The usual analytic approach to differential geometry was explored in the early 18th century using the defining notion of limit.With the advent of topos theory in the late 20th century, it has been possible to rediscover without the use of limits the differential and integral calculus, and its development into Synthetic Differential Geometry (SDG), in mathematical universes other than that represented by set theory.These universes obey a subtle but well understood generalisation of classical logic: intuitionistic logic.This thesis presents an exploration of the construction of quantum mechanics in the context of SDG.Chapter 1 introduces SDG, intuitionistic logic, topos theory and a particular topos used in the thesis.Chapter 2 explores the new logical perspective afforded by topos theory and SDG in describing the quantum world from an intuitionistic viewpoint.We build a working quantum-mechanical model using topos theory and analyse the physical and philosophical relevance of the logical steps taken, and account for the consistency of the model with its set-theoretic predecessor by suggesting a certain physical interpretation for the mathematics.In chapter 3, we analyse the extension of this quantum mechanics to state spaces with an infinite number of dimensions, and the associated use of various mathematical constructions that are taken for granted in set theory.Chapter 4 reaches an important proof of the lack of validity of the Spectral 4 The Spectral Theorem 108 4.1 Proof of Failure 4.2 The Internal Logic of the Spectral Theorem 4.3 Measurement-Enforced Validity 5 Conclusions 118 5.
The Quantum Fisher Information (QFI) metric governs a fundamental duality: it quantifies both how precisely a parameter can be estimated (metrology) and how distinguishable two quantum states are (privacy). We exploit this duality to establish a geometry-aware framework for quantum differential privacy (DP) that replaces isotropic depolarizing noise with direction-dependent noise aligned to the QFI eigenstructure of the quantum embedding. We prove six principal theorems: (1) the minimax-optimal mechanism concentrates the noise budget in the dominant QFI eigenmode, achieving ε=(Δ2/2)λmax(1−cγ) with O(d/λmax) advantage; (2) mixed-state QFI decomposition reveals that dephasing in the adversary's basis increases accessible information, while misaligned-basis dephasing provides constructive privacy amplification from hardware noise; (3) a tight privacy − utility uncertainty relation ε⋅(1−F)≥2Δ2dTr(F); (4) adaptive QFI estimation converging at O(1/n) yields 1.92× tighter bounds; (5) QFI-aligned composition saturates at O(1) versus O(k) for standard composition; and (6) hardware noise can be harnessed for privacy amplification. Adversarial vulnerabilities, Wasserstein guarantees, subspace projection, and a zero-knowledge audit protocol follow as corollaries. Results are validated on Qiskit Aer GPU simulations, IBM Quantum hardware (ibm_fez, 156 qubits), and against classical DP baselines, achieving equivalent utility at ε≈0.001 versus ε≈4800 for classical DP.
Summary form only given.We report on a general approach to coherent control in semiconductors, based on optical waveforms with synthesized intensity and phase versus time. This technique is limited only by the bandwidth of the initial laser pulses and the spectral resolution of the pulse-shaping apparatus. It is therefore especially suitable to investigate effects such as off-resonant Rabi oscillations and polaritons in multiple quantum wells. Experiments were performed in four-wave mixing (FWM) and differential reflection geometry based on a mode-locked Ti:sapphire laser emitting 100-fs pulses.