Notation
Section author: Nicolás Quesada <nicolas@xanadu.ai>
Matrices and vectors
In this section we introduce the notation that is used in the rest of the documentation.
We deal with so-called bivectors in which the second half of their components is the complex conjugate of the first half. Thus if \(\bm{u} = (u_1,\ldots u_\ell) \in \mathbb{C}^{\ell}\) then \(\vec{\alpha} = (\bm{u},\bm{u}^*) = (u_1,\ldots,u_\ell,u_1^*,\ldots,u_\ell^*)\) is a bivector. We use uppercase letters for (multi)sets such as \(S = \{1,1,2\}\).
Following standard Python and C convention, we index starting from zero, thus the first \(\ell\) natural numbers are \([\ell]:=\{0,1,\ldots,\ell-1\}\).
We define the \(\text{diag}\) function as follows: When acting on a vector \(\bm{u}\) it returns a square diagonal matrix \(\bm{u}\). When acting on a square matrix it returns a vector with its diagonal entries.
Finally, we define the vector in diagonal (\(\text{vid}\)) operation, that takes a matrix \(\bm{A}\) of size \(n \times n\) and a vector \(\bm{u}\) of size \(n\) and returns the matrix
\[\text{vid}(\bm{A},\bm{u}) = \bm{A} - \text{diag}(\text{diag}( \bm{A})) + \text{diag}(\bm{u}),\]
which is simply the matrix \(\bm{A}\) with the vector \(\bm{u}\) placed along its diagonal.
The reduction operation
It is very useful to have compact notation to deal with matrices that are constructed by removing or repeating rows and column of a given primitive matrix.
Imagine for example a given \(4 \times 4\) matrix
\[\begin{split}\bm{A} = \left(
\begin{array}{cccc}
A_{0,0} & A_{0,1} & A_{0,2} & A_{0,3} \\
A_{1,0} & A_{1,1} & A_{1,2} & A_{1,3} \\
A_{2,0} & A_{2,1} & A_{2,2} & A_{2,3} \\
A_{3,0} & A_{3,1} & A_{3,2} & A_{3,3} \\
\end{array}
\right),\end{split}\]
and that you want to construct the following matrix
\[\begin{split}\bm{A}'= \left(
\begin{array}{c|ccc||cc}
A_{0,0} & A_{0,1} & A_{0,1} & A_{0,1} & A_{0,3} & A_{0,3} \\
\hline
A_{1,0} & A_{1,1} & A_{1,1} & A_{1,1} & A_{1,3} & A_{1,3} \\
A_{1,0} & A_{1,1} & A_{1,1} & A_{1,1} & A_{1,3} & A_{1,3} \\
A_{1,0} & A_{1,1} & A_{1,1} & A_{1,1} & A_{1,3} & A_{1,3} \\
\hline
\hline
A_{3,0} & A_{3,1} & A_{3,1} & A_{3,1} & A_{3,3} & A_{3,3} \\
A_{3,0} & A_{3,1} & A_{3,1} & A_{3,1} & A_{3,3} & A_{3,3} \\
\end{array}
\right),\end{split}\]
where the first row and column have been kept as they were, the second row and column have been repeated 3 times, the third row and column have been eliminated and the fourth and the last row and column have been repeated twice. To specify the number of repetitions (or elimination) of a given row-column we simply specify a vector of integers where each value tells us the number of repetitions, and we use the value 0 to indicate that a given row-column is removed. Thus defining \(\bm{m}=(1,3,0,2)\) we find its reduction by \(\bm{m}\) to be precisely the matrix in the last equation
\[\bm{A}' = \bm{A}_{\bm{n}}.\]
One can also define the reduction operation on vectors. For instance if \(\bm{u}=(u_0,u_1,u_2,u_3)\) and \(\bm{m}=(1,3,0,2)\) then \(\bm{u}_\bm{n} = (u_0,u_1,u_1,u_1,u_3,u_3)\).
Reduction on block matrices
When dealing with Gaussian states one typically encounters \(2\ell \times 2 \ell\) block matrices of the following form
\[\begin{split}\bm{A} = \left(\begin{array}{c|c}
\bm{C} & \bm{D} \\
\hline
\bm{E} & \bm{F} \\
\end{array}
\right),\end{split}\]
where \(\bm{C},\bm{D},\bm{E},\bm{F}\) are of size \(\ell \times \ell\).
