Iterative construction of $\mathfrak{S}_p \times \mathfrak{S}_p$ group-adapted irreducible matrix units for the walled Brauer algebra

1. Problem Statement

The paper addresses the representation theory of the algebra of partially transposed permutation operators, denoted as $\mathcal{A}^d_{p,p}$. This algebra serves as a concrete matrix representation of the abstract walled Brauer algebra ($B_{\delta, p, p}$) acting on the tensor product space $(\mathbb{C}^d)^{\otimes 2p}$.

The core challenges identified are:

• Lack of Group-Adaptedness: Existing constructions of matrix units for the walled Brauer algebra (typically based on Young-Yamanouchi bases or branching graphs) are not "group-adapted" with respect to the subalgebra $\mathbb{C}[S_p] \times \mathbb{C}[S_p]$. This means the basis elements do not explicitly respect the symmetry of the product of two symmetric groups acting on the left and right sides of the "wall."
• Nested vs. Direct Sum Structure: Previous methods often treated the algebra's ideals in a nested inclusion structure ($M(p) \subset M(p-1) \subset \dots$). The authors seek a construction that yields a direct sum decomposition of the algebra into orthogonal ideals ($\tilde{M}(p) \oplus \tilde{M}(p-1) \oplus \dots$), which is crucial for efficient computation and physical interpretation in quantum information tasks (e.g., port-based teleportation).
• Scalability: While group-adapted bases exist for the highest ideal $M(p)$ and the second-highest $\tilde{M}(p-1)$, a general, systematic algorithm for constructing these bases for all ideals in the algebra was missing.

2. Methodology

The authors propose an iterative algorithm to construct orthonormal, group-adapted irreducible matrix units. The method relies on the following key components:

A. Recursive Orthogonalization

The algorithm constructs the basis layer by layer, starting from the smallest sub-algebra ($\mathcal{A}^d_{1,1}$) up to the full algebra ($\mathcal{A}^d_{p,p}$).

1. Base Case: Construct the orthonormal basis for $\mathcal{A}^d_{1,1}$ (spanned by the identity and the projector onto the maximally entangled state).
2. Iterative Step: For a given layer $k$ (constructing $\mathcal{A}^d_{k,k}$), the algorithm:
   • Takes the previously constructed orthonormal basis for the highest ideal $M(k)$.
   • Constructs a basis for the quotient layer $\tilde{M}(k-1)$ by orthogonalizing $(k-1)$-arc operators against the $k$-arc operators from $M(k)$ and against each other.
   • Repeats this process for lower layers down to $\tilde{M}(0)$.

B. Use of Orthogonal Projectors ($Q(k)$)

To achieve the direct sum decomposition, the authors introduce a new set of operators $Q(k)$ derived from the essential projectors $V(r)$ (which define the ideals $M(r)$).

• $Q(k)$ are defined as linear combinations of $V(k)$ and $V(k+1)$ such that they form a complete set of orthogonal projectors ($Q(k)Q(l) = \delta_{kl}Q(k)$).
• This allows the algebra to be decomposed as $\mathcal{A}^d_{p,p} = \bigoplus_{k=0}^p \text{span}\{ \dots Q(k) \dots \}$.

C. Contraction and Trace Techniques

A critical technical hurdle is calculating the overlaps (Gram matrix elements) between operators with different numbers of arcs.

• Lemma 1 (Contraction): The authors utilize a "squeezing" lemma where $V(r) X V(r) = X_{p\setminus r, p\setminus r} \otimes V(r)$. This reduces the calculation of overlaps in $\mathcal{A}^d_{p,p}$ to calculations in smaller algebras $\mathcal{A}^d_{p-r, p-r}$.
• Partial Trace Handling: To handle partial traces required for Gram matrices, the authors employ a "ping-pong" trick involving the transpose of operators and the properties of maximally entangled states. They also utilize a specific permutation $\sigma$ to align the basis construction direction (Young-Yamanouchi) with the partial trace requirements.

