I’m trying to simulate the time evolution of a 3-qubit density matrix under a master equation with intrinsic decoherence (à la Milburn). The evolution is given by:
$
\frac{d\rho(t)}{dt} = -i[H(t), \rho(t)] - \frac{\gamma}{2} [H(t), [H(t), \rho(t)]]
$
where \rho(t) is an 8*8 desity matrix.
my code gives alot of errors and doesnt give any result. here is my code. can somebody help me to fix it?
Your definition
\[Rho] = Array[r, {n, n}][t]
does not make any sense this way. The dimension n must be a number. The final [t] does not accomplish anything with this syntax.
i set n = 8 () for 3 qubits, still errors and no result.
You must check every step to see if it does what you mean. You may try this, which allows for derivatives of the entries with respect to t:
\[Rho] = Table[r[i, j][t], {i, 8}, {j, 8}]
NDSolve supports matrix-valued dependent variables. So after removing the extra semicolon in the definition of H[t] we can do:
(*Commutator function*)
Comm[A_, B_] := A . B - B . A;
(*Define the master equation:dρ/dt=-i[H,ρ]-Γ/2[H,[H,ρ]]*)
eq := -I Comm[H[t], ρ[t]] - (Γ/2) Comm[H[t],
Comm[H[t], ρ[t]]] == ρ'[t]
(*Initial conditions from ρ(0)*)
initConds = ρ[0] == ρinitial;
Tmax = 10;
ρsol = NDSolveValue[{eq, initConds}, ρ, {t, 0, Tmax}]