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Physica Scripta

Research on quantum correlations and Uhlmann phase in two-fermion mixed state system

Wei-Min Yao, Xiao‐Gang Fan, Xue‐Ke Song, Dong Wang, Liu Ye

Published Jun 3, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ad5386

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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 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.
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.
Abstract Quantum entanglement and steering are both fundamental concepts for describing quantum correlations, with steerable states forming a strict subset of entangled states. In general two-qubit systems, the relation between entanglement and steering can be characterized by an inequality, referred to as the constraint relation. In this work, we investigate a two-fermion spin system to explore the influence of static magnetic fields on quantum entanglement and steering. Our results demonstrate that entanglement and steering are equivalent in this system. The finding provides new perspectives for understanding quantum correlations in two-fermion spin systems. Furthermore, we examine the dynamic behavior of entanglement, showing that the maximum entanglement of this system affected by both external and internal factors.
In this paper a system consisting of two electron spins has been prepared initially in a singlet state using the chemical compass model is considered. It is assumed that each electron spin interacts symmetrically and/or asymmetrically with its respective private nuclear environment in the presence of an external magnetic field. We discussed the effect of the interaction parameters and the external magnetic field on some quantifiers of quantum correlations as entanglement, coherence, Bell inequality, as well as the steerability inequality. It is shown that within a certain range of external magnetic fields, the quantum coherence and entanglement behave similarly. The Bell and the steerable inequalities predicted a similar behavior for symmetric and asymmetric interactions. Moreover, as one increases the external magnetic field, the lower bounds of both inequalities have improved. The usefulness of using the spin state as quantum channel to teleport a two-qubit system has examined where the Bell inequality could be above its classical bounds by controlling the interaction parameters. It is shown that by tuning the coupling parameters the fidelity of the teleported state exceeds the classical bounds, as well as the long-lived stationary fidelity could be achieved during the interaction time.
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.
We address the dynamics of quantum coherence, quantum correlations, quantum steering and Bell nonlocality in a model of two interacting qubits, initially prepared in a thermal state. The two qubit are coupled via a dipolar interaction and exposed to Dzyaloshinskii–Moriya (DM) interaction and classical fluctuating fields influenced by Ornstein–Uhlenbeck (OU)-Static (ST) mixed noise in common environments. We analyse the dynamics effects on quantum resources, as Bell nonlocality, quantum steering, quantum entanglement, quantum discord and quantum coherence. We find that the dynamics of quantum correlations are not only strongly affected by the type of system-environment coupling but also by the kind of the considered noise. The entanglement diminution is over time, accompanied by damped oscillations, indicative of coherent dynamics. The DM interaction induces a decrease in quantum coherence and correlations with increasing values, leading to complete maximizing at critical thresholds. The effects of thermal effects destroy the quantum correlation. The variations in state noise disorder parameters reveal enhanced preservation and reduced oscillations as the noise disorder parameter decreases, emphasizing its crucial role in shaping quantum correlations.