Projection operators of complex Hilbert space

ITensor Julia Questions
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Hello everyone!

Here, I want to ask a question about projection operators to a complex Hilbert space constructed by basis of two nearest sites.

Suppose there is a three sites spin-1/2 chain. The wavefunction is a superposition of two states, which is \ket{\psi}=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow\downarrow}+\ket{\downarrow\uparrow\downarrow})

Thus, thus the first site and the second site construct a tripplet in the complex Hilbert space of \mathcal{H}=\mathcal{H}_1\otimes\mathcal{H}_2 , \ket{\psi}=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})\otimes\ket{\downarrow}.

One can detect such a triplet in the complex Hilbert space of 1-st site and 2-nd by projector operator \hat{P}=\frac{1}{2}(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})(\bra{\uparrow\downarrow}+\bra{\downarrow\uparrow})

Here is my code

using ITensors
using LinearAlgebra
L=3
sites=siteinds("S=1/2",L;conserve_qns=true)
states=["0","1","1"]
psi=MPS(sites,states)
states1=["1","0","1"]
psi1=MPS(sites,states1)
psi_mixed=normalize!(psi+psi1)

##projection operator
ij_updn=state("Up",sites[1])*state("Dn",sites[2])
ij_dnup=state("Dn",sites[1])*state("Up",sites[2])
plus_ij=1/sqrt(2)*(ij_updn+ij_dnup)
# plus_ij=(ij_updn+ij_dnup)
proij_plus=dag(plus_ij)*prime(plus_ij)
orthogonalize!(psi_mixed,1)
psi_12=psi_mixed[1]*psi_mixed[2]
psi_pr=prime(psi_mixed,sites)
psi_12_1=dag(psi_pr[1]*psi_pr[2])
@show scalar(psi_12_1*proij_plus*psi_12)
###0.9999999999999997


##reduced density matrix
psi_pr=prime(psi_mixed,sites)
dagpsi=prime(dag(psi_mixed),linkinds(psi_mixed))
rho_12=psi_pr[1]*dagpsi[1]
rho_12=rho_12*prime(psi_pr[2],linkinds(psi_pr,2))
rho_12=rho_12*dagpsi[2]
@show scalar(dag(proij_plus)*rho_12)
###0.9999999999999996

I have also checked it by reduced density matrix.
It is quite good to see a return value of 1.

My question is pretty trivial.

Is it a reasonable process that multiplying local tensors of site A and B or obtaining the reduced density matrix to project the wavefunction to the complex Hilbert space constructed by site A and B?

Is that a right way to construct a projection operator of complex Hilbert space?

Thanks!

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