# Supplemental Material for “The quantum beam splitter with many partially
indistinguishable photons: multiphotonic interference and asymptotic classical
correspondence”

##  1 Preliminaries

This section provides some required background on the combinatorial and group
theoretical techniques techniques that are necessary for the derivation of the
main results of the paper.

###  1.1 Product basis expansion of the $n$-boson Fock basis

Consider a system of $n$ identical particles with single-particle Hilbert
space $\mathcal{H}_{1}$ of finite dimension $d$. Given some standard basis for
$\mathcal{H}_{1}$ with basis elements $\ket{i}$, the product basis for the
$n$-particle Hilbert space $\mathcal{H}_{1}^{\otimes n}$ can be labeled by
sequences $s=(i_{1}i_{2}\ldots i_{n})\in\\{1,\ldots d\\}^{n}$ of index values:

$\ket{s}=\ket{i_{1}i_{2},...i_{n}}=\ket{i_{1}}\otimes\ket{i_{2}}\otimes...\otimes\ket{i_{n}}.$
(1)

If the particles are indistinguishable bosons, the $n$-particle Hilbert space
is further restricted to be the symmetric subspace of
$\mathcal{H}_{1}^{\otimes n}$,

$\mathcal{H}_{n}=\mathrm{Sym}_{n}\mathcal{H}_{1}.$

A natural basis for $\mathcal{H}_{n}$ can be obtained from projecting product
basis states to the symmetric subspace using the projector operator

$\Pi_{\mathrm{Sym}}=\frac{1}{n!}\sum_{\pi\in S_{n}}U(\pi),$ (2)

where $\pi$ runs over all permutations of $n$ symbols and $U(\pi)$ acts on a
product basis state as

$U(\pi)\ket{s}=\ket{\pi s}=\ket{i_{\pi^{-1}(1)}\ldots i_{\pi^{-1}(n)}}.$ (3)

From the definition of $\Pi_{\mathrm{Sym}}$, we see that two product basis
elements labeled by sequences that differ from each other by a permutation
project to the same state. Hence, the possible states obtained from projecting
the product basis states are in one-to-one correspondence with the equivalence
classes of sequences in $\\{1,\ldots d\\}^{n}$ under the equivalence relation
$s\sim s^{\prime}$ iff $s^{\prime}=\pi s$ for some $\pi\in S_{n}$. Each class
is composed of sequences of the same _type_ , that is, sequences with the same
set of occupation numbers $(n_{1},\ldots n_{d})$, where $n_{i}$ counts the
number of occurrences of the value $i$ in the sequence. Hence, images of the
product basis elements can be labelled by the set of occupation numbers
$(n_{1},\ldots n_{d})$ and, once normalized, are given by the superposition

$\ket{n_{1},n_{2},\ldots n_{d}}=\left(\frac{n!}{n_{1}!n_{2}!\ldots
n_{d}!}\right)^{-1/2}\sum_{s\sim(n_{1},n_{2},\ldots n_{d})}\ket{s}$ (4)

where $s\sim(n_{1},n_{2},\ldots n_{d})$ means that $s$ is a sequence of type
$(n_{1},\ldots n_{d})$, i.e., equivalent under permutations to the sequence

$(\underbrace{11\ldots 1}_{n_{1}\mathrm{\ times}}\,\underbrace{22\ldots
2}_{n_{2}\mathrm{\ times}}\,\ldots\underbrace{dd\ldots d}_{n_{d}\mathrm{\
times}}).$

The set of all basis elements given by Eq. (4), with $\sum_{i}n_{i}=n$, spans
the symmetric space $\mathcal{H}_{n}$ of $n$ bosons [1]. Expansion (4) is the
first-quantization representation of the second-quantization $n$-particle Fock
basis states $\ket{n_{1},\ldots n_{d}}$ for bosons.

###  1.2 $GL(d,\mathbb{C})$ symmetric representations and $SU(2)$ IRRs as
bosonic Hilbert spaces

Since $\mathcal{H}_{1}$ is $d$-dimensional, it is the defining representation
of the group $GL(d,\mathbb{C})$ of $d\times d$ complex invertible matrices.
This group acts naturally on $\mathcal{H}_{1}^{\otimes n}$ by the
representation $\rho(g)$ corresponding to the application of the same group
element to every copy of $\mathcal{H}_{1}$, that is

$\rho(g)\ket{\phi_{1}}\otimes\ket{\phi_{2}}\otimes...\otimes\ket{\phi_{n}}=g^{\otimes
n}\ket{\phi_{1}}\otimes\ket{\phi_{2}}\otimes...\otimes\ket{\phi_{n}}=g\ket{\phi_{1}}\otimes
g\ket{\phi_{2}}\otimes...\otimes g\ket{\phi_{n}},$ (5)

for any $g\in GL(d,\mathbb{C})$. Furthermore $GL(d,\mathbb{C})$ acts
irreducibly when restricted to the symmetric subspace. Hence, the bosonic
spaces $\mathcal{H}_{n}$ are carrier spaces of the so-called _symmetric IRR_ s
of $GL(d,\mathbb{C})$, which are labelled by Young diagrams of $n$ boxes with
only one row, i.e., $\lambda=(n)$. From expression (4), the matrix elements of
$\rho(g)$ in the Fock basis are given by

$\bra{n_{1}^{\prime},n_{2}^{\prime},\ldots
n_{d}^{\prime}}\rho(g)\ket{n_{1},n_{2},\ldots
n_{d}}=\frac{\left(\prod_{i=1}^{d}n_{i}!n_{i}^{\prime}!\right)^{1/2}}{n!}\sum_{s^{\prime}\sim(n_{1}^{\prime},n_{2}^{\prime},\ldots
n_{d}^{\prime})}\sum_{s\sim(n_{1},n_{2},\ldots
n_{d})}\bra{s^{\prime}}g^{\otimes n}\ket{s}.$ (6)

Expanding $\bra{s^{\prime}}g^{\otimes n}\ket{s}$ as a product of matrix
elements

$\bra{s^{\prime}}g^{\otimes
n}\ket{s}=g_{i_{1}^{\prime}i_{1}}g_{i_{2}^{\prime}i_{2}}\ldots
g_{i_{n}^{\prime}i_{n}}$ (7)

and collecting all monomials in the matrix elements $g_{ij}$, one obtains the
explicit expression for the symmetric representation matrices as polynomial in
the matrix elements $g_{ij}$

$\bra{n_{1}^{\prime},n_{2}^{\prime},\ldots
n_{d}^{\prime}}\rho(g)\ket{n_{1},n_{2},\ldots
n_{d}}=\left(\prod_{i=1}^{d}n_{i}!n_{i}^{\prime}!\right)^{1/2}\sum_{W}^{*}\prod_{ij}\frac{g_{ij}^{W_{ij}}}{W_{ij}!},$
(8)

where the sum runs over all $d\times d$ matrices $W$ with non-negative
integer-valued elements such that the rows sum up to the occupations
$n_{i}^{\prime}$ and the columns to the occupations $n_{i}$[2].

