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Nielsen's geometric approach to quantum circuit complexity provides a Riemannian framework for quantifying the cost of implementing unitary (closed--system) dynamics. For open dynamics, however, the reduced evolution is described by quantum…
We develop a resource-theoretic framework for quantum coherence directly in continuous basis, with emphasis on the position representation. Since position eigenstates are non-normalizable generalized eigenstates, the standard…
Using the earlier developed classical Hamiltonian framework as the point of departure, we carry out a non-perturbative quantization of the sector of general relativity, coupled to matter, admitting non-rotating isolated horizons as inner…
Recent advancements in the discipline of quantum algorithms have displayed the importance of the geometry of quantum operators. Given this thrust, this paper develops a rigorous geometric framework to analyze how the Riemannian structure of…
We develop a detector-based framework in which quantum theory and spacetime geometry arise within a common inferential structure. Detector states and a detector kernel assign amplitudes to measurement events, allowing quantum theory to be…
In this paper, geometric characterizations of conformally flat and radially flat hypersurfaces in $\mathbb{S}^n \times \mathbb{R}$ and $\mathbb{H}^n \times \mathbb{R}$ are given by means of their extrinsic geometry. Under suitable…
The unitary dynamics of quantum systems can be modeled as a trajectory on a Riemannian manifold. This theoretical framework naturally yields a purely geometric interpretation of computational complexity for quantum algorithms, a notion…
We establish formulas that give the intrinsic volumes, or curvature measures, of sublevel sets of functions defined on Riemannian manifolds as integrals of functionals of the function and its derivatives. For instance, in the Euclidean…
A non-Hermitian operator $H$ defined in a Hilbert space with inner product $\langle\cdot|\cdot\rangle$ may serve as the Hamiltonian for a unitary quantum system, if it is $\eta$-pseudo-Hermitian for a metric operator (positive-definite…
In this manuscript, we show how conformal invariance can be incorporated in a classical theory of gravitation, in the context of metric measure space. Metric measure space involves a geometrical scalar $f$, dubbed as density function, which…
This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at "quantum…
Topological insulators have been studied intensively over the last decades. Earlier research focused on Hermitian Hamiltonians, but recently, peculiar and interesting properties were found by introducing non-Hermiticity. In this work, we…
Integrability, one of the classic issues in galactic dynamics and in general in celestial mechanics, is here revisited in a Riemannian geometric framework, where newtonian motions are seen as geodesics of suitable ``mechanical'' manifolds.…
The interplay between quantum geometry and electron correlation has emerged as a compelling paradigm in quantum many-body physics. Recent studies have highlighted the diagnostic utility of quantum geometry in identifying magnetic…
Here shape space is either the manifold of simple closed smooth unparameterized curves in $\mathbb R^2$ or is the orbifold of immersions from $S^1$ to $\mathbb R^2$ modulo the group of diffeomorphisms of $S^1$. We investige several…
From the perspective of the network theory, the present work illustrates how the parametric intrinsic geometric description exhibits an exact set of pair correction functions and global correlation volume with and without the inclusion of…
This work presents a differentiable geometric parameterization of quantum channels in Kraus representation, which can be efficiently probed to find an unknown quantum channel. We explore its feasibility in finding the quasi inverse…
We analyze the properties of foliations in presence of non-metricity, deriving the generalized Gauss-Codazzi relations in full generality. These results are employed to study the teleparallel framework of non-metric geometry, obtaining…
For particles constrained on a curved surface, how to perform quantization within Dirac's canonical quantization scheme is a long-standing problem. On one hand, Dirac stressed that the Cartesian coordinate system has fundamental importance…
Postulates which lead to Minkowski spacetime are amended in a subtle way, and used to construct a consistent flat spacetime geometry with intrinsic quantum character. Events in the new quantum geometry are described by labels of the form…