A restricted signature normal form for Hermitian matrices, quasi-spectral decompositions, and applications

In recent years, a number of results on the relationships between the inertias of Hermitian matrices and the inertias of their principal submatrices appeared in the literature. We study restricted congruence transformation of Hermitian matrices M which, at the same time, induce a congruence transformation of a given principal submatrix A of M. Such transformations lead to concept of the restricted signature normal form of M. In particular, by means of this normal form, we obtain short proofs of most of the known inertia theorems and also derive some new results of this type. For some applications, a special class of almost unitary restricted congruence transformations turns out to be useful. We show that, with such transformations, M can be reduced to a quasi-diagonal form which, in particular, displays the eigenvalues of A. Finally, applications of this quasi-spectral decomposition to generalize inverses and Hermitian matrix pencils are discussed

A Note on the correspondence between Qubit Quantum Operations and Special Relativity

We exploit a well-known isomorphism between complex hermitian $2\times 2$ matrices and $\mathbb{R}^4$, which yields a convenient real vector representation of qubit states. Because these do not need to be normalized we find that they map onto a Minkowskian future cone in $\mathbb{E}^{1,3}$, whose vertical cross-sections are nothing but Bloch spheres. Pure states are represented by light-like vectors, unitary operations correspond to special orthogonal transforms about the axis of the cone, positive operations correspond to pure Lorentz boosts. We formalize the equivalence between the generalized measurement formalism on qubit states and the Lorentz transformations of special relativity, or more precisely elements of the restricted Lorentz group together with future-directed null boosts. The note ends with a discussion of the equivalence and some of its possible consequences.Comment: 6 pages, revtex, v3: revised discussio

A restricted signature normal form for Hermitian matrices, quasi-spectral decompositions, and applications

AbstractIn recent years, a number of results on the relationships between the inertias of Hermitian matrices and the inertias of their principal submatrices have appeared in the literature. In this paper, we study restricted congruence transformations of Hermitian matrices M that, at the same time, induce a congruence transformation of a given principal submatrix A of M. Such transformations lead to the concept of the restricted signature normal form of M. In particular, by means of this normal form, we obtain new and shorter proofs for several known inertia theorems and also derive some new results of this type. For some applications, a special class of “almost” unitary restricted congruence transformations turns out to be useful. We show that, with such transformations, M can be reduced to a quasi-diagonal form, which, in particular, displays the eigenvalues of A. Moreover, this quasi-spectral decomposition is used to derive a generalized signature formula and to study Hermitian matrix pencils

Local U(2,2) Symmetry in Relativistic Quantum Mechanics

Local gauge freedom in relativistic quantum mechanics is derived from a measurement principle for space and time. For the Dirac equation, one obtains local U(2,2) gauge transformations acting on the spinor index of the wave functions. This local U(2,2) symmetry allows a unified description of electrodynamics and general relativity as a classical gauge theory.Comment: 18 pages, LaTeX, typo in second formula on page 6 corrected (published version

M 5-brane and superconformal (0,2) tensor multiplet in 6 dimensions

We present a gauge-fixed M 5-brane action: a 6-dimensional field theory of a self-interacting (0,2) tensor multiplet with 32 worldvolume supersymmetries. The quadratic part of this action is shown to be invariant under rigid OSp(6,2|4) superconformal symmetry, with 16 supersymmetries and 16 special supersymmetries. We explore a deep relation between the superconformal symmetry on the worldvolume of the brane and symmetry of the near horizon anti-de Sitter infinite throat geometry of the M 5-brane in space-time.Comment: 42 pages, latex, no figures; v2: some formulas corrected:v3: version to be published in Nuclear Physics B; v4:The name of the conformal algebra OSp(6,2|4) has been replaced by the correct name OSp(8*|4

A special simplex in the state space for entangled qudits

Focus is on two parties with Hilbert spaces of dimension d, i.e. "qudits". In the state space of these two possibly entangled qudits an analogue to the well known tetrahedron with the four qubit Bell states at the vertices is presented. The simplex analogue to this magic tetrahedron includes mixed states. Each of these states appears to each of the two parties as the maximally mixed state. Some studies on these states are performed, and special elements of this set are identified. A large number of them is included in the chosen simplex which fits exactly into conditions needed for teleportation and other applications. Its rich symmetry - related to that of a classical phase space - helps to study entanglement, to construct witnesses and perform partial transpositions. This simplex has been explored in details for d=3. In this paper the mathematical background and extensions to arbitrary dimensions are analysed.Comment: 24 pages, in connection with the Workshop 'Theory and Technology in Quantum Information, Communication, Computation and Cryptography' June 2006, Trieste; summary and outlook added, minor changes in notatio