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How to numerically solve a master equation for a time-dependent 3-qubit density matrix in Mathematic

Posted 3 days ago

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?

POSTED BY: Reza rho
4 Replies

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}]
POSTED BY: David Trimas

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}]
POSTED BY: Gianluca Gorni

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.

POSTED BY: Gianluca Gorni
Posted 3 days ago

i set n = 8 () for 3 qubits, still errors and no result.

POSTED BY: Reza rho
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