Now imagine that one applies the reduction operation by a vector \(\bm{n} \in \mathbb{N}^{\ell}\) to each of the blocks. We introduce the following notation
\[\begin{split}\bm{A}_{(\bm{n})} = \left(\begin{array}{c|c}
\bm{C}_{\bm{n}} & \bm{D}_{\bm{n}} \\
\hline
\bm{E}_{\bm{n}} & \bm{F}_{\bm{n}} \\
\end{array}
\right),\end{split}\]
where we have used the notation \((\bm{n})\) with the round brackets \(()\) to indicate that the reduction is applied to the blocks of the matrix \(\bm{A}\).
Similarly to block matrices, one can also define a reduction operator for bivectors. Thus if \(\vec \beta = (u_0,u_1,u_2,u_3,u_0^*,u_1^*,u_2^*,u_3^*)\) and \(\bm{m}=(1,3,0,2)\), then
\[\vec \beta_{(\bm{n} ) } = (u_0,u_1,u_1,u_1,u_3,u_3,u_0^*,u_1^*,u_1^*,u_1^*,u_3^*,u_3^*).\]
The reduction operation in terms of sets
A different way of specifying how many times a given row and column must me repeated is by giving a set in which we simply list the columns to be repeated. Thus for example the reduction index vector \(\bm{n} = (1,3,0,2)\) can alternatively be given as the multiset \(S=\{0,1,1,1,3,3 \}\) where the element 0 appears once to indicate the first row and column is repeated once, the index 1 appears three times to indicate that this row and column are repeated three times, etcetera.
Similarly for matrices of even size for which the following partition makes sense
\[\begin{split}\bm{A} = \left(\begin{array}{c|c}
\bm{C} & \bm{D} \\
\hline
\bm{E} & \bm{F} \\
\end{array}
\right),\end{split}\]
where \(\bm{A}\) is of size \(2\ell \times 2\ell\) and \(\bm{C},\bm{D},\bm{E},\bm{F}\) are of size \(\ell \times \ell\) we define
\[\begin{split}\bm{A}_{(S)} = \left(\begin{array}{c|c}
\bm{C}_S & \bm{D}_S \\
\hline
\bm{E}_S & \bm{F}_S \\
\end{array}
\right).\end{split}\]
This implies that if the index \(i\) appears \(m_i\) times in \(S\) then the columns \(i\) and \(i+\ell\) of \(\bm{A}\) will be repeated \(m_i\) times in \(\bm{A}_S\).
For instance if
\[\begin{split}\bm{A} = \left(
\begin{array}{ccc|ccc}
A_{0,0} & A_{0,1} & A_{0,2} & A_{0,3} & A_{0,4} & A_{0,5} \\
A_{1,0} & A_{1,1} & A_{1,2} & A_{1,3} & A_{1,4} & A_{1,5} \\
A_{2,0} & A_{2,1} & A_{2,2} & A_{2,3} & A_{2,4} & A_{2,5} \\
\hline
A_{3,0} & A_{3,1} & A_{3,2} & A_{3,3} & A_{3,4} & A_{3,5} \\
A_{4,0} & A_{4,1} & A_{4,2} & A_{4,3} & A_{4,4} & A_{4,5} \\
A_{5,0} & A_{5,1} & A_{5,2} & A_{5,3} & A_{5,4} & A_{5,5} \\
\end{array}
\right),\end{split}\]
and \(S=\{0,0,2,2,2\}\) one finds
\[\begin{split}\bm{A}_{(S)} = \left(
\begin{array}{cc|ccc|cc|ccc}
A_{0,0} & A_{0,0} & A_{0,2} & A_{0,2} & A_{0,2} & A_{0,3} & A_{0,3} & A_{0,5} & A_{0,5} & A_{0,5} \\