We note that $\rho(g)$ is not generally a unitary IRR for $g\in
GL(d,\mathbb{C})$, but is when restricted to $g\in U(d)$ where $U(d)\subset
GL(d,\mathbb{C})$ is the group of $d\times d$ unitary matrices. In such a
case, $g$ represents a basis change to a new orthonormal basis in the single-
particle Hilbert space, and its action on $\mathcal{H}_{n}$ can be seen in
second-quantized language as the transformation of creation/annhiliation
operators

$\rho^{\dagger}(g)a_{i}\rho(g)=g_{ij}a_{j},\qquad\rho^{\dagger}(g)a_{i}^{\dagger}\rho(g)=g_{ij}^{*}a_{j}^{\dagger},$
(9)

which by unitarity of $g$ commutes with the total number operator
$N=\sum_{i=}^{d}a_{i}^{\dagger}a_{i}$.

When $d=2$, $n+1$-dimensional bosonic Hilbert spaces $\mathcal{H}_{n}$ are in
one-to one correspondence with the $2j+1$-dimensional spin-$j$ representations
of $SU(2)\subset GL(2,\mathbb{C})$, for $j=n/2$. Thinking of the index values
$i=1$ and $i=2$ as respectively labeling spin-up ($m=1/2)$ and spin-down
$(m=1/2)$ states of the $j=1/2$ representation, the Fock state
$\ket{n_{1},n_{2}}$ with $n_{1}+n_{2}=n$ corresponds to the spin-$j$ angular
momentum state $\ket{j,m}$ with

$j=\frac{n_{1}+n_{2}}{2},\qquad m=\frac{n_{1}-n_{2}}{2},$ (10)

or equivalently

$n_{1}=j+m,\qquad n_{2}=j-m.$ (11)

Imposing the row and sum conditions for the $2\times 2$ matrix $W$ in the
specialization of expansion (8) to $d=2$ leaves only one free element; letting
this element be $x:=n_{12}$, we have

$W(x)=\left(\begin{array}[]{cc}j+m^{\prime}-x&x\\\
m-m^{\prime}+x&j-m-x\end{array}\right),$ (12)

and consequently expansion of the spin-$j$ representation matrices as
polynomials in the matrix elements of $g\in SU(2)$:

$D^{j}_{m^{\prime},m}(g)=\sqrt{(j+m)!(j-m)!(j+m^{\prime})!(j-m^{\prime})!}\sum_{x}\frac{g_{11}^{j+m^{\prime}-x}\,g_{12}^{x}\,g_{21}^{m-m^{\prime}+x}\,g_{22}^{j-m-x}}{(j+m^{\prime}-x)!x!(m-m^{\prime}+x)!(j-m-x)!}$
(13)

where the sum runs over all values of $x$ for which the factorials in the
summand are non-negative. Taking $g$ to be the matrix

$g=\left(\begin{array}[]{cc}\cos\frac{\theta}{2}&\sin\frac{\theta}{2}\\\
-\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{array}\right),$ (14)

and substituting into Eq. (13) gives the explicit expression for the Wigner
small-$d$ matrices.

###  1.3 $GL(2,\mathbb{C})$ IRRs

As discussed in the paper, the $2j+1$-dimensional spin-$j$ representation
spaces are also carrier spaces of the $2j+1$-dimensional IRRs of
$GL(2,\mathbb{CC})$, the group of $2\times 2$ complex invertible matrices.
These representations are labelled by two-row Young diagrams
$\lambda=(\lambda_{1},\lambda_{2})$, with $\lambda_{1}\geq\lambda_{2}$, and
include the symmetric representations just discussed when $\lambda_{2}=0$.
Alternatively, we can label the IRRs using the pairs $(n,j)$ and the
correspondence between $(\lambda_{1},\lambda_{2})$ and $(n,j)$ depicted in
Fig. 2 of the paper, namely

$\lambda_{1}=\frac{n}{2}+j,\qquad\lambda_{2}=\frac{n}{2}-j,$ (15)

or conversely,

$n=\lambda_{1}+\lambda_{2},\qquad j=\frac{\lambda_{1}-\lambda_{2}}{2}.$ (16)

The IRRs are obtained by a symmetrization-antisymmetrization algorithm encoded
in the Young diagram (see e.g., [3, 4]). For a given pair $(n,j)$ the elements
of the representation matrices for $g\in GL(2,\mathbb{C})$ are given by the
same polynomials $D^{j}(g)$ of the spin-$j$ representation matrices, as per
Eq. (13), but multiplied by the determinant of $g$ to the power
$\lambda_{2}=\frac{n}{2}-j$. Labeling the matrix elements by
$\mathcal{D}^{j,n}_{m^{\prime}m}(g)$, they are therefore given by

$\mathcal{D}^{j,n}_{m^{\prime}m}(g)=\det(g)^{\frac{n}{2}-j}\
D^{j}_{m^{\prime}m}(g)$ (17)

where $D^{j}_{m^{\prime}m}(g)$ are as given in Eq. (13). The matrix element
$\mathcal{D}^{j,n}_{m^{\prime}m}(g)$ admits an expansion as a polynomial in
the matrix elements $g_{ij}$ similar to that of Eq. (13). Using the latter
expansion and multiplying with the expansion of $\det(g)^{\frac{n}{2}-j}$, one
arrives at the expansion of the form

$\mathcal{D}_{m^{\prime}m}^{j,n}(g)=\sum_{W}^{*}C^{j,n}_{m^{\prime}m}(W)\prod_{ij}\frac{n!}{W_{ij}!}g_{ij}^{W_{ij}},$
(18)

where the matrix $W$ is again a $2\times 2$ integer-valued matrix with non-
negative entries, but satisfying the condition that the rows sum to
$\frac{n}{2}\pm m^{\prime}$ and the columns to $\frac{n}{2}\pm m$, and
$C^{j,n}_{m^{\prime}m}(W)$ are matrix valued coefficients also known in the
literature as Louck polynomials ([5, 2]). With a suitable parametrization of
$W$, these polynomials can be expressed in terms of hypergeometric functions
(see the appendix of [6]).

###  1.4 Schur transform for $(\mathbb{C}^{2})^{\otimes n}$

Schur-Weyl duality[7] states that for single-particle Hilbert space
$\mathcal{H}_{1}$, its $n$-fold tensor product can be decomposed as the direct
sum

$\mathcal{H}_{1}^{\otimes n}=\oplus_{\lambda}V_{\lambda}\otimes[\lambda],$
(19)

where $\lambda$ are partitions (i.e., Young diagrams) of $n$ of at most
$d=\mathrm{dim}\mathcal{H}_{1}$ parts, and $V_{\lambda}$ and $[\lambda]$ are
respectively the carrier spaces for IRRs of $GL(d,\mathbb{C})$ and the
permutation group $S_{n}$, both labeled by $\lambda$ [3, 4, 8]. Thus,
$[\lambda]$ acts as the multiplicity space of the representation $V_{\lambda}$
and vice versa. In particular, the symmetric representation is the
$V_{\lambda}$ IRR for the one-row partition $(n)$, and is multiplicity-free
given that $[(n)]$ is one-dimensional.

