Related papers: Quantum Geometry of Expectation Values
Based on empirical evidence, quantum systems appear to be strictly linear and gauge invariant. This work uses concise mathematics to show that quantum eigenvalue equations on a one dimensional ring can either be gauge invariant or have a…
We develop a notion of quantum observable for the general boundary formulation of quantum theory. This notion is adapted to spacetime regions rather than to hypersurfaces and naturally fits into the topological quantum field theory like…
We construct in this article a new realization of quantum geometry, which is obtained by quantizing the recently-introduced flux formulation of loop quantum gravity. In this framework, the vacuum is peaked on flat connections, and states…
Relaxing the postulates of an axiomatic theory is a natural way to find more general theories, and historically, the discovery of non-Euclidean geometry is a famous example of this procedure. Here, we use this way to extend quantum…
The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical operators are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then,…
By invoking quantum estimation theory we formulate bounds of errors in quantum measurement for arbitrary quantum states and observables in a finite-dimensional Hilbert space. We prove that the measurement errors of two observables satisfy…
Most approaches towards a quantum theory of gravitation indicate the existence of a minimal length scale of the order of the Planck length. Quantum mechanical models incorporating such an intrinsic length scale call for a deformation of…
We introduce a geometric formulation of quantum indeterminacy from which the standard uncertainty inequalities emerge as necessary consequences. Our approach is based on convex geometry in phase space and on methods from symplectic…
A geometric approach to formulate the uncertainty principle between quantum observables acting on an $N$-dimensional Hilbert space is proposed. We consider the fidelity between a density operator associated with a quantum system and a…
A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces…
Estimating the eigenvalues of non-normal matrices is a foundational problem with far-reaching implications, from modeling non-Hermitian quantum systems to analyzing complex fluid dynamics. Yet, this task remains beyond the reach of standard…
We study Heisenberg's matrix mechanics within an algebraic pre-Hilbert framework of arbitrary finite dimension. The commutator of the position and momentum matrices naturally generates a third Hermitian operator whose unbounded character…
The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this…
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…
Dynamic equations concerning physical expectation values have been examined in terms of the real Hilbert space approach to quantum mechanics. The considered cases involve complex wave functions, as well as quaternionic wave functions. The…
We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along…
We identify quantum geometric bounds for observables in non-Hermitian systems. We find unique bounds on non-Hermitian quantum geometric tensors, generalized two-point response correlators, conductivity tensors, and optical weights. We…
We elaborate on the proposed general boundary formulation as an extension of standard quantum mechanics to arbitrary (or no) backgrounds. Temporal transition amplitudes are generalized to amplitudes for arbitrary spacetime regions. State…
Uncertainty principle is one of the cornerstones of quantum theory. In the literature, there are two types of uncertainty relations, the operator form concerning the variances of physical observables and the entropy form related to entropic…
A generalized formulation of non-relativistic quantum mechanics is developed within multidimensional geometric (NG) frameworks characterized by a power-law dispersion relation \(E \propto |p|^{j}\), where \(j = N - 1\). Starting from the…