Extreme expected values and their applications in quantum metrology
作者：互联网
DOI: 10.1103/PhysRevA.105.023718
Eq.(48):
\[\left< \left( n_a-n_b \right) ^2 \right> -\left< \left( n_a-n_b \right) \right> ^2 \\ c\left( |\phi 0\rangle +|0\phi \rangle \right) \\ |\phi \rangle =\sqrt{p_n}|n\rangle \\ c^2\left( \langle \phi 0|+\langle 0\phi | \right) \left( n_a-n_b \right) ^2\left( |\phi 0\rangle +|0\phi \rangle \right) -c^2\left[ \left( \langle \phi 0|+\langle 0\phi | \right) \left( n_a-n_b \right) \left( |\phi 0\rangle +|0\phi \rangle \right) \right] ^2 \\ n|\phi \rangle =\sqrt{p_n}n|n\rangle \\ c^2\left[ \left( \langle \phi 0|+\langle 0\phi | \right) \left( n_a-n_b \right) \left( |\phi 0\rangle +|0\phi \rangle \right) \right] ^2 \\ =c^2\left[ \left( \langle \phi 0|+\langle 0\phi | \right) \left( \sqrt{p_n}n|n\rangle |0\rangle -\sqrt{p_n}n|0\rangle |n\rangle \right) \right] ^2 \\ =c^2\left[ \left( \langle \phi 0|\sqrt{p_n}n|n\rangle |0\rangle -\langle \phi 0|\sqrt{p_n}n|0\rangle |n\rangle +\langle 0\phi |\sqrt{p_n}n|n\rangle |0\rangle -\langle 0\phi |\sqrt{p_n}n|0\rangle |n\rangle \right) \right] ^2 \\ =c^2\left[ \sum_{n\ne 0}{p_nn}-\sum_{n\ne 0}{p_nn} \right] ^2=0 \\ c^2\left( \langle \phi 0|+\langle 0\phi | \right) \left( n_a-n_b \right) ^2\left( |\phi 0\rangle +|0\phi \rangle \right) \\ \left( n_a-n_b \right) \left( |\phi 0\rangle +|0\phi \rangle \right) \\ =\sum_{n\ne 0}{n\sqrt{p_n}|n\rangle |0\rangle}-|0\rangle \sum_{n\ne 0}{n\sqrt{p_n}|n\rangle} \\ \therefore c^2\left( \langle \phi 0|+\langle 0\phi | \right) \left( n_a-n_b \right) ^2\left( |\phi 0\rangle +|0\phi \rangle \right) \\ =c^2\left( \sum_{n\ne 0}{n\sqrt{p_n}\langle n|\langle 0|}-\langle 0|\sum_{n\ne 0}{n\sqrt{p_n}\langle n|} \right) \left( \sum_{n\ne 0}{\sqrt{p_n}n|n\rangle |0\rangle}-|0\rangle \sum_{n\ne 0}{\sqrt{p_n}n|n\rangle} \right) \\ =2c^2\sum_{n\ne 0}{p_nn^2} \\ c^2=\frac{1}{2\left( 1+p_0 \right)} \\ 2c^2\sum_{n\ne 0}{p_nn^2} \\ =\frac{\sum_n{p_nn^2}}{1+p_0} \\ \]标签：phi,right,metrology,sqrt,applications,langle,quantum,rangle,left 来源： https://www.cnblogs.com/zhaoyizhou/p/15942992.html