For the case $d=2$, the basis adapted to the Schur-Weyl duality of
$\mathcal{H}_{1}^{\otimes n}$ can be conveniently labelled by three quantum
numbers: the spin-$j$ quantum number $j$, which together with $n$ gives the
partition $\lambda_{j,n}=(\frac{n}{2}+j,\frac{n}{2}-j)$; the “magnetic”
quantum number $m$ labeling the basis elements of the spin-$j$ IRR
$V_{\lambda_{j,n}}$; and a third label $\tau$ for the multiplicity of the
spin-$j$ representation, which acts as a label for the $S_{n}$ IRR
$[\lambda_{j,n}]$ basis elements. The elements can thus be written as

$\ket{j,m,\tau}.$

The so-called _Schur transform_ [9] implements the basis change from the
product basis $\ket{s}$ to a Schur-Weyl basis $|j,m,\tau\rangle$ in a
particularly transparent way. The idea is to build the basis progressively by
performing angular momentum addition one spin-1/2 at a time, with the spin to
be added at the $k$-th corresponding to the $k$-th element of the sequence
$s$. The index $\tau$ therefore runs over all admissiible sequences of
intermediate angular momentum numbers up to the final angular momentum $j$:

$\tau=(j_{1}j_{2},\ldots,j_{n-1},j_{n}),\qquad j_{k+1}=|j_{k}\pm 1/2|,\ \
j_{0}:=0,\ \ j_{n}:=j$ (20)

The transformation matrix element from the product basis to the Schur-Weyl
basis can then be written as a product of Clebsch-Gordan coefficients

$\langle j,m,\tau|s\rangle=\prod_{k=1}^{n}\left\langle
j_{k},m_{k}\left|j_{k-1},m_{k-1};\frac{1}{2},\frac{3}{2}-i_{k}\right.\right\rangle,\qquad
m_{k}=m_{k-1}+\frac{3}{2}-i_{k},\ \ m_{0}:=0$ (21)

where here we assume that $i_{k}\in\\{1,2\\}$, as before. We note that from
the reality of the Clebsch-Gordan coefficients,

$\langle j,m,\tau|s\rangle=\langle j,m,\tau|s\rangle^{*}=\langle
s|j,m,\tau\rangle,$

a property that will prove useful shortly. Another consequence that follows
from this construction is the reality of the representation matrices of the
permutations, that is, the matrices $S_{\tau^{\prime},\tau}^{j,n}(\pi)$ such
that

$U(\pi)\ket{j,m,\tau}=\sum_{\tau^{\prime}}S_{\tau^{\prime},\tau}^{j,n}(\pi)\ket{j,m,\tau}.$
(22)

Several useful expansions involving the Schur transform are worth noting.
First, In the case of binary sequences, the quantum number $m$ suffices to
determine the type of the sequence according to

$n_{1}=\frac{n}{2}+m,\ \ n_{2}=\frac{n}{2}-m,$ (23)

so the projection onto the subspace of $(\mathbb{C}^{2})^{\otimes n}$ with
$J_{z}=m$ is the projector onto the subspace of sequences of type
$(\frac{n}{2}+m\,,\frac{n}{2}-m)$:

$\Pi_{m}=\sum_{j\geq|m|}\sum_{\tau=1}^{\gamma_{j,n}}\ket{j,m,\tau}\bra{j,m,\tau}\
\ \ =\sum_{s\sim(\frac{n}{2}+m\,,\frac{n}{2}-m)}\ket{s}\bra{s}.$ (24)

Expanding one basis in terms of the other in the above expression, we obtain
the relations

$\sum_{j\geq|m|}\sum_{\tau=1}^{\gamma_{j,n}}\langle s|j,m,\tau\rangle\langle
j,m,\tau|s^{\prime}\rangle=\left\\{\begin{array}[]{cc}\delta_{s,s^{\prime}},&s\sim(\frac{n}{2}+m\,,\frac{n}{2}-m)\\\
0,&s\nsim(\frac{n}{2}+m\,,\frac{n}{2}-m),\end{array}\right.$ (25)

and

$\sum_{s\sim(\frac{n}{2}+m\,,\frac{n}{2}-m)}\langle
j^{\prime\prime},m^{\prime\prime},\tau^{\prime\prime}|s\rangle\langle
s|j^{\prime},m^{\prime},\tau^{\prime}\rangle=\left\\{\begin{array}[]{lc}\delta_{j^{\prime},j^{\prime\prime}}\delta_{m,m^{\prime}}\delta_{m,m^{\prime\prime}}\delta_{\tau^{\prime},\tau^{\prime\prime}},&j^{\prime}\geq|m|\\\
0,&j^{\prime}<|m|,\end{array}\right.$ (26)

as well as all associated relations obtained by interchanging any of the
$\langle j,m,\tau|s\rangle$ for $\langle s|j,m,\tau\rangle$.

Finally, it is worth mentioning that by Schur’s Lemma [8], there are no non-
zero permutationally invariant linear operators mapping one IRR $[\lambda]$ of
$S_{n}$ to an inequivalent one $[\lambda^{\prime}]$, and within a given IRR
$[\lambda]$ of $S_{n}$, the only invariant operators are those that are
multiples of the identity. This implies that the space of $S_{n}$ invariant
operators in $(\mathbb{C}^{2})^{\otimes n}$ is spanned by operators of the
form

$X_{j,m,m^{\prime}}=\sum_{\tau=1}^{\gamma_{j,n}}\ket{j,m,\tau}\bra{j,m^{\prime},\tau},$
(27)

which, when restricted to $V_{j,n}\otimes[\lambda_{j,n}]$ are equivalent to
the operator

$\ket{j,m}\bra{j,m^{\prime}}\otimes\mathds{1}_{[\lambda_{j,n}]}.$

Since the matrix element $\bra{s}X_{j,m,m^{\prime}}\ket{s^{\prime}}$ has the
property that

$\bra{s}X_{j,m,m^{\prime}}\ket{s^{\prime}}=\bra{\pi
s}X_{j,m,m^{\prime}}\ket{\pi s^{\prime}},$ (28)

it only depends on the equivalence class of sequence pairs $(s,s^{\prime})$,
under the equivalence relation
$(s,s^{\prime})\sim(s^{\prime\prime},s^{\prime\prime\prime})\Leftrightarrow(s,s^{\prime})=(\pi
s^{\prime\prime},\pi s^{\prime\prime\prime})$ for some $\pi\in S_{n}$. The
invariant property of these equivalence classes are the joint types of the
sequence pair $(s,s^{\prime})$. These types are the occurences (or occupation
numbers) $n_{ij}$ of the pair $(i,j)$ in the sequence of pairs
$((i_{1},i_{1}^{\prime})(i_{s},i_{2}^{\prime})\ldots(i_{n},i_{n}^{\prime}))$
formed from pairing up the elements of the two sequences $s$ and $s^{\prime}$.
Remarkably, if we form the matrix $W(s,s^{\prime})$ with its elements given by
the joint weights $n_{ij}$ of $(s,s^{\prime})$, the matrix element
$\bra{s}X_{j,m,m^{\prime}}\ket{s^{\prime}}$ is given by

$\bra{s}X_{j,m,m^{\prime}}\ket{s^{\prime}}={\gamma_{j,n}}C^{j,n}_{mm^{\prime}}(W(s,s^{\prime}))$
(29)

where $C^{j,n}_{mm^{\prime}}(W(s,s^{\prime}))$ are the Louck polynomials in
expansion (18). This identity can be proved from the identity

$\frac{1}{\gamma_{j,n}}\sum_{\tau}\bra{j,m,\tau}g^{\otimes
n}\ket{j,m,\tau}=\mathcal{D}^{j,n}_{m,m^{\prime}}(g)$ (30)

by expanding the left-hand side in the product basis, the right hand side
using (18), and after using Eq. (28), comparing coefficients of the monomials
in the matrix elements $g_{ij}$.

