State-dependent approach to entropic measurement–disturbance relations

• We present a novel state-dependent entropic measurement–disturbance relation.
• The relation provides a trade-off between disturbance and predictive measurement error.
• The disturbance can further include the effect on an isolated memory system.
• We show tightness for qubit as well as position–momentum measurements.

Heisenberg's intuition was that there should be a trade-off between measuring a particle's position with greater precision and disturbing its momentum. Recent formulations of this idea have focused on the question of how well two complementary observables can be jointly measured. Here, we provide an alternative approach based on how enhancing the predictability of one observable necessarily disturbs a complementary one. Our measurement–disturbance relation refers to a clear operational scenario and is expressed by entropic quantities with clear statistical meaning. We show that our relation is perfectly tight for all measurement strengths in an existing experimental setup involving qubit measurements.