Deformations of the symmetric subspace of qubit chains

The symmetric subspace of multi-qubit systems, that is, the space of states invariant under permutations, is commonly encountered in applications in the context of quantum information and communication theory. It is known that the symmetric subspace can be described in terms of irreducible representations of the group $SU(2)$, whose representation spaces form a basis of symmetric states, the so-called Dicke states. In this work, we present deformations of the symmetric subspace as deformations of this group structure, which are promoted to a quantum group $\mathcal{U}_q(\mathfrak{su}(2))$. We see that deformations of the symmetric subspace obtained in this manner correspond to local deformations of the inner product of each spin, in such a way that departure from symmetry can be encoded in a position-dependent inner product. The consequences and possible extensions of these results are also discussed.

💡 Research Summary

The paper investigates how the symmetric subspace of an N‑qubit chain—traditionally defined as the set of states invariant under all permutations of the qubits—can be continuously deformed by exploiting the well‑known q‑deformation of the Lie algebra su(2). The authors start by recalling that the symmetric subspace is naturally described by the Schur‑Weyl duality between the symmetric group Sₙ and the unitary group U(2). In this picture, Dicke states form an orthonormal basis for the irreducible representation of SU(2) with total spin j = N/2, and the whole symmetric subspace is generated by acting with the collective SU(2) operators on any reference symmetric vector (e.g., the all‑down state).

The core of the work is the promotion of the universal enveloping algebra U(su(2)) to its quantum group counterpart U₍q₎(su(2)). The deformation modifies the coproduct of the raising and lowering operators to Δ₍q₎(J±)=J±⊗q^{J₃}+q^{-J₃}⊗J± while leaving Δ₍q₎(J₃) unchanged. This introduces non‑commutativity (the Hopf algebra becomes non‑cocommutative) and replaces the ordinary braid relations of the adjacent transpositions t_i∈Sₙ by the Hecke algebra relations t_i t_{i+1} t_i = t_{i+1} t_i t_{i+1} and t_i² = (q−1) t_i + q. Consequently, the representation theory of the deformed algebra yields a new family of collective states, the q‑Dicke states |j,m⟩₍q₎, which reduce to ordinary Dicke states when q→1.

The q‑Dicke states are constructed by applying the deformed collective operators to the all‑down reference state. Their coefficients involve q‑integers