##  2 Derivations of results involving the CSB basis

In this section we use results from the previous section to derive results for
the symmetric space $\mathcal{H}_{n}=Sym_{n}\mathcal{H}_{1}$ associated to a
single-particle Hilbert space that is the tensor product of two two-
dimesnional Hilbert spaces spaces $\mathcal{H}_{p}$ and $\mathcal{H}_{q}$:

$\mathcal{H}_{1}=\mathcal{H}_{p}\otimes\mathcal{H}_{q}=\mathbb{C}^{2}\otimes\mathbb{C}^{2}.$
(31)

We shall exploit the fact that

$\mathcal{H}_{1}^{\otimes n}=(\mathcal{H}_{p}\otimes\mathcal{H}_{q})^{\otimes
n}\cong(\mathcal{H}_{p})^{\otimes n}\otimes(\mathcal{H}_{q})^{\otimes n},$
(32)

so states in $\mathcal{H}_{1}^{\otimes n}$ can be expanded in a basis that is
a tensor product of the product bases for
$\mathcal{H}_{P}:=(\mathcal{H}_{p})^{\otimes n}$ and
$\mathcal{H}_{Q}:=(\mathcal{H}_{q})^{\otimes n}$:

$\ket{\psi}=\sum_{s,s^{\prime}}\psi(s,s^{\prime})\ket{s}_{P}\ket{s^{\prime}}_{Q}^{\prime}.$
(33)

Equivalently, transforming to the Schur-Weyl bases of $\mathcal{H}_{P}$ and
$\mathcal{H}_{Q}$, the same state can be expanded as

$\ket{\psi}=\sum_{j,m,\tau,j^{\prime},m^{\prime},\tau^{\prime}}\widetilde{\psi}(j,m,\tau,j^{\prime},m^{\prime},\tau^{\prime\prime})\ket{j,m,\tau}_{P}\ket{j^{\prime},m^{\prime},\tau^{\prime}}_{Q}^{\prime}.$
(34)

Since the Schur transform coefficients $\langle j,m,\tau|s\rangle$ are real,
we can associate to $\ket{\psi}$ an operator $\mathcal{O}_{\psi}$ acting on
$(\mathbb{C}^{2})^{\otimes n}$ by

$\mathcal{O}_{\psi}:=\sum_{s,s^{\prime}}\psi(s,s^{\prime})\ket{s}\bra{s^{\prime}},$
(35)

and the coefficients
$\widetilde{\psi}(j,m,\tau,j^{\prime},m^{\prime},\tau^{\prime\prime})$ in (34)
can be understood as the matrix elements of $\mathcal{O}_{\psi}$ but in the
Schur-Weyl basis:

$\widetilde{\psi}(j,m,\tau,j^{\prime},m^{\prime},\tau^{\prime\prime})=\bra{j,m,\tau}\mathcal{O}_{\psi}\ket{j^{\prime},m^{\prime},\tau^{\prime}}.$
(36)

Furthermore, if we restrict ourselves to operations on $\mathcal{H}_{n}$ of
the form $\ c_{i}A_{i}\otimes B_{i}$ where $A_{i}$ and $B_{i}$ have _real_
representation matrix elements in either basis (and hence in both), we can
move back and forth between representations using

$\ket{\psi^{\prime}}=\sum_{i}A_{i}\otimes
B_{i}|\psi\rangle\leftrightarrow\mathcal{O}_{\psi^{\prime}}=\sum_{i}c_{i}A_{i}\mathcal{O}_{\psi}B_{i}^{\dagger}=\sum_{i}c_{i}A_{i}\mathcal{O}_{\psi}B_{i}^{T}$
(37)

where $\leftrightarrow$ is the linear extension of the bijection
$\ket{s}_{P}\ket{s^{\prime}}_{Q}\leftrightarrow\ket{s}\bra{s^{\prime}}$. We
can then exploit this bijection to connect properties of linear operators in
$(\mathcal{C}^{2})^{\otimes n}$ to properties of entangled states in
$\mathcal{H}_{P}\otimes\mathcal{H}_{Q}$.

###  2.1 Derivation of Eq (4)

The first derivation is that of the CSB basis of
$\mathcal{H}_{n}=\mathrm{Sym}_{N}\mathcal{H}_{p}\otimes\mathcal{H}_{q}$ in
terms of the Schur-Weyl bases of $\mathcal{H}_{P}$ and $\mathcal{H}_{Q}$. We
recall from subsection…, that the algebra of permutationally-invariant
operators in $(\mathcal{C}^{2})^{\otimes n}$ is spanned by the operators

$X_{j,m,m^{\prime}}=\sum_{\tau=1}^{\gamma_{j,n}}\ket{j,m,\tau}\bra{j,m^{\prime},\tau}.$
(38)

Permutational invariance implies that

$U(\pi)X_{j,m,m^{\prime}}U(\pi)^{\dagger}=X_{j,m,m^{\prime}}.$ (39)

Since $U(\pi)$ has real matrix elements in the Schur transform basis
$\ket{j,m\tau}$, this implies that the under the the bijection
$\ket{s}_{P}\ket{s^{\prime}}_{Q}\leftrightarrow\ket{s}\bra{s^{\prime}}$, the
entangled states

$\widetilde{\ket{j,m,m^{\prime}}}:=\sum_{\tau}\ket{j,m,\tau}\ket{j,m^{\prime},\tau}\leftrightarrow
X_{j,m,m^{\prime}}$ (40)

are invariant under the same permutation applied to the $P$ and $Q$ spaces:

$U(\pi)\otimes
U(\pi)\widetilde{\ket{j,m,m^{\prime}}}=\widetilde{\ket{j,m,m^{\prime}}},$ (41)

and hence are permutationally invariant under permutations of the different
copies of $\mathcal{H}_{p}\otimes\mathcal{H}_{q}$. The CSB basis elements are
simply the states $\widetilde{\ket{j,m,m^{\prime}}}$ when normalized. Finally,
since we know from Schur’s lemma that no non-zero permutationally invariant
operators exist connecting different $j$ sectors in
$(\mathcal{C}^{2})^{\otimes n}$, the bijection
$\ket{s}_{P}\ket{s^{\prime}}_{Q}\leftrightarrow\ket{s}\bra{s^{\prime}}$
implies that no permutationally invariant states exist in
$\mathcal{H}_{P}\otimes\mathcal{H}_{Q}$ involving different values of $j$ for
$P$ and $Q$. Therefore, the CSB states span the permutationally symmetric
subspace of $\mathcal{H}_{1}^{\otimes n}$.

