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Physical Review A

Bell correlations of a thermal fully connected spin chain in the vicinity of a quantum critical point

Danish Ali Hamza, Jan Chwedeńczuk

Published Jul 11, 2024• 5 citations
Publisher

Abstract

Bell correlations are among the most exotic of the phenomena through which quantum mechanics manifests itself. Their presence signals that the system may violate the postulates of local realism---conjectures once thought to be to be a non-negotiable property of the physical world. The importance of Bell correlations from this fundamental point of view is even further clarified by their applications, ranging from quantum cryptography to quantum metrology and quantum computing. It is therefore of growing interest to characterize the ``Bell content'' of complex, scalable many-body systems. Here, we perform a detailed analysis of the character and strength of many-body Bell correlations in interacting multiqubit systems with particle exchange symmetry. Such a configuration can be described by an effective Schr\"odinger-like equation, which allows precise analytical predictions. We show that in the vicinity of the quantum critical point, these correlations quickly become so strong that only a fraction of the qubits remain uncorrelated. We also identify the threshold temperature above which thermal fluctuations destroy Bell correlations. We hope that the approach presented here, due to its universality, will be useful for upcoming research on genuinely nonclassical Bell-correlated complex systems.

DOI
10.1103/physreva.110.012210

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Bell correlations are among the most exotic of the phenomena through which quantum mechanics manifests itself. Their presence signals that the system may violate the postulates of local realism---conjectures once thought to be to be a non-negotiable property of the physical world. The importance of Bell correlations from this fundamental point of view is even further clarified by their applications, ranging from quantum cryptography to quantum metrology and quantum computing. It is therefore of growing interest to characterize the ``Bell content'' of complex, scalable many-body systems. Here, we perform a detailed analysis of the character and strength of many-body Bell correlations in interacting multiqubit systems with particle exchange symmetry. Such a configuration can be described by an effective Schr"odinger-like equation, which allows precise analytical predictions. We show that in the vicinity of the quantum critical point, these correlations quickly become so strong that only a fraction of the qubits remain uncorrelated. We also identify the threshold temperature above which thermal fluctuations destroy Bell correlations. We hope that the approach presented here, due to its universality, will be useful for upcoming research on genuinely nonclassical Bell-correlated complex systems.
Bell correlations are among the most exotic phenomena through which quantum mechanics manifests itself. Their presence signals that the system can violate the postulates of local realism, once believed to be the nonnegotiable property of the physical world. The importance of Bell correlations from this fundamental point of view is even straightened by their applications -- ranging from quantum cryptography through quantum metrology to quantum computing. Hence it is of growing interest to characterize the ``Bell content'' of complex, scalable many-body systems. Here we perform the detailed analysis of the character and strength of many-body Bell correlations in interacting multi-qubit systems with particle-exchange symmetry. Such configuration can be mapped onto an effective Schrödinger-like equation, which allows for precise analytical predictions. We show that in the vicinity of the quantum critical point, these correlations quickly become so strong that only a fraction of qubits remains uncorrelated. We also identify the threshold temperature, which, once overpassed, empowers thermal fluctuations that destroy Bell correlations in the system. We hope that the approach presented here, due to its universality, could be useful for the upcoming research on genuinely nonclassical Bell-correlated complex systems.
Non-classical correlations arise in various quantum mechanical systems. Characterization and quantification of these correlations is an important and active branch of research in the field of quantum information theory. Investigation of non-classical correlations in condensed matter systems gives important insights about the characteristics of these systems. In particular, systems possessing a quantum critical point in their phase diagrams have attracted much attention due to the peculiar behavior of correlations near these points. In this thesis, we have investigated two distinct quantum spin models from the perspective of correlations and, we have discussed the correlation content of an important subclass of bipartite states. We start by an analytical calculation of the quantum discord for a system composed of spin-j and spin-1/2 subsystems possessing rotational symmetry. We have compared our results with the quantum discord of states having similar symmetries and seen that in rotationally invariant states the amount of quantum discord is much higher. Moreover, using the well known entanglement properties of these states, we have compared their quantum discord with entanglement and seen that quantum discord is higher than the entanglement. Next, we have investigated the thermal quantum correlations and entanglement in spin-1 Bose-Hubbard model with two and three particles. We have demonstrated that the energy level crossings in the ground state of the system are signalled by both the behavior of thermal quantum correlations and entanglement. Finally, we have investigated various thermal quantum and total correlations in the anisotropic XY spin-chain with transverse magnetic field. We have shown that the ability of the considered measures to estimate the critical points of this system at finite temperature strongly depends on the anisotropy parameter of the Hamiltonian. Furthermore, we have studied the effect of temperature on long-range correlations of the XY chain.
When a collection of distant observers share an entangled quantum state, the statistical correlations among their measurements may violate a many-body Bell inequality, demonstrating a nonlocal behavior. Focusing on the Ising model in a transverse field with power-law (1/r^{α}) ferromagnetic interactions, we show that a permutationally invariant Bell inequality based on two-body correlations is violated in the vicinity of the quantum-critical point. This observation, obtained via analytical spin-wave calculations and numerical density-matrix renormalization group computations, is traced back to the squeezing of collective-spin fluctuations generated by quantum-critical correlations. We observe a maximal violation for infinite-range interactions (α=0), namely, when interactions and correlations are themselves permutationally invariant.
The interplay of quantum and thermal fluctuations in the vicinity of a quantum critical point characterizes the physics of strongly correlated systems. Here we investigate this interplay from a quantum information perspective presenting the universal phase diagram of the quantum Fisher information at a quantum phase transition. Different regions in the diagram are identified by characteristic scaling laws of the quantum Fisher information with respect to temperature. This feature has immediate consequences on the thermal robustness of quantum coherence and multipartite entanglement. We support the theoretical predictions with the analysis of paradigmatic spin systems showing symmetry-breaking quantum phase transitions and free-fermion models characterized by topological phases. In particular we show that topological systems are characterized by the survival of large multipartite entanglement, reaching the Heisenberg limit at finite temperature.
We find non-localities, violation of local realism, in the many-body ground states of spin-1 XXZ chain with on-site anisotropy. In order to identify the non-localities in higher spin systems, we exploit the generalized version of multipartite Bell-type inequalities which characterize symmetric entangled states under the most general settings via combination of high-order correlations. For a given set of unbiased measurements, we obtain a sharp violation of the multipartite Bell-type inequality at the vicinity of the quantum criticality, a type of the first-order, in the regime of large exchanges and strong on-site anisotropies. It signifies that impossibility of local realistic picture is manifested when a system is subjected to quantum phase transition between weekly entangled states via GHZ-like state. Our results provide the first extendible picture on the relationship between the impossibility of local realistic model and many-body quantum phases in higher-spin system as the observable identifies measurable quantities to detect the non-locality on a particular many-body quantum state.
Since John Bell formulated his paramount inequality for a pair of spin-1/2 particles, quantum mechanics has been confronted with the postulates of local realism with various equivalent configurations. Current technology, with its advanced manipulation and detection methods, allows to extend the Bell tests to more complex structures. The aim of this work is to analyze a set of Bell inequalities suitable for a possibly broad family of many-body systems with the focus on bosonic qubits. We develop a method that allows for a step-by-step study of the many-body Bell correlations, for instance among atoms forming a two-mode Bose-Einstein condensate or between photons obtained from the parametric-down conversion. The presented approach is valid both for cases of fixed and non-fixed number of particles, hence it allows for a thorough analysis of quantum correlations in a variety of many-body systems.