A SU(2) recipe for mutually unbiased bases

A simple recipe for generating a complete set of mutually unbiased bases in dimension (2j+1)**e, with 2j + 1 prime and e positive integer, is developed from a single matrix acting on a space of constant angular momentum j and defined in terms of the irreducible characters of the cyclic group C(2j+1). As two pending results, this matrix is used in the derivation of a polar decomposition of SU(2) and of a FFZ algebra.Comment: v2: abstract enlarged, a corollary added, acknowledgments added, one reference added, presentation improved; v3: two misprints correcte

Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?

We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension $q$, containing a square, into its factors. Illustrative low dimensional examples are the quartit ($q=4$) and two-qubit ($q=2^2$) systems, the octit ($q=8$), qubit/quartit ($q=2\times 4$) and three-qubit ($q=2^3$) systems, and so on. In the single qudit case, e.g. $q=4,8,12,...$, one defines a bijection between the $\sigma (q)$ maximal commuting sets [with $\sigma[q)$ the sum of divisors of $q$] of Pauli observables and the maximal submodules of the modular ring $\mathbb{Z}_q^2$, that arrange into the projective line $P_1(\mathbb{Z}_q)$ and a independent set of size $\sigma (q)-\psi(q)$ [with $\psi(q)$ the Dedekind psi function]. In the multiple qudit case, e.g. $q=2^2, 2^3, 3^2,...$, the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if $q=2^2$) and GQ(3,3) (if $q=3^2$). More precisely, in dimension $p^n$ ($p$ a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the $2n$-dimensional vector space over the field $\mathbb{F}_p$. In this space, one makes use of the commutator to define a symplectic polar space $W_{2n-1}(p)$ of cardinality $\sigma(p^{2n-1})$, that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of $W_{2n-1}(p)$ are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function $\psi(p^{2n-1})$. For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, ponctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.Comment: 18 pages, version submiited to J. Phys. A: Math. Theo

Quantum Entanglement and Projective Ring Geometry

The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15 × 15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two. In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states. In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube. A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of n copies of the Galois field GF(2), with n = 2, 3 and 4

Bases for qudits from a nonstandard approach to SU(2)

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.Comment: From a talk presented at the 13th International Conference on Symmetry Methods in Physics (Dubna, Russia, 6-9 July 2009) organized in memory of Prof. Yurii Fedorovich Smirnov by the Bogoliubov Laboratory of Theoretical Physics of the JINR and the ICAS at Yerevan State University

Weak mutually unbiased bases

Quantum systems with variables in ${\mathbb Z}(d)$ are considered. The properties of lines in the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space of these systems, are studied. Weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases, is equal to $d^{-1/2}$ or alternatively to one of the $d_i^{-1/2},0$ (where $d_i$ is a divisor of $d$ apart from $d,1$). They are designed for the geometry of the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space, in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin. In the special case of prime $d$, there are no divisors of $d$ apart from $1,d$ and the weak mutually unbiased bases are mutually unbiased bases

Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces

The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this essay, the geometry of a Clifford group G is expressed as a BN-pair, i.e. a pair of subgroups B and N that generate G, is such that intersection H = B \cap N is normal in G, the group W = N/H is a Coxeter group and two extra axioms are satisfied by the double cosets acting on B. The BN-pair used in this decomposition relies on the swap and match gates already introduced for classically simulating quantum circuits (R Jozsa and A Miyake, Preprint arXiv:0804.4050 [quant-ph]). The two- and three-qubit steps are related to the configuration with 27 lines on a smooth cubic surface.Comment: 7 pages, version to appear in Journal of Physics A: Mathematical and Theoretical (fast track communications

A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two more references adde

Variations on a theme of Heisenberg, Pauli and Weyl

The parentage between Weyl pairs, generalized Pauli group and unitary group is investigated in detail. We start from an abstract definition of the Heisenberg-Weyl group on the field R and then switch to the discrete Heisenberg-Weyl group or generalized Pauli group on a finite ring Z_d. The main characteristics of the latter group, an abstract group of order d**3 noted P_d, are given (conjugacy classes and irreducible representation classes or equivalently Lie algebra of dimension d**3 associated with P_d). Leaving the abstract sector, a set of Weyl pairs in dimension d is derived from a polar decomposition of SU(2) closely connected to angular momentum theory. Then, a realization of the generalized Pauli group P_d and the construction of generalized Pauli matrices in dimension d are revisited in terms of Weyl pairs. Finally, the Lie algebra of the unitary group U(d) is obtained as a subalgebra of the Lie algebra associated with P_d. This leads to a development of the Lie algebra of U(d) in a basis consisting of d**2 generalized Pauli matrices. In the case where d is a power of a prime integer, the Lie algebra of SU(d) can be decomposed into d-1 Cartan subalgebras.Comment: Dedicated to the memory of Mosh\'e Flato on the occasion of the tenth anniversary of his deat

Partial ordering of weak mutually unbiased bases

YesA quantum system (n) with variables in Z(n), where n = Qpi (with pi prime numbers), is considered. The non-near-linear geometry G(n) of the phase space Z(n) × Z(n), is studied. The lines through the origin are factorized in terms of ‘prime factor lines’ in Z(pi)×Z(pi). Weak mutually unbiased bases (WMUB) which are products of the mutually unbiased bases in the ‘prime factor Hilbert spaces’ H(pi), are also considered. The factorization of both lines and WMUB is analogous to the factorization of integers in terms of prime numbers. The duality between lines and WMUB is discussed. It is shown that there is a partial order in the set of subgeometries of G(n), isomorphic to the partial order in the set of subsystems of (n)