Uncertainty conservation relations: theory and experiment

TL;DR

This paper derives and experimentally verifies an uncertainty conservation relation in quantum physics, showing that the total measurement uncertainty in a bipartite system remains constant when quantum memory is considered, contrasting with traditional uncertainty inequalities.

Contribution

It introduces a new uncertainty conservation relation based on an information measure, and experimentally confirms it without requiring quantum state tomography.

Findings

Uncertainty sum over mutually unbiased bases equals a total uncertainty from the initial state.

Total uncertainty is conserved in the presence of quantum memory.

Experimental verification performed on a 5-qubit spin system.

Abstract

As a very fundamental principle in quantum physics, uncertainty principle has been studied intensively via various uncertainty inequalities. Based on the information measure introduced by Brukner and Zeilinger in [Phys. Rev. Lett. 83, 3354 (1999)], we derive an uncertainty conservation relation in the presence of quantum memory. We show that the sum of measurement uncertainties over a complete set of mutually unbiased bases on a subsystem is equal to a total uncertainty determined by the initial bipartite state. Hence for a given bipartite system the uncertainty is conserved and the uncertainty relation is given by an equality. When the total uncertainty vanishes, all the uncertainties related to every mutually unbiased base measurement are zero, which is substantially different from the uncertainty relation given by Berta et. al. [Nat. Phys. 6, 659 (2010)] where an uncertainty inequality is presented. By directly measuring the corresponding quantum mechanical observables on a 5-qubit spin system, we experimentally verify the uncertainty conservation relation in the presence of quantum memory without quantum state tomography.