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Deformations of the symmetric subspace of qubit chains

Angel Ballesteros, Ivan Gutierrez-Sagredo, Jose de Ramon, J. Javier Relancio

TL;DR

This work analyzes the permutation-invariant subspace of $N$ qubits, traditionally described by Dicke states and Schur-Weyl duality with $SU(2)$, and extends it via the quantum group $\mathcal{U}_q(\mathfrak{su}(2))$. The authors construct the $q$-symmetric subspace and its basis of $q$-Dicke states, showing that deformation preserves much of the representation structure while encoding symmetry-breaking through the deformation parameter $q$ and the Hecke algebra dual. They reveal a novel, non-unitary $S_N$-action on the $q$-symmetric subspace and demonstrate how it can be unitarized by a locally deformed inner product, effectively interpreting the deformation as a change of Hilbert-space geometry. The study connects Hopf-algebra deformations, Hecke algebra representations, and Schur-Weyl duality to provide a framework for modeling imperfect symmetry and potential fault-tolerant or sensing applications, with open directions toward higher-dimensional systems, density-matrix subspaces, and root-of-unity regimes.

Abstract

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.

Deformations of the symmetric subspace of qubit chains

TL;DR

This work analyzes the permutation-invariant subspace of qubits, traditionally described by Dicke states and Schur-Weyl duality with , and extends it via the quantum group . The authors construct the -symmetric subspace and its basis of -Dicke states, showing that deformation preserves much of the representation structure while encoding symmetry-breaking through the deformation parameter and the Hecke algebra dual. They reveal a novel, non-unitary -action on the -symmetric subspace and demonstrate how it can be unitarized by a locally deformed inner product, effectively interpreting the deformation as a change of Hilbert-space geometry. The study connects Hopf-algebra deformations, Hecke algebra representations, and Schur-Weyl duality to provide a framework for modeling imperfect symmetry and potential fault-tolerant or sensing applications, with open directions toward higher-dimensional systems, density-matrix subspaces, and root-of-unity regimes.

Abstract

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 , 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 . 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.

Paper Structure

This paper contains 12 sections, 168 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The symmetric subspace
  3. The $\mathfrak{su} (2)$ Hopf algebra
  4. The $\mathcal{U}(\mathfrak{su} (2))$ Hopf algebra and Dicke states
  5. $q$-Deformation of the $\mathfrak{su}(2)$ algebra
  6. The $q$-symmetric subspace
  7. A new symmetry of the $q$-Dicke states
  8. A $q$-representation of the symmetric group
  9. Deformation of the symmetric subspace as a deformation of the Hilbert space's product
  10. Concluding remarks and open problems
  11. Proof of $\mathrm{span } \rho^N_s(U(2))=\Delta^{(N)}_{\frac{1}{2}} (\mathcal{U}(\mathfrak{su}(2)))$
  12. Deformation of the symmetric group and the Hecke algebra

Figures (5)

  • Figure 1: Identity diagram
  • Figure 2: Elementary transposition
  • Figure 3: Squared $q$-transposition
  • Figure 4: Braid relations
  • Figure 5: Total inversion in terms of adjacent transpositions

Theorems & Definitions (9)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9