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Journal of Physics A Mathematical and Theoretical

Dipole–quadrupole decomposition of two coupled spin 1 systems

Yao-Min Di, Yan Wang, Hai‐Rui Wei

Published Jan 19, 2010• 10 citations
Publisher

Abstract

Using dipole and quadrupole operators as the orthogonal basis of su (3) algebra and successive Cartan decompositions, the decomposition of matrices for two coupled spin 1 systems is investigated so as to meet the requirements of some realistic quantum systems. Finally, this kind of decomposition for the ternary SWAP gate is given specifically. This method can be used to investigate the realization of two-qutrit logic gates.

DOI
10.1088/1751-8113/43/6/065303

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Using dipole and quadrupole operators as the orthogonal basis of su (3) algebra and successive Cartan decompositions, the decomposition of matrices for two coupled spin 1 systems is investigated so as to meet the requirements of some realistic quantum systems. Finally, this kind of decomposition for the ternary SWAP gate is given specifically. This method can be used to investigate the realization of two-qutrit logic gates.
We present three different matrix bases that can be used to decompose density matrices of d -dimensional quantum systems, so-called qudits: the generalized Gell–Mann matrix basis , the polarization operator basis and the Weyl operator basis . Such a decomposition can be identified with a vector—the Bloch vector, i.e. a generalization of the well-known qubit case—and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We present a new method to decompose density matrices via so-called standard matrices, consider the important case of an isotropic two-qudit state and decompose it according to each basis. In the case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin-1 measurements, which is appropriate for an experimental realization.
This paper addresses the question of how to implement a desired two-qubit gate U using a given tunable two-qubit entangling interaction . We present a general method which is based on the K 1 AK 2 decomposition of unitary matrices ∊ SU (4) to calculate the smallest number of two-qubit gates U int ( t ) (based on ) and single-qubit rotations, and the explicit sequence of these operations that are required to implement U . We illustrate our protocol by calculating the implementation of (1) the transformation from standard basis to Bell basis, (2) the CNOT-gate and (3) the quantum Fourier transform for two kinds of interaction—Heisenberg exchange interaction and quantum inductive coupling—and discuss the relevance of our results for solid-state qubits.
We investigate the realization of 2-qutrit logic gate in a bipartite 3-level system with qusi-Ising interaction. On the basis of Cartan decomposition of matrices,the unitary matrices of 2-qutrit are factorized into products of a series of realizable matrices.It is equivalent to exerting a certain control field on the system,and the control goal is usually gained by a sequence of control pulses.The general discussion on the realization of 2-qutrit logic gate is made first,and then the realization of the ternary SWAP gate and the ternry SWAP~(1/2)gate are discussed specifically,and the sequences of control pulses and drift processes implementing these gates are given.
The relationship between quantum state control and matrix decomposition is reviewed.The matrix decomposition newly applied in quantum information science,i.e.Cartan decomposition based on group theory and cosine-sine decomposition based on numerical linear algebra,is introduced firstly.Then the current research situation and the results in this area are reviewed.These results play important roles in quantum entanglement dynamics,the control of quantum states and optimization of quantum circuit Finally,the Cartan decomposition of a 2-qutrit logic gate is given.
In NMR quantum computing, spin states of spin-1/2 nuclei are called qubits. Quantum logic gates are represented by unitary matrices. As a universal gate, controlled-NOT (CNOT) is a two-qubit gate. For the IS (I = 1/2 and S = 1/2) spin system, two-qubit CNOT gate is represented by a 4 × 4 matrix. SWAP logic gate, which exchanges two quantum states, is constructed by CNOT gates. In this study, first, four-qubit CNOT gates are constructed for the IS (I = 3/2, S = 3/2) spin system. Then, by using these CNOT gates, a four-qubit SWAP logic gate is found. As an application and verification, an obtained SWAP logic gate is applied to the matrix representation of product operators for the IS (I = 3/2, S = 3/2) spin system. SWAP logic gate can also be presented by an NMR pulse sequence. By using the product operator theory, the pulse sequence of the SWAP logic gate is applied to product operators of the IS (I = 3/2, S = 3/2) spin system. The expected exchange results are obtained for both the matrix representation and the pulse sequence of SWAP logic gate.