###  2.2 Derivation of Eq (5)

A similar trick can be used to derive the expansion (5) of the Fock state
$\ket{n_{1},0,0,n_{2}}$ in terms of the CSB basis. For this we first turn to
the first quantized representation of the Fock states
$\ket{n_{11},n_{12},n_{21},n_{22}}$ in terms of a product basis of
$\mathcal{H}_{1}^{\otimes n}$. Thinking of pairs $(a,b)$ with
$a,b\in\\{1,2\\}$ as symbols of a four-symbol alphabet, the Fock state can be
written as

$\ket{n_{11},n_{12},n_{21},n_{22}}=\sqrt{\frac{n_{11}!n_{12}!n_{21}!n_{22}!}{n!}}\sum_{S\sim(n_{11},n_{12},n_{21},n_{22})}\ket{S},$
(42)

where $S$ runs over all sequences of pairs $(a,b)$ where the pair $(a,b)$
occurs $n_{ab}$ times. It is easier, however to think of $S$ as composed from
a pair of binary sequences $(s,s^{\prime})$, where $s=(a_{1}a_{2}\ldots
a_{n})$ and $s^{\prime}=(b_{1}b_{2}\ldots b_{n})$. When represented in this
way, and decomposing $\mathcal{H}_{n}$ as

$\mathcal{H}_{n}=\mathcal{H}_{P}\otimes\mathcal{H}_{Q},$ (43)

we have

$\ket{n_{11},n_{12},n_{21},n_{22}}=\sqrt{\frac{n_{11}!n_{12}!n_{21}!n_{22}!}{n!}}\sum_{(s,s^{\prime})\sim(n_{11},n_{12},n_{21},n_{22})}\ket{s}_{P}\ket{s^{\prime}}_{Q}.$
(44)

The case in which $n_{11}=n_{1}$ and $n_{22}=n_{2}$ with $n_{12}=n_{21}=0$
corresponds to the situation where $s=s^{\prime}$ and both sequences are
binary sequences of type $(n_{1},n_{2})$. Hence, we have

$\ket{n_{1},0,0,n_{2}}=\frac{1}{\sqrt{\binom{n}{n_{1}}}}\sum_{s\sim(n_{1},n_{2})}\ket{s}_{P}\ket{s}_{Q}.$
(45)

Using the bijection
$\ket{s}_{P}\ket{s^{\prime}}_{Q}\leftrightarrow\ket{s}\bra{s^{\prime}}$, the
state maps to the projector operator onto the $J_{z}=m$ subspace of
$(\mathbb{C}^{2})^{\otimes n}$ and from (24) we have the correspondence

${\sqrt{\binom{n}{n_{1}}}}\ket{n_{1},0,0,n_{2}}\leftrightarrow\Pi_{m}=\sum_{j\geq|m|}\sum_{\tau=1}^{\gamma_{j,n}}\ket{j,m,\tau}\bra{j,m,\tau}\leftrightarrow\sum_{j\geq|m|}\sum_{\tau=1}^{\gamma_{j,n}}\ket{j,m,\tau}_{P}\ket{j,m,\tau}_{Q}$
(46)

Finally, using the expansion (Eq. (4) in the paper) that was just derived for
the CSB elemeents, we obtain expansion (5). Equivalently, we can derive (5)
starting from (45), expanding the product bases $\ket{s}$ in terms of the
Schur transform bases, and use the relation (26) together with the reality of
the transformation coefficients $\langle j,m,\tau|s\rangle$.

###  2.3 General expansion of the Fock basis in terms of the CSB basis

Although not needed for the results of the paper, it is worthwhile to complete
the connection between the Fock basis and the CSB basis. For this we start
with the the inner product products between the basis elements of the two
bases

$\langle j,m,m^{\prime}|n_{11},n_{12},n_{21},n_{22}\rangle$ (47)

and expand $\bra{j,m,m^{\prime}}$ using Eq. (…) and
$\ket{n_{11},n_{12},n_{21},n_{22}}$ using expansion (44), to obtain

$\langle
j,m,m^{\prime}|n_{11},n_{12},n_{21},n_{22}\rangle=\sqrt{\frac{n_{11}!n_{12}!n_{21}!n_{22}!}{n!\
\gamma_{j,n}}}\sum_{(s,s^{\prime})\sim(n_{11},n_{12},n_{21},n_{22})}\langle
j,m,\tau|s\rangle\langle j,m^{\prime},\tau|s^{\prime}\rangle.$ (48)

Next we note from the reality of the transformation coefficients $\langle
j,m,\tau|s\rangle$ that the sum is precisely the matrix element
$\bra{s}X_{j,m,m^{\prime}}\ket{s^{\prime}}$ of Eq. (29); hence, from that same
equation,

$\langle
j,m,m^{\prime}|n_{11},n_{12},n_{21},n_{22}\rangle=\sqrt{\frac{\gamma_{j,n}}{\frac{n!}{n_{11}!n_{12}!n_{21}!n_{22}!}}}C_{m,m^{\prime}}^{j,n}(W)$
(49)

where $C_{m,m^{\prime}}^{j,n}(W)$ is the Louck polynomial as defined by Eq.
(18), and $W$ is the integer-values matix with elements $W_{ij}=n_{ij}$.

##  3 Channel probability and its asymptotics

From Eq. (10) of the paper, the channel probability is given by

$p(j|m)=\frac{\gamma_{j,n}}{\binom{n}{\frac{n}{2}+m}}\mathcal{D}^{j,n}_{m,m}(B_{q}^{\dagger}B_{q}).$
(50)

When expressed in terms of $\eta$, $p(j|m)$ can be cast in terms of
hypergeometric functions or other well-known orthogonal polynomials. Moreover,
it exhibits large deviation behavior asymptotically, leading to the
concentration-of-measure about the value $j^{*}$ discussed in the paper.

###  3.1 Derivation of Eq. (12)

From the definition of the basis change matrix

$B_{q}=\ket{\Gamma_{1}}\bra{1}+\ket{\Gamma_{1}}\bra{2},$ (51)

we obtain in the orthogonal basis $\ket{1},\ket{2}$ of $\mathcal{H}_{q}$ the
matrix representation

$B_{q}^{\dagger}B_{q}\dot{=}\left(\begin{array}[]{cc}1&\langle\Gamma_{1}|\Gamma_{2}\rangle\\\
\langle\Gamma_{2}|\Gamma_{1}\rangle&1\end{array}\right),$ (52)

and note that

$\det(B_{q}^{\dagger}B_{q})=1-\eta^{2}.$ (53)

Defining $\chi=\arg\langle\Gamma_{2}|\Gamma_{1}\rangle$ and, recalling the
definition of $\eta=|\langle\Gamma_{2}|\Gamma_{1}\rangle|)$, we can factorize
the matrix as

$B_{q}^{\dagger}B_{q}\dot{=}\left(\begin{array}[]{cc}e^{-i\chi/2}&0\\\
0&e^{i\chi/2}\end{array}\right)\left(\begin{array}[]{cc}1&\eta\\\
\eta&1\end{array}\right)\left(\begin{array}[]{cc}e^{i\chi/2}&0\\\
0&e^{-i\chi/2}\end{array}\right)=e^{-i\chi\sigma_{z}/2}(\mathds{1}+\eta\sigma_{x})e^{i\chi\sigma_{z}/2}$
(54)

Since $\mathcal{D}^{j,n}_{m,m}(g)$ is a representation,

$\mathcal{D}^{j,n}_{m,m}(B_{q}^{\dagger}B_{q})=\left[\mathcal{D}^{j,n}(e^{-i\chi\sigma_{z}/2})\mathcal{D}^{j,n}(\mathds{1}+\eta\sigma_{x})\mathcal{D}^{j,n}(e^{i\chi\sigma_{z}/2})\right]_{mm}\,.$
(55)

