Questions at the intersection of the AdS/CFT correspondence and quantum information theory motivate the study of projectors in sequences of subalgebras of finite-dimensional commutative associative semisimple algebras $\mathcal{A}$, obtained by incrementally adjoining one generator at each step to produce a non-linear generating set for $\mathcal{A}$. We define degeneracy graphs, which are finite layered tree graphs whose nodes represent projectors in the successive subalgebras. Using combinatorial properties of the degeneracy graph, we give a simple formula for constructing a linear basis of $\mathcal{A}$ in terms of monomials in the generators.The nodes can be labelled by formal variables corresponding to the eigenvalues of the generators added at each layer. We prove that the construction is compatible with the required counting of projectors in $\mathcal{A}$, and give explicit constructions of the projectors in terms of the monomials, in the cases of one- and two-layer degeneracy graphs with arbitrary numbers of nodes. More generally, we provide extensive computational evidence for the invertibility of the matrix relating the proposed monomial basis to the projector basis, by evaluating its determinant. In the 1-layer case, this is a Vandermonde determinant. A simple formula for the non-vanishing determinant in the general layer case is conjectured and supported by the computational data. The construction is illustrated with examples including centres of symmetric group algebras and maximally commuting subalgebras generated by JucysMurphy elements. We outline applications of the monomial basis to algorithms for constructing matrix units in non-commutative semisimple algebras, with relevance to orthogonal bases of multi-matrix gauge-invariant operators and to quantum information theory.
Questions at the intersection of the AdS/CFT correspondence and quantum information theory motivate the study of projectors in sequences of subalgebras of finite-dimensional commutative associative semisimple algebras A, obtained by incrementally adjoining one generator at each step to produce a non-linear generating set for A. We define degeneracy graphs, which are finite layered tree graphs whose nodes represent projectors in the successive subalgebras. Using combinatorial properties of the degeneracy graph, we give a simple formula for constructing a linear basis of A in terms of monomials in the generators.The nodes can be labelled by formal variables corresponding to the eigenvalues of the generators added at each layer. We prove that the construction is compatible with the required counting of projectors in A, and give explicit constructions of the projectors in terms of the monomials, in the cases of one- and two-layer degeneracy graphs with arbitrary numbers of nodes. More generally, we provide extensive computational evidence for the invertibility of the matrix relating the proposed monomial basis to the projector basis, by evaluating its determinant. In the 1-layer case, this is a Vandermonde determinant. A simple formula for the non-vanishing determinant in the general layer case is conjectured and supported by the computational data. The construction is illustrated with examples including centres of symmetric group algebras and maximally commuting subalgebras generated by JucysMurphy elements. We outline applications of the monomial basis to algorithms for constructing matrix units in non-commutative semisimple algebras, with relevance to orthogonal bases of multi-matrix gauge-invariant operators and to quantum information theory.
Kemp and Ramgoolam proposed a basis of monomials for a commutative semisimple algebra with generators C1,…,CL, constructed from a degeneracy graph which records how the eigenspaces of C1 are successively split by C2,…,CL. They showed that the basis property is equivalent to the non-vanishing of the determinant of the matrix of values of these monomials on the projector basis, and conjectured a factorised formula for this determinant, as a product of powers of eigenvalue differences. We prove this formula for any number of generators. The proof uses graph analogues of Newton polynomials. Their evaluation matrix is not triangular in general, but expanding the rows of the children of each node of the penultimate layer in a Newton basis of their eigenvalues makes it block triangular, with diagonal blocks given by smaller degeneracy graphs in which the final layer is removed. The formula then follows by induction on the number of layers. We show that the exponents in the determinant are overlap volumes of higher-dimensional Young diagrams attached to the nodes of the graph: intervals, ordinary Young diagrams, plane partitions and their higher analogues. Finally, the block triangular structure gives an explicit recursive inversion of the evaluation matrix, expressing the projectors in the monomial basis.
The walled Brauer algebras BN(m,n) govern Schur--Weyl duality for unitary groups U(N) acting on mixed tensor spaces VN⊗m⊗VN⊗n and play an important role in applications ranging from AdS/CFT to quantum information theory. In the stable regime N≥m+n the algebra is semisimple and its representation theory is well understood. For N<m+n, however, BN(m,n) becomes non-semisimple. The representation of the algebra on tensor space has a non-trivial kernel and the corresponding quotient algebra is semisimple, with representation dimensions differing from those in the stable regime.
We introduce \emph{restricted Bratteli diagrams}, obtained by modifying the standard Bratteli diagrams for BN(m,n). This construction provides a systematic way to use representation-theoretic data from the stable regime to compute the dimension modifications arising in the non-semisimple regime. In the regime N=m+n−l, with l small compared to m,n, we show that the restricted diagrams exhibit a stability property and enable an efficient counting of the paths responsible for these dimension corrections.
Remarkably, the resulting generating functions are governed by the partition function of an infinite tower of simple harmonic oscillators. We briefly discuss implications for the construction of orthogonal bases of matrix invariants in gauge theory and related applications in quantum information theory.
