= Solution
For an orthogonal $R$, unitary invariance of the <Hilbert-Schmidt norm> gives
$$
\lVert L_1-L_2R\rVert_{\rm HS}^2
=\lVert L_1\rVert_{\rm HS}^2+\lVert L_2\rVert_{\rm HS}^2
-2\operatorname{Re}\operatorname{tr}(R^*L_2^*L_1).
$$
The product $L_2^*L_1$ is trace-class. Its <polar decomposition of a bounded operator> and <trace duality> give
$$
\sup_{R\in\mathcal O(H)}
\operatorname{Re}\operatorname{tr}(R^*L_2^*L_1)
=\lVert L_2^*L_1\rVert_1
=\sum_{k=1}^{\infty}\sigma_k,
$$
where $(\sigma_k)$ are its <singular values>. Taking the infimum proves
$$
d_P(C_1,C_2)^2
=\lVert L_1\rVert_{\rm HS}^2+\lVert L_2\rVert_{\rm HS}^2
-2\sum_{k=1}^{\infty}\sigma_k.
$$
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