Wilczek-Zee Realization of Uhlmann Parallel Transport

• URL: http://arxiv.org/abs/2607.05591v1
• Date: Mon, 06 Jul 2026 19:42:15 GMT
• Title: Wilczek-Zee Realization of Uhlmann Parallel Transport
• Abstract summary: The Uhlmann phase extends geometric phases to mixed quantum states via a parallel-transport condition on purification amplitudes.<n>We establish a four-level Hermitian parent Hamiltonian whose doubly degenerate ground-state subspace carries a Wilczek--Zee connection exactly equal to the Uhlmann connection.
• Score: 3.035804245322184
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: The Uhlmann phase extends geometric phases to mixed quantum states via a parallel-transport condition on purification amplitudes, yet its direct implementation under standard Hamiltonian dynamics is obstructed by the non-Hermitian nature of the purification. We establish that for any smooth one-dimensional closed loop of full-rank qubit density matrices, there exists a four-level Hermitian parent Hamiltonian whose doubly degenerate ground-state subspace carries a Wilczek--Zee connection exactly equal to the Uhlmann connection. Consequently, the Uhlmann holonomy is faithfully reproduced by adiabatic evolution in the enlarged system. We further prove that this auxiliary-field construction is obstructed in generic two-dimensional parameter spaces by a Frobenius integrability condition, which we derive explicitly. The one-dimensional Uhlmann phase is thus placed on the same footing as the non-Abelian Berry phase, offering a purely Hermitian, Hamiltonian-based route to simulating mixed-state geometric phases. Numerical integration of the adiabatic dynamics confirms the exact correspondence and validates the convergence to the Uhlmann holonomy in the large-gap limit.

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