$\DeclareMathOperator\diag{diag} \newcommand\Id{\mathbb I}$
Since I’m a physicist — and this question was originally migrated from physics.sx (Why? It’s a standard quantum optic question to me!) — I’ll answer to this question using physics techniques.
I. Symplectic operations in quantum optics
Symplectic algebra offers a way to simply encode the Heisenberg uncertainty principle for multimode quantum optics: for a $n$ modes $2n×2n$ covariance matrix $σ$ where the quadatures are alternating ($Q₁, P₁, Q₂, P₂, …$) the Heisenberg uncertainty principle is
\begin{align}
σ + iΩ &≥0 & \text{with }& Ω=ω^{⊕n}= \begin{bmatrix}0&1\\-1&0\end{bmatrix}^{⊕n}\\
&& &\text{in units where } \frac\hbar2≝1.
\end{align}
Physical transformations are the one which preserve this relations.
Let us restrict ourselves to transformations $M$ linear in quadratures.The covariance matrix transforms as $σ↦M^TσM$. If we want $M$ to preserve Heisenberg uncertainty principle, it should preserve $Ω$: it is a symplectic transformations, defined by $M^TΩM=Ω.$
Some example of symplectic and non symplectic operations:
- Special orthogonal operations on a single mode, defined by rotation matrices $R⊕\mathbb I_{2(n-1)}$, corresponds to a dephasing in this mode. This operation is both orthogonal and symplectic.
- Squeezing operations,
$M=\diag\left(s_1,\frac1{s_1},s_2,\frac1{s_2}, \dots\right)$, are symplectic, but not orthogonal.
- A beamsplitter between modes $1$ and $1$ is corresponds to the following symplectic and orthogonal operation $M=\begin{bmatrix}\Id_2\cosθ & -\Id_2\sinθ \\ \Id_2\sinθ & \Id_2\cosθ\end{bmatrix}⊕\Id_{2(n-2)}$.
The above operation generates the whole symplectic group. Note that it contains a non-orthogoal operations (squeezing operations) and that some orthogonal operations are excluded (like rotations involving $P_1$ and $Q_2$: e.g. $\begin{bmatrix} 1& 0& 0& 0\\ 0&0&1&0\\0&1&0&0\\0&0&0&1 \end{bmatrix}⊕\Id_{2(n-2)}$). Therefore, symplectic diagonalizations is not SVD, since the set of allowed transformations is different.
Another set of orthogonal but not symplectic operations are orthogonal operations of determinant $-1$ (reflections), which is essentially the answer to your question about why $c$ and $d$ cannot always be made positive.
II. Local transformation of a 2-mode covariance matrix $σ_{AB}$ in its standard form
The task here is to transform a a two mode covariance matrix $σ_{AB}$
$$ σ_{AB}=\begin{bmatrix} α & γ \\ γ^T & β\end{bmatrix}$$
in a standard form where the $2×2$ submatrices $α$, $β$ and $γ$ are diagonal and, if possible, $∝\Id_2$, using local operations, that is dephasing and squeezing.
Since $α$ is symmetric, it can be diagonalized into $\diag(a_1, a_2)$ using the local special orthogonal transformation $R_a⊕\Id_2$. This operation changes $γ$, but not $β$. Applying the squeezing operation
$S_a=\diag\left(\sqrt{\frac{a_2}{a_1}}, \sqrt{\frac{a_1}{a_2}}, 1, 1\right)$ finally transforms $α$ into $a\Id_2$, with $a=\sqrt{a_1a_2}$, further changing $γ$, but not transforming $β$. The same method, using $\Id_2⊕S_bR_b$, transforms $β$ into $b\Id_2$ without changing $a\Id_2$.
Our covariance matrix is now under the form
$\begin{bmatrix} a\Id_2 & γ' \\ γ'^T & b\Id_2\end{bmatrix}$.The SVD of $γ'$ gives us $c$,$d$ and the orthogonal operations $O_1$ and $O_2$ s.t.
$\diag(|c|,|d|)=O_1^T γ' O_2$.
Let us define the rotations $R_i≝\diag(1,\det(O_i))O_i$. We have then
$\diag(c,d)=O_1^T γ' O_2$.
Let us now apply $R_1⊕R_2$ to our covariance matrix:
- It diagonalizes $γ'$ and $γ'^T$, as expected.
- On the mode $A$, it has the effect of the rotation $R_1$, but since the covaraince matrix of this mode is $a\Id_2$, it is invariant by rotation, and it does not change.
- Similarly, the mode $B$ does not change.
The matrix is now under the standard form
$\begin{bmatrix} a& &c\\ &a& &d\\c& &b\\ &d& &b\end{bmatrix}$.
One could be tempted to change the balance beween $c$ and $d$ by squeezing operations, but it would destroy the balance between $a_1$ and $a_2$. All we can do is exhanging $c$ and $d$, and change both their signs simultaneously.