A_{0,0} & A_{0,0} & A_{0,2} & A_{0,2} & A_{0,2} & A_{0,3} & A_{0,3} & A_{0,5} & A_{0,5} & A_{0,5} \\
\hline
A_{2,0} & A_{2,0} & A_{2,2} & A_{2,2} & A_{2,2} & A_{2,3} & A_{2,3} & A_{2,5} & A_{2,5} & A_{2,5} \\
A_{2,0} & A_{2,0} & A_{2,2} & A_{2,2} & A_{2,2} & A_{2,3} & A_{2,3} & A_{2,5} & A_{2,5} & A_{2,5} \\
A_{2,0} & A_{2,0} & A_{2,2} & A_{2,2} & A_{2,2} & A_{2,3} & A_{2,3} & A_{2,5} & A_{2,5} & A_{2,5} \\
\hline
A_{3,0} & A_{3,0} & A_{3,2} & A_{3,2} & A_{3,2} & A_{3,3} & A_{3,3} & A_{3,5} & A_{3,5} & A_{3,5} \\
A_{3,0} & A_{3,0} & A_{3,2} & A_{3,2} & A_{3,2} & A_{3,3} & A_{3,3} & A_{3,5} & A_{3,5} & A_{3,5} \\
\hline
A_{5,0} & A_{5,0} & A_{5,2} & A_{5,2} & A_{5,2} & A_{5,3} & A_{5,3} & A_{5,5} & A_{5,5} & A_{5,5} \\
A_{5,0} & A_{5,0} & A_{5,2} & A_{5,2} & A_{5,2} & A_{5,3} & A_{5,3} & A_{5,5} & A_{5,5} & A_{5,5} \\
A_{5,0} & A_{5,0} & A_{5,2} & A_{5,2} & A_{5,2} & A_{5,3} & A_{5,3} & A_{5,5} & A_{5,5} & A_{5,5} \\
\end{array}
\right).\end{split}\]
The notation also extends in a straightforward fashion for bivectors. For example \(\vec \beta = (u_0,u_1,u_2,u_3,u_0^*,u_1^*,u_2^*,u_3^*)\) and \(S=\{1,1,2\}\) then
\(\vec \beta_{(S)} = (u_1,u_1,u_2,u_1^*,u_1^*,u_2^*)\).
This notation becomes handy when describing certain algorithms for the calculation of the hafnian and torontonian introduced in the rest of the documentation.
Combining reduction and vector in diagonal
Here we show some basic examples of how the reduction and vector in diagonal operations work together
Consider the following matrix
\[\begin{split}\Sigma = \left(
\begin{array}{ccc|ccc}
A_{0,0} & A_{0,1} & A_{0,2} & B_{0,0} & B_{0,1} & B_{0,2} \\
A_{1,0} & A_{1,1} & A_{1,2} & B_{1,0} & B_{1,1} & B_{1,2} \\
A_{2,0} & A_{2,1} & A_{2,2} & B_{2,0} & B_{2,1} & B_{2,2} \\
\hline
C_{0,0} & C_{0,1} & C_{0,2} & D_{0,0} & D_{0,1} & D_{0,2} \\
C_{1,0} & C_{1,1} & C_{1,2} & D_{1,0} & D_{1,1} & D_{1,2} \\
C_{2,0} & C_{2,1} & C_{2,2} & D_{2,0} & D_{2,1} & D_{2,2} \\
\end{array}
\right),\end{split}\]
and bivector \(\vec{\beta} = (\beta_0,\beta_1,\beta_2,\beta_0^*,\beta_1^*,\beta_2^*)\) and we are given the index vector \(\bm{u} = (1,0,3)\). Then we can calculate the following
\[\begin{split}\Sigma_{(\bm{u})} &= \left(
\begin{array}{cccc|cccc}
A_{0,0} & A_{0,2} & A_{0,2} & A_{0,2} & B_{0,0} & B_{0,2} & B_{0,2} & B_{0,2} \\
A_{2,0} & A_{2,2} & A_{2,2} & A_{2,2} & B_{2,0} & B_{2,2} & B_{2,2} & B_{2,2} \\
A_{2,0} & A_{2,2} & A_{2,2} & A_{2,2} & B_{2,0} & B_{2,2} & B_{2,2} & B_{2,2} \\
A_{2,0} & A_{2,2} & A_{2,2} & A_{2,2} & B_{2,0} & B_{2,2} & B_{2,2} & B_{2,2} \\
\hline
C_{0,0} & C_{0,2} & C_{0,2} & C_{0,2} & D_{0,0} & D_{0,2} & D_{0,2} & D_{0,2} \\
