Uncertainty relation in Schwarzschild spacetime

Jun Feng, Yao-Zhong Zhang, Mark D. Gould, Heng Fan

Published 2015-01-08Version 1

We explore the entropic uncertainty relation in the curved background outside a Schwarzschild black hole, and find that Hawking radiation introduce a nontrivial modification on the uncertainty bound for particular observer, therefore could be witnessed by proper uncertainty game experimentally. We first investigate an uncertainty game between a free falling observer and his static partner holding a quantum memory initially entangled with the quantum system to be measured. Due to the information loss from Hawking decoherence, we find an inevitably increase of the uncertainty on the outcome of measurements in the view of static observer, which is dependent on the mass of the black hole, the distance of observer from event horizon, and the mode frequency of quantum memory. In an alternative game between two static players, we show that quantum information of qubit can be transferred to quantum memory through a bath of fluctuating quantum fields outside the black hole, which triggers an effectively reduced uncertainty bound that violate the intrinsic limit $-\log_2c$. Numerically estimation for a proper choice of initial state shows that the analysis is comparable with possible real experiments. Moreover, the relation between our results and other uncertainty probe, e.g., time-energy uncertainty, is also discussed.