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I've seen various different matrices used to represent beam splitters, and am wondering which is the "right" one. Alternatively, are there various kinds of beam splitters but everyone just ambiguously calls them the same thing?
The matrices I've seen are the square-root-of-not-with-extra-phase-factor matrix ##A = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & i \\ i & 1 \end{bmatrix}## here and here, and the hadamard matrix ##H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}## here.
A major difference between the square root of NOT gate and the Hadamard gate is that ##H^2 = I## whereas ##A^2 \propto X##. Also, if you have a 180-degree phase shift gate ##Z## then ##HZH = X## whereas ##AZA = A^2 e^{i \pi Z/2}##.
Which operation should I be thinking of, when an article says "beam splitter"?
The matrices I've seen are the square-root-of-not-with-extra-phase-factor matrix ##A = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & i \\ i & 1 \end{bmatrix}## here and here, and the hadamard matrix ##H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}## here.
A major difference between the square root of NOT gate and the Hadamard gate is that ##H^2 = I## whereas ##A^2 \propto X##. Also, if you have a 180-degree phase shift gate ##Z## then ##HZH = X## whereas ##AZA = A^2 e^{i \pi Z/2}##.
Which operation should I be thinking of, when an article says "beam splitter"?