Classical spin Hamiltonians are a powerful tool to model complex systems, characterized by a local structure given by the local Hamiltonians. One of the best understood local structures is the grammar of formal languages, which are central in computer science and linguistics, and have a natural complexity measure given by the Chomsky hierarchy. If we see classical spin Hamiltonians as languages, what grammar do the local Hamiltonians correspond to? Here, we cast classical spin Hamiltonians as formal languages, and classify them in the Chomsky hierarchy. We prove that the language of (effectively) zero-dimensional spin Hamiltonians is regular, one-dimensional spin Hamiltonians is deterministic context-free, and higher-dimensional and all-to-all spin Hamiltonians is context-sensitive. This provides a new complexity measure for classical spin Hamiltonians, which captures the hardness of recognizing spin configurations and their energies. We compare it with the computational complexity of the ground state energy problem, and find a different easy-to-hard threshold for the Ising model. We also investigate the dependence on the language of the spin Hamiltonian. Finally, we define the language of the time evolution of a spin Hamiltonian and classify it in the Chomsky hierarchy. Our work suggests that universal spin models are weaker than universal Turing machines.Classical spin Hamiltonians are a powerful tool to model complex systems, characterized by a local structure given by the local Hamiltonians. One of the best understood local structures is the grammar of formal languages, which are central in computer science and linguistics, and have a natural complexity measure given by the Chomsky hierarchy. If we see classical spin Hamiltonians as languages, what grammar do the local Hamiltonians correspond to? Here, we cast classical spin Hamiltonians as formal languages, and classify them in the Chomsky hierarchy. We prove that the language of (effectively) zero-dimensional spin Hamiltonians is regular, one-dimensional spin Hamiltonians is deterministic context-free, and higher-dimensional and all-to-all spin Hamiltonians is context-sensitive. This provides a new complexity measure for classical spin Hamiltonians, which captures the hardness of recognizing spin configurations and their energies. We compare it with the computational complexity of the ground state energy problem, and find a different easy-to-hard threshold for the Ising model. We also investigate the dependence on the language of the spin Hamiltonian. Finally, we define the language of the time evolution of a spin Hamiltonian and classify it in the Chomsky hierarchy. Our work suggests that universal spin models are weaker than universal Turing machines.

arXiv:2006.03529Proceedings of the Royal Society A Mathematical Physical and Engineering SciencesJun 1, 20236 citations

About Spin

WORLD SCIENTIFIC eBooksFeb 1, 20220 citations

Tensor Representation of Spin States

O. Giraud, Daniel Braun, Dorian Baguette +2 more

We propose a generalization of the Bloch sphere representation for arbitrary spin states. It provides a compact and elegant representation of spin density matrices in terms of tensors that share the most important properties of Bloch vectors. Our representation, based on covariant matrices introduced by Weinberg in the context of quantum field theory, allows for a simple parametrization of coherent spin states, and a straightforward transformation of density matrices under local unitary and partial tracing operations. It enables us to provide a criterion for anticoherence, relevant in a broader context such as quantum polarization of light.We propose a generalization of the Bloch sphere representation for arbitrary spin states. It provides a compact and elegant representation of spin density matrices in terms of tensors that share the most important properties of Bloch vectors. Our representation, based on covariant matrices introduced by Weinberg in the context of quantum field theory, allows for a simple parametrization of coherent spin states, and a straightforward transformation of density matrices under local unitary and partial tracing operations. It enables us to provide a criterion for anticoherence, relevant in a broader context such as quantum polarization of light.

arXiv:1409.1106Physical Review LettersFeb 25, 201557 citations

Relational Objectivity in the Presence of Finite Quantum Resources

Luis C. Barbado, Časlav Brukner

Abstract The no-go theorems of Bell and Kochen and Specker could be interpreted as implying that the notions of system and experimental context are fundamentally inseparable. In this interpretation, statements such as ‘spin is “up” along direction $x$ ’ are relational statements about the configurations of macroscopic devices which are mediated by the spin and not about any intrinsic properties of the spin. The operational meaning of these statements is provided by the practically infinite resources of macroscopic devices that serve to define the notion of a direction in three-dimensional space. This is the subject of ‘textbook quantum mechanics’: the description of quantum systems in relation to an experimental context. Therefore, the subject that textbook quantum mechanics treats as a formally closed system is not a physically isolated system. The subject rather describes the physical relation that a microscopic system holds with some macroscopic environment. Can one go beyond that? Relational quantum mechanics endeavours to provide a relational description between any quantum systems without the necessity of involving macroscopic devices. However, by applying ‘textbook quantum mechanics’ in such situations, it implicitly assumes infinite resources, even for simple quantum systems such as spins, which have no capacity to define an experimental context. This leads to conceptual difficulties. As an alternative, we analyse Penrose’s spin network proposal as a potential formalisation of quantum theory that goes beyond the textbook framework: a description in presence of finite resources, which is inherently relational and inseparable in the system-context entity.Abstract The no-go theorems of Bell and Kochen and Specker could be interpreted as implying that the notions of system and experimental context are fundamentally inseparable. In this interpretation, statements such as ‘spin is “up” along direction $x$’ are relational statements about the configurations of macroscopic devices which are mediated by the spin and not about any intrinsic properties of the spin. The operational meaning of these statements is provided by the practically infinite resources of macroscopic devices that serve to define the notion of a direction in three-dimensional space. This is the subject of ‘textbook quantum mechanics’: the description of quantum systems in relation to an experimental context. Therefore, the subject that textbook quantum mechanics treats as a formally closed system is not a physically isolated system. The subject rather describes the physical relation that a microscopic system holds with some macroscopic environment. Can one go beyond that? Relational quantum mechanics endeavours to provide a relational description between any quantum systems without the necessity of involving macroscopic devices. However, by applying ‘textbook quantum mechanics’ in such situations, it implicitly assumes infinite resources, even for simple quantum systems such as spins, which have no capacity to define an experimental context. This leads to conceptual difficulties. As an alternative, we analyse Penrose’s spin network proposal as a potential formalisation of quantum theory that goes beyond the textbook framework: a description in presence of finite resources, which is inherently relational and inseparable in the system-context entity.

arXiv:2407.04391Apr 15, 20260 citations

Introduction to spin

Giampiero V. M. Esposito, Giuseppe Marmo, George Sudarshan

Cambridge University Press eBooksMar 11, 20040 citations

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’ are relational statements about the configurations of macroscopic devices which are mediated by the spin and not about any intrinsic properties of the spin. The operational meaning of these statements is provided by the practically infinite resources of macroscopic devices that serve to define the notion of a direction in three-dimensional space. This is the subject of ‘textbook quantum mechanics’: the description of quantum systems in relation to an experimental context. Therefore, the subject that textbook quantum mechanics treats as a formally closed system is not a physically isolated system. The subject rather describes the physical relation that a microscopic system holds with some macroscopic environment. Can one go beyond that? Relational quantum mechanics endeavours to provide a relational description between any quantum systems without the necessity of involving macroscopic devices. However, by applying ‘textbook quantum mechanics’ in such situations, it implicitly assumes infinite resources, even for simple quantum systems such as spins, which have no capacity to define an experimental context. This leads to conceptual difficulties. As an alternative, we analyse Penrose’s spin network proposal as a potential formalisation of quantum theory that goes beyond the textbook framework: a description in presence of finite resources, which is inherently relational and inseparable in the system-context entity.