add note about symplectic forms in docs

apkille · Nov 18, 2024 · d9580fc · d9580fc
1 parent 6a02a85
commit d9580fc
Showing 1 changed file with 9 additions and 1 deletion.
diff --git a/docs/src/manual.md b/docs/src/manual.md
@@ -74,8 +74,16 @@ an `N`-mode Gaussian bosonic system. As long as we have a displacement vector of
 !!! note
     A matrix $\mathbf{S}$ of size $2N\times 2N$ is symplectic when it satisfies the following relation:
 
+    $$\mathbf{S} \mathbf{\Omega} \mathbf{S}^{\text{T}} = \mathbf{\Omega}, \qquad \mathbf{\Omega} = \bigoplus_{i=1}^{N} \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}.$$
+
+    In this library, we define symplectic matrices with respect to $\Omega$, the *symplectic form*, which satisfies the canonical
+    commutation relation $[\hat{x}_i, \hat{x}_j] = 2i\Omega_{ij}$, where $\hat{x}_i$ and $\hat{x}_j$
+    are quadrature field operators. However, a different basis can of course be used for a symplectic matrix. 
+    Another common way of expressing symplectic matrices is with the block diagonal symplectic form:
 
-    $$\mathbf{S} \mathbf{\Omega} \mathbf{S}^{\text{T}} = \mathbf{\Omega}, \qquad \mathbf{\Omega} \equiv \bigoplus_{i=1}^{N} \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}.$$
+    $$\mathbf{J} = \begin{pmatrix} 0 & \mathbf{I}_N \\ -\mathbf{I}_N & 0 \end{pmatrix},$$
+
+    where $\mathbf{I}_N$ is the identity matrix of size $N \times N$.
 
 This library has a number of predefined Gaussian unitaries, which are listed below: