- #1

Haorong Wu

- 417

- 90

- TL;DR Summary
- I have a matrix with negative eigenvalues. This makes it inproper for a density matrix. Hence I am trying to correct it.

Hi, there. I am working with a model, in which the dimension of the Hilbert space is infinite. But Since only several states are directly coupled to the initial state and the coupling strength are weak, then I only consider a subspace spanned by these states.

The calculation shows that the trace of the matrix remains 1 which is good for a density matrix. However, the final matrix has a negative eigenvalue, -0.0661. Other eigenvalues are 0.5496, 0.4671, 0.0493. This means that the matrix will not be semi-positive. Hence it can not be a density matrix.

I guess this is due to the fact that I only consider a small fraction of the full-space.

Then in order to construct a density matrix, I try the following procedure. First, I eigendecompose the matrix. Second, I throw away the eigenvector with negative eigenvalue since the negative eigenvalue is small. Then, I reconstruct a matrix by using the remaining eigenvectors and eigenvalues. Finally, I normalize the matrix by imposing the condition ##\rm{tr} (\rho)=1##.

This makes it to be a density matrix. However, I am not sure whether this procedure reasonable.

The calculation shows that the trace of the matrix remains 1 which is good for a density matrix. However, the final matrix has a negative eigenvalue, -0.0661. Other eigenvalues are 0.5496, 0.4671, 0.0493. This means that the matrix will not be semi-positive. Hence it can not be a density matrix.

I guess this is due to the fact that I only consider a small fraction of the full-space.

Then in order to construct a density matrix, I try the following procedure. First, I eigendecompose the matrix. Second, I throw away the eigenvector with negative eigenvalue since the negative eigenvalue is small. Then, I reconstruct a matrix by using the remaining eigenvectors and eigenvalues. Finally, I normalize the matrix by imposing the condition ##\rm{tr} (\rho)=1##.

This makes it to be a density matrix. However, I am not sure whether this procedure reasonable.