Fine-grained uncertainty relation and entropic uncertainty relation in the presence of quantum memory

In presence of quantum memory [M. Berta, M. Christandl, R. Colbeck, J.M. Renes, and R. Renner, Nature Phys. 6, 659 (2010)] the lower bound of entropic uncertainty relation depends on amount of entanglement between the particle (on which two incompatible observable are measured) and the quantum memory. On other hand, fine grained uncertainty relation [J. Oppenheim and S. Wehner, Science 330, 1072 (2010)] gives the upper bound of winning non-local (CHSH) game. We construct similar kind of quantum game and relate two uncertainty relation for measurement of any two observable on the quantum system in presence of quantum memory and show there are certain region where the lower bound of entropic uncertainty relation in presents of quantum memory given by Berta et. al. should not be achieve by spin projective measurement.

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