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Physical Review A

Algebraic approch to master equations with superoperator generators of su(1,1) and su(2) Lie algebras

Huai-Xin Lu, Jie Yang, Yong-De Zhang, Zeng‐Bing Chen

Published Feb 24, 2003• 39 citations
Publisher

Abstract

In this paper, using the decomposition theorem of superoperators, we present a general method to solve master equations with superoperator generators of su(1,1) and su(2) Lie algebras. With the help of this method, the time evolution of the density operators for certain systems, which have SU(1,1) and SU(2) symmetry, can be easily obtained from any given initial state.

DOI
10.1103/physreva.67.024101

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In this paper, using the decomposition theorem of superoperators, we present a general method to solve master equations with superoperator generators of su(1,1) and su(2) Lie algebras. With the help of this method, the time evolution of the density operators for certain systems, which have SU(1,1) and SU(2) symmetry, can be easily obtained from any given initial state.
A scheme to perform the Cartan decomposition for the Lie algebra su(N) of arbitrary finite dimensions is introduced. The schme is based on two algebraic structures, the conjugate partition and the quotient algebra, that are easily generated by a Cartan subalgebra and generally exist in su(N). In particular, the Lie algebras su(2^p) and every su(2^{p-1} < N < 2^p) share the isomorphic structure of the quotient algebra. This structure enables an efficient algorithm for the recursive and exhaustive construction of Cartan decompositions. Further with the scheme, a unitary transformation in SU(N) can be recursively decomposed into a product of certain designated operators, e.g., local and nonlocal gates. Such a recursive decomposition of a transformation implies an evolution path on the manifold of the group.
Understanding the Lie algebraic structure of a physical problem often makes it easier to find its solution. In this paper, we focus on the Lie algebra of Gaussian-conserving superoperators. We construct a Lie algebra of nnn-mode states, go(n)\mathfrak{go}(n)go(n), composed of all superoperators conserving Gaussianity, and we find it isomorphic to R2n2+3n⊕Sgl(2n,R)\mathbb{R}^{2n^2+3n}\oplus_{\mathrm{S}}\mathfrak{gl}(2n,\mathbb{R})R2n2+3n⊕S​gl(2n,R). This allows us to solve the quadratic-order Redfield equation for any, even non-Gaussian, state. We find that the algebraic structure of Gaussian operations is the same as that of super-Poincaré algebra in three-dimensional spacetime, where the CPTP condition corresponds to the combination of causality and directionality of time flow. Additionally, we find that a bosonic density matrix satisfies both the Klein-Gordon and the Dirac equations. Finally, we expand the algebra of Gaussian superoperators even further by relaxing the CPTP condition. We find that it is isomorphic to a superconformal algebra, which represents the maximal symmetry of the field theory. This suggests a deeper connection between two seemingly unrelated fields, with the potential to transform problems from one domain into another where they may be more easily solved.