationalist copings within a system with no clearly identifiable nemesis. Here, the act of resistance to whatever is playing us, also enacts the playing of that which plays us. The heinous power of family and narrative insists. Two lead narratives are interwoven during the course of time it took for a pizza to be delivered (from pizzeria Marco Polo): Looping on a Panasonic Viera TX-55AX630E, the  search for a/my lost and/or hidden dad arrives through the mispronunciation of the word “that” during a negotiation in a park. Where a tangled hypothesis, along the lines of “what if the plot device of a shot of a moon in a video used to forward time (as in an alternative to ‘later that night’) was a graphic representation of an analog clock recognized in a to-be sliced pizza that is eventually consumed as a means to induce a food coma?” takes place (and time, as saved time is always already lost). It’s almost dark already. Also it's not, and the future held hostage by the river of the forest of the woods is followed by the sense of not being able to tell if there’s an emotion or a feeling present. Visits from passed pasts. It all starts to feel originatory. Thanks to X, this keeps on happening, thoughts and behaviours about suspicious timing and holistic systems keep on happening. This keeps on happening, and it works its way into physical works in an exhibition space with no reference to an infirmary, where parrot tulips are emotional flowers, a print is printed using a temperamental printer, sawdust creates glossy eyes, and all locked doors are eventually keyed out. An investigation into conditions of influence, co-authorship, connectivity, all within a space cohabited by comedy, sentimentality, and other states subject to flatness.

FAPEMIG - Fundação de Amparo a Pesquisa do Estado de Minas Gerais / Neste trabalho, investigamos propriedades estáticas e dinâmicas de sistemas fortemente correlacionados quase-unidimensionais. A principal técnica utilizada no estudo de tais sistemas foi o grupo de renormalização da matriz de densidade. Neste contexto, um dos sistemas que consideramos foram as escadas de Heisenberg de N pernas com spin-s. Para estas escadas, investigamos propriedades estáticas, tais como energia por sítio no limite termodinâmico e gap de spin. Em particular, verificamos a validade da conjectura de Haldane-Sénéchal-Sierra para o comportamento do gap de spin das escadas de Heisenberg. Ainda para sistemas com geometria de escadas, outro problema que analisamos foi a entropia de emaranhamento de escadas quânticas críticas. Neste caso, propusemos uma conjectura para o comportamento de escala desta entropia. A fim de verificar nossa conjectura, consideramos as escadas férmions livres, de Heisenberg e escadas de Ising quânticas. Por fim, investigamos o comportamento das correlações dinâmicas de sistemas fortemente correlacionados unidimensionais. Para este caso, apresentamos um estudo detalhado do comportamento assintótico das autocorrelações de spin dinâmicas no bulk e na borda de tais sistemas. / In this work, we investigated static and dynamical properties of quasi-one-dimensional strongly correlated systems. The main technique used in the study of such systems was the density matrix renormalization group. In this context, one of the systems that we considered were the spin-s N-leg Heisenberg ladders. For these ladders, we investigated static properties, such as the energy per site in the thermodynamic limit and the spin gap. In particular, we checked the validity of the Haldane-Sénéchal-Sierra's conjecture for the spin gap behavior of the Heisenberg ladders. Also for systems with ladders geometry, another problem that we analyzed was the entanglement entropy of quantum critical ladders. In this case, we proposed a conjecture for the scaling behavior of this entropy. In order to check our conjecture, we consider free fermions, Heisenberg ladders and quantum Ising ladders. Finally, we investigated the behavior of the dynamical correlations in one-dimensional strongly correlated systems. For this case, we presented a detailed study of the asymptotic behavior of the dynamical spin autocorrelations at the bulk and the boundary of such systems. / Tese (Doutorado)

Universidade Federal de Uberlândia / In this work we investigate theoretically the dissipative dynamics of exciton states in a system constituted by coupled quantum dots, which in turn exhibit a great flexibility and experimental ability to change their energy spectrum and structural geometry through of external electric fields. In this way, the optical coherent control of charge carriers enables the investigation of several quantum interference process, such as tunneling induced transparency. We investigate the optical response of the quantum dot molecule considering different optical regimes and electric field values, where the tunneling between the dots can establish efficiently quantum destructive interference paths causing significant changes on the optical spectrum. Using realistic experimental parameters we show that the excitons states coupled by tunneling exhibit a controllable and enriched optical response. In this system, we also investigate the entanglement