Minimum Uncertainty and Entanglement

TL;DR

This paper investigates how entanglement affects the Heisenberg uncertainty relation, showing that maximal entanglement prevents the equality from being reached for certain observables in both finite and infinite-dimensional systems.

Contribution

It demonstrates that entanglement imposes fundamental constraints on the attainability of the minimum uncertainty bound in quantum systems.

Findings

Maximum entanglement prevents the uncertainty relation from reaching equality in finite dimensions.

Entangled qubits cannot achieve the lower bound of the uncertainty relation.

Certain entangled states in infinite dimensions also cannot attain the minimum uncertainty equality.

Abstract

We address the question, does a system A being entangled with another system B, put any constraints on the Heisenberg uncertainty relation (or the Schrodinger-Robertson inequality)? We find that the equality of the uncertainty relation cannot be reached for any two noncommuting observables, for finite dimensional Hilbert spaces if the Schmidt rank of the entangled state is maximal. One consequence is that the lower bound of the uncertainty relation can never be attained for any two observables for qubits, if the state is entangled. For infinite-dimensional Hilbert space too, we show that there is a class of physically interesting entangled states for which no two noncommuting observables can attain the minimum uncertainty equality.