quantum mechanics - Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix?

1. 
   

   Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix?

   
   
   


2. 
   

   Does a density matrix always have a a basis of eigenvectors?  

   
   

Answer

In general, the density matrix of a given system can always be written in the form

$$\rho = \sum_i p_i |\phi_i\rangle\langle\phi_i|, \tag 1$$ representing among other things a probabilistic mixture in which the pure state $|\phi_i\rangle$ is prepared with probability $p_i$, but this decomposition is generally not unique. The clearest example of this is the maximally mixed state on, say, a two-level system with orthonormal basis $\{|0⟩,|1⟩\}$, $$\rho = \frac12\bigg[|0⟩⟨0|+|1⟩⟨1|\bigg],$$ which has exactly the same form on any orthonormal basis for the space.

Generally speaking, though, the eigenvalues and eigenvectors of a given density matrix $\rho$ provide a set of states and weights such that $\rho$ can be written as in $(1)$ - but with the added guarantee that the $|\phi_i⟩$ are orthogonal.

This does not uniquely specify the states in question, because if any eigenvalue $p_i$ is degenerate then there will be a two-dimensional (or bigger) subspace within which any orthonormal basis is equally valid, but that kind of undefinedness is just an intrinsic part of the structure.