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arXiv:2106.15879 · Physical Review A

Uhlmann phase in composite systems with entanglement

Jorge Villavicencio, E. Cota, F. Rojas, Jesús A. Maytorena, Diego Morachis Galindo

Published Oct 6, 2021• 7 citations
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Abstract

We study the geometric Uhlmann phase of entangled mixed states in a composite system made of two coupled spin-$\frac{1}{2}$ particles with a magnetic field acting on one of them. Within a uniform noisy quantum channel setup, an exact analytical expression for this geometric phase in each subsystem is derived. We find an explicit connection to the concurrence of the density matrix of such a channel, which allows us to characterize the features of the Uhlmann phase in terms of the degree of entanglement in the system. In the space of field direction and coupling parameter, it exhibits a phase singularity revealing a topological transition between orders characterized by different winding numbers. The transition occurs for fields lying in the equator of the sphere of directions and at critical values of the coupling which can be controlled by tuning the depolarization strength. Notably, under these conditions the concurrence of the composite system is bounded to the range $[0,1/2]$. We also compare the calculated Uhlmann phase to an interferometric phase, which has been formulated as an alternative for density matrices. The latter does not present a phase vortex, although they coincide in the weak entanglement regime for vanishing depolarization (pure states). Otherwise, they behave clearly different in the strong-entanglement regime.

