Symmetry and Liouville Space Formulation of Decoherence-Free Subsystems

Abstract: We propose a generic and systematic decoherence-free scheme to encode quantum information into an open quantum system based focusing on symmetry. Under a given symmetry, the Liouville space is decomposed into invariant subspaces characterized by a tensor-product structure. A decoherence-free subsystem is then identified as a factor of the tensor product. Unlike decoherence-free subspaces, which typically require strong symmetries, decoherence-free systems are permitted under less restrictive weak symmetries. Specifically, we primarily concern the permutation symmetry in conjunction with the unitary symmetry and utilize the Schur-Weyl duality, which facilitates numerous efficient and systematic calculations based on the well-established group representation theory. Employing the isomorphism between the Liouville space and the fictitious Hilbert space, we construct a super-Schur basis, which block-diagonalizes the super-operators that describe the noisy quantum channels, both in the Kraus representation and in terms of the quantum master equation. Each block reveals the tensor-product structure and facilitates the identification of physically relevant decoherence-free subsystems under the specified weak symmetry.