Group Functions

The basis states and GT patterns page introduced these labels for basis states, with different irreps corresponding to different exchange symmetries. Here we sketch the mathematical framework behind the library; actual code is shown in the applications pages on HOM effect, qubit transmission with entangled light, sum rules, as well as the related notes on characters and immanants.

The problem: mode mixing in quantum mechanics

Consider a system of $n$ modes, each described by a creation operator $a^\dagger_i$. A Fock state is built by applying creation operators to the vacuum:

\[|m_1, m_2, \ldots, m_n\rangle = \frac{(a_1^\dagger)^{m_1} \cdots (a_n^\dagger)^{m_n}}{\sqrt{m_1! \cdots m_n!}} |0\rangle.\]

Now suppose the modes get mixed by a unitary transformation $U \in \mathrm{U}(n)$: $a_i^\dagger \mapsto \sum_j U_{ji} a_j^\dagger.$

What is the output state? Each creation operator in the original Fock state transforms according to the rule above. Expanding the product, we obtain a superposition of Fock states with coefficients that are polynomials in the matrix elements $U_{ij}$.

The function group_function computes these coefficients — the transition amplitudes $\langle m' | D^{(\lambda)}(U) | m \rangle$, where $\lambda$ labels the space the operator is acting on (the irreducible representation). The presentation here is slightly more mathematical than a typical quantum optics treatment, for consistency with representation theory literature and to handle more general cases beyond bosons. The partition $\lambda$ labels the symmetry type: $[N,0,0,\ldots]$ for bosons, $[1,1,\ldots,1]$ for fermions, mixed shapes for particles with mixed exchange symmetry. In particular, we keep the label $\lambda$ in the equations to accommodate the more general cases.

Bosons: symmetric representation

For bosonic systems, consider the transition amplitude between an input Fock state $|m\rangle$ and an output Fock state $|m'\rangle$, both having $N$ total particles. We follow Scheel's derivation.^[1]

Label the $N$ particles by $\alpha = 1, \ldots, N$, assigning each to its input mode via $\alpha \mapsto i_\alpha$ such that mode $i$ appears $m_i$ times (so, e.g. for $\ket{2,1,0}$, $(i_1, i_2, i_3)=(1,1,2)$). Then:

\[D^{(\lambda)}(U)|m\rangle = \frac{1}{\sqrt{\prod_i m_i!}} \prod_{i=1}^{n} \left(\sum_j U_{ij} a_j^\dagger\right)^{m_i} |0\rangle = \frac{1}{\sqrt{\prod_i m_i!}} \sum_{j_1, \ldots, j_N} \left(\prod_{\alpha=1}^{N} U_{i_\alpha, j_\alpha}\right) \prod_{\alpha=1}^N a_{j_\alpha}^\dagger |0\rangle\]

The first step unfolds powers into labeled factors; the second is distributivity.

So now we have products of operators acting on the vacuum $\ket{0}$; we wish to close it with a bra $\bra{m'}$ to compute the matrix element. Let us act with the creation operators on the vacuum and identify terms proportional to $\ket{m'}$ there; then, the matrix element will be a sum of prefactors leading to $\ket{m'}$. For terms where the tuple $(j_1, \ldots, j_N)$ contains mode $k$ exactly $m'_k$ times, the creation operators produce:

\[\prod_{\alpha=1}^N a_{j_\alpha}^\dagger |0\rangle = \sqrt{\prod_j m'_j!}\, |m'\rangle\]

The $\sqrt{m'_j!}$ arises because $m'_j$ identical creation operators acting on vacuum give $(a_j^\dagger)^{m'_j}|0\rangle = \sqrt{m'_j!}|m'_j\rangle_j$. Denoting the constraint on $(j_1, \ldots, j_N)$ as $\sim m'$:

\[\langle m' | D^{(\lambda)}(U) | m \rangle = \frac{\sqrt{\prod_j m'_j!}}{\sqrt{\prod_i m_i!}} \sum_{(j_1,\ldots,j_N) \sim m'} \prod_{\alpha=1}^{N} U_{i_\alpha, j_\alpha}\]

The above expression can be shown to be expressible as a permanent of a properly constructed matrix. This is, by the way, the basis for boson sampling problems; computing the permanent is #P-hard,^[2] making bosonic transition amplitudes classically intractable.