Moreover, recalling that for $GL(2,\mathbb{C})$ matrices

$\mathcal{D}^{j,n}_{m,m}(g)=\det(g)^{\frac{n}{2}-j}{D}^{j}_{m,m}(g),$ (56)

where ${D}^{j}_{m,m}(g)$ are the $SU(2)$ representation matrix elements, we
have $\mathcal{D}_{mm^{\prime}}^{j,n}(e^{\pm i\chi\sigma_{z}/2})=e^{\pm
im\chi}\delta_{mm^{\prime}}$ and therefore

$\mathcal{D}^{j,n}_{m,m}(B_{q}^{\dagger}B_{q})=\mathcal{D}_{mm}^{j,n}(\mathds{1}+\eta\sigma_{x}).$
(57)

Next using expansion (13) with $g=\mathds{1}+\eta\sigma_{x}$ (i.e.,
$g_{11}=g_{22}=1$ and $g_{12}=g_{21}=\eta$), we obtain

$\displaystyle\mathcal{D}^{j,n}_{m,m}(B_{q}^{\dagger}B_{q})$ $\displaystyle=$
$\displaystyle(1-\eta^{2})^{\frac{n}{2}-j}\sum_{x=0}^{j-|m|}\frac{(j+m)!(j-m)!\eta^{2x}}{(x!)^{2}(j+m-x)!(j-m-x)!}$
(58) $\displaystyle=$
$\displaystyle(1-\eta^{2})^{\frac{n}{2}-j}\sum_{x=0}^{j-|m|}\binom{j+m}{x}\binom{j-m}{x}\eta^{2x}$
(59)

The sum on the right-hand side can be written in terms of Gauss hypergeometric
functions as

${}_{2}F_{1}(-j-m,-j+m;1;\eta^{2})=\sum_{x=0}^{j-|m|}\binom{j+m}{x}\binom{j-m}{x}\eta^{2x},$
(60)

yielding finally equation (12) in the paper. With appropriate parameter
transformations, the sum can also be expressed in terms of Meixner polynomials
or Kravtchuk polynomials, which are also given in terms of ${}_{2}F_{1}$
hypergeometric functions (see e.g. [10]).

Putting all the terms together, the final exact expression for the channel
probability in terms of Gauss hypergeometric functions reads

$p(j|m)=\frac{2j+1}{\frac{n}{2}+j+1}\binom{n}{\frac{n}{2}+j}{\binom{n}{\frac{n}{2}+m}}^{\\!-1}(1-\eta^{2})^{\frac{n}{2}-j}\
{}_{2}F_{1}(-j-m,-j+m;1;\eta^{2}).$ (61)

###  3.2 Derivation of Eq. (20)

We now turn to the asymptotics of $p(j|m)$ and the determination of the
critical value $j^{*}$, which is interpreted asymptotically as half the number
of effectively indistinguishable photons interacting at the BS. In the paper,
we showed that the OID admits the expansion

$p(m^{\prime}|m)=\sum_{j}p(j|m)p(m^{\prime}|j,m),$ (62)

where $p(j|m)p(m^{\prime}|j,m)$ comes from the product rule
$p(j,m^{\prime}|m)=p(j|m)p(m^{\prime}|j,m)$ and $p(j|m)$ is the channel OID
for $2j$ indistinguishable photons. We now show that $p(j|m)$ exhibits
asymptotic concentration precisely around the value
$j^{*}=n\overline{\jmath}^{*}$, where

$\overline{\jmath}^{*}=\sqrt{\frac{\eta^{2}}{4}+(1-\eta^{2})\overline{m}^{2}},$
(63)

and $\overline{m}=m/n$, so that in terms of the scaled variables
$\overline{m}$ and $\overline{\jmath}=j/n$, the sequence of distributions

$p_{n}(\overline{j}|\overline{m})=\sum_{j=0}^{n/2}p(j|\lceil
n\overline{m}\rceil)\delta(\overline{\jmath}-j/n)$ (64)

satisfies

$\lim_{n\rightarrow\infty}p_{n}(\overline{\jmath}|\overline{m})=\delta(\overline{\jmath}-\overline{\jmath}^{*}).$
(65)

This concentration is a consequence of the fact that
$p(\overline{\jmath}|\overline{m})$ satisfies a large deviation principle [11,
12]

$p(\overline{\jmath}|\overline{m})\sim
e^{-nr(\overline{\jmath}|\overline{m})},$ (66)

with a convex (in $\overline{\jmath}$) rate (or Cramér) function,

$r(\overline{\jmath}|,\overline{m})=-\lim_{n\rightarrow\infty}\frac{1}{n}\ln
p(\,j=n\overline{j}\,|\,m=n\overline{m}\,),$ (67)

for which $\overline{\jmath}^{*}$ is its minimum.

To establish the rate function, we turn to the expression for $p(j|m)$
obtained by using (59), namely

$p(j|m)=\frac{2j+1}{\frac{n}{2}+j+1}\binom{n}{\frac{n}{2}+j}{\binom{n}{\frac{n}{2}+m}}^{\\!-1}(1-\eta^{2})^{\frac{n}{2}-j}\sum_{x=0}^{j-|m|}\binom{j+m}{x}\binom{j-m}{x}\eta^{2x}\,.$
(68)

Letting $B(k|N,p)$ be the binomial distribution with parameters $(n,p)$ for a
discrete variable $k$

$B(k|n,p)=\binom{n}{k}p^{k}(1-p)^{n-k}\,\qquad k,\in\\{0,\ldots 1\\}$ (69)

we can rewrite the factor $(1-\eta)^{\frac{n}{2}-j}$ as

$(1-\eta)^{\frac{n}{2}-j}=\Big{(}\frac{1-\eta}{2}\Big{)}^{\frac{n}{2}-j}\Big{(}\frac{1+\eta}{2}\Big{)}^{\frac{n}{2}+j}2^{\frac{n}{2}+m}2^{\frac{n}{2}-m}\frac{1}{(1+\eta)^{j+m}(1+\eta)^{j-m}},$
(70)

to obtain an expression for $p(j|m)$ in terms of binomial distributions:

$p(j|m)=\kappa\times
B\left(\left.\frac{n}{2}+j\right|n,\frac{1+\eta}{2}\right)B\left(\left.\frac{n}{2}+m\right|n,\frac{1}{2}\right)^{\\!-1}\sum_{x=0}^{j-|m|}B\left(x\left|j+m,\frac{\eta}{1+\eta}\right.\right)B\left(x\left|j-m,\frac{\eta}{1+\eta}\right.\right)$
(71)

where $\kappa=\frac{2j+1}{\frac{n}{2}+j+1}$ is $O(1)$ asymptotically. The
convenience of this expression is that the binomial distribution has a well-
known large deviation behavior; namely [11]

$B(k|n,p)\sim e^{-nD_{2}(\overline{k}\;\|\;p)}$ (72)

where the rate exponent
$D_{2}(\overline{k}\;\|\;p)=\lim_{n\rightarrow\infty}\frac{1}{n}\ln
B(n\overline{k}|n,p)$ is the binary relative entropy