DOI
10.1103/physreva.104.042204

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We study the geometric Uhlmann phase of entangled mixed states in a composite system made of two coupled spin-12\frac{1}{2}21​ particles with a magnetic field acting on one of them. Within a uniform noisy quantum channel setup, an exact analytical expression for this geometric phase in each subsystem is derived. We find an explicit connection to the concurrence of the density matrix of such a channel, which allows us to characterize the features of the Uhlmann phase in terms of the degree of entanglement in the system. In the space of field direction and coupling parameter, it exhibits a phase singularity revealing a topological transition between orders characterized by different winding numbers. The transition occurs for fields lying in the equator of the sphere of directions and at critical values of the coupling which can be controlled by tuning the depolarization strength. Notably, under these conditions the concurrence of the composite system is bounded to the range [0,1/2][0,1/2][0,1/2]. We also compare the calculated Uhlmann phase to an interferometric phase, which has been formulated as an alternative for density matrices. The latter does not present a phase vortex, although they coincide in the weak entanglement regime for vanishing depolarization (pure states). Otherwise, they behave clearly different in the strong-entanglement regime.
Abstract Geometric phase, characterized by its unique properties has been an interesting subject since the discovery of Berry phase. A state inevitably becomes a mixed state in the environment. Uhlmann phase can be applied to mixed states and may be the generalization of Berry phase. In this paper, we use a pair of interacting spin-1/2 fermions, where one particle is influenced by an external magnetic field, while the other remains unaffected, to investigate quantum correlations and Uhlmann phase, we define g, p as the ratio of the particle coupling intensity to the magnetic field, and q as the depolarization parameter. Through calculation, we get the specific parameter range q of steering and Bell nonlocality under the depolarized noise channel, q ∈ 0 , 1 − 4 + g 2 4 + 2 g 2 (steering) and q ∈ 0 , 1 − 4 + g 2 4 + 3 g 2 (Bell nonlocality). Moreover, by using Uhlmann phase, we obtain a simple discrimination criterion of the steerability (Bell nonlocality) of the mixed system.
We study the geometric Uhlmann phase of mixed states at finite temperature in a system of two coupled spin-12\frac{1}{2}21​ particles driven by a magnetic field applied to one of the spins. In the parameter space of temperature and coupling, we show the emergence of two topological Uhlmann phase transitions when the magnetic field evolves around the equator, where a winding number can characterize each temperature range. For small couplings, the width of the temperature gap of the nontrivial phase is roughly the critical temperature Tc{T}_{c}Tc​ of one-dimensional fermion systems with two-band Hamiltonians. The first phase transition in the low-temperature regime and small values of the coupling corresponds to the peak of the Schottky anomaly of the heat capacity, typical of a two-level system in solid-state physics involving the ground and first excited states. The second phase transition occurs at temperatures very close to the second maximum of the heat capacity associated with a multilevel system. We also derive analytical expressions for the thermal Uhlmann phase for both subsystems, showing that they exhibit phase transitions. In the driven subsystem, for minimal ggg, a topological phase transition appears at Tc{T}_{c}Tc​ again. However, for larger values of ggg, the transitions occur at lower-temperature values, and they disappear when the coupling reaches a critical value gc{g}_{c}gc​. The latter is not the case for the undriven subsystem, where at low temperatures, a single phase transition occurs at gc{g}_{c}gc​. Nevertheless, as the temperature rises, we demonstrate the emergence of two phase transitions defining a coupling gap, where the phase is nontrivial and vanishes as the temperature reaches a critical value.
Geometric phases have been extensively investigated in a wide range of quantum systems, often revealing deep connections to the underlying topology of many-body states. In this work, we examine two geometric phases defined for mixed quantum states−-−the interferometric geometric phase and the Uhlmann phase−-−extracted from two-site reduced density matrices of the transverse-field Ising model with nearest-neighbor interacting spins. By applying coordinated local unitary rotations to the spins, we compute the geometric phases associated with the two-site states across the critical point. We find that the interferometric phase is a more reliable indicator of the quantum phase transition in this model than the Uhlmann phase.
The generalization of the geometric phase to the realm of mixed states is known as the Uhlmann phase. Recently, applications of this concept to the field of topological insulators have been made and an experimental observation of a characteristic critical temperature at which the topological Uhlmann phase disappears has also been reported. Here we study the case of the Uhlmann phase of a paradigmatic system such as the spin-jjj particle in the presence of a slowly rotating magnetic field at finite temperature in an exact analytical form. We find that the Uhlmann phase is given by the argument of a complex-valued second-kind Chebyshev polynomial of order 2j2j2j. Correspondingly, the Uhlmann phase displays 2j2j2j singularities, occurring at the roots of such polynomials which define the critical temperatures at which the system undergoes topological order transitions. Appealing to the argument principle of complex analysis, each topological order is characterized by a winding number, which happens to be 2j2j2j for the ground state and decreases by unity each time the increasing temperature passes through a critical value. We hope this study encourages experimental verification of this phenomenon of thermal control of topological properties, as has already been done for the spin-1/21/21/2 particle.
The Uhlmann process is built on the density matrix of a mixed quantum state and offers a way to characterize topological properties at finite temperatures. We analyze an ideal spin-jjj quantum paramagnet in a magnetic field undergoing an Uhlmann process and derive general formulas of the Uhlmann phase and Loschmidt amplitude for arbitrary jjj as the system traverses a great circle in the parameter space. A quantized jump of the Uhlmann phase signifies a topological quantum phase transition (TQPT) of the underlying process, which is accompanied by a zero of the Loschmidt amplitude. The exact results of j=12j=\frac{1}{2}j=21​ and j=1j=1j=1 systems show topological regimes that survive only at finite temperatures but not at zero temperature, and the number of TQPTs is associated with the winding number in the parameter space. Our results pave the way for future studies on finite-temperature topological properties, and possible experimental protocols and implications for atomic simulators and digital simulations are discussed.
Two geometric phases of mixed quantum states, known as the interferometric phase and Uhlmann phase, are generalizations of the Berry phase of pure states. After reviewing the two geometric phases and examining their parallel-transport conditions, we specify a class of cyclic processes that are compatible with both conditions and therefore accumulate both phases through their definitions, respectively. Those processes then facilitate a fair comparison between the two phases. We present exact solutions of two-level and three-level systems to contrast the two phases. While the interferometric phase exhibits finite-temperature transitions only in the three-level system but not the two-level system, the Uhlmann phase shows finite-temperature transitions in both cases. Thus, using the two geometric phases as finite-temperature topological indicators demonstrates the rich physics of topology of mixed states.