C_{2,0} & C_{2,2} & C_{2,2} & C_{2,2} & D_{2,0} & D_{2,2} & D_{2,2} & D_{2,2} \\
C_{2,0} & C_{2,2} & C_{2,2} & C_{2,2} & D_{2,0} & D_{2,2} & D_{2,2} & D_{2,2} \\
C_{2,0} & C_{2,2} & C_{2,2} & C_{2,2} & D_{2,0} & D_{2,2} & D_{2,2} & D_{2,2} \\
\end{array}
\right),\\
\vec \beta_{(\bm{u})} &= (\beta_0,\beta_2,\beta_2,\beta_2,\beta_0^*,\beta_2^*,\beta_2^*,\beta_2^*),\end{split}\]
and finally write
\[\begin{split}\text{vid}(\Sigma_{(\bm{u})},\vec \beta_{(\bm{u})})= \left(
\begin{array}{cccc|cccc}
\beta_{0} & A_{0,2} & A_{0,2} & A_{0,2} & B_{0,0} & B_{0,2} & B_{0,2} & B_{0,2} \\
A_{2,0} & \beta_{2} & A_{2,2} & A_{2,2} & B_{2,0} & B_{2,2} & B_{2,2} & B_{2,2} \\
A_{2,0} & A_{2,2} & \beta_{2} & A_{2,2} & B_{2,0} & B_{2,2} & B_{2,2} & B_{2,2} \\
A_{2,0} & A_{2,2} & A_{2,2} & \beta_{2} & B_{2,0} & B_{2,2} & B_{2,2} & B_{2,2} \\
\hline
C_{0,0} & C_{0,2} & C_{0,2} & C_{0,2} & \beta_{0}^* & D_{0,2} & D_{0,2} & D_{0,2} \\
C_{2,0} & C_{2,2} & C_{2,2} & C_{2,2} & D_{2,0} & \beta_{2}^* & D_{2,2} & D_{2,2} \\
C_{2,0} & C_{2,2} & C_{2,2} & C_{2,2} & D_{2,0} & D_{2,2} & \beta_{2}^* & D_{2,2} \\
C_{2,0} & C_{2,2} & C_{2,2} & C_{2,2} & D_{2,0} & D_{2,2} & D_{2,2} & \beta_{2}^* \\
\end{array}
\right).\end{split}\]
Note that because there are repetitions, the diagonal elements of the matrix \(\bm{A}\) appear off diagonal in \(\bm{A}_{(\bm{n})}\) and also in \(\text{vid}(\bm{A}_{(\bm{n})},\vec{\beta}_{\bm{n}})\).
One can ignore the block structure of the matrix \(A\) and bivector \(\vec{\beta}\), and treat them as 6 dimensional objects and use an index vector of the same length. If we now define \(\bm{p} = (1,0,3,0,2,2)\) one finds
\[\begin{split}\Sigma_{\bm{p}} &= \left(
\begin{array}{cccccccc}
A_{0,0} & A_{0,2} & A_{0,2} & A_{0,2} & B_{0,1} & B_{0,1} & B_{0,2} & B_{0,2} \\
A_{2,0} & A_{2,2} & A_{2,2} & A_{2,2} & B_{2,1} & B_{2,1} & B_{2,2} & B_{2,2} \\
A_{2,0} & A_{2,2} & A_{2,2} & A_{2,2} & B_{2,1} & B_{2,1} & B_{2,2} & B_{2,2} \\
A_{2,0} & A_{2,2} & A_{2,2} & A_{2,2} & B_{2,1} & B_{2,1} & B_{2,2} & B_{2,2} \\
C_{1,0} & C_{1,2} & C_{1,2} & C_{1,2} & D_{1,1} & D_{1,1} & D_{1,2} & D_{1,2} \\
C_{1,0} & C_{1,2} & C_{1,2} & C_{1,2} & D_{1,1} & D_{1,1} & D_{1,2} & D_{1,2} \\
C_{2,0} & C_{2,2} & C_{2,2} & C_{2,2} & D_{2,1} & D_{2,1} & D_{2,2} & D_{2,2} \\
C_{2,0} & C_{2,2} & C_{2,2} & C_{2,2} & D_{2,1} & D_{2,1} & D_{2,2} & D_{2,2} \\
\end{array}
\right),\\
\vec{\beta}_{\bm{p}}&=(\beta_0,\beta_2,\beta_2,\beta_2,\beta_1^*,\beta_1^*,\beta_2^*,\beta_2^*).\end{split}\]
Note that we wrote \(\Sigma_{\bm{p}}\) and not \(\Sigma_{(\bm{p})}\) to indicate that we ignore the block structure of the matrix \(\Sigma\).