#229 Add a transformation into the dual space

[Ulf Lorenz]

Update: * Dual space is bad. In particular, when you transform a linear operator: What comes out? I cannot represent an operator by a dual-space state and vice versa * Operating in the dual space requires to fulfill consistency conditions to ensure that you stay within the manifold of physically sensible states. * Needs sparse matrices to work at all, for which there is currently not even a ticket.

What and Why

While working on [#158], it turned out that it is pretty difficult to set up a prototypical Lindblad equation from a simple model (Fermi's golden rule in this case), except for simple toy systems.

A simpler approach, more in line with the Matlab code, would be to go to the dual space. Density operators become simple "wavefunctions" in the dual space, while Liouvillians become linear operators.
Instead of a Lindblad jump operator, we can then set up a quantum master equation by directly specifying the linear operator, which is much simpler to reason about, e.g. ,for a direct rate equation (pure relaxation without dephasing).

The drawback of the dual space is the high dimensionality of all operations: For a representation of size N, Applying a Liouvillian becomes a multiplication with a matrix of dimension N^2 x N^2, for a total of N^4 operations. In contrast, a Liouvillian in the worst case is a set of matrix multiplications between operators and density matrices that scale with N^3 each. This can be alleviated by small dimensions or possibly by sparse matrices for the operators in the dual space.

Todo

• Implement a transformation that can lift density operators from the normal space to the dual space
• It should also be possible to lift Liouvillians from the normal to the dual space
  If this can be easily done by the user, documentation may suffice, otherwise add utilities or member functions for that.
• Add an operator similar to GenericOperator, but which uses a sparse matrix in the background
• add a demo that shows something, maybe in conjunction with [#158]?