degree between the electron and hole, and we demonstrate through of the control of parameters such as, the applied gate voltage, the incident laser frequency and intensity, the system goes to an asymptotic state with a high entanglement degree, which is robust to decoherence process. / Neste trabalho investigamos teoricamente a dinâmica dissipativa de estados de éxcitons em um sistema formado por pontos quânticos duplos, que por sua vez apresentam uma grande flexibilidade e capacidade experimental em alterar seu espectro de energia juntamente com sua forma estrutural por meio de campos elétricos externos. Deste modo, o controle óptico coerente de portadores de carga nessas nanoestruturas permite a investigação de diversos processos de interferência quântica, tais como transparência induzida por tunelamento. Investigamos a resposta óptica da molécula quântica considerando diferentes regimes ópticos e valores de campo elétrico, nos quais o tunelamento entre os pontos pode estabelecer eficientemente caminhos de interferência quântica destrutiva provocando mudanças significativas no espectro óptico. Usando parâmetros experimentais realísticos mostramos que os estados excitônicos acoplados por tunelamento exibem uma resposta óptica controlável e bastante enriquecida. Neste mesmo sistema, investigamos o emaranhamento entre elétron e buraco, e demonstramos que através do controle de parâmetros tais como, a barreira de potencial aplicada, a frequência e intensidade do laser incidente, o sistema evolui para um estado assintótico com um alto grau de emaranhamento, que se apresenta robusto a processos de decoerência. / Doutor em Física

On a compris ces dernières années que certaines mesures d'intrications sont un outil efficace pour la compréhension et la caractérisation de phases nouvelles et exotiques de la matière, en particulier lorsque les méthodes traditionnelles basées sur l'identification d'un paramètre d'ordre sont insuffisantes. Cette thèse porte sur l'étude de quelques systèmes quantiques à basse dimension où un telle approche s'avère fructueuse. Parmi ces mesures, l'entropie d'intrication, définie via une bipartition du système quantique, est probablement la plus populaire, surtout à une dimension. Celle-ci est habituellement très difficile à calculer en dimension supérieure, mais nous montrons ici que le calcul se simplifie drastiquement pour une classe particulière de fonctions d'ondes, nommées d'après Rokhsar et Kivelson. L'entropie d'intrication peut en effet s'exprimer comme une entropie de Shannon relative à la distribution de probabilité générée par les composantes de la fonction d'onde du fondamental d'un autre système quantique, cette fois-ci unidimensionnel. Cette réduction dimensionnelle nous permet d'étudier l'entropie aussi bien par des méthodes numériques (fermions libres, diagonalisations exactes, ...) qu'analytiques (théories conformes). Nous argumentons aussi que cette approche permet d'accéder facilement à certaines caractéristiques subtiles et universelles d'une fonction d'onde donnée en général.Une autre partie de cette thèse est consacrée aux trempes quantiques locales dans des systèmes critiques unidimensionnels. Nous insisterons particulièrement sur une quantité appelée écho de Loschmidt, qui est le recouvrement entre la fonction d'onde avant la trempe et la fonction d'onde à temps t après la trempe. En exploitant la commensurabilité du spectre de la théorie conforme, nous montrons que l'évolution temporelle doit être périodique, et peut même être souvent obtenue analytiquement. Inspiré par ces résultats, nous étudions aussi la contribution de fréquence nulle à l'écho de Loschmidt après la trempe. Celle-ci s'exprime comme un simple produit scalaire -- que nous nommons fidélité bipartie -- et est une quantité intéressante en elle-même. Malgré sa simplicité, son comportement se trouve être très similaire à celui de l'entropie d'intrication. Pour un système critique unidimensionnel en particulier, notre fidélité décroît algébriquement avec la taille du système, un comportement rappelant la célèbre catastrophe d'Anderson. L'exposant est universel et relié à la charge centrale de la théorie conforme sous-jacente. / In recent years, it has been understood that entanglement measures can be useful tools for the understanding and characterization of new and exotic phases of matter, especially when the study of order parameters alone proves insufficient. This thesis is devoted to the study of a few low-dimensional quantum systems where this is the case. Among these measures, the entanglement entropy, defined through a bipartition of the quantum system, has been perhaps one of the most heavily studied, especially in one dimension. Such a quantity is usually very difficult to compute in dimension larger than one, but we show that for a particular class of wave functions, named after Rokhsar and Kivelson, the entanglement entropy of an infinite cylinder cut into two parts simplifies considerably. It can be expressed as the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum system. This dimensional reduction allows for a detailed numerical study (free fermion, exact diagonalizations, \ldots) as well as