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.
Implementing quantum gates as non-Abelian holonomies, a class of topologically protected unitary operators, is a particularly promising paradigm for the design of intrinsically stable quantum computers [1]. In contrast to dynamic phases, the geometric phase accumulated by a quantum system propagating through a Hilbert space
Hdepends exclusively on its path. In general, geometric phases can exhibit arbitrary dimensionality. Wilczek and Zee introduced the idea of multi-dimensional, non-Abelian geometric phases - so called holonomies [2]. Anandan later dropped the requirement of adiabaticity to create holonomies, that are truly time-independent [3]. Non-adiabatic holonomies rely on a subspace
Hgeoof the Hilbert-space that is spanned by states
{∣Φk⟩}kthat fulfill
(Φk∣H^∣Φj⟩=0, where
H^is the system's Hamiltonian. Restricting the propagation to
Hgeoensures parallel transport and, thus, a purely geometric phase (see Fig. 1a) [4], [5]. Quantum optics constitutes a particularly versatile platform for quantum information processing, and in particular for the construction of non-adiabatic holonomic quantum computers: In addition to integration and miniaturization provided by the platform, the bosonic nature of photons also conveniently allows for multiple excitations of the same mode, readily expanding
Hgeoand enabling the synthesis of holonomies from higher symmetry groups
U(N)as larger and more capable computational units [6], [7].
Non-Abelian holonomy, a noncommutative process that measures the parallel transport of non-Abelian gauge fields, has so far been realized in degenerate Hermitian systems with degenerate eigenstates or nondegenerate non-Hermitian systems with exceptional points. Here, we introduce non-Abelian holonomy into degenerate non-Hermitian systems possessing degenerate exceptional points and degenerate energy topologies. The interplay between energy degeneracy and energy topology around exceptional points leads to a non-Abelian holonomy with multiple energy levels and multiple degenerate levels simultaneously, going beyond that in degenerate Hermitian systems with a single energy level, or in nondegenerate non-Hermitian systems with a single degenerate level. We exploit an on-chip photonic platform to experimentally demonstrate the holonomy induced non-Abelian phenomenon, including the switching of eigenstates associated with different degenerate exceptional points and sequence-dependent holonomic outcomes. Our work shifts the paradigm of non-Abelian holonomy and adds new degrees of freedom for non-Abelian applications.
In 1984 Berry pointed out that abelian gauge structures exist naturally in slowly varying quantum system. Since then, a great deal of interest both theoretically and experimentally has been devoted to the interpretation, generalization and detection of geometric phases. In particular, relevant for the present work, it has been shown that a non-Abelian gauge structure emerges if a set of quantum states remains degenerate as the Hamiltonian varies. Non-Abelian phases have been originally investigated in Nuclear Quadrupole Resonance. More recently, they have found a renewed interest within the field of Quantum Computation, since it has been proved that quantum gates can be also implemented by geometrical means using both Abelian and non-Abelian phases. In this novel geometric approach, known as Holonomic Quantum Computation (HQC), information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and the computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops, the non trivial curvature of the geometry of the control manifold gives rise to unitary transformations, i.e. holonomies, that implement desired quantum gates. Until now only one scheme has been proposed to realize HQC in trapped ions by means of pulsed coherent excitations. On the other hand, it would be of great interest to have an implementation for solid-state systems, in view of their scalability and integrability. In particular, the impressing development of nanotechnologies with the consequent control of artificial two-level system in superconducting devices stimulates a more detailed investigation of geometric interference in mesoscopic systems. Here we show that non-Abelian phases can appear in the quantum charge dynamics of Josephson networks. Besides their possible detection, we shall see how they may be employed both in adiabatic charge pumping and as an implementation of HQC with Josephson junctions. 2. Josephson network for non abelian phases
We study the Uhlmann holonomy [Rep. Math. Phys. 24, 229 (1986)] of quantum states for hydrogenlike atoms where the intrinsic spin and orbital angular momentum are coupled by the spin-orbit interaction and are subject to a slowly varying magnetic field. We show that the holonomy for the orbital angular momentum and spin subsystems is non-Abelian while the holonomy of the whole system is Abelian. Quantum entanglement in the states of the whole system is crucially related to the non-Abelian gauge structure of the subsystems. We analyze the phase of the Wilson loop variable associated with the Uhlmann holonomy and find a relation between the phase of the whole system and corresponding marginal phases. Based on the results for the model system, we provide evidence that the phase of the Wilson loop variable and the mixed-state geometric phase [E. Sj\"oqvist et al., Phys. Rev. Lett. 85, 2845 (2000).] are generally inequivalent.