Abstract We define the computational task of detecting projectors in finite dimensional associative algebras with a combinatorial basis, labelled by representation theory data, using combinatorial central elements in the algebra. In the first example, the projectors belong to the centre of a symmetric group algebra and are labelled by Young diagrams with a fixed number of boxes n. We describe a quantum algorithm for the task based on quantum phase estimation (QPE) and obtain estimates of the complexity as a function of n. We compare to a classical algorithm related to the projector identification problem by the AdS/CFT correspondence. This gives a concrete proof of concept for classical/quantum comparisons of the complexity of a detection task, based in holographic correspondences. A second example involves projectors labelled by triples of Young diagrams, all having n boxes, with non-vanishing Kronecker coefficient. The task takes as input the projector, and consists of identifying the triple of Young diagrams. In both of the above cases the standard QPE complexities are polynomial in n. A third example of quantum projector detection involves projectors labelled by a triple of Young diagrams, with m, n and m + n boxes respectively, such that the associated Littlewood-Richardson coefficient is non-zero. The projector detection task is to identify the triple of Young diagrams associated with the projector which is given as input. This is motivated by a two-matrix model, related via the AdS/CFT correspondence, to systems of strings attached to giant gravitons. The QPE complexity in this case is polynomial in m and n.
We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra A^1, defined as the real subalgebra of the Weyl algebra A1 consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for A^1, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in A^1 generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in A^1 that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in A^1? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of A^1 that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in A^1, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.
In this dissertation, we investigate algebraic structures in finite-dimensional Hilbert spaces, as concerns quantum computation and quantum information, as well as these structures' applications to lattices. On the quantum computation side, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies these axioms. Subsequently, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, using the algebraic framework developed. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many algebraic identities, including a novel algebraic proof of a Yang-Baxter equation. We also derive a new identity for the braid elements, which is key to our proofs. We then use the Yang-Baxter equation proof to resolve an open question of Cobanera and Ortiz. We demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. In addition, we show how to explicitly compute various vector states in an efficient manner using algebraic methods. On the quantum information side, we introduce a new decomposition of quantum channels acting on group algebras, which we term Kraus-like operator decompositions (Kraus-like decompositions for short). An important motivation for this new decomposition is a general nonexistence result that we show for Kraus operator decompositions for quantum channels in this setting. We show that the notion of convex Kraus-like operator decompositions (in which the coefficients in the sum decomposition are nonnegative and satisfy a sum rule) that are induced by the irreducible characters of a finite group is equivalent to the notion of a conditionally negative-definite length when the length is a class function. For a general finite group G, we prove a stability condition which shows that if the semigroup associated with a length has a convex Kraus-like operator decomposition for all t>0 small enough, then it has a convex Kraus-like operator decomposition for all time t>0. Using the stability condition, we show that for a general finite group, conditional negativity of the length function is equivalent to a set of semidefinite linear constraints on the length function. By a result of Schoenberg, our result implies that in the group algebra setting, a semigroup P_t induced by a length function which is a class function is a quantum channel for all t greater than or equal to 0 if and only if it possesses a convex Kraus-like operator decomposition for all t>0. Finally, motivated by the importance of lattice problems in quantum cryptography, we extend the algebraic framework for multi-qudit representations of generalized Clifford algebras to d-dimensional lattices mod P. We show that under suitable number-theoretic conditions, the subalgebra induced by a lattice has trivial center. Under the trivial center constraint, we construct for pairs of lattice vectors satisfying an algebraic constraint a unitary operator based on the product of generalized Clifford algebra generators associated to each lattice vector.
In quantum information theory and statistical physics, symmetries of multiple copies, or replicas, of a system play a pivotal role. For unitary ensembles, these symmetries are encoded in the replicated commutant: the algebra of operators commuting with the ensemble across k replicas. Determining the commutant is straightforward for the full unitary group, but remains a major obstacle for structured, computationally relevant circuit families. We solve this problem for matchgate circuits, which prepare fermionic Gaussian states on n qubits. Using a Majorana fermion representation, we show that operators coupling different system copies generate the orthogonal Lie algebra so(k), endowing the space of invariants with rich and tractable structure. This underlying symmetry decomposes the matchgate commutant into irreducible sectors, which we completely resolve via a Gelfand--Tsetlin construction. We provide an explicit orthonormal basis of the matchgate commutant for all k and n, together with a formula for its dimension that grows polynomially in n. Furthermore, we characterize the commutant of the Clifford--matchgate subgroup, showing that restricting to signed permutations of Majorana modes yields a commutant that qualitatively diverges from the matchgate case for k≥4 replicas. Ultimately, our orthonormal basis turns algebraic classification into a working toolbox. Using it, we derive closed-form expressions for matchgate twirling channels and a fermionic analogue of Weingarten calculus, the projector encoding all moments of the Gaussian state orbit, state and unitary frame potentials, the average nonstabilizerness of fermionic Gaussian states, a systematic hierarchy of non-Gaussianity measures, and a fermionic de Finetti theorem.