an analytic treatment, using conformal field theory (CFT) techniques. We also argue that this approach can give an easy access to some refined universal features of a given wave function in general.Another part of this thesis deals with the study of local quantum quenches in one-dimensional critical systems. The emphasis is put on the Loschmidt echo, the overlap between the wave function before the quench and the wave function at time t after the quench. Because of the commensurability of the CFT spectrum, the time evolution turns out to be periodic, and can be obtained analytically in various cases. Inspired by these results, we also study the zero-frequency contribution to the Loschmidt echo after such a quench. It can be expressed as a simple overlap -- which we name bipartite fidelity -- and can be studied in its own right. We show that despite its simple definition, it mimics the behavior of the entanglement entropy very well. In particular when the one-dimensional system is critical, this fidelity decays algebraically with the system size, reminiscent of Anderson's celebrated orthogonality catastrophe. The exponent is universal and related to the central charge of the underlying CFT.

In this thesis, we investigate the entanglement and magnetic properties of frustrated spin systems and correlated electronic properties of conjugated carbon systems. In chapter 1, we present different approaches to solve the time-independent, nonrelativistic Schr¨odinger equation for a many-body system. We start with the full non-relativistic Hamiltonian of a multi nuclear system to describe the Born - Oppenheimer approximation which allows the study of electronic Hamiltonian which treats nuclear positions parametrically. We then also describe ab initio techniques such as the Hartree-Fock Method and density functional theories. We then introduce model Hamiltonians for strongly correlated systems such as the Hubbard, Pariser-Parr-Pople and Heisenberg models, and show how they result from the noninteracting one-band tight-binding model. In chapter 2, we discuss various numerical techniques like the exact diagonalization methods and density matrix renormalization group (DMRG) method. We also discuss quantum entanglement and the success of DMRG which can be attributed to the area law of entanglement entropy. In chapter 3, we study here different regions in phase diagrams of the spin-1/2, spin-1 and spin-3/2 one-dimensional antiferromagnetic Heisenberg systems with nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions and dimerization (d ). Frustration arises for specific relative signs of the interactions J1 and J2. In particular, we analyze the behavior of the bipartite entanglement entropy and fidelity at the gapless to gapped phase transitions and across the lines separating different phases in the J2−d plane. All the calculations in this work are based on exact diagonalizations of finite systems. In chapter 4, we study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (JF 1 ) and antiferromagnetic (JA 1 ) nearest-neighbor interactions and a ferromagnetic next-nearest-neighbor interaction (JF 2 ). In this model frustration is present due to non-zero JF 2 . The model with site spin s behaves like a Haldane spin chain with site spin 2s in the limit of vanishing JF 2 and large JF 1 /JA 1 . We show that the exact ground state of the model can be found along a line in the parameter space. For fixed JF 1 , the phase diagram in the space of JA 1 −JF 2 is determined using numerical techniques complemented by analytical calculations. A number of quantities, including the structure factor, energy gap, entanglement entropy and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and potentially important feature of this model is that it can exhibit a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian. In chapter 5, we study correlated electronic properties of zigzag and armchair fused naphthalenes and polyperylene systems in the presence of long-range electronelectron interactions. We find that the ground state of zigzag fused naphthalene system is a higher spin state, while the ground state of armchair fused naphthalene is a singlet. The spin gap of polyperylene is unusually small and the ground state is a singlet. Our calculations of optical gap and two-photon gap suggest that polyperylene should exhibit fluorescence. From the charge gap calculation, we predict that in zigzag fused naphthalene and polyperylene systems, excitons are weakly binding. Peierls type of distortion is negligible in zigzag fused naphthalene and polyperylene systems, however, in armchair fused naphthalene system, interior bonds have tendency to distort in low-lying excited states. In chapter 6, we study the ground state spin of the Heisenberg spin-1/2 nearestneighboring antiferromagnetic exchange models of systems with fused odd member rings. In particular, we compute the ground state spin of fused three and five membered rings as well as fused five membered rings. In the thermodynamic limit, the ground state of the fused three and five membered system is a higher spin state, while fused five membered system shows a singlet ground state, for all system sizes.

