Experimental investigation of quantum uncertainty relations with classical shadows

TL;DR

This paper experimentally investigates quantum uncertainty relations using classical shadows to measure quantum coherence, revealing how lower bounds depend on basis choice and state purity, without full quantum state tomography.

Contribution

It introduces the use of classical shadow algorithms for experimentally testing quantum uncertainty bounds related to coherence, avoiding traditional tomography methods.

Findings

Lower bounds' tightness varies with reference basis.

Quantum state purity influences coherence bounds.

Classical shadows effectively detect uncertainty bounds.

Abstract

The quantum component in uncertainty relation can be naturally characterized by the quantum coherence of a quantum state, which is of paramount importance in quantum information science. Here, we experimentally investigate quantum uncertainty relations construed with relative entropy of coherence, $l_1$ norm of coherence and coherence of formation. In stead of quantum state tomographic technology, we employ the classical shadow algorithm for the detection of lower bounds in quantum uncertainty relations. With an all optical setup, we prepare a family of quantum states whose purity can be fully controlled. We experimentally explore the tightness of various lower bounds in different reference bases on the prepared states. Our results indicate the tightness of quantum coherence lower bounds dependents on the reference bases as well as the purity of quantum state.