# Newton’s Cradle Spectra

### APA

Soda, B. (2022). Newton’s Cradle Spectra. Perimeter Institute. https://pirsa.org/22110081

### MLA

Soda, Barbara. Newton’s Cradle Spectra. Perimeter Institute, Nov. 09, 2022, https://pirsa.org/22110081

### BibTex

@misc{ pirsa_PIRSA:22110081, doi = {10.48660/22110081}, url = {https://pirsa.org/22110081}, author = {Soda, Barbara}, keywords = {Quantum Information}, language = {en}, title = {Newton{\textquoteright}s Cradle Spectra}, publisher = {Perimeter Institute}, year = {2022}, month = {nov}, note = {PIRSA:22110081 see, \url{https://pirsa.org}} }

Barbara Soda Perimeter Institute for Theoretical Physics

## Abstract

We present broadly applicable nonperturbative results on the behavior of eigenvalues and eigenvectors under the addition of self-adjoint operators and under the multiplication of unitary operators, in finite-dimensional Hilbert spaces. To this end, we decompose these operations into elementary 1-parameter processes in which the eigenvalues move similarly to the spheres in Newton's cradle. As special cases, we recover level repulsion and Cauchy interlacing. We discuss two examples of applications. Applied to adiabatic quantum computing, we obtain new tools to relate algorithmic complexity to computational slowdown through gap narrowing. Applied to information theory, we obtain a generalization of Shannon sampling theory, the theory that establishes the equivalence of continuous and discrete representations of information. The new generalization of Shannon sampling applies to signals of varying information density and finite length.

Zoom link: https://pitp.zoom.us/j/94120657832?pwd=SmpsWFhhVCtyeXM3a0pVQU9lMGFLdz09