We introduce the first complete and approximatively universal diagrammatic language for quantum mechanics. We make the ZX-Calculus, a diagrammatic language introduced by Coecke and Duncan, complete for the so-called Clifford+T quantum mechanics by adding four new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open questions in ca...

This lecture takes a closer look at Hermitian matrices and at their eigenvalues. After a few generalities about Hermitian matrices, we prove a minimax and maximin characterization of their eigenvalues, known as Courant–Fischer theorem. We then derive some consequences of this characterization, such as Weyl theorem for the sum of two Hermitian matrices, an interlacing theorem for the sum of two ...

It is known that the Williamson construction for Hadamard matrices can be generalized to constructions using sums of tensor products. This paper describes a specific construction using real monomial representations of Clifford algebras, and its connection with graphs of amicability and anti-amicability. It is proven that this construction works for all such representations where the order of th...

We consider a straightforward extension of the 4-dimensional spacetime M4 to the space of extended events associated with strings/branes, corresponding to points, lines, areas, 3-volumes, and 4-volumes in M4. All those objects can be elegantly represented by the Clifford numbers X ≡ xγA ≡ x γa1...ar , r = 0, 1, 2, 3, 4. This leads to the concept of the so-called Clifford space C, a 16-dimension...

We consider the problem of realizing hyperbolicity cones as spectrahedra, i.e. as linear slices of cones of positive semidefinite matrices. The generalized Lax conjecture states that this is always possible. We use generalized Clifford algebras for a new approach to the problem. Our main result is that if −1 is not a sum of hermitian squares in the Clifford algebra of a hyperbolic polynomial, t...

We review a rough scheme of quantum mechanics using the Clifford algebra. Following the steps previously published in a paper by another author [31], we demonstrate that quantum interference arises in a Clifford algebraic formulation of quantum mechanics. In 1932 J. von Neumann showed that projection operators and, in particular, quantum density matrices can be interpreted as logical statements...

, 773 (2006); 314 Science et al. Clifford B. Saper, Body Temperature BIOMEDICINE: Enhanced: Life, the Universe, and www.sciencemag.org (this information is current as of February 7, 2007 ): The following resources related to this article are available online at http://www.sciencemag.org/cgi/content/full/314/5800/773 version of this article at: including high-resolution figures, can be found in ...