15,655 research outputs found

    Quantum retrodiction in Gaussian systems and applications in optomechanics

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    What knowledge can be obtained from the record of a continuous measurement about the quantum state of the measured system at the beginning of the measurement? The task of quantum state retrodiction, the inverse of the more common state prediction, is rigorously addressed in quantum measurement theory through retrodictive positive operator-valued measures (POVMs). This introduction to this general framework presents its practical formulation for retrodicting Gaussian quantum states using continuous-time homodyne measurements and applies it to optomechanical systems. We identify and characterize achievable retrodictive POVMs in common optomechanical operating modes with resonant or off-resonant driving fields and specific choices of local oscillator frequencies in homodyne detection. In particular, we demonstrate the possibility of a near-ideal measurement of the quadrature of the mechanical oscillator, giving direct access to the position or momentum distribution of the oscillator at a given time. This forms the basis for complete quantum state tomography, albeit in a destructive manner

    Opening the system to the environment: new theories and tools in classical and quantum settings

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    The thesis is organized as follows. Section 2 is a first, unconventional, approach to the topic of EPs. Having grown interest in the topic of combinatorics and graph theory, I wanted to exploit its very abstract and mathematical tools to reinterpret something very physical, that is, the EPs in wave scattering. To do this, I build the interpretation of scattering events from a graph theory perspective and show how EPs can be understood within this interpretation. In Section 3, I move from a completely classical treatment to a purely quantum one. In this section, I consider two quantum resonators coupled to two baths and study their dynamics with local and global master equations. Here, the EPs are the key physical features used as a witness of validity of the master equation. Choosing the wrong master equation in the regime of interest can indeed mask physical and fundamental features of the system. In Section 4, there are no EPs. However I transition towards a classical/quantum framework via the topic of open systems. My main contribution in this work is the classical stochastic treatment and simulation of a spin coupled to a bath. In this work, I show how a natural quantum--to--classical transition occurs at all coupling strengths when certain limits of spin length are taken. As a key result, I also show how the coupling to the environment in this stochastic framework induces a classical counterpart to quantum coherences in equilibrium. After this last topic, in Section 5, I briefly present the key features of the code I built (and later extended) for the latter project. This, in the form of a Julia registry package named SpiDy.jl, has seen further applications in branching projects and allows for further exploration of the theoretical framework. Finally, I conclude with a discussion section (see Sec. 5) where I recap the different conclusions gathered in the previous sections and propose several possible directions.Engineering and Physical Sciences Research Council (EPSRC

    Koopman Kernel Regression

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    Many machine learning approaches for decision making, such as reinforcement learning, rely on simulators or predictive models to forecast the time-evolution of quantities of interest, e.g., the state of an agent or the reward of a policy. Forecasts of such complex phenomena are commonly described by highly nonlinear dynamical systems, making their use in optimization-based decision-making challenging. Koopman operator theory offers a beneficial paradigm for addressing this problem by characterizing forecasts via linear time-invariant (LTI) ODEs, turning multi-step forecasts into sparse matrix multiplication. Though there exists a variety of learning approaches, they usually lack crucial learning-theoretic guarantees, making the behavior of the obtained models with increasing data and dimensionality unclear. We address the aforementioned by deriving a universal Koopman-invariant reproducing kernel Hilbert space (RKHS) that solely spans transformations into LTI dynamical systems. The resulting Koopman Kernel Regression (KKR) framework enables the use of statistical learning tools from function approximation for novel convergence results and generalization error bounds under weaker assumptions than existing work. Our experiments demonstrate superior forecasting performance compared to Koopman operator and sequential data predictors in RKHS.Comment: Accepted to the thirty-seventh Conference on Neural Information Processing Systems (NeurIPS 2023

