What question did this study set out to answer?

To prove the existence of palindromic symmetry in the Liouvillian spectrum under local Z-dephasing and its implications for quantum state transfer.

March 16, 2026Open Access

Palindromic Liouvillian symmetry under dephasing: proof, spectral decomposition, and quantum state transfer

Key Points

To prove the existence of palindromic symmetry in the Liouvillian spectrum under local Z-dephasing and its implications for quantum state transfer.
Examined N-qubit systems with Heisenberg coupling on various graph topologies.
Analyzed the impact of local Z-dephasing on the Liouvillian spectrum.
Verified findings numerically using a 7-qubit model with 16384 dimensions.
Evaluated decay rates of observable components and their palindromic pairings.
Confirmed exact palindromic symmetry in the Liouvillian spectrum for N-qubit systems with Z-dephasing.
Demonstrated that individual observables decay at specific palindromic rates, such as purity at 2γ.
Showed a star topology with a 2:1 receiver-to-sender coupling ratio achieved F_avg = 0.888, outperforming standard configurations.

Abstract

We prove that for N-qubit systems with Heisenberg (XXZ) coupling on any graph topology, subject to local Z-dephasing, the Liouvillian spectrum has an exact palindromic symmetry. The conjugation operator Π acts per site on Pauli indices: I → X (+1), X → I (+1), Y → iZ (+i), Z → iY (+i) It satisfies Π·L·Π⁻¹ = -L - 2Σγᵢ·I. Every decay rate d is paired with a partner at 2Σγ - d. Verified numerically through N=7 (16384×16384 Liouvillian) across star, chain, ring, complete graph, and binary tree topologies. Individual observable components decay at specific palindromic rates: purity near 2γ, coherence near 10γ/3, concurrence envelope near 8γ/3. Applied to quantum state transfer: a star topology with 2: 1 receiver-to-sender coupling ratio achieves Fₐvg = 0. 888, outperforming standard chain configurations. The palindrome theorem applies to every Heisenberg spin chain with Z-dephasing studied in the QST literature. Authors: Thomas Wicht (independent researcher, Krefeld, Germany) and Claude (Anthropic, AI research partner).

Palindromic Liouvillian symmetry under dephasing: proof, spectral decomposition, and quantum state transfer

Key Points

Abstract

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