The relation is as follows: construct an $N \times N$ matrix $M$ using output labeling $\beta \mapsto j_\beta$ (analogous to input): $M_{\alpha\beta} = U_{i_\alpha, j_\beta}$. The permanent sums over all permutations $\sigma \in S_N$:

\[\mathrm{perm}(M) = \sum_{\sigma \in S_N} \prod_{\alpha=1}^{N} U_{i_\alpha, j_{\sigma(\alpha)}}\]

Each valid $(j_1, \ldots, j_N) \sim m'$ corresponds to $\prod_j m'_j!$ permutations (permuting indices within each output mode). Therefore: $\sum_{(j_1,\ldots,j_N) \sim m'} \prod_{\alpha} U_{i_\alpha, j_\alpha} = \frac{\mathrm{perm}(M)}{\prod_j m'_j!}$

As a result, we have

\[\langle m' | D^{(\lambda)}(U) | m \rangle = \frac{\mathrm{perm}(M)}{\sqrt{\prod_i m_i!}\sqrt{\prod_j m'_j!}}\]

Example: 3 modes, input $|m\rangle = |2,0,1\rangle$, output $|m'\rangle = |1,1,1\rangle$.

Input labeling ($N=3$ particles): $(i_1, i_2, i_3) = (1, 1, 3)$ — two particles from mode 1, one from mode 3.

Output labeling: $(j_1, j_2, j_3) = (1, 2, 3)$ — one particle into each mode.

The matrix $M$ has entries $M_{\alpha\beta} = U_{i_\alpha, j_\beta}$:

\[M = \begin{pmatrix} U_{11} & U_{12} & U_{13} \\ U_{11} & U_{12} & U_{13} \\ U_{31} & U_{32} & U_{33} \end{pmatrix}\]

Note the repeated rows from $m_1 = 2$.

In representation-theoretic terms, bosonic Fock states live in the symmetric subspace, the irrep $\lambda = [N, 0, \ldots, 0]$.

Fermions: anti-symmetric representation

For fermions, the derivation parallels the bosonic case with three modifications: occupation numbers are restricted to $m_i, m'_j \in \{0,1\}$ (Pauli exclusion), so all normalization factors become $\sqrt{0!}=\sqrt{1!} = 1$, and anticommutation introduces signs.

The expansion step has the same structure as bosons:

\[D^{(\lambda)}(U) |m\rangle = \sum_{j_1, \ldots, j_N} \left(\prod_{\alpha=1}^{N} U_{i_\alpha, j_\alpha}\right) a_{j_1}^\dagger \cdots a_{j_N}^\dagger |0\rangle\]

To project the result to final bra $\bra{m'}$, consider the following. For a term to contribute to $|m'\rangle$, the tuple $(j_1, \ldots, j_N)$ must have all distinct entries (otherwise $a_j^\dagger a_j^\dagger = 0$), forming a permutation of the occupied output modes. Write $j_\alpha = j'_{\sigma(\alpha)}$ where $(j'_1, \ldots, j'_N)$ is the sorted list and $\sigma \in S_N$. Reordering to standard form introduces signs from the anticommutation relation (rectifying every transposition changes it: $a_2^\dagger a_1^\dagger=-a_1^\dagger a_2^\dagger$):

\[a_{j_1}^\dagger \cdots a_{j_N}^\dagger |0\rangle = \mathrm{sgn}(\sigma) \cdot a_{j'_1}^\dagger \cdots a_{j'_N}^\dagger |0\rangle = \mathrm{sgn}(\sigma) |m'\rangle\]

As a result,

\[\langle m' | D^{(\lambda)}(U) | m \rangle = \sum_{\sigma \in S_N} \mathrm{sgn}(\sigma) \prod_{\alpha=1}^{N} U_{i_\alpha, j'_{\sigma(\alpha)}} = \det(M)\]

where $M_{\alpha\beta} = U_{i_\alpha, j'_\beta}$ is the submatrix of $U$ with rows = occupied input modes, columns = occupied output modes.

