Quantum Information Processing with Low-Dimensional Systems.pdf

a r X i v : q u a n t - p h / 0 5 1 2 0 3 4 v 1 5 D e c 2 0 0 5 Quantum Information Processing with Low-Dimensional Systems Alexander Yu. Vlasov February 1, 2008 Abstract A ‘register’ in quantum information processing — is composition of k quantum systems, ‘qudits’. The dimensions of Hilbert spaces for one qudit and whole quantum register are d and dk respectively, but we should have possibility to prepare arbitrary entangled state of these k systems. Preparation and arbitrary transformations of states are possible with universal set of quantum gates and for any d may be suggested such gates acting only on single systems and neighbouring pairs. Here are revisited methods of construction of Hamiltonians for such universal set of gates and as a concrete new example is considered case with qutrits. Quantum tomography is also revisited briefly. 1 Introduction Discrete quantum variables — are basic resource in quantum computing. A qubit is described by two-dimensional Hilbert space and systems with higher dimensions are also widely used [1]. Quantum mechanics with continuous variables may be more understand- ing due to a correspondence principle. For example, after change of classical momentum q and coordinate p to quantum operators q?, p? in simple Hamil- tonians we almost directly may produce correct quantum description. On the other hand, it is impossible to introduce the p?, q? operators for system with finite-dimensional Hilbert space. Even if for large dimensions d ? 2 the continuous case could be used as an approximate model of a discrete system, it does not seem possible for low dimensions. 1 In 1928 Weyl suggested a method of quantization, appropriate both for finite and infinite-dimensional case [2]. The basic idea — is to use instead of operators of coordinate q? and momentum p? they exponents with pure imaginary multipliers and instead of Heisenberg commutation relations to write Weyl system U? = eiαp?, V? = eiβq?, U? V? = eiαβV? U? . (1) An analogue of