# Find eigenvectors/eigenvalues for this density matrix [closed]

Consider the following density matrix defined on two coherent states

$$\rho = a\vert\alpha\rangle\langle\alpha\vert + b\vert\alpha\rangle\langle\beta\vert + + b^*\vert\beta\rangle\langle\alpha\vert + c\vert\beta\rangle\langle\beta\vert,$$

where $$\vert\alpha\rangle$$ is a coherent state with amplitude $$\alpha$$. Note that the coherent states are not orthogonal such that

$$\langle\beta\vert\alpha\rangle = \exp\left[-\frac{1}{2}(\vert\alpha\vert^2 + \vert\beta\vert^2 - 2\beta^*\alpha)\right].$$

The eigenvalues of this state can be used to determine different properties. However, to determine the eigenvalues of this state, an orthonormal basis must be determined. How can I determine this basis? The Gram matrix could be useful, but I'm not sure how to apply this.

• @CosmasZachos you may be able to help with this
– Sid
Commented Jul 7, 2021 at 18:16
• Hi Sid. Just as an FYI, you can only ping users if they have previously commented on the same post, so Cosmas was not notified by your comment. Further, if you comment on a user's post (either question or answer) then they are automatically notified, which is why I didn't need to ping you. Commented Jul 7, 2021 at 20:15
• @J.Murray, got you, thanks for the tip!
– Sid
Commented Jul 7, 2021 at 20:17

$$e_a |\alpha \rangle + e_b | \beta \rangle.$$
Then, allow the density operator to act on your eigenfunction and you'll get a new vector that's some new combination of the two coherent states. By requiring that the new eigenvector is proportional to the old one, you should be able to determine a relationship between $$e_a, e_b$$ and the constants of proportionality.