On squares of representations of compact Lie algebras

Abstract

We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems

Symmetry criteria for quantum simulability of effective interactions

“…Example 1 below illustrates why matching the linear symmetries does not suffice to ensure simulability. As shown in a companion paper [45], one can decide if a subalgebra h ⊆ g of a compact semisimple Lie algebra g actually fulfills h = g (e.g., P = P∪Q ) just by analyzing quadratic symmetries. But Example 2 elucidates why condition (A) alone does not, in the general compact case, imply simulability.…”

“…So Ref. [45] secondly verifies that the semisimple parts of P and P∪Q have to agree if dim[P (2) ] = dim[(P∪Q) (2) ]. When generalizing from semisimple to arbitrary compact Lie algebras, the equality of the two tensor-square commutants implies that P and P∪Q agree-except for the central elements (commuting with all the other ones).…”

“…On a much more general scale, condition (B) closes the gap to completely characterizing equality in h ⊆ g now for all compact Lie algebras (generated by skewHermitian interactions) beyond the semisimple ones of [45]. Simplifying within the Lie-algebraic frame, our symmetry approach to decide simulability and the membership Q ⊆ P can thus be seen as a major step beyond the well-established Lie-algebra rank condition [14,16,17] and beyond the limited first use of quadratic symmetries to establish full controllability in [19].…”

“…Remarkably, symmetries beyond quadratic ones (i.e., those of the tensor square) are not required for a necessary and sufficient condition for simulability [58]. Concerning the tensor-square commutant, we build on two important classification-free results of [45] for compact Lie algebras [59,60] (as generated by skew-Hermitian matrices iH ν ): For P∪Q being semisimple (and compact), Ref. [45] first shows that P = P∪Q holds if and only if dim[P (2) ] = dim[(P∪Q) (2) ].…”

“…Concerning the tensor-square commutant, we build on two important classification-free results of [45] for compact Lie algebras [59,60] (as generated by skew-Hermitian matrices iH ν ): For P∪Q being semisimple (and compact), Ref. [45] first shows that P = P∪Q holds if and only if dim[P (2) ] = dim[(P∪Q) (2) ]. Beyond the semisimple case, any compact Lie algebra g can be uniquely decomposed as g = s ⊕ c into its semisimple part s and its center c (where s := [g, g] and [g, c] = 0 [44]).…”