38 research outputs found
A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n)
Employing the fact that the geometry of the n-qubit (n≥2) Pauli group is embodied in the structure of the symplectic polar space W(2n−1,2) and using properties of the Lagrangian Grassmannian LGr(n,2n) defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the n-qubit Pauli group and a certain subset of elements of the 2ⁿ⁻¹-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases n=3 (also addressed recently by Lévay, Planat and Saniga [J. High Energy Phys. 2013 (2013), no. 9, 037, 35 pages]) and n=4 are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space PG(2n−1,2) of the 2ⁿ⁻¹-qubit Pauli group in terms of G-orbits, where G≡SL(2,2)×SL(2,2)×⋯×SL(2,2)⋊Sn, to decompose π(LGr(n,2n)) into non-equivalent orbits. This leads to a partition of LGr(n,2n) into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups
Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions
We present a systematic approach to studying the geometric aspects of Vinberg
theta-representations. The main idea is to use the Borel-Weil construction for
representations of reductive groups as sections of homogeneous bundles on
homogeneous spaces, and then to study degeneracy loci of these vector bundles.
Our main technical tool is to use free resolutions as an "enhanced" version of
degeneracy loci formulas. We illustrate our approach on several examples and
show how they are connected to moduli spaces of Abelian varieties. To make the
article accessible to both algebraists and geometers, we also include
background material on free resolutions and representation theory.Comment: 41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the
occasion of his 65th birthday; v2: fixed some typos and added reference
Geometric descriptions of entangled states by auxiliaries varieties
The aim of the paper is to propose geometric descriptions of multipartite
entangled states using algebraic geometry. In the context of this paper,
geometric means each stratum of the Hilbert space, corresponding to an
entangled state, is an open subset of an algebraic variety built by classical
geometric constructions (tangent lines, secant lines) from the set of separable
states. In this setting we describe well-known classifications of multipartite
entanglement such as , for , quantum systems and
a new example with the quantum system.
Our description completes the approach of Miyake and makes stronger
connections with recent work of algebraic geometers. Moreover for the quantum
systems detailed in this paper we propose an algorithm, based on the classical
theory of invariants, to decide to which subvariety of the Hilbert space a
given state belongs.Comment: 32 pages, 15 Tables, 5 Figures. References and remarks adde