= Solution
In <quantum dense coding>, a shared <Bell pair> lets Alice send two classical <bits> by transmitting one <qubit>. Use <photon polarization> as the <computational basis>, $|0\rangle=|H\rangle$, $|1\rangle=|V\rangle$, and define the <Bell states>
$$
|\phi^\pm\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt2},\qquad
|\psi^\pm\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt2}.
$$
Alice and Bob initially share $|\phi^+\rangle_{AB}$. Alice encodes a two-<bit> message $(m,n)$ by applying the <Pauli operator> $X^nZ^m$ to her <qubit>:
$$
\begin{array}{c|c|c}
(m,n)&\text{Alice's operation}&\text{shared state}\\\hline
(0,0)&I&|\phi^+\rangle\\
(1,0)&Z&|\phi^-\rangle\\
(0,1)&X&|\psi^+\rangle\\
(1,1)&XZ&-|\psi^-\rangle
\end{array}
$$
The last minus sign is a <global phase>. Alice transmits her <photon> to Bob, who now holds both <qubits> and uses a <Bell-basis measurement> to identify the message. The four <Bell states> are <orthogonal>, so an ideal <Bell state analyzer> decodes them without error. The resource cost is one previously shared <Bell pair> and one transmitted <qubit> per two classical <bits>; the shared <quantum entanglement> is consumed. Before transmission Bob's <reduced density matrix> is $I/2$ for every message, consistent with the <no-communication theorem>.
The <BBO> crystal supplies the <polarization-entangled photon pair>. An ultraviolet <laser> pumps this <nonlinear optical crystal>; <spontaneous parametric down-conversion> produces signal and idler <photons> with $\omega_p=\omega_s+\omega_i$ and <phase matching> $\mathbf k_p=\mathbf k_s+\mathbf k_i$. In a noncollinear <type-II phase matching> arrangement, collect the intersections of the two emission cones. The alternatives $|H\rangle_A|V\rangle_B$ and $|V\rangle_A|H\rangle_B$ must be indistinguishable in arrival time, spectrum and spatial mode. Compensation of <birefringence> and adjustment of the <relative quantum phase> then give $(|HV\rangle+|VH\rangle)/\sqrt2$; a local <Pauli X gate> converts it to $|\phi^+\rangle$. <Half-wave plates> and other <waveplates> implement Alice's <Pauli gates>. The <BBO> is the pair source; the message is encoded subsequently by the local optical elements.<New High-Intensity Source of Polarization-Entangled Photon Pairs by Paul G. Kwiat and others>[1995, Figures 1 and 2: intersecting type-II emission cones and compensation; Table I: four Bell states]{ref}
A basic passive <linear optics> <Bell state analyzer> with no extra resources identifies only two of the four polarization <Bell states> unambiguously. Thus the ideal two-<bit> capacity presupposes complete <Bell-basis measurement>; that basic optical implementation alone does not attain it deterministically.
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