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International Conference on Micro- and Nano-Electronics (ICMNE)

Bifurcations and catastrophes at the boundaries separating the phases of quantum correlations

Mikhail A. Yurischev

Published Jan 31, 2022• 0 citations
Publisher

Abstract

The spin-1/2 XXZ chain in a uniform magnetic field at thermal equilibrium is considered. For this model, we give phase diagrams for the one-way quantum work deficit (thermal discord). The diagrams can contain regions with both stationary and variable (state-dependent) angles of optimal measurement. We also established a relationship between the behavior of optimal measurement angles near the boundaries separated different regions and Landau’s theory of phase transitions of the second and first kind. Namely, when crossing the boundary from a region with a stationary optimal measurement angle (0 or π/2) to the region with a variable angle of optimal measurement (0 < ϑ < π/2), quantum discord and one-way quantum work deficit experience a sudden change (“catastrophe”) in their behavior. The sudden change is accompanied by a splitting of the minimum characterizing stationary point of quantum correlation into two minima (i.e., bifurcation of the initial single minimum occurs). Such phenomena are analogous to phase transitions described within the Landau theory, or, more generally, by the mathematical formalism of catastrophe theory.

DOI
10.1117/12.2622367

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The spin-1/2 XXZ chain in a uniform magnetic field at thermal equilibrium is considered. For this model, we give phase diagrams for the one-way quantum work deficit (thermal discord). The diagrams can contain regions with both stationary and variable (state-dependent) angles of optimal measurement. We also established a relationship between the behavior of optimal measurement angles near the boundaries separated different regions and Landau’s theory of phase transitions of the second and first kind. Namely, when crossing the boundary from a region with a stationary optimal measurement angle (0 or π/2) to the region with a variable angle of optimal measurement (0 < ϑ < π/2), quantum discord and one-way quantum work deficit experience a sudden change (“catastrophe”) in their behavior. The sudden change is accompanied by a splitting of the minimum characterizing stationary point of quantum correlation into two minima (i.e., bifurcation of the initial single minimum occurs). Such phenomena are analogous to phase transitions described within the Landau theory, or, more generally, by the mathematical formalism of catastrophe theory.
Abstract Sudden changes of quantum correlations in the Bell-diagonal states are well-known effects. They occur when the set of optimal parameters that determine the quantum correlation consists of isolated points and optimal parameters during the evolution of the system jump from one such point to another (e.g., the optimal measurement angle of the quantum discord changes discontinuously from zero to π /2 or vice versa). However, when considering more general X quantum states, we found that quantum discord and one-way quantum work deficit can experience sudden changes of other kinds. Namely, the optimal measurement angle may suddenly start to shift continuously from its stationary value 0 or π /2 to an intermediate optimal measurement angle ϑ ∈ (0, π /2). This leads to a new behavior of quantum correlations, which is mathematically described by the Landau phenomenological theory of second-order phase transitions. In addition, for the one-way quantum work deficit, we found cases where the optimal measurement angle jumps from zero to a nonzero step less than π /2, and then continuously changes its value. This behavior of quantum correlation is similar to a first-order phase transition in Landau’s theory. Dependencies of quantum discord and one-way quantum work deficit near the boundaries, which separate regions with state-dependent (variable) and state-independent (stationary, constant) optimal measurement angles, are examined in detail on an example of the XXZ spin model in an external field at thermal equilibrium.
We examine the behavior of quantum correlations of spin pairs in a finite anisotropic XY spin chain immersed in a transverse magnetic field, through the analysis of the quantum discord and the conventional and quadratic one-way information deficits. We first provide a brief review of these measures, showing that the last ones can be obtained as particular cases of a generalized information deficit based on general entropic forms. All of these measures coincide with an entanglement entropy in the case of pure states, but can be non-zero in separable mixed states, vanishing just for classically correlated states. It is then shown that their behavior in the exact ground state of the chain exhibits similar features, deviating significantly from that of the pair entanglement below the critical field. In contrast with entanglement, they reach full range in this region, becoming independent of the pair separation and coupling range in the immediate vicinity of the factorizing field. It is also shown, however, that significant differences between the quantum discord and the information deficits arise in the local minimizing measurement that defines them. Both analytical and numerical results are provided.
We examine the behavior of quantum correlations of spin pairs in a finite anisotropic XY spin chain immersed in a transverse magnetic field, through the analysis of the quantum discord and the conventional and quadratic one way-information deficits. We first provide a brief review of these measures, showing that the last ones can be obtained as particular cases of a generalized information deficit based on general entropic forms. All these measures coincide with an entanglement entropy in the case of pure states, but can be non-zero in separable mixed states, vanishing just for classically correlated states. It is then shown that their behavior in the exact ground state of the chain exhibits similar features, deviating significantly from that of the pair entanglement below the critical field. In contrast with entanglement, they reach full range in this region, becoming independent of the pair separation and coupling range in the immediate vicinity of the factorizing field. It is also shown, however, that significant differences between the quantum discord and the information deficits arise in the local minimizing measurement that defines them. Both analytical and numerical results are provided.
The one-way quantum work deficit, a measure of quantum correlation, exhibits the regions with the bimodal behavior of post-measurement entropy versus the measurement angle θ ε (0, π/2). Under certain conditions this can lead to the finite jumps Δν> 0 of optimal measurement angle ν from the endpoint 0 or π/2 to the interior minimum and vice versa. In turn, such sudden jumps lead to the fractures, i.e., non-analytical points on the dependencies of quantum correlation upon the state parameters. We examine a possibility to experimentally observe these phenomena on an example of three-component mixture consisting of one pair of Bell-diagonal states and a basis state orthogonal to the Bell ones. Observation of named effects, beyond their intrinsic interest, could demonstrate the experimental ability in controlling and manipulating the quantum states.
We examine the quantum correlations of spin pairs in the cyclic XXXXXX spin 1/2 chain in a transverse field, through the analysis of the quantum discord, the geometric discord and the information deficit. It is shown that while these quantities provide the same qualitative information, being non-zero for all temperatures and separations and exhibiting the same type of asymptotic behavior for large temperatures or separations, important differences arise in the minimizing local measurement that defines them. Whereas the quantum discord prefers a spin measurement perpendicular to the transverse field, the geometric discord and information deficit exhibit a perpendicular to parallel transition as the field increases, which subsists at all temperatures and for all separations. Moreover, it is shown that such transition signals the change from a Bell state to an aligned separable state of the dominant eigenstate of the reduced density matrix of the pair. Full exact results for both the thermodynamic limit and the finite chain are provided, through the Jordan-Wigner fermionization.
We examine the quantum correlations of spin pairs in the cyclic XX\mathit{XX}XX spin-1/21/21/2 chain in a transverse field through the analysis of the quantum discord, the geometric discord, and the information deficit. It is shown that while these quantities provide the same qualitative information, being nonzero for all temperatures and separations and exhibiting the same type of asymptotic behavior for large temperatures or separations, important differences arise in the minimizing local measurement that defines them. Whereas the quantum discord prefers a spin measurement perpendicular to the transverse field, the geometric discord and information deficit exhibit a perpendicular-to-parallel transition as the field increases, which subsists at all temperatures and for all separations. Moreover, it is shown that such transition signals the change from a Bell state to an aligned separable state of the dominant eigenstate of the reduced density matrix of the pair. Full exact results for both the thermodynamic limit and the finite chain are provided through the Jordan-Wigner fermionization.