Theoretical formulation of finite-dimensional discrete phase spaces: II. On the uncertainty principle for Schwinger unitary operators

Abstract

We introduce a self-consistent theoretical framework associated with the Schwinger unitary operators whose basic mathematical rules embrace a new uncertainty principle that generalizes and strengthens the Massar-Spindel inequality. Among other remarkable virtues, this quantum-algebraic approach exhibits a sound connection with the Wiener-Kinchin theorem for signal processing, which permits us to determine an effective tighter bound that not only imposes a new subtle set of restrictions upon the selective process of signals and wavelets bases, but also represents an important complement for property testing of unitary operators. Moreover, we establish a hierarchy of tighter bounds, which interpolates between the tightest bound and the Massar-Spindel inequality, as well as its respective link with the discrete Weyl function and tomographic reconstructions of finite quantum states. We also show how the Harper Hamiltonian and discrete Fourier operators can be combined to construct finite ground states which yield the tightest bound of a given finite-dimensional state vector space. Such results touch on some fundamental questions inherent to quantum mechanics and their implications in quantum information theory.

31 pages, 3 figures, comments welcome

References in corpus (9)

• Discreteness of Space from the Generalized Uncertainty Principle
• Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects
• Jacobi Theta-functions and Discrete Fourier Transforms
• Discrete phase-space structure of $n$-qubit mutually unbiased bases
• Spin squeezing and entanglement via finite-dimensional discrete phase-space description
• Quasiprobability distribution functions for finite-dimensional discrete phase spaces: Spin-tunneling effects in a toy model
• A new description of spin tunneling in magnetic molecules
• Finite oscillator obtained through finite frame quantization
• Factorization Properties of Finite Spaces

Cited by in corpus (8)

• Uncertainty relations for general unitary operators
• Strong unitary uncertainty relations
• From the discrete Weyl -- Wigner formalism for symmetric ordering to a number -- phase Wigner function
• On the discrete Wigner function for SU(N)
• An improved bound for strong unitary uncertainty relations with refined sequence
• Mixed discrete variable Gaussian states
• Uncertainty relations in terms of the Gini index for finite quantum systems
• Minimum uncertainty states and completeness of non-negative quasi probability of finite-dimensional quantum systems