$D_{2}(\overline{k}\;\|\;p):=\overline{k}\ln\frac{\overline{k}}{p}+(1-\overline{k})\ln\frac{1-\overline{k}}{1-p}.$
(73)

Therefore the rate function can be written as

$r(\overline{\jmath}|,\overline{m})=D_{2}\bigg{(}\frac{1}{2}+\overline{\jmath}\;\bigg{\|}\;\frac{1+\eta}{2}\bigg{)}-D_{2}\bigg{(}\frac{1}{2}+\overline{m}\;\bigg{\|}\;\frac{1}{2}\bigg{)}-\lim_{n\rightarrow\infty}\frac{1}{n}\ln\left[\
\sum_{x=0}^{\lfloor n(\overline{\jmath}-|\overline{m}|)\rfloor}T(x,j)\right]$
(74)

where

$T(x,j)=B\left(x\left|j+m,\frac{\eta}{1+\eta}\right.\right)B\left(x\left|j-m,\frac{\eta}{1+\eta}\right.\right).$
(75)

The last term of Eq. (74) can be computed using Laplace ’s principle: In the
sum, all terms are positive and the number of terms in the sum is $j-|m|+1$;
therefore the sum has the two-sided bound

$T_{\max}(j)\leq\sum_{x=0}^{j-|m|}T(x,j)\leq(j-|m|+1)T_{\max}(j)\,,\qquad
T_{\max}(j)=\max_{0\leq x\leq j-|m|}T(x,j).$ (76)

Applying the logarithm on both sides of this inequality, dividing by $n$, and
noting that $n(j-|m|+1)$ is $O(\ln(n)/n)$, we get

$-\lim_{n\rightarrow\infty}\frac{1}{n}\ln\left[\ \sum_{x=0}^{\lfloor
n(\overline{\jmath}-|\overline{m}|)\rfloor}T(x,j)\right]=-\lim_{n\rightarrow\infty}\frac{\ln(T_{\max}(j))}{n}=-\inf_{\overline{x}\in[0,\overline{\jmath}+\overline{m}]}\lim_{n\rightarrow\infty}\frac{1}{n}\ln
T(n\overline{x},n\overline{\jmath})$ (77)

which is the minimum value of the rate exponent of $T(x,j)$. Since the
binomial distributions in $T(x,j)$ are parameterized by $j\pm m$ and not $n$,
the rate exponent needs to be recast in terms of the growth rate $n$. This
gives

$-\lim_{n\rightarrow\infty}\frac{1}{n}\ln
T(n\overline{x},n\overline{\jmath})=(\overline{\jmath}+\overline{m})D_{2}\bigg{(}\frac{\overline{x}}{\overline{\jmath}+\overline{m}}\;\bigg{\|}\;\frac{\eta}{1+\eta}\bigg{)}+(\overline{\jmath}-\overline{m})D_{2}\bigg{(}\frac{\overline{x}}{\overline{\jmath}-\overline{m}}\;\bigg{\|}\;\frac{\eta}{1+\eta}\bigg{)}\,.$
(78)

Minimizing the right hand side with respect to $\overline{x}$ gives the
equation for the location of the minimum:

$\ln\left(\frac{\overline{x}^{2}}{(\overline{\jmath}-\overline{m}-\overline{x})(\overline{\jmath}+\overline{m}-\overline{x})\eta^{2}}\right)=0.$
(79)

Denoting the minimizer by $\overline{x}_{\overline{\jmath}}$, we readily find
that

$\overline{x}_{\overline{\jmath}}=\frac{\eta}{1-\eta^{2}}\left(\sqrt{\overline{\jmath}^{2}-(1-\eta^{2})\overline{m}^{2}}-\eta\,\overline{\jmath}\right).$
(80)

Our final expression for the rate function of the channel probability is then:

$r(\overline{\jmath}|\overline{m})=D_{2}\bigg{(}\frac{1}{2}+\overline{\jmath}\;\bigg{\|}\;\frac{1+\eta}{2}\bigg{)}-D_{2}\bigg{(}\frac{1}{2}+\overline{m}\;\bigg{\|}\;\frac{1}{2}\bigg{)}+(\overline{\jmath}+\overline{m})D_{2}\bigg{(}\frac{\overline{x}_{\overline{\jmath}}}{\overline{\jmath}+\overline{m}}\;\bigg{\|}\;\frac{\eta}{1+\eta}\bigg{)}+(\overline{\jmath}-\overline{m})D_{2}\bigg{(}\frac{\overline{x}_{\overline{\jmath}}}{\overline{\jmath}-\overline{m}}\;\bigg{\|}\;\frac{\eta}{1+\eta}\bigg{)}.$
(81)

where $\overline{x}_{\overline{\jmath}}$ is given by Eq. (80).

To study the concentration of measure of $p(j|m)$ we look at the first two
derivatives of $r(\overline{\jmath}|\overline{m})$ with respect to
$\overline{\jmath}$. The first derivative can be expressed in terms of
$\overline{x}_{\overline{\jmath}}$ as

$r^{\prime}(\overline{\jmath}|\overline{m})=\ln\left[\frac{1-\eta^{2}}{\eta^{2}}\,\frac{1+2\overline{\jmath}}{1-2\overline{\jmath}}\,\frac{x_{\overline{\jmath}}^{2}}{(\overline{\jmath}^{2}-\overline{m}^{2})}\right]$
(82)

while the second derivative is more transparently expressed as

$r^{\prime\prime}(\overline{\jmath}|\overline{m})=\frac{4}{1-4\overline{\jmath}^{2}}+\frac{2\eta\,\overline{m}^{2}}{(\overline{\jmath}^{2}-\overline{m}^{2})\sqrt{\overline{\jmath}^{2}-(1-\eta^{2})\overline{m}^{2}}}.$
(83)

From the second derivative we verify that
$r^{\prime\prime}(\overline{\jmath}|\overline{m})>0$ in the interval
$[|\overline{m}|,1/2]$, and therefore that $r(\overline{\jmath}|\overline{m})$
is convex. Inserting $\overline{x}_{\overline{\jmath}}$ into Eq. (82), and
solving for $r^{\prime}(\overline{\jmath}|\overline{m})=0$ finally gives one
sole root in that interval, which is now guaranteed to be a minimum. The root
is precisely

$\overline{\jmath}^{*}=\sqrt{\frac{\eta^{2}}{4}+(1-\eta^{2})\overline{m}^{2}},$
(84)

as advertised.

