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IEEE Conference on Decision and Control (CDC)

Decompositions of unitary evolutions and dynamics of quantum systems

Domenico D’Alessandro, Raffaele Romano

Published Dec 1, 2006• 0 citations
Publisher

Abstract

Decompositions of the Lie group of unitary matrices are very useful tools in the control and analysis of quantum dynamics. In this paper, we survey recent results on decompositions, concerning their significance in terms of symmetries, entanglement, and their applications to control and dynamics. Several decompositions can be obtained by recursively applying the Cartan classification of the symmetric spaces of the classical Lie groups. The emphasis is on a novel recursive procedure to decompose the unitary evolution of bipartite quantum systems of arbitrary dimensions, in simpler factors. This procedure makes transparent the contributions of the entangling and non entangling transformations

DOI
10.1109/cdc.2006.377387

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Decompositions of the Lie group of unitary matrices are very useful tools in the control and analysis of quantum dynamics. In this paper, we survey recent results on decompositions, concerning their significance in terms of symmetries, entanglement, and their applications to control and dynamics. Several decompositions can be obtained by recursively applying the Cartan classification of the symmetric spaces of the classical Lie groups. The emphasis is on a novel recursive procedure to decompose the unitary evolution of bipartite quantum systems of arbitrary dimensions, in simpler factors. This procedure makes transparent the contributions of the entangling and non entangling transformations
Decompositions of the unitary group U ( n ) are useful tools in quantum information theory as they allow one to decompose unitary evolutions into local evolutions and evolutions causing entanglement. Several recursive decompositions have been proposed in the literature to express unitary operators as products of simple operators with properties relevant in entanglement dynamics. In this paper, using the concept of grading of a Lie algebra, we cast these decompositions in a unifying scheme and show how new recursive decompositions can be obtained. In particular, we propose a new recursive decomposition of the unitary operator on N qubits, and give a numerical example.
We investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition there corresponds a quantum symmetry which is the identity when applied twice. As an application, we describe a new and general method to obtain Cartan decompositions of the unitary group of evolutions of multipartite systems from Cartan decompositions on the single subsystems. The resulting decomposition, which we call of the odd-even type, contains, as a special case, the concurrence canonical decomposition (CCD) presented in the context of entanglement theory. The CCD is therefore extended from the case of a multipartite system of n qubits to the case where the component subsystems have arbitrary dimension.
The purpose of controlling a quantum system is to steer the system from an initial state to a target state. Such a control problem can be changed into the problem of generating the desired unitary evolution operators by using an appropriate decomposition. By means of the desired state's exponential factor product theorem in the right-invariant system, the desired unitary evolution operators of a quantum system can be obtained. This paper investigates the functions and features of the Magnus decomposition, the Wei-Norman decomposition, the Lie Group decomposition and the Cartan decomposition contrastively and respectively. Properties and realization performance of those decompositions are analyzed, too. Each type of decomposition applied to different situations is also given.
This chapter discusses in detail the case of two-level systems and the associated Lie group. It gives an example of a general decomposition of SU(n), namely the decomposition into planar rotation, which is quite simple and can be obtained by straightforward computations. This decomposition gives a direct way to parametrize the special unitary group. The chapter presents the basics of Cartan decompositions of semisimple Lie groups. It gives some examples of applications of Lie group decompositions and introduces some notions of Lie group and Lie algebra theory. The chapter discusses Cartan decompositions of Lie groups. Cartan decompositions appear to be ubiquitous in quantum control and dynamics and in quantum information theory.
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.