I am working with an infinite-dimensional density matrix in Mathematica, where the basis states are:
∣000n⟩, ∣100n⟩, ∣010n⟩, ∣001n⟩, ∣110n⟩, ∣101n⟩, ∣000(n+1)⟩, ∣100(n+1)⟩
The density matrix is given by:
X = Sum[ (tanh(r)^(2n) / cosh(r)^2) [
|100n⟩ ⟨100n| +
|010n⟩ ⟨010n| +
|001n⟩ ⟨001n| +
|001n⟩ ⟨010n| +
|010n⟩ ⟨001n| +
|110n⟩ ⟨000n| +
|101n⟩ ⟨000n| +
|000n⟩ ⟨110n| +
|000n⟩ ⟨101n| +
(sqrt(n+1) / cosh(r)) [
|000n⟩ ⟨100(n+1)| +
|010n⟩ ⟨000(n+1)| +
|001n⟩ ⟨000(n+1)| +
|100(n+1)⟩ ⟨000n| +
|000(n+1)⟩ ⟨010n| +
|000(n+1)⟩ ⟨001n|
] +
(n+1 / cosh(r)^2) |000(n+1)⟩ ⟨000(n+1)|
]
,{n,0,Infinity}]
where n runs from 0 to infinity, meaning the full matrix is infinite-dimensional.
My Questions:
1- How can I define this density matrix in Mathematica?
2- Is there a more efficient way to represent it symbolically?
3-What is the best approach to compute its eigenvalues numerically?