Strong unitary and overlap uncertainty relations: theory and experiment

TL;DR

This paper develops a new strong uncertainty relation for multiple unitary operators, linking geometric phases and quantum state overlaps, and experimentally verifies it using photonic qubits, enhancing understanding of quantum uncertainties.

Contribution

It introduces a novel strong uncertainty relation applicable to any number of unitary operators, connecting geometric phases with quantum state overlaps, and demonstrates experimental validation with photonic systems.

Findings

The uncertainty relation is saturated by all pure states in any dimension.

It provides a tight overlap uncertainty relation for multiple pure states.

Experimental tests confirm the theoretical predictions using photonic polarization qubits.

Abstract

We derive and experimentally investigate a strong uncertainty relation valid for any $n$ unitary operators, which implies the standard uncertainty relation as a special case, and which can be written in terms of geometric phases. It is saturated by every pure state of any $n$-dimensional quantum system, generates a tight overlap uncertainty relation for the transition probabilities of any $n+1$ pure states, and gives an upper bound for the out-of-time-order correlation function. We test these uncertainty relations experimentally for photonic polarisation qubits, including the minimum uncertainty states of the overlap uncertainty relation, via interferometric measurements of generalised geometric phases.