## Learn to Use ITensor

main / formulas / mps_two_rdm C++v3 | C++v2

# Compute a two-site reduced density matrix from an MPS

Say we have an MPS representing a quantum wavefunction $\Psi$ , and wish to compute the reduced density matrix

$$\rho_{s_i s_j}^{s'_i s'_j} = \text{Tr}_{\{s \neq s_i,s_j\}}[|\Psi\rangle\langle\Psi|]$$

In diagrammatic form, what we want to compute is Then we can use the following sample code to compute this quantity for fixed i and j.

### Sample code:

//Given an MPS called "psi",
//and assuming j > i

//'gauge' the MPS to site i
//any 'position' between i and j, inclusive, would work here
psi.position(i);

//index linking i to i+1:
auto ir = commonIndex(psi.A(i),psi.A(i+1));

auto rho = psi.A(i)*dag(prime(psi.A(i),Site,ir));
for(int k = i+1; k < j; ++k)
{
rho *= psi.A(k);
rho *= dag(prime(psi.A(k),Link));
}
rho *= psi.A(j);
rho *= dag(prime(psi.A(j)));
for(int k = j+1; k <= psi.N(); ++k)
{
rho *= psi.A(k);
rho *= dag(prime(psi.A(k),Link));
} Back to Formulas Back to Main