Extreme expected values and their applications in quantum metrology
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
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c\left( |\phi 0\rangle +|0\phi \rangle \right)
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|\phi \rangle =\sqrt{p_n}|n\rangle
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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
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n|\phi \rangle =\sqrt{p_n}n|n\rangle
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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
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=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
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=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
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=c^2\left[ \sum_{n\ne 0}{p_nn}-\sum_{n\ne 0}{p_nn} \right] ^2=0
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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)
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\left( n_a-n_b \right) \left( |\phi 0\rangle +|0\phi \rangle \right)
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=\sum_{n\ne 0}{n\sqrt{p_n}|n\rangle |0\rangle}-|0\rangle \sum_{n\ne 0}{n\sqrt{p_n}|n\rangle}
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\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)
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=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)
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=2c^2\sum_{n\ne 0}{p_nn^2}
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c^2=\frac{1}{2\left( 1+p_0 \right)}
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2c^2\sum_{n\ne 0}{p_nn^2}
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=\frac{\sum_n{p_nn^2}}{1+p_0}
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\]