Exercise 1: Density Matrices
Consider a quantum system formed by N subsystems (spins, atoms, particles etc..) each described by its wave function ψi ∈ HD where HD is a D-dimensional Hilbert space.
- (a) How do you write the total wave function of the system Ψ(ψ1, . . . ψN )? Write a Fortran code to 1) describe such a system (N-body non interacting, separable pure state) and
2) a general N-body pure wave function Ψ ∈ HDN .
Comment on their efficiency. - (b) Given N=2, write the density matrix of a pure state Ψ, ρ = |Ψ⟩⟨Ψ|.
- (c) Given a generic density matrix in HD2 compute the reduce density matrix of either the left or the
right system, e.g. ρ1 = Tr2 ρ.
- (d) Test the functions described before (and all others needed) on two-spin one-half (qubits) with different states.
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