Notes on qubit phase space and discrete symplectic structures

Symplectic cohomologies on phase space

The phase space of a particle or a mechanical system contains an intrinsic symplectic structure, and hence, it is a symplectic manifold. Recently, new invariants for symplectic manifolds in terms of cohomologies of differential forms have been introduced by Tseng and Yau. Here, we discuss the physical motivation behind the new symplectic invariants and analyze these invariants for phase space, ...

Discrete Structures Lecture Notes

A Weyl Calculus on Symplectic Phase Space

We study the twisted Weyl symbol of metaplectic operators; this requires the definition of an index for symplectic paths related to the Conley–Zehnder index. We thereafter define a metaplectically covariant algebra of pseudo-differential operators acting on functions on symplectic space.

Discrete phase-space structures and Wigner functions for N qubits

We further elaborate on a phase-space picture for a system of N qubits and explore the structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves satisfying certain additional properties and different entanglement properties. We discuss the construction of discrete covariant Wigner functions for these bundles and provide several illuminating examples.

Notes on Symplectic Geometry

1. Week 1 1 1.1. The cotangent bundle 1 1.2. Geodesic flow as Hamiltonian flow 4 2. Week 2 7 2.1. Darboux’s theorem 7 3. Week 3 10 3.1. Submanifolds of symplectic manifolds 10 3.2. Contact manifolds 12 4. Week 4 15 4.1. Symplectic linear group and linear complex structures 15 4.2. Symplectic vector bundles 18 5. Week 5 21 5.1. Almost complex manifolds 21 5.2. Kähler manifolds 24 6. Week 6 26 6....