Orthogonal Projections on Hyperplanes Intertwined With Unitaries

TL;DR

This paper investigates the behavior of iterates formed by composing a unitary transformation with orthogonal projections in complex vector spaces, revealing a dimension-dependent sum and proposing a quantum dimension witness based on measurement probabilities.

Contribution

It introduces a novel analysis of iterated projections and unitaries, leading to a new device-dependent quantum dimension witness based on sequential measurement outcomes.

Findings

Sum of squared norms equals space dimension in generic cases

Provides exact formulas for non-generic cases

Proposes a quantum dimension witness based on measurement probabilities

Abstract

Fix a point in a finite-dimensional complex vector space and consider the sequence of iterates of this point under the composition of a unitary map with the orthogonal projection on the hyperplane orthogonal to the starting point. We prove that, generically, the series of the squared norms of these iterates sums to the dimension of the underlying space. This leads us to construct a (device-dependent) dimension witness for quantum systems which involves the probabilities of obtaining certain strings of outcomes in a sequential yes-no measurement. The exact formula for this series in non-generic cases is provided as well as its analogue in the real case.