Error Analysis on Photonic Qubit Rotations Implemented by Wave Plates

Abstract

Optical Poincare sphere rotations $e^{-i\theta\sigma_{x}/2}$, $e^{-i\theta\sigma_{y}/2}$ and $e^{-i\theta\sigma_{z}/2}$ can be realized by wave-plate combinations. Errors due to combinations with non-ideal wave plates are discussed for three specific combinations (θ=π) by trace distance. The result shows that different settings of combinations affect trace distance: (i) trace distance for $e^{-i\pi\sigma_{x}/2}$ equals that for $e^{-i\pi\sigma_{z}/2}$, but both of them are smaller than that for $e^{-i\pi\sigma_{y}/2}$, when optics-axis random errors are considered; (ii) trace distance for $e^{-i\pi\sigma_{x}/2}$ also equals that for $e^{-i\pi\sigma_{z}/2}$, but both of them are larger than that for $e^{-i\pi\sigma_{y}/2}$, when phase-shift random errors are considered. The method outlined in this paper is general and is useful to analyze other combinations.

Appendix: Derivation

This appendix describes the derivation of (27) and (28) in detail. First, we prove (27) from

$$ 1-2\exp \bigl( -2\sigma^{2} \bigr) +\exp \bigl( -4 \sigma^{2} \bigr) = \bigl[ 1-\exp \bigl( -2\sigma^{2} \bigr) \bigr] ^{2}>0 $$

(29)

Then it follows

$$ 1+\exp \bigl( -4\sigma^{2} \bigr) >2\exp \bigl( -2 \sigma^{2} \bigr) $$

(30)

So it yields

$$ -\exp \bigl( -2\sigma^{2} \bigr) -\exp \bigl( -6\sigma^{2} \bigr) <-2\exp \bigl( -4\sigma^{2} \bigr) $$

(31)

Therefore, we obtain

$$ 2-\exp \bigl( -2\sigma^{2} \bigr) -\exp \bigl( -6 \sigma^{2} \bigr) <2-2\exp \bigl( -4\sigma^{2} \bigr) $$

(32)

According to (23) and (24), we have (27).

Next, we prove (28) from

$$ \frac{\exp ( -\frac{3\sigma^{2}}{8} ) }{\exp ( -\frac {2\sigma^{2}}{8} ) }=\exp \biggl( -\frac{\sigma^{2}}{8} \biggr) <1 $$

(33)

Then it follows

$$ \exp \biggl( -\frac{3\sigma^{2}}{8} \biggr) <\exp \biggl( -\frac{\sigma^{2}}{4} \biggr) $$

(34)

Therefore, we obtain

$$ 2-2\exp \biggl( -\frac{3\sigma^{2}}{8} \biggr) >2-2\exp \biggl( -\frac {\sigma^{2}}{4} \biggr) $$

(35)

According to (25) and (26), we have (28).