Related papers: Geometric uncertainty relation for mixed quantum s…
The concept of fixed-area states has proven useful for recent studies of quantum gravity, especially in connection with gravitational holography. We explore the Lorentz-signature spacetime geometry intrinsic to such fixed-area states in…
The Robertson's formulation of the uncertainty relation is the most widely accepted form of the Heisenberg uncertainty relation (HUR). It gets modified when we consider it for entangled particles. But this formulation does not consider the…
The two-dimensional cluster state, a universal resource for measurement-based quantum computation, is also the gapped ground state of a short-ranged Hamiltonian. Here, we examine the effect of perturbations to this Hamiltonian. We prove…
An algebraic framework for noncommutative bundles with (quantum) homogeneous fibres is proposed. The framework relies on the use of principal coalgebra extensions which play the role of principal bundles in noncommutative geometry which…
In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum $\bsA+\bsB$ of two independent random Hermitian…
We show that the geometry of the set of quantum states plays a crucial role in the behavior of entanglement in different physical systems. More specifically it is shown that singular points at the border of the set of unentangled states…
The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an…
This paper concerns the relationship between locally homogeneous geometric structures on topological surfaces and the moduli of polystable Higgs bundles on Riemann surfaces, due to Hitchin and Simpson. In particular we discuss the…
Natural metric structures on the tangent bundle and tangent sphere bundles $S_rM$ of a Riemannian manifold $M$ with radius function $r$ enclose many important unsolved problems. Admitting metric connections on $M$ with torsion, we deduce…
We present a geometric formulation of quantum mechanics based on the symplectic structure of the projective Hilbert space. Building upon the standard K\"ahler framework, we introduce an extension in which the symplectic structure is allowed…
It has been proposed that on (anti)-de Sitter background, the Heisenberg uncertainty principle should be modified by the introduction of a term proportional to the cosmological constant. We show that this modification of the uncertainty…
A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on…
A geometric framework for quantum statistical estimation is used to establish a series of higher order corrections to the Heisenberg uncertainty relations associated with pairs of canonically conjugate variables. These corrections can be…
The Heisenberg-Robertson uncertainty relation quantitatively expresses the impossibility of jointly sharp preparation of incompatible observables. However it does not capture the concept of incompatible observables because it can be trivial…
These are the expanded notes of a course given at the Summer school "Geometric, topological and algebraic methods for quantum field theory" held at Villa de Leyva, Colombia in July 2015. We first give an introduction to non-commutative…
We have previously presented a version of the Weak Equivalence Principle for a quantum particle as an exact analog of the classical case, based on the Heisenberg picture analysis of free particle motion. Here, we take that to a full…
We propose a version of the non-relativistic quantum mechanics in which the pure states of a quantum system are described as sections of a Hilbert (generally infinitely-dimensional) fibre bundle over the space-time. There evolution is…
In this paper we give a method to associate a graph with an arbitrary density matrix referred to a standard orthonormal basis in the Hilbert space of a finite dimensional quantum system. We study the related issues like classification of…
We show that the Euclidean Snyder non-commutative space implies infinitely many different physical predictions. The distinct frameworks are specified by generalized uncertainty relations underlying deformed Heisenberg algebras. Considering…
Certain many-particle Hardy inequalities are derived in a simple and systematic way using the so-called ground state representation for the Laplacian on a subdomain of $\mathbb{R}^n$. This includes geometric extensions of the standard Hardy…