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Journal of Physics A Mathematical and General

G-invariant polynomial extensions of Lie algebras in quantum many-body physics

V. P. Karassiov

Published Jan 7, 1994• 58 citations
Publisher

Abstract

A new class of Lie-algebraic structures g d is revealed in some multi-particle processes of quantum physics having internal symmetry groups G inv . They are extensions of some Lie algebras h (via coset construction) by G-invariant h-tensors v which are polynomials in boson operators. Applications of these algebras are briefly discussed for solving spectral and evolution tasks with g d as their dynamic symmetry algebras.

DOI
10.1088/0305-4470/27/1/009

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A new class of Lie-algebraic structures g d is revealed in some multi-particle processes of quantum physics having internal symmetry groups G inv . They are extensions of some Lie algebras h (via coset construction) by G-invariant h-tensors v which are polynomials in boson operators. Applications of these algebras are briefly discussed for solving spectral and evolution tasks with g d as their dynamic symmetry algebras.
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A general algebraic approach incorporating both invariance groups and dynamical symmetry algebras is developed to reveal hidden coherent structures (closed complexes and configurations) in quantum manybody physics models due to symmetries of their Hamiltonians H . Its general ideas are manifested on some recent new examples: (1) G-invariant biphotons and a related SU (2)-invariant treatment of unpolarized light; (2) quasispin clusters in nonlinear models of quantum optics; and (3) construction of composite particles and (para)fields from G -invariant clusters due to internal symmetries.
We develop some calculation schemes to determine dynamics of a wide class of integrable quantum-optical models using their symmetry adapted reformulation in terms of polynomial Lie algebras supd(2)su_{pd}(2)supd​(2). These schemes, based on "diagonal" representations of model evolution operators (via diagonalizing Hamiltonians with the help of the supd(2)su_{pd}(2)supd​(2) defining relations), are implemented in the form adapted for numerical calculations. Their efficiency is demonstrated on the example of the second-harmonic-generation model.
Understanding the dynamical properties of quantum systems is an essential task in quantum computing, quantum control, and many-body physics. Tools such as representation theory and Lie theory provide crucial information on reachability and computational power. However, this information can be difficult to access exactly or compute efficiently for arbitrary generating sets. Here we focus on the setting of Pauli strings, which satisfy numerous exceptional properties that simplify the problem. We find deep connections between Pauli Lie algebras and certain subgroups of the Clifford group generated by transvections, through the symplectic properties of the Pauli strings. This allows us to give an invariant-based perspective on these objects and their reachability, in the language of Pauli orbits, symmetries, and invariant subspaces. The invariant-based approach provides efficient algorithms for identifying Lie algebras and orbits, as well as a simple framework for analyzing structured Pauli generating sets. We also show in an elementary way that Clifford subgroups generated by transvections provide 3-designs for the corresponding Pauli Lie groups. We illustrate the framework through structured examples from variational quantum algorithms, restricted quantum computation, many-body systems, and random circuits.
We study the algebra of complex polynomials which remain invariant under the action of the local Clifford group under conjugation. Within this algebra, we consider the linear spaces of homogeneous polynomials degree by degree and construct bases for these vector spaces for each degree, thereby obtaining a generating set of polynomial invariants. Our approach is based on the description of Clifford operators in terms of linear operations over GF(2). Such a study of polynomial invariants of the local Clifford group is mainly of importance in quantum coding theory, in particular in the classification of binary quantum codes. Some applications in entanglement theory and quantum computing are briefly discussed as well.
Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical tools (qqq-numbers, qqq-analysis, qqq-oscillators, qqq-algebras), the suq_qq​(2) rotator model and its extensions, the construction of deformed exactly soluble models (Interacting Boson Model, Moszkowski model), the use of deformed bosons in the description of pairing correlations, and the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underly the structure of superdeformed and hyperdeformed nuclei, are discussed in some detail. A brief description of similar applications to molecular structure and an outlook are also given.