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AIP conference proceedings

Bases for Spin Systems and Qudits

M. Kibler, Mădălin O. Bunoiu, Iosif Malaescu

Published Jan 1, 2009• 0 citations
Publisher

Abstract

There is a growing interest these days for the field of quantum information and quantum computation (for which classical bits are replaced by qubits in dimension 2 and qudits in dimension d). This field is at the crossing of mathematics, informatics and quantum physics. In this work, bases of relevance for spin systems with cyclic symmetry as well as for quantum information and quantum computation are discussed from the theory of angular momentum and group‐theoretical methods. This approach is connected to the use of generalized Pauli matrices (in dimension d) arising from a polar decomposition of the group SU2. Examples are given for d = 2, 3 and 4.

DOI
10.1063/1.3153451

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There is a growing interest these days for the field of quantum information and quantum computation (for which classical bits are replaced by qubits in dimension 2 and qudits in dimension d). This field is at the crossing of mathematics, informatics and quantum physics. In this work, bases of relevance for spin systems with cyclic symmetry as well as for quantum information and quantum computation are discussed from the theory of angular momentum and group‐theoretical methods. This approach is connected to the use of generalized Pauli matrices (in dimension d) arising from a polar decomposition of the group SU2. Examples are given for d = 2, 3 and 4.
Symmetry-adapted bases in quantum chemistry and bases adapted to quantum information share a common characteristics: both of them are constructed from subspaces of the representation space of the group SO(3) or its double group (i.e., spinor group) SU(2). We exploit this fact for generating spin bases of relevance for quantum systems with cyclic symmetry and equally well for quantum information and quantum computation. Our approach is based on the use of generalized Pauli matrices arising from a polar decomposition of SU(2). This approach leads to a complete solution for the construction of mutually unbiased bases in the case where the dimension d of the considered Hilbert subspace is a prime number. We also give the starting point for studying the case where d is the power of a prime number. A connection of this work to the unitary group U(d) and the Pauli group is briefly underlined.
The aim of the present paper is twofold. First, to give the main ideas behind quantum computing and quantum information, a field based on quantum-mechanical phenomena. Therefore, a short review is devoted to (i) quantum bits or qubits (and more generally qudits), the analogues of the usual bits 0 and 1 of the classical information theory, and to (ii) two characteristics of quantum mechanics, namely, linearity, which manifests itself through the superposition of qubits and the action of unitary operators on qubits, and entanglement of certain multi-qubit states, a resource that is specific to quantum mechanics. A, second, focus is on some mathematical problems related to the so-called mutually unbiased bases used in quantum computing and quantum information processing. In this direction, the construction of mutually unbiased bases is presented via two distinct approaches: one based on the group SU(2) and the other on Galois fields and Galois rings.
The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of SU (2). A single formula for the bases is obtained from a polar decomposition of SU (2) and is analyzed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.
Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for quantum information and quantum computation are constructed from angular momentum theory and su(2) Lie algebraic methods. We report on a formula for deriving in one step the (1 + p)p qupits (i.e., qudits with d = p a prime integer) of a complete set of 1 + p mutually unbiased bases in C p . Repeated application of the formula can be used for generating mutually unbiased bases in C d with d = p e (e ≥ 2) a power of a prime integer. A connection between mutually unbiased bases and the unitary group SU(d) is briefly discussed in the case d = p e .
Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for quantum information and quantum computation are constructed from angular momentum theory and su(2) Lie algebraic methods. We report on a formula for deriving in one step the (1+p)p qupits (i.e., qudits with d = p a prime integer) of a complete set of 1+p mutually unbiased bases in C^p. Repeated application of the formula can be used for generating mutually unbiased bases in C^d with d = p^e (e > or = 2) a power of a prime integer. A connection between mutually unbiased bases and the unitary group SU(d) is briefly discussed in the case d = p^e.
We show that higher-dimensional versions of qubits, or qudits, can be encoded into spin systems and into harmonic oscillators, yielding important advantages for quantum computation. Whereas qubit-based quantum computation is adequate for analyses of quantum vs classical computation, in practice qubits are often realized in higher-dimensional systems by truncating all but two levels, thereby reducing the size of the precious Hilbert space. We develop natural qudit gates for universal quantum computation, and exploit the entire accessible Hilbert space. Mathematically, we give representations of the generalized Pauli group for qudits in coupled spin systems and harmonic oscillators, and include analyses of the qubit and the infinite-dimensional limits.