D. Gram Matrix Diagonalization

The construction of the orthonormal basis for a specific ideal $\tilde{M}(k)$ involves:

1. Generating a spanning set of operators.
2. Computing the Gram matrix of overlaps between these operators.
3. Diagonalizing this Gram matrix (which is block-diagonal due to symmetry) to extract the orthonormal irreducible matrix units.

3. Key Contributions

1. General Algorithm for Group-Adapted Bases: The paper provides the first systematic algorithm to generate irreducible matrix units for the entire walled Brauer algebra that are adapted to the $\mathbb{C}[S_p] \times \mathbb{C}[S_p]$ action. Previous works were limited to the top two ideals.
2. Direct Sum Decomposition: The method explicitly constructs a basis that decomposes the algebra into a direct sum of ideals ($\tilde{M}(p) \oplus \tilde{M}(p-1) \oplus \dots$), resolving the nested structure issue found in prior literature.
3. Explicit Construction for $p=2$: The authors fully derive and present the matrix units for the algebra $\mathcal{A}^d_{2,2}$. They demonstrate how the structure of the algebra (specifically the dimensions of the irreps) depends on the local dimension $d$ (e.g., distinguishing cases $d=2$, $d=3$, and $d>3$).
4. Novel Contraction Theorem: For $p=3$, the authors prove a contraction theorem showing how an operator sandwiched by a single arc $V^{t_{3'}}_{(3,3')}$ can be decomposed into a linear combination of the group-adapted matrix units of $\mathcal{A}^d_{2,2}$.
5. Orthogonal Projectors $Q(k)$: The introduction of the $Q(k)$ operators provides a clean algebraic tool for separating the ideals, which is essential for the iterative construction.

4. Results

• Algorithmic Framework: The paper details Algorithm 1, which recursively builds the basis. It highlights that while the general closed-form expressions become complex for large $p$, the procedure is computationally feasible for small system sizes and arbitrary local dimensions $d$.
• Structure of $\mathcal{A}^d_{2,2}$:
  • The algebra decomposes into $\tilde{M}(2) \oplus \tilde{M}(1) \oplus \tilde{M}(0)$.
  • $\tilde{M}(2)$ is a 2-dimensional matrix algebra (independent of $d$).
  • $\tilde{M}(1)$ is a matrix algebra of dimension 4 (or 3 if $d=2$).
  • $\tilde{M}(0)$ is a direct sum of 1-dimensional algebras (dimension varies with $d$).
  • The total dimension of $\mathcal{A}^d_{2,2}$ is shown to be 16 for $d=2$, 23 for $d=3$, and 24 for $d>3$.
• Contraction Theorem (Theorem 39): The paper provides explicit coefficients ($\lambda$) for decomposing the squeezed operator $V^{t_{3'}}_{(3,3')} E^\mu_{ij} \otimes E^\nu_{kl} V^{t_{3'}}_{(3,3')}$ into the basis of $\mathcal{A}^d_{2,2}$. This result is a prerequisite for constructing bases for higher $p$.

5. Significance and Applications

• Quantum Information Theory: The algebra $\mathcal{A}^d_{p,p}$ is central to the study of Port-Based Teleportation (PBT) and other quantum protocols involving symmetry. A group-adapted basis simplifies the analysis of these protocols, particularly in calculating entanglement measures and channel capacities.
• Computational Efficiency: By providing a basis that respects the $\mathbb{C}[S_p] \times \mathbb{C}[S_p]$ symmetry, the method allows for the reduction of complex semi-definite programs (SDPs) and optimization problems in quantum information to smaller, block-diagonal problems.
• Invariant Theory: The results contribute to the understanding of invariants under unitary groups of the form $U^{\otimes n} \otimes U^{\otimes n}$, which are fundamental in quantum many-body physics and quantum chemistry.
• Future Directions: The authors suggest applying this framework to higher-order quantum operations and parallel configurations (e.g., transforming unknown unitary programs), where the specific symmetry structure is naturally present.

In summary, this work bridges the gap between abstract diagrammatic algebras and concrete, symmetry-adapted matrix representations, offering a powerful computational tool for analyzing quantum systems with specific permutation symmetries.