    Quantum-Classical hybrid systems and their quasifree transformations

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    The focus of this work is the description of a framework for quantum-classical hybrid systems. The main emphasis lies on continuous variable systems described by canonical commutation relations and, more precisely, the quasifree case. Here, we are going to solve two main tasks: The first is to rigorously define spaces of states and observables, which are naturally connected within the general structure. Secondly, we want to describe quasifree channels for which both the Schrödinger picture and the Heisenberg picture are well defined. We start with a general introduction to operator algebras and algebraic quantum theory. Thereby, we highlight some of the mathematical details that are often taken for granted while working with purely quantum systems. Consequently, we discuss several possibilities and their advantages respectively disadvantages in describing classical systems analogously to the quantum formalism. The key takeaway is that there is no candidate for a classical state space or observable algebra that can be put easily alongside a quantum system to form a hybrid and simultaneously fulfills all of our requirements for such a partially quantum and partially classical system. Although these straightforward hybrid systems are not sufficient enough to represent a general approach, we use one of the candidates to prove an intermediate result, which showcases the advantages of a consequent hybrid ansatz: We provide a hybrid generalization of classical diffusion generators where the exchange of information between the classical and the quantum side is controlled by the induced noise on the quantum system. Then, we present solutions for our initial tasks. We start with a CCR-algebra where some variables may commute with all others and hence generate a classical subsystem. After clarifying the necessary representations, our hybrid states are given by continuous characteristic functions, and the according state space is equal to the state space of a non-unital C*-algebra. While this C*-algebra is not a suitable candidate for an observable algebra itself, we describe several possible subsets in its bidual which can serve this purpose. They can be more easily characterized and will also allow for a straightforward definition of a proper Heisenberg picture. The subsets are given by operator-valued functions on the classical phase space with varying degrees of regularity, such as universal measurability or strong*-continuity. We describe quasifree channels and their properties, including a state-channel correspondence, a factorization theorem, and some basic physical operations. All this works solely on the assumption of a quasifree system, but we also show that the more famous subclass of Gaussian systems fits well within this formulation and behaves as expected

    On the Normal Sheaf of Gorenstein Curves

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    We show that any tetragonal Gorenstein integral curve is a complete intersection in its respective 33-fold rational normal scroll S, implying that the normal sheaf on CC embedded in S, and in Pg1\mathbb{P}^{g-1} as well, is unstable for g5g\geq 5, provided that SS is smooth. We also compute the degree of the normal sheaf of any singular reduced curve in terms of the Tjurina and Deligne numbers, providing a semicontinuity of the degree of the normal sheaf over suitable deformations, revisiting classical results of the local theory of analytic germs

    Sumsets and Veronese varieties

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    In this paper, to any subset AZn\mathcal{A} \subset \mathbb{Z}^n we explicitly associate a unique monomial projection Yn,dAY_{n, d_{\mathcal{A}}} of a Veronese variety, whose Hilbert function coincides with the cardinality of the tt-fold sumsets tAt \mathcal{A}. This link allows us to tackle the classical problem of determining the polynomial pAQ[t]p_{\mathcal{A}} \in \mathbb{Q}[t] such that tA=pA(t)|t \mathcal{A}|=p_{\mathcal{A}}(t) for all tt0t \geq t_0 and the minimum integer n0(A)t0n_0(\mathcal{A}) \leq t_0 for which this condition is satisfied, i.e. the so-called phase transition of tA|t \mathcal{A}|. We use the Castelnuovo-Mumford regularity and the geometry of Yn,dAY_{n, d_{\mathcal{A}}} to describe the polynomial pA(t)p_{\mathcal{A}}(t) and to derive new bounds for n0(A)n_0(\mathcal{A}) under some technical assumptions on the convex hull of A\mathcal{A}; and vice versa we apply the theory of sumsets to obtain geometric information of the varieties Yn,dAY_{n, d_{\mathcal{A}}}

    Higher gauge theories: Extended Chern-Simons forms and L∞ formulation

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    The Tracy-Singh product of solutions of the Yang-Baxter equation

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    Let VV and VV' be vector spaces over the same field with dim(V)=n\operatorname{dim}(V )=n and dim(V)=m\operatorname{dim}(V')=m. Let c:VVVVc:V \otimes V \rightarrow V \otimes V and d:VVVVd:V'\otimes V' \rightarrow V' \otimes V' be solutions of the Yang-Baxter equation. We show that the Tracy-Singh (or block Kronecker) product of the matrices cc and dd with a particular partition into blocks of cc and dd is the representing matrix of a solution of the Yang-Baxter equation, cd~:WWWW\tilde{ c_d}:\mathcal{ W} \otimes \mathcal{ W} \rightarrow \mathcal{ W} \otimes \mathcal{ W} , with dim(W)=nm\operatorname{dim}(\mathcal{ W} )=nm. Iteratively, it is possible to construct from cc and dd an infinite family of solutions of the Yang-Baxter equation.Comment: arXiv admin note: text overlap with arXiv:2212.1380

    Carath\'eodory Theory and A Priori Estimates for Continuity Inclusions in the Space of Probability Measures

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    In this article, we extend the foundations of the theory of differential inclusions in the space of probability measures with compact support, laid down recently in one of our previous work, to the setting of general Wasserstein spaces. Anchoring our analysis on novel estimates for solutions of continuity equations, we propose a new existence result ``\`a la Peano'' for this class of dynamics, under mere Carath\'eodory regularity assumptions. The latter is based on a set-valued generalisation of the semi-discrete Euler scheme proposed by Filippov to study ordinary differential equations with measurable right-hand sides. We also bring substantial improvements to the earlier versions of the Filippov theorem, compactness and relaxation properties of the solution sets of continuity inclusions which are derived in the more restrictive Cauchy-Lipschitz setting
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