Without seeing the full paper, I'm almost certain that the following is what is being done.

Let $\{\hat{X}_j\}_{j=1}^N$ be the chosen orthonormal basis vectors for our state space $\mathbb{C}^N$ with respect to which we wish to write the matrices representing our linear, unitary transformations.

Any homogeneous linear transformation $T:\mathbb{C}^N\to\mathbb{C}^N$ is defined wholly by the $N$ values $T(\hat{X}_j)$.

Now let's think about the authors' $T_{p,q}$. What they mean is $T_{p,q}(\hat{X}_j) = \hat{X}_j$ **unless** $j=p$ or $j=q$. In this special case, $T_{p,q}$ restricted to the vector space spanned by $\hat{X}_p,\,\hat{X}_q$ has an image in that same vector space, *i.e.* $T_{p,q}$ maps the space $\operatorname{span}(\{\hat{X}_p,\,\hat{X}_q\})$ to itself:

$$T_{p,q}:\operatorname{span}(\{\hat{X}_p,\,\hat{X}_q\})\to \operatorname{span}(\{\hat{X}_p,\,\hat{X}_q\})$$

and this restricted linear operator is wholly defined by

$$T_{p,q}(\hat{X}_p) = \ell_{p,p} \hat{X}_p+\ell_{q,p} \hat{X}_q$$
$$T_{p,q}(\hat{X}_q) = \ell_{p,q} \hat{X}_p+\ell_{q,q} \hat{X}_q$$

and, moreover, the $2\times 2$ matrix $\Lambda_{p,q}=\left(\begin{array}{cc}\ell_{p,p}&\ell_{q,p}\\\ell_{p,q}&\ell_{q,q}\end{array}\right)\in U(2)$ and is unitary. So the images of $\hat{X}_p$ and $\hat{X}_q$ define the matrix $T_{p,q}$. (The weird ordering of the indices in the off diagonal terms will become clearer below).

So, how do we write the matrix of $T_{p,q}$? Well, we write the image of $\hat{X}_j$ in the $j^{th}$ column. So all the columns of $T_{p,q}$ aside from the $p^{th}$ and $q^{th}$ column are simply what they would be in the identity matrix, since the image of $\hat{X}_j$ is $\hat{X}_j$. This leaves the last two columns, the columns $p$ and $q$.

For column $p$:

These are all noughts aside from the element at position $(p,p)$, which is the element $\ell_{p,p}$ in our restricted $2\times 2$ matrix $\Lambda_{p,q}$. Likewise, the element at position $(q,p)$ in the $p^{th}$ column is $\ell_{q,p}$ in our restricted $2\times 2$ matrix $\Lambda_{p,q}$.

For column $q$, the analogous thing happens. All noughts aside from $\ell_{p,q}$ and $\ell_{q,q}$

Now, I am not sure whether the authors allow the restricted $2\times 2$ matrix $\Lambda_{p,q}$ to be *any* unitary $2\times 2$ matrix. They mention *beamsplitter* matrices, which could be construed as *any* unitary $2\times2$ matrix, or they could have the matrices I describe in this answer here in mind. I would guess the former, *i.e.* any unitary matrix.

## Best Answer

Both operations are equivalent, up to a local phase in the second mode. In particular, if you shift the second basis vector's phase by $i$, then you will turn $H$ into $A$. In a beam splitter this is perfectly natural, because the phases of the output modes are not particularly well defined, and you can always model the difference between the two operations as an extra phase plate on one of the output modes. In any case, the phase difference between the two modes is not an experimental observable unless you bring the two modes together and interfere them, in which case you will want to introduce a variable phase delay between them to control the interference. This extra controlled phase delay will eat up this static par difference.