Pour d non puissance d’un nombre premier, le nombre maximal de bases deux à deux décorrélées d’un espace de Hilbert de dimension d n’est pas encore connu. Dans ce mémoire, nous commençons par donner une construction de bases décorrélées en lien avec une famille de représentations irréductibles de l'algèbre de Lie su(2) et faisant appel aux sommes de Gauss.Puis nous étudions de façon systématique la possibilité de construire de telle bases au moyen des opérateurs de Pauli. 1) L’étude de la droite projective sur Zdm montre que, pour obtenir des ensembles maximaux de bases décorrélées à l’aide d'opérateurs de Pauli, il est nécessaire de considérer des produits tensoriels de ces opérateurs. 2) Les sous-modules lagrangiens de Zd2n, dont nous donnons une classification complète, rendent compte des ensembles maximalement commutant d'opérateurs de Pauli. Cette classification permet de savoir lesquels de ces ensembles sont susceptibles de donner des bases décorrélées : ils correspondent aux demi-modules lagrangiens, qui s'interprètent encore comme les points isotropes de la droite projective (P(Mat(n, Zd)²),ω). Nous explicitons alors un isomorphisme entre les bases décorrélées ainsi obtenues et les demi-modules lagrangiens distants, ce qui précise aussi la correspondance entre sommes de Gauss et bases décorrélées. 3) Des corollaires sur le groupe de Clifford et l’espace des phases discret sont alors développés.Enfin, nous présentons quelques outils inspirés de l’étude précédente. Nous traitons ainsi du rapport anharmonique sur la sphère de Bloch, de géométrie projective en dimension supérieure, des opérateurs de Pauli continus et nous comparons l'entropie de von Neumann à une mesure de l'intrication par calcul d'un déterminant. / For d not a power of a prime, the maximal number of mutually unbiased bases (MUBs) in a d-dimensional Hilbert space is still unknown. In this thesis, we begin by an original building of MUBs by means of Gauss sums, in relation with a family of irreducible representations of the Lie algebra su(2).Then, we sytematically study the possibility of building such bases by means of Pauli operators. 1) The study of the projective line on Zdm shows that, in order to obtain maximal sets of MUBs, tensorial products of these operators are in order. 2) Lagrangian submodules of Zd2n, of which we give a complete classification, account for maximally commuting sets of Pauli operators. This classification enables to know which of these sets are likely to yield unbiased bases. They correspond to Lagrangian half-modules that can be interpreted as the isotropic points of the projective line (P(Mat(n, Zd)²),ω). Hence, we establish an isomorphism between the unbiased bases thus obtained and distant Lagrangian half-modules, which precises by the way the correspondance between Gauss sums and MUBs. 3) Corollaries on the Clifford group and the finite phase space are then developed.Finally, we present some tools inspired by the previous study. We deal with the cross-ratio on the Bloch sphere and projective geometry in higher dimension, Pauli operators with continuous exponents and we compare von Neumann entropy with a determinantal measure of entanglement