Generalized entropic uncertainty relation (EUR) near a Schwarzschild black hole.
Imagine both Alice and Bob free fall into a Schwarzschild black hole. They share a particular quantum state, like Bell state. Then Alice stays in that free-falling reference frame, but Bob hoves near the event horizon. By approximation, Alice can be described in Minkowski reference frame, while Bob can be described in Rindler (uniformly accelerating) reference frame. It is found that quantum entanglement between this two parties will be reduced by Unruh effect[1].
Adapting entropy to quantify uncertainties in quantum system, the entropic uncertainty relation captures the essence of the uncertainty caused by imcompatible measurements better than Heisenberg one. When the measured system is correlated with a other quantum system (called memory), the information in memory can reduce uncertainty in measured system through quantum correlation between them[2]. In [2] they used mutual information as a measure to quantify the quantum correlation between Alice and Bob. However, there is a better way to clarify the effect of quantum correlation in the uncertainty relation[3]. In many cases, lower bounds in [3] are tighter than [2]. But the exact relation between these two bound is unkonwn.
The first investigation of uncertainty relation in curved spacetime is conducted by Feng et al[4]. As the memory Bob goes near to event horizon, the more Hawking radiation he will feel. The quantum correlation will drop and the uncertainty bound will increase. Other than simply examining the effect of black hole on uncertainty relation, EUR can be served as an experimental method to detect the existance of firewall[4].
In our work[5], we firstly proved the new relation in [3] is equivalent to relation without quantum memory. This gives us a quantity to clarify how total uncertainty is influenced by quantum correlation, that is Holevo quantity.
Reference:
[1] Fuentes-Schuller, I., & Mann, R. B. (2005). Alice falls into a black hole: entanglement in noninertial frames. Physical review letters, 95(12), 120404.
[2] Berta, M., Christandl, M., Colbeck, R., Renes, J. M., & Renner, R. (2010). The uncertainty principle in the presence of quantum memory. Nature Physics, 6(9), 659.
[3] Xiao, Yunlong, Naihuan Jing, and Xianqing Li-Jost. "Uncertainty under quantum measures and quantum memory." Quantum Information Processing 16.4 (2017): 104.
[4] Feng, Jun, et al. "Uncertainty relation in Schwarzschild spacetime." Physics Letters B 743 (2015): 198-204.
[5] Huang, J. L., Gan, W. C., Xiao, Y., Shu, F. W., & Yung, M. H. (2017). Holevo Bound of Entropic Uncertainty in Schwarzschild Spacetime. arXiv preprint arXiv:1712.04287.
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