= Solution
Use $Z|H\rangle=|H\rangle$, $Z|V\rangle=-|V\rangle$. Choose the <eigenstate> phases
$$
\boxed{|H'\rangle=|+\rangle_X=\frac{|H\rangle+|V\rangle}{\sqrt2},\quad
|V'\rangle=|-\rangle_X=\frac{|H\rangle-|V\rangle}{\sqrt2},}
$$
$$
\boxed{|R\rangle=|+\rangle_Y=\frac{|H\rangle+i|V\rangle}{\sqrt2},\quad
|L\rangle=|-\rangle_Y=\frac{|H\rangle-i|V\rangle}{\sqrt2}.}
$$
The inverse relations are $|H\rangle=(|H'\rangle+|V'\rangle)/\sqrt2=(|R\rangle+|L\rangle)/\sqrt2$ and $|V\rangle=(|H'\rangle-|V'\rangle)/\sqrt2=-i(|R\rangle-|L\rangle)/\sqrt2$. The X states are <linear polarization>[linearly polarized] at $+45^\circ$ and $-45^\circ$ to horizontal, while the Y states have equal orthogonal amplitudes in quadrature and hence opposite <circular polarization>[circular polarizations]. We define R as the $+Y$ state; descriptions as clockwise or anticlockwise must specify the observation direction and time convention.
A <polarizing beam splitter> and two <photon counting>[single-photon detectors] directly measure Z, routing H and V to separate detectors. For X, place a <half-wave plate> with its axis at $22.5^\circ$ before that splitter; it implements a Hadamard transformation up to a <global phase>. For Y, first apply a <quarter-wave plate>[quarter-wave] retardance $Q=\operatorname{diag}(1,-i)$ in the H/V axes and then the same <half-wave plate>. The total analyzer transformation is $HQ$, with $HQ|R\rangle=|H\rangle$ and $HQ|L\rangle=|V\rangle$. The physical fast/slow-axis assignment is selected to realize this stated retardance.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2019-polarization-analyzers.png]
{title=Single-photon analyzers for the Z, X and Y eigenbases}
{description=Each polarizing beam splitter routes horizontal and vertical output components to separate photon detectors. The X analyzer uses a half-wave plate at 22.5 degrees; the Y analyzer first uses $Q=\operatorname{diag}(1,-i)$.}
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