Resolve inconsistencies in the XY Hamiltonian

[HGSilveri]

There are currently some inconsistencies regarding the XY Hamiltonian, both in the documentation and in the implementation itself.

What's wrong

1. Contradicting state vector definitions in the Conventions page: The conventions page defines the XY basis state vectors as

At the same time, it states that

• Combining these two statements, we must conclude that |0> is the higher energy state, but this is not true.
• Therefore, we must change one of these to lift the contradiction.

2. Incorrect driving Hamiltonian definition: Regardless of the statevector definitions, the incorrect assumption that |0>=|u> is the higher energy state is reflected in the construction of the driving Hamiltonian terms. This is incorrect and must be fixed.

3. Confusing definitions of the $\sigma^+$ and $\sigma^-$ operators: These operators appear (at least) in the toggled section here, here and here. Depending on the state vector definitions we fall on, the way these operators are defined is either incorrect or confusing. Since they can just be unambiguously replaced by projectors, we should just do that and get rid of them altogether.

How to fix it

Points 2. and 3. are straightforward, so let's focus on point 1.

Option A: Modify the XY basis state vectors

• We could just say that $\ket{0}=(0,1)^T$ and $\ket{1}=(1,0)^T$, but this goes against a very well established convention so it's a bad idea.
• The same argument holds for renaming $\ket{0} -> \ket{\uparrow}$, since there is also a strong expectation that $\ket{\uparrow}$ is the +1 eigenstate of $\sigma_{z}$, so there is a strong expectation that $\ket{\uparrow}=(1,0)^T$
• The best way to do this might be to rename the basis states to something without any prior expectation on the state vector. For example, we could say that $\ket{s}=(0,1)^T$ and $\ket{p}=(1,0)^T$.

Expected impact

• Widespread adjustment of the documentation to use the new nomenclature
• Modification of the state vectors used in simulation: The initial state vector (|00000...>) and the state vectors coming from simulations will have the order of their entries changed

Option B: Make an exception in the Pulser state vector convention for the XY Hamiltonian

• We keep the $\ket{0}$ and $\ket{1}$ state vector definitions intact and instead drop

Expected impact

• Breaking of an established and uniform convention:
  • Defining the $\ket{a}$ and $\ket{b}$ based on their relative energies will no longer be valid.
  • The definition of the driving Hamiltonian will now depend on the basis
  • Will require special treatment of the XY Hamiltonian everywhere.
• Widespread adjustment of the documentation to reflect the new convention