Authors: Koji Nagata, Tadao Nakamura
Recently, violation of Heisenberg's uncertainty relation in spin measurements is discussed {[J. Erhart {\it et al.}, Nature Physics {\bf 8}, 185 (2012)]} and {[G. Sulyok {\it et al.}, Phys. Rev. A {\bf 88}, 022110 (2013)]}. We derive the optimal limitation of Heisenberg's uncertainty principle in a specific two-level system (e.g., electron spin, photon polarizations, and so on). Some physical situation is that we would measure $\sigma_x$ and $\sigma_y$, simultaneously. The optimality is certified by the Bloch sphere. We show that a violation of Heisenberg's uncertainty principle means a violation of the Bloch sphere in the specific case. Thus, the above experiments show a violation of the Bloch sphere when we use $\pm 1$ as measurement outcome. This conclusion agrees with recent researches {[K. Nagata, Int. J. Theor. Phys. {\bf 48}, 3532 (2009)]} and {[K. Nagata {\it et al.}, Int. J. Theor. Phys. {\bf 49}, 162 (2010)]}.
Comments: 3 Pages. Open Access Library Journal, Volume 2, e1797.
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