Group-theoretical approach to the construction of bases in 2 n -dimensional Hilbert space
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ELEMENTARY PARTICLES AND FIELDS Theory
Group-Theoretical Approach to the Construction of Bases in 2n -Dimensional Hilbert Space* A. Garcia, J. L. Romero, and A. B. Klimov** Departamento de Fisica, Universidad de Guadalajara, Mexico Received December 17, 2010
Abstract—We propose a systematic procedure to construct all the possible bases with definite factorization structure in 2n -dimensional Hilbert space and discuss an algorithm for the determination of basis separability. The results are applied for classification of bases for an n-qubit system. DOI: 10.1134/S1063778811060093
1. INTRODUCTION The entanglement quantification in multipartite quantum states [1], methods of quantum error corrections [2] and measurement-based quantum computation [3] are intimately related with the problem of characterization and classification of bases for nqubit systems. Especial interest represent bases entirely formed by the so-called stabilizer states [2]. In the 2n dimensional Hilbert space the stabilizer states can be obtained from the standard logical (computational) basis by applying only local Clifford transformations and CNOT gates and thus, they can be efficiently applied for the quantum tomography purposes [4– 7]. According to the general principles of the stabilizer formalism [2] such bases are constructed as eigenstates of n commuting monomials formed by products of n Pauli operators, which generate an Abelian group of order 2n . Nevertheless, the stabilizer formalism does not provide any systematic method for defining sets of such commuting monomials. An elegant way for bases generation in small dimensions is offered by the graph-state approach [8, 9]. Unfortunately, the graph states are extremely difficult to classify for large number of qubits [10]. From the general point of view the problem of bases generation is related to the classification of symplectic group elements. Nevertheless, since the order of the symplectic group Sp2n (Z2 ) rapidly grows with n, such classification becomes a challenging algebraic problem. In this paper we propose a systematic procedure to determine symplectic matrices which generate all ∗ **
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possible different sets of commuting monomials and thus, all possible bases for an n-qubit system. We will also discuss algorithms for the determination of the basis separability. 2. COMMUTATIVITY CONDITION AND SYMPLECTIC GROUP Following the stabilizer formalism, we codify elements of the Pauli group, P 1 , in terms of twodimensional vectors as follows. T T (1) I = 0 0 , σz = 1 0 , T T σx = 0 1 , σy = 1 1 , so that any element of P n = P 1 ⊗ P 1 ⊗ · · · ⊗ P 1 can be represented as an 2n × 1 vector Cab , (Cab )T = an an−1 · · · a1 b1 b2 · · · bn , where aj , bj ∈ Z2 , which represent a monomial Yab = Za Xb ∈ P n , Za = σza1 ⊗ · · · ⊗ σzan ,
(2) (3)
Xb = σxb1 ⊗ · · · ⊗ σxbn . The commutativity condition between two monomials Ya1 b1 and Ya2 b2 can be rewritten in an explicitly symplectic form [11]: 2 T 1 JCab1 = 0, (4
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