On uniqueness of density matrix description as mixed state

[stevendaryl]

If you have a density matrix $\rho$, there is a basis $|\psi_j\rangle$ such that
$\rho$ is diagonal in that basis. What are the conditions on $\rho$ such that the basis that diagonalizes it is unique?

It's easy enough to work out the answer in the simplest case, of a two-dimensional basis: Then $\rho$ can be represented as a 2x2 matrix. It will have two eigenvalues, $p$ and $q$. If $p = q$, then $\rho$ is diagonal in every basis. If $p \neq q$, then there is a unique basis (up to permutations of elements and overall phase factor) that diagonalizes $\rho$.

Is there some result that is similar for bases of arbitrary dimension? What's the condition on $\rho$ such that there is a unique way to diagonalize it?

 

[A. Neumaier]

The basis is never unique as one can always change the length or the phase of the basis vectors. The necessary and sufficient condition for this being the only freedom is that the matrix has no multiple eigenvalues.

 

[zonde]

The basis is never unique as one can always change the length or the phase of the basis vectors.

Aren't entries of density matrix real valued? Then there is no phase.
And what do you mean by changing length of basis vectors? If intensity in one output of PBS is twice the intensity of other output then it is so. You can not change that by mathematical manipulations.

 

[A. Neumaier]

Aren't entries of density matrix real valued?

No. Any positive semidefinite Hermitian matrix of trace 1 qualifies as a density matrix.

And what do you mean by changing length of basis vectors?

Changing the length of a basis vector preserves linear independence, hence we get another basis. This is a mathematical fact independent of physics. (Maybe stevendaryl intended to have an orthonormal basis - where lengths are fixed at 1 - but he didn't say so.)

 

[zonde]

No. Any positive semidefinite Hermitian matrix of trace 1 qualifies as a density matrix.

Okay. So let me restate my question:
Aren't entries of diagonal density matrix real valued?

This is a mathematical fact independent of physics.

If a mathematical fact disagrees with empirical fact then this particular mathematical fact does not describe particular physical situation and is irrelevant.

Changing the length of a basis vector preserves linear independence, hence we get another basis.

Does it preserve condition that the trace is fixed at 1?

 

[A. Neumaier]

Aren't entries of diagonal density matrix real valued?

Yes, but they do not contain the phase information.

Does it preserve condition that the trace is fixed at 1?

The trace is a basis-independent concept.