Unlike permanents, determinants can be computed efficiently in $O(N^3)$ time, which underlies the tractability of free-fermion systems.

The general formula

Both results share a common structure: a sum over permutations, weighted by representation-dependent coefficients, times a monomial in matrix elements. For bosons:

\[\langle m' | D^{(\lambda)}(U) | m \rangle \propto \sum_{\sigma \in S_N} 1 \cdot \prod_{\alpha} U_{i_\alpha, j_{\sigma(\alpha)}}\]

and for fermions:

\[\langle m' | D^{(\lambda)}(U) | m \rangle \propto \sum_{\sigma \in S_N} \mathrm{sgn}(\sigma) \cdot \prod_{\alpha} U_{i_\alpha, j_{\sigma(\alpha)}}\]

The weights $1$ and $\mathrm{sgn}(\sigma)$ are the matrix elements of the trivial and sign representations of $S_N$, both one-dimensional.

For a general irrep $\lambda$, the picture complicates a bit: the basis vectors can no longer be uniquely indexed by occupation numbers, and semistandard Young tableaux are used instead (equivalent to GT patterns), here denoted by $A$ and $B$. A semistandard tableau is a filling of the Young diagram with numbers nondecreasing along each row and increasing down each column. The end formula takes a similar form:

\[\langle A \vert D^{(\lambda)}(U) \vert B\rangle = \sum (\text{coefficient}) (\text{monomial in matrix elements of }U),\]

but the construction of coefficients and monomials requires explanation of the notation. First, let us consider the basis states: the user-facing functions mostly use Gelfand-Tsetlin patterns; which are translated in the code to semistandard Young tableaux. For an example of such a tableau, let us take $A$ in the irrep $\lambda=[4,1]$ to be

The monomials are still constructed as products of matrix elements of $U$, but the indexing depends nontrivially on $A$ and $B$, and the expression involves the Young orthogonal representation of the permutation group $S_N$ too, where $N$ is the number of particles ($N=5$ in the example above). Standard Young tableaux enumerate the basis vectors of this representation: fillings that increase both along rows and down columns, using each of $1,\ldots,N$ exactly once. Each semistandard $A$ has an associated standard $\bar A$; for the $A$ above, the associated $\bar A$ is

The monomial is built using the fills of these tableaux. A fill is the sequence of entries read as concatenated rows; for the $A$ above,

\[\phi_A(k) \coloneqq k\text{-th element of }(1,2,3,4,2),\]

and similarly the fill $\phi_{\bar A}(k)$ of $\bar A$ is the $k$-th element of $(1,2,4,5,3)$; fills of $B$ and $\bar B$ are denoted in the same way. The fill of a standard tableau is always a permutation of $1,\ldots,N$, and in particular has an inverse. Define the compositions $f \coloneqq \phi_A \circ \phi_{\bar A}^{-1}$ and $g \coloneqq \phi_B \circ \phi_{\bar B}^{-1}$. The monomial then takes the form $\prod_k U_{f \circ \gamma^{-1}(k),\, g(k)}$, summed over selected permutations $\gamma$ with proper coefficients.

We now have the pieces needed to state the general formula, due to Grabmeier and Kerber:^[3]

\[\langle A \vert D^{(\lambda)}(U) \vert B \rangle = \frac{1}{\sqrt{\Theta^\lambda_A \Theta^\lambda_B}} \sum_{\gamma} \left( \sum_{\sigma \in S_A \gamma S_B} \omega^\lambda_{\bar A,\bar B}(\sigma) \right) \prod_{k} U_{f \circ \gamma^{-1}(k),\; g(k)}.\]

Some new symbols appear in the above formula, and they require careful explanation.

