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Journal of High Energy Physics

A thermofield-double model of Uhlmann’s anholonomy

Péter Lévay, Csaba Velich

Published Mar 24, 2026• 0 citations
Publisher

Abstract

A bstract Recently it has been shown that certain aspects of changing states in holography can be studied by considering the Berry Phase properties of simple entangled qubit systems. In this context the term wormhole quantum mechanics has been coined. In particular an interesting interplay between entanglement and Berry Phase was found. Here we present a mixed state generalization of this connection illustrated for a simple parametrized family of entangled quantum systems in the thermofield double state. Our state is purifying two subsystems dubbed the left and the right ones and evolves in a purely geometric manner based on a parallel transport condition due to Uhlmann. We show that different interpretations of this evolution relative to observers either coupled to the left or to the right exist. When the evolution on say the left manifests itself via local operations of non-unitary optimal measurements then on the right the evolutionary steps are organized into a sequence of unitary operations of a holonomic quantum computation. For closed curves featuring geodesic segments with respect to the Bures metric we calculate the anholonomy of Uhlmann’s connection which turns out to be related to the one of higher dimensional instantons. The initial and final states are having the same entanglement but different Uhlman’s Phase. We show that by conducting an interference experiment one can observe the physical effects of the resulting anholonomic quantum computation. The consequences of our findings in the holographic context are discussed.

DOI
10.1007/jhep03(2026)226

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A bstract Recently it has been shown that certain aspects of changing states in holography can be studied by considering the Berry Phase properties of simple entangled qubit systems. In this context the term wormhole quantum mechanics has been coined. In particular an interesting interplay between entanglement and Berry Phase was found. Here we present a mixed state generalization of this connection illustrated for a simple parametrized family of entangled quantum systems in the thermofield double state. Our state is purifying two subsystems dubbed the left and the right ones and evolves in a purely geometric manner based on a parallel transport condition due to Uhlmann. We show that different interpretations of this evolution relative to observers either coupled to the left or to the right exist. When the evolution on say the left manifests itself via local operations of non-unitary optimal measurements then on the right the evolutionary steps are organized into a sequence of unitary operations of a holonomic quantum computation. For closed curves featuring geodesic segments with respect to the Bures metric we calculate the anholonomy of Uhlmann’s connection which turns out to be related to the one of higher dimensional instantons. The initial and final states are having the same entanglement but different Uhlman’s Phase. We show that by conducting an interference experiment one can observe the physical effects of the resulting anholonomic quantum computation. The consequences of our findings in the holographic context are discussed.
A simple parametrized family of quantum systems consisting of two entangled subsystems, dubbed left and right ones, both of them featuring N qubits is considered in the thermofield double formalism. We assume that the system evolves in a purely geometric manner based on the parallel transport condition due to Uhlmann. We explore the different interpretations of this evolution relative to observers either coupled to the left or to the right subsystems. The Uhlmann condition breaks the symmetry between left and right by regarding one of the two possible sets of local unitary operations as gauge degrees of freedom. Then gauging the right side we show that the geometric evolution on the left manifests itself via certain local operations reminiscent of non-unitary filtering measurements. On the other hand on the right the basic evolutionary steps are organized into a sequence of unitary operations of a holonomic quantum computation. We calculate the Uhlmann connection governing the transport for our model which turns out to be related to higher dimensional instantons. Then we evaluate the anholonomy of the connection for geodesic triangles with geodesic segments defined with respect to the Bures metric. By analysing the explicit form of the local filtering measurements showing up on the left side we realize that they are also optimal measurements for distinguishing two given mixed states in the statistical sense. We also point out that by conducting an interference experiment on the right side one can observe the physical effects of the anholonomic quantum computation. We demonstrate this by calculating explicit examples for phase shifts and visibility patterns arising in such interference experiments. Finally a sequence of geodesic triangles producing the iSWAP gate via anholonomy needed for computational universality is presented.
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.
We establish a relation between entanglement in simple quantum mechanical qubit systems and in wormhole physics as considered in the context of the AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT correspondence. We show that in both cases, states with the same entanglement structure, indistinguishable by any local measurement, nevertheless are characterized by a different Berry phase. This feature is experimentally accessible in coupled qubit systems where states with different Berry phase are related by unitary transformations. In the wormhole case, these transformations are identified with a time evolution of one of the two throats.
The Berry phase arises naturally from the adiabatic dynamics of a pure quantum state, whereas the Uhlmann phase is usually formulated kinematically for a prescribed path of mixed states. For a qubit subject to a uniformly gapped conical drive, we obtain the Uhlmann connection and holonomy in closed form for the equilibrium Gibbs cycle. We then solve the corresponding Lindblad equation exactly in the rotating frame and show that its steady state forms a nonequilibrium limit cycle that lags the instantaneous Gibbs cycle. As the driving becomes slow, the Uhlmann phase approaches the equilibrium Uhlmann phase, which cooling then reduces to the Berry phase: in the joint adiabatic and low-temperature limit, the dynamical Uhlmann phase therefore converges to the Berry phase. This grounds the Uhlmann--Berry correspondence in the physical dynamics of an open system. We also characterize the finite-driving corrections, together with the discontinuous πππ jump that the Uhlmann phase undergoes on the equatorial cycle as the temperature is varied. Finally, we examine an isolated transversal gap closing, where this zero-temperature correspondence can break down: the pure-state path becomes open, its closure is ambiguous, and gap-opening regularizations close it along a geodesic set by the direction of the bias field, splitting the Berry phase into a one-parameter family of values. The Gibbs path instead closes smoothly through the maximally mixed state, so the Uhlmann holonomy requires no regularization and remains unique and continuous at every finite temperature. In the low-temperature limit the Uhlmann phase selects a single member of the Berry family---the geodesic closure in the osculating plane, fixed by the velocity and acceleration of the driving field. The Uhlmann--Berry correspondence of the gapped regime thus survives the gap closing as a geodesic selection rule.
It is shown that Uhlmann's parallel transport of purifications along a path of mixed states represented by 2 × 2 density matrices is just the path-ordered product of Thomas rotations. These rotations are invariant under hyperbolic translations inside the Bloch sphere that can be regarded as the Poincaré ball model of hyperbolic geometry. A general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the Bures fidelities is derived. The formula gives back the solid angle result well known from studies of the pure state geometric phase. It is also shown that this mixed state anholonomy can be reinterpreted as the pure state non-Abelian anholonomy of entangled states living in a suitable restriction of the quaternionic Hopf bundle. In this picture Uhlmann's parallel transport is just the Pancharatnam transport of quaternionic spinors.
The geometric phase for a pure quantal state undergoing an arbitrary evolution is a "memory" of the geometry of the path in the projective Hilbert space of the system. We find that Uhlmann's geometric phase for a mixed quantal state undergoing unitary evolution depends not only on the geometry of the path of the system alone but also on a constrained bilocal unitary evolution of the purified entangled state. We analyze this in general, illustrate it for the qubit case, and propose an experiment to test this effect. We also show that the mixed state geometric phase proposed recently in the context of interferometry requires unilocal transformations and is therefore essentially a property of the system alone.