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In the study of d-dimensional quantum channels $(d \geq 2)$, an assumption which is not very restrictive, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra.…
The generalized Bloch decomposition of a bipartite quantum state gives rise to a correlation matrix whose singular values provide rich information about non-local properties of the state, such as the dimensionality of entanglement. While…
We present a simple geometric construction linking geometric to deformation quantization. Both theories depend on some apparently arbitrary parameters, most importantly a polarization and a symplectic connection, and for real polarizations…
In this paper we investigate the possibility of constructing a complete quantization procedure consisting of geometric and deformation quantization. The latter assigns a noncommutative algebra to a symplectic manifold, by deforming the…
Quantum mechanics is sensitive to the geometry of the underlying space. Here, we present a framework for quantum scattering of a non-relativistic particle confined to a two-dimensional space. When the motion manifold hosts localized…
We analyze qubit decoherence in the framework of geometric quantum mechanics. In this framework the qubit density operators are represented by probability distributions which are also the K\"ahler functions on the Bloch sphere.…
We develop a wave mechanics formalism for qubit geometry using holomorphic functions and Mobius transformations, providing a geometric perspective on quantum computation. This framework extends the standard Hilbert space description,…
In this paper, we study quantization on a compact integral symplectic manifold $X$ with transversal real polarizations. In the case of complex polarizations, namely $X$ is K\"ahler equipped with transversal complex polarizations $T^{1, 0}X,…
The formulation of Geometric Quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the…
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…
The depolarization channel is usually modelled as a quantum operation that destroys all input information, replacing it by a completely chaotic state. For qubits this has a quite intuitive interpretation as a shrinking of the Bloch sphere.…
The mechanism of describing quantum states by standard probability (tomographic one) instead of wave function or density matrix is elucidated. Quantum tomography is formulated in an abstract Hilbert space framework, by means of the identity…
In this paper, we explore the quantization of K\"ahler manifolds, focusing on the relationship between deformation quantization and geometric quantization. We provide a classification of degree 1 formal quantizable functions in the…
A generalized depolarizing channel acts on an N-dimensional quantum system to compress the ``Bloch ball'' in N^2-1 directions; it has a corresponding compression vector. We investigate the geometry of these compression vectors and prove a…
We give a simple geometric description of all formal deformation quantizations on a K\"ahler manifold $M$ which enjoy the following property of separation of variables into holomorphic and antiholomorphic ones. For each open subset…
Given n qubits prepared according to the same unknown density operator, we propose a nondestructive measuring method which approximately yields the eigenstates. It is shown that, for any plane which passes through the center point of the…
Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both,…
Quantum state space is endowed with a metric structure and Riemannian monotone metric is an important geometric entity defined on such a metric space. Riemannian monotone metrics are very useful for information-theoretic and statistical…
Deformation quantization and geometric quantization on K\"ahler manifolds give the mathematical description of the algebra of quantum observables and the Hilbert spaces respectively, where the later forms a representation of quantum…
This paper presents an introduction to geometric representations of quantum states in which each distinct quantum state, pure and mixed, corresponds to a unique point in a Euclidean space. Beginning with a review of some underappreciated…