Application Of Machine Learning To Nonlocality In Quantum Many-body Systems

Nonlocality is the strongest possible nonclassical correlations,and it represents the most profound departure of quantum from classical physics.However,nonlocality in quantum many-body systems has been rarely explored.The reasons are that the complete characterization of classical correlations for a generic many-body systems is a NPhard problem,and most known Bell inequalities involve high-order(the product of local observables of multiple observers)correlations,so that it is hard for the measurement experimentally.A central problem in quantum many-body systems is how to relate the global property to the underlying correlations among particles in the subsystems.This dissertation just uses local near-neighbor information in one-dimensional(1D)many-body systems,constructs the translation invariance(TI)Bell inequalities that only involve oneand two-body correlations,and thus obtains three types of Hamiltonians of 1D TI quantum many-body systems with near-neighbor(nearest-neighbor and next-to-nearest-neighbor)interactions.Subsequently,the problem of detecting nonlocality in quantum many-body systems is transformed into the problem of finding the minimum ground state energy density of the TI Hamiltonian.Furthermore,based on the NPA(Navascués-Pironio-Acín)hierarchy method,machine learning and the tools of Matrix Product States(MPS),this dissertation develops methods for detecting nonlocality in 1D quantum many-body systems,and then systematically analyzes the three types of quantum many-body systems.Since the assumption space-like separation is difficult to overcome in quantum manybody systems,this dissertation explores the contextuality of three types of 1D TI quantum many-body systems.Specifically,the results and innovations of this dissertation are as follows:(1)Treating each site in many-body systems as one observer in Bell scenarios,consider the following three Bell scenarios: nearest-neighbor with two dichotomic observables per site? nearest-and next-to-nearest neighbor with two dichotomic observables per site and nearest-neighbor with three dichotomic observables per site.In each Bell scenario,this dissertation utilizes the equivalence relationship between the global TI distribution and the local translation invariance(LTI)marginal distribution in 1D infinite system to fully characterize the set of LTI near-neighbor correlations admitting a local hidden variable(LHV),and the set turns out to be a convex polytope.Using the PANDA software,all facets of the this kind of polytope are obtained,and these facets are TI Bell inequalities,that is,contextuality witness.These Bell inequalities provide available models for Bell test experimentally.(2)The symmetries of TI Bell inequalities are studied by relabeling,and an equivalence classification of these Bell inequalities is obtained based on group theory.This reduces the complexity of the subsequent analysis of the three types of 1D TI quantum systems.Treating any measurement of each observer in the Bell scenario as a local observable in the quantum many-body system,and a TI Bell inequality is transformed into a 1D infinite TI Hamiltonian with fixed couplings and free local observables.Therefore,the problem of finding the quantum limit of the Bell functional in the TI Bell inequality is transformed into the problem of finding the minimum ground state energy density of TI Hamiltonian.If the ground state energy density violates the classical bound of the TI Bell inequality,then the ground state must be contextual.(3)To explore the contextuality of three types of 1D infinite TI quantum many-body systems derived from TI Bell inequalities,this dissertation develops three methods to find or approximate the quantum limit.One is that using linear programming to obtain the lower bound of the Bell functional quantum limit over the no-signaling set with LTI constraints.The second method is that using the NPA method and LTI constraints to compute the lower bounding of the quantum limit over the supra-quantum set with LTI constraints via semi-definite programming.Finally,based on MPS tools,as well as gradient descent and Momentum methods in machine learning,this dissertation designs a gradient descent algorithm based on MPS when the measurements are projective measurements,and a Momentum-based projected gradient descent algorithm when the local observables are POVMs.These two algorithms are used to optimize the ground state energy density of TI Hamiltonians,thus obtaining the upper bound of the minimum of ground state energy density of TI Hamiltonian.The above methods are able to find the maximum quantum violation of TI Bell inequality.(4)Using the methods proposed in(3),the contextuality of three types of 1D infinite TI quantum many-body systems is systematically studied.For the simplest scenario of quantum many-body systems: nearest-neighbor with two dichotomic observables per site,this dissertation finds that they do not exhibit any contextuality.Thus,violation of contextuality witnesses are only possible when either increase the interaction distance to include next-to-nearest neighbor terms or have three dichotomic observables per site.(5)For the quantum systems with nearest-and next-to-nearest neighbor and two dichotomic observables per site,this dissertation identifies several ground-state energy densities of low-dimensional Hamiltonians that reach the ultimate quantum limits,which means that the maximum violation of Bell inequality has been found.(6)For the quantum systems with nearest-neighbor and three dichotomic observables per site,this dissertation finds that the violation of contextuality witness requires that the dimension of the local observables at least be 3,which excludes the usual Heisenberg-type models where local observables are Pauli matrices.All above results on the contextuality of 1D infinite TI quantum many-body systems are also adapted to 1D large(finite,not TI)quantum many-body systems.The quantum many-body systems considered in this dissertation are TI,which enables the contextuality of the global quantum many-body systems are certified by the contextuality of quantum states of local subsystems.Since contextual Hamiltonians are guaranteed to have entangled ground states,which makes the methods in the dissertation suitable for benchmarking future quantum simulators.In this dissertation,some Hamiltonians with nextnearest neighbor interactions whose ground state energy density can reach the ultimate quantum limits are found.These results pave the way for self-testing ground states of quantum many-body systems in the thermodynamic limit? provide good candidate models for quantum simulation? provide ample motivation for future research into less common of quantum models with higher-dimensional local observables and longer interaction ranges,and potentially allow one to falsify quantum theory in the many-body systems and lowtemperature regime.