A special irreducible matrix representation of the real Clifford algebra C(3,1)

4x4 Dirac (gamma) matrices (irreducible matrix representations of the Clifford algebras C(3,1), C(1,3), C(4,0)) are an essential part of many calculations in quantum physics. Although the final physical results do not depend on the applied representation of the Dirac matrices (e.g. due to the invariance of traces of products of Dirac matrices), the appropriate choice of the representation used may facilitate the analysis. The present paper introduces a particularly symmetric real representation of 4x4 Dirac matrices (Majorana representation) which may prove useful in the future. As a byproduct, a compact formula for (transformed) Pauli matrices is found. The consideration is based on the role played by isoclinic 2-planes in the geometry of the real Clifford algebra C(3,0) which provide an invariant geometric frame for it. It can be generalized to larger Clifford algebras.Comment: 23 pages LaTeX, to appear in the J. Math. Phys. (v2: appendix B on Pauli matrices and references are added, minor other changes

Symmetric Informationally Complete Measurements of Arbitrary Rank

There has been much interest in so-called SIC-POVMs: rank 1 symmetric informationally complete positive operator valued measures. In this paper we discuss the larger class of POVMs which are symmetric and informationally complete but not necessarily rank 1. This class of POVMs is of some independent interest. In particular it includes a POVM which is closely related to the discrete Wigner function. However, it is interesting mainly because of the light it casts on the problem of constructing rank 1 symmetric informationally complete POVMs. In this connection we derive an extremal condition alternative to the one derived by Renes et al.Comment: Contribution to proceedings of International Conference on Quantum Optics, Minsk, 200

Tight informationally complete quantum measurements

We introduce a class of informationally complete positive-operator-valued measures which are, in analogy with a tight frame, "as close as possible" to orthonormal bases for the space of quantum states. These measures are distinguished by an exceptionally simple state-reconstruction formula which allows "painless" quantum state tomography. Complete sets of mutually unbiased bases and symmetric informationally complete positive-operator-valued measures are both members of this class, the latter being the unique minimal rank-one members. Recast as ensembles of pure quantum states, the rank-one members are in fact equivalent to weighted 2-designs in complex projective space. These measures are shown to be optimal for quantum cloning and linear quantum state tomography.Comment: 20 pages. Final versio

Symmetric Informationally Complete Quantum Measurements

We consider the existence in arbitrary finite dimensions d of a POVM comprised of d^2 rank-one operators all of whose operator inner products are equal. Such a set is called a ``symmetric, informationally complete'' POVM (SIC-POVM) and is equivalent to a set of d^2 equiangular lines in C^d. SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.Comment: 8 page

Optimizing quantum process tomography with unitary 2-designs

We show that weighted unitary 2-designs define optimal measurements on the system-ancilla output state for ancilla-assisted process tomography of unital quantum channels. Examples include complete sets of mutually unbiased unitary-operator bases. Each of these specifies a minimal series of optimal orthogonal measurements. General quantum channels are also considered.Comment: 28 page