• The $\Theta$ symbols are normalization factors generalizing $\sqrt{m!}$.
• The outer sum runs over representatives of the double coset decomposition $S_A \backslash S_N / S_B$. Here, $S_A$ is the stabilizer of $A$; these are the permutations which do not change the fills. For the semistandard Young tableau $A$ above, the only repeated entry is $2$ at positions $2$ and $5$ of the fill and the stabilizer group is generated by the transposition $(2,5)$ alone. The double coset structure then partitions $S_N$: for any $\gamma, \gamma' \in S_N$, the sets $S_A \gamma S_B$ and $S_A \gamma' S_B$ are either equal or disjoint. Hence, each double coset has well-defined representatives; and the sum runs over these; $\gamma$ denotes the representative.
• The inner sum runs over the coset $S_A \gamma S_B$ defined by $\gamma$, with the summand $\omega^\lambda_{\bar A,\bar B}(\sigma)$ being the matrix element of the Young orthogonal representation between the basis vectors labelled by $\bar A$ and $\bar B$.

The main functionality of this library – the function group_function(λ, ...) – evaluates this formula. For the tableau $A$ above (as a GT pattern) and a symbolic $U$, the diagonal matrix element is $\langle A\vert D^{(\lambda)}(U) \vert A\rangle$:

julia> using GroupFunctions
julia> # The GT pattern equivalent to the semistandard tableau A above
       A = GTPattern([[4, 1, 0, 0, 0], [4, 1, 0, 0], [3, 1, 0], [2, 1], [1]]);
julia> group_function([4, 1, 0, 0, 0], A, A)2*((1/6)*u_4_1*u_2_2^2*u_1_3*u_3_4 + (1/6)*u_4_1*u_2_2^2*u_1_4*u_3_3 + (1/2)*u_4_3*u_2_2^2*u_1_1*u_3_4 + (1/6)*u_4_3*u_2_2^2*u_1_4*u_3_1 + (1/2)*u_4_4*u_2_2^2*u_1_1*u_3_3 + (1/6)*u_4_4*u_2_2^2*u_1_3*u_3_1 + (-1/6)*u_4_1*u_2_2*u_2_3*u_1_2*u_3_4 + (1/6)*u_4_1*u_2_2*u_2_3*u_1_4*u_3_2 + (-1/6)*u_4_1*u_2_2*u_2_4*u_1_2*u_3_3 + (1/6)*u_4_1*u_2_2*u_2_4*u_1_3*u_3_2 + (-1/3)*u_4_1*u_2_4*u_2_3*u_1_2*u_3_2 + (-2/3)*u_4_2*u_2_1*u_2_3*u_1_2*u_3_4 + (-1/3)*u_4_2*u_2_1*u_2_3*u_1_4*u_3_2 + (-2/3)*u_4_2*u_2_1*u_2_4*u_1_2*u_3_3 + (-1/3)*u_4_2*u_2_1*u_2_4*u_1_3*u_3_2 + (-1/6)*u_4_2*u_2_2*u_2_1*u_1_3*u_3_4 + (-1/6)*u_4_2*u_2_2*u_2_1*u_1_4*u_3_3 + (5/6)*u_4_2*u_2_2*u_2_3*u_1_1*u_3_4 + (1/6)*u_4_2*u_2_2*u_2_3*u_1_4*u_3_1 + (5/6)*u_4_2*u_2_2*u_2_4*u_1_1*u_3_3 + (1/6)*u_4_2*u_2_2*u_2_4*u_1_3*u_3_1 + (2/3)*u_4_2*u_2_4*u_2_3*u_1_1*u_3_2 + (-1/3)*u_4_2*u_2_4*u_2_3*u_1_2*u_3_1 + (-2/3)*u_4_3*u_2_1*u_2_4*u_1_2*u_3_2 + (-1/2)*u_4_3*u_2_2*u_2_1*u_1_2*u_3_4 + (-1/6)*u_4_3*u_2_2*u_2_1*u_1_4*u_3_2 + (5/6)*u_4_3*u_2_2*u_2_4*u_1_1*u_3_2 + (-1/6)*u_4_3*u_2_2*u_2_4*u_1_2*u_3_1 + (-2/3)*u_4_4*u_2_1*u_2_3*u_1_2*u_3_2 + (-1/2)*u_4_4*u_2_2*u_2_1*u_1_2*u_3_3 + (-1/6)*u_4_4*u_2_2*u_2_1*u_1_3*u_3_2 + (5/6)*u_4_4*u_2_2*u_2_3*u_1_1*u_3_2 + (-1/6)*u_4_4*u_2_2*u_2_3*u_1_2*u_3_1)