For completeness, it is worthwhile to derive the Gaussian approximation to
$p(\overline{\jmath}|\overline{m})$ about $j^{*}$. Expanding the rate function
about $\overline{\jmath}^{*}$ to second order:

$r(\overline{\jmath}|\overline{m})=r(\overline{\jmath}^{*}|\overline{m})+\frac{r^{\prime\prime}(\overline{\jmath}^{*}|\overline{m})}{2}(\overline{\jmath}-\overline{\jmath}^{*})^{2}+O((\overline{\jmath}-\overline{\jmath}^{*})^{3}),$
(85)

one can verify that $r(\overline{\jmath}^{*}|\overline{m})=0$ and that

$r^{\prime\prime}(\overline{\jmath}^{*}|\overline{m})=\frac{16(\overline{\jmath}^{*})^{2}}{\eta^{2}(1-\eta^{2})(1-4\overline{m}^{2})}.$
(86)

Hence, from $p(\overline{\jmath}|\overline{m})\sim
e^{-nr(\overline{\jmath}|\overline{m})}$, we find that asympotically
$p(\overline{\jmath}|\overline{m})$ is Gaussian with

$\Delta\overline{\jmath}=\
\frac{\eta\sqrt{1-\eta^{2}}\sqrt{1-4m^{2}}}{4\,\sqrt{n}\,\overline{\jmath}}.$
(87)

##  4 The OID for perfectly distinguishable photons and the binary summetric
channel

When the photons are perfectly distinguishable ($\eta=0$), the output
statistics is consistent with a process in which the photons pass
independently through the BS (as for instance when the photons are fed one at
a time with no overlap in their temporal wave functions). In this case the
statistics can be understood as sampling from a binary symmetric channel
(BSC). The correspondence is most transparently established by working with
the product basis expansion of $\mathcal{H}_{1}^{\otimes
n}\cong\mathcal{H}_{P}\otimes\mathcal{H}_{Q}$ of the Fock states.

As discussed in the paper, when $\eta=0$, the basis transformation matrix
$B_{q}$ is the identity and the initial state is precisely the Fock state
$\ket{n_{1},0,0,n_{2}}$, with product basis expansion (45). The output state
is therefore

$\ket{\psi}_{out}=\frac{1}{\sqrt{\binom{n}{n_{1}}}}\sum_{s\sim(n_{1},n_{2})}U_{p}^{\otimes
n}\ket{s}_{P}\ket{s}_{Q}.$ (88)

Since we are only interested in port counts, we can work with the partial
density matrix of output the ports,

$\rho_{P,out}=\tr_{Q}(\ket{\psi}\bra{\psi}_{out}).$ (89)

Since the product basis states are orthonormal, this partial state becomes a
mixture single particle product states:

$\rho_{P,out}=\frac{1}{\binom{n}{n_{1}}}\sum_{s\sim(n_{1},n_{2})}U_{p}^{\otimes
n}\ket{s}\bra{s}(U_{p}^{\dagger})^{\otimes n}.$ (90)

The probability of obtaining
$(n_{1}^{\prime},n_{2}^{\prime}=n-n_{1}^{\prime})$ counts at the detectors is
then given by adding all diagonal elelements in the product basis of type
$(n_{1}^{\prime},n_{2}^{\prime}=n-n_{1})$:

$p(n_{1}^{\prime}|n_{1},n_{2})=\sum_{s^{\prime}\sim(n_{1}^{\prime},n_{2}^{\prime})}\bra{s^{\prime}}\rho_{P,out}\ket{s}=\frac{1}{\binom{n}{n_{1}}}\sum_{s^{\prime}\sim(n_{1}^{\prime},n_{2}^{\prime})}\sum_{s\sim(n_{1},n_{2})}|\bra{s^{\prime}}U_{p}^{\otimes
n}\ket{s}|^{2},$ (91)

This sum is similar to the one encountered in Eq. (6), except that it involves
$|\bra{s^{\prime}}U_{p}^{\otimes n}\ket{s}|^{2}$ as opposed to
$\bra{s^{\prime}}U_{p}^{\otimes n}\ket{s}$; the monomials to collect are
therefore in the elements

$Q_{ij}=|g_{ij}|^{2}$ (92)

which are precisely the elements of the transition probability $p(i|j)$ for a
single photon that enters port $j$ to exit at port $i$. Collecting monomials
in the $Q_{ij}$, we obtain

$p(n_{1}^{\prime}|n_{1},n_{2})=n_{1}!n_{2}!\sum_{W}^{*}\prod_{ij}\frac{Q_{ij}^{W_{ij}}}{W_{ij}!},$
(93)

where the rows of $W$ add to the $n_{i}^{\prime}$ and its columns to the
$n_{j}$. Incorporating these constraints, the sum can finally be written as

$\displaystyle p(n_{1}^{\prime}|n_{1},n_{2})$ $\displaystyle=$
$\displaystyle\sum_{k}\left[\binom{n_{1}}{k}Q_{11}^{k}Q_{21}^{n_{1}-k}\right]\left[\binom{n_{2}}{n_{1}^{\prime}-k}Q_{12}^{n_{1}^{\prime}-k}Q_{22}^{n_{2}-n_{1}^{\prime}+k}\right]$
(94) $\displaystyle=$
$\displaystyle\sum_{k}B(k|n_{1},Q_{11})B(n_{1}^{\prime}-k|n_{2},Q_{12}),$ (95)

where $B(k|n,p)$ stands for the binomial distribution as before and we use the
fact that since $U_{p}$ unitary, the row and column sums of $Q_{ij}$ are equal
to $1$. Therefore we obtain the discrete convolution of two binomials, which
is consistent with interpreting the counts $n_{1}^{\prime}$ as the sum of two
independent variables $k_{1}$ and $k_{2}$:

$n_{1}^{\prime}=k_{1}+k_{2},\qquad k_{1}\sim Bin(n_{1},1-f),\ \ k_{2}\sim
Bin(n_{2},f)$ (96)

where

$f=Q_{12}=Q_{21}=|\bra{1}U_{p}\ket{2}|^{2}\,.$ (97)

This is precisely the distribution obtained from sampling the BSC $n_{1}$
times through input $1$ and $n_{2}$ times from input $2$, where $f$ is the BSC
error (or flip) probability. The resulting distribution therefore has mean and
variance

$\displaystyle\langle n_{1}^{\prime}\rangle$ $\displaystyle=$ $\displaystyle
n_{1}(1-f)+n_{2}f=\frac{n}{2}$ (98) $\displaystyle\langle(\Delta
n_{1}^{\prime})^{2}\rangle$ $\displaystyle=$ $\displaystyle nf(1-f)$ (99)

or in terms of imbalances, and using $f=\sin\theta/2^{2}$ as in the paper,
gives

$\displaystyle\langle m^{\prime}\rangle$ $\displaystyle=$ $\displaystyle
m\cos\theta$ (100) $\displaystyle\langle(\Delta m)^{2}\rangle$
$\displaystyle=$ $\displaystyle\frac{n}{4}\sin^{2}\theta.$ (101)

Finally, from Eq. (95) and the Laplace principle, it follows that for
$\eta=0$, the OID satisfies a large deviation principle

$p(m^{\prime}|m)_{\eta=0}\sim e^{-nr(\overline{m}^{\prime}|\overline{m})}$
(102)

with rate function

$r(\overline{m}^{\prime}|\overline{m})=\inf_{\overline{k}\in[0,\frac{1}{2}+\overline{m}]}\left[\left(\frac{1}{2}+\overline{m}\right)D_{2}\bigg{(}\frac{\overline{k}}{\frac{1}{2}+\overline{m}}\;\bigg{\|}\;1-f\bigg{)}+\left(\frac{1}{2}-\overline{m}\right)D_{2}\bigg{(}\frac{\frac{1}{2}+\overline{m}^{\prime}-\overline{k}}{\frac{1}{2}-\overline{m}}\;\bigg{\|}\;f\bigg{)}\right]\,.$
(103)

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