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The Lie algebra structure and nonlinear controllability of spin systems
 Linear Algebra Appl
, 2002
"... In this paper, we provide a complete analysis of the Lie algebra structure of a system of n interacting spin 1 2 particles with different gyromagnetic ratios in an electromagnetic field. We relate the structure of this Lie algebra to the properties of a graph whose nodes represent the particles and ..."
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In this paper, we provide a complete analysis of the Lie algebra structure of a system of n interacting spin 1 2 particles with different gyromagnetic ratios in an electromagnetic field. We relate the structure of this Lie algebra to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active. We prove that for these systems all the controllability notions, including the possibility of driving the state or the evolution operator of the system, are equivalent. We also provide a necessary and sufficient condition for controllability in terms of the properties of the above described graph. We analyze low dimensional problems (number of particles less then or equal to three) with possibly equal gyromagnetic ratios. This provides an example of quantum mechanical systems where controllability of the state is verified while controllability of the evolution operator is not.
A constructive algorithm for the Cartan decomposition of SU(2 N)
, 2008
"... We present an explicit numerical method to obtain the CartanKhanejaGlaser decomposition of a general element G ∈ SU(2 N) in terms of its ‘Cartan ’ and ‘nonCartan ’ components. This effectively factors G in terms of group elements that belong in SU(2 n) with n < N, a procedure that an be iterat ..."
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We present an explicit numerical method to obtain the CartanKhanejaGlaser decomposition of a general element G ∈ SU(2 N) in terms of its ‘Cartan ’ and ‘nonCartan ’ components. This effectively factors G in terms of group elements that belong in SU(2 n) with n < N, a procedure that an be iterated down to n = 2. We show that every step reduces to solving the zeros of a matrix polynomial, obtained by truncation of the BakerCampbellHausdorff formula, numerically. All computational tasks involved are straightforward and the overall truncation errors are well under control. 1
Aparametrization of bipartite systems based on SU(4) Euler angles
, 2002
"... In this paper we give an explicit parametrization for all twoqubit density matrices. This is important for calculations involving entanglement and many other types of quantum information processing. To accomplish this we present a generalized Euler angle parametrization for SU(4) and all possible t ..."
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In this paper we give an explicit parametrization for all twoqubit density matrices. This is important for calculations involving entanglement and many other types of quantum information processing. To accomplish this we present a generalized Euler angle parametrization for SU(4) and all possible twoqubit density matrices. The important grouptheoretical properties of such a description are then manifest. We thus obtain the correct Haar (Hurwitz) measure and volume element for SU(4) which follows from this parametrization. In addition, we study the role of this parametrization in the Peres–Horodecki criteria for separability and its corresponding usefulness in calculating entangled twoqubit states as represented through the parametrization. PACS numbers: 03.67.−a, 02.20.−a, 03.65.Ud 1.
SU(4) Euler Angle Parameterization and Bipartite Density Matrices
, 2008
"... In quantum mechanics, sets of density matrices are important for numerous reasons. For example, their compact notation make them useful for describing decoherence and entanglement properties of multiparticle quantum systems. In particular, two twostate density matrices, otherwise known as two qubi ..."
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In quantum mechanics, sets of density matrices are important for numerous reasons. For example, their compact notation make them useful for describing decoherence and entanglement properties of multiparticle quantum systems. In particular, two twostate density matrices, otherwise known as two qubit density matrices, are important for their role in explaining quantum teleportation, dense coding, and computation theorems. The aim of this paper is to show an explicit parameterization for the Hilbert space of all two qubit density matrices. Such a parameterization would be extremely useful for numerical calculations concerning entanglement and other quantum information parameters. We would also like to know the properties of such parameterized two qubit density matrices; in particular, the representation of their convex sets, their subsets, and their set boundaries in terms of our parameterization. Here we present a generalized Euler angle parameterization for SU(4) and all possible two qubit density matrices as well as the corrected Haar Measures for SU(3) and SU(4) from this parameterization. The role of the parameterization in the PeresHorodecki criteria will also be introduced as well as its usefulness in calculating entangled two qubit states.
NTT Communication Science Laboratories, Nippon Telegraph and Telephone Corporation
, 2005
"... using KAK decompositions ..."
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NTT Communication Science Laboratories, NTT Corporation
, 2005
"... using KAK decompositions ..."
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Supported in Part by nsfdms 9803186
, 2000
"... Constructive techniques for controlling a coupled, heteronuclear spin system, via bounded amplitude sinusoidal pulses are presented. The technique prepares exactly any desired unitary generator in a rotating frame, through constant controls. Passage to the original coordinates provides a procedure t ..."
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Constructive techniques for controlling a coupled, heteronuclear spin system, via bounded amplitude sinusoidal pulses are presented. The technique prepares exactly any desired unitary generator in a rotating frame, through constant controls. Passage to the original coordinates provides a procedure to prepare arbitrary unitary generators, via bounded amplitude piecewise sinusoidal pulses whose frequency is one of the two Larmor frequencies and whose phase takes one of two values. The techniques are based on a certain Cartan decomposition of SU(4) available in the literature. A method for determining the parameters entering this Cartan decomposition, in terms of the entries of the target unitary The goal of this paper is to provide, constructively and exactly (i.e., without any approximations) a decomposition: e −iLIij = Π Q k=1 e(−iakI1zI2z−ibkIij) , i = 1, 2, j = x, y (1.1)
A Scheme of Cartan Decomposition for su(N)
, 2006
"... A scheme to perform the Cartan decomposition for the Lie algebra su(N) of arbitrary finite dimensions is introduced. The scheme 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 pa ..."
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A scheme to perform the Cartan decomposition for the Lie algebra su(N) of arbitrary finite dimensions is introduced. The scheme 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. 1
On the Quantum Circuit Complexity Equivalence
, 2007
"... Nielsen [3] recently asked the following question: ”What is the minimal size quantum circuit required to exactly implement a specified nqubit unitary operation U, without the use of ancilla qubits? ” Nielsen was able to prove that a lower bound on the minimal size circuit is provided by the length ..."
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Nielsen [3] recently asked the following question: ”What is the minimal size quantum circuit required to exactly implement a specified nqubit unitary operation U, without the use of ancilla qubits? ” Nielsen was able to prove that a lower bound on the minimal size circuit is provided by the length of the geodesic between the identity I and U, where the length is defined by a suitable Finsler metric on SU(2n). We prove that the minimum circuit size that simulates U is in linear relation with the geodesic length and simulation parameters, for the given Finsler structure F. As a corollary we prove the highest lower bound of O ( n4 p d2Fp (I, U)LFp(I, Ũ))and the lowest upper bound of Ω(n4d3 (I, U)), for the Fp standard simulation technique. Therefore, our results show that by standard simulation one can not expect a better then n2 times improvement in the upper bound over the result from Nielsen, Dowling, Gu and Doherty [4]. Moreover, our equivalence result can be applied to the arbitrary path on the manifold including the one that is generated adiabatically. 1