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note
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A direct comparison between arbitrary polarization qubits, μ | H + ν | V, and coherent-state qubits, μ | α + ν | - α, is not straightforward when α is small due to the nonzero overlap between the two component coherent states. It should be noted that we use the orthogonal basis, | + and | -, for coherent-state qubits in order to calculate the average fidelity for a "fair comparison" with the polarization-qubit case. In this sense, the input coherent-state qubit in comparison to the polarization qubit, μ | H + ν | V, should be considered to be μ | + + ν | -, rather than μ | α + ν | - α, when the overlap | α | - α | 2 is non-negligible.
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