Finding Weights for Pure States in a Mixed State

[genericusrnme]

Homework Statement

Let q be a mixed state which we mix from pure states.
What are the weights we must take for the pure states, respectively? Let us
start the solution with the two-state system.

The Attempt at a Solution

My problem is I can't decypher what the problem actually is, from my knowledge a mixed state is any combination of (assume there's a discrete collection of them) pure states summed with weights that add to 1 so at best we get a n-1 dimensional plane of solutions (for n pure states).
So I think I'm seeing this problem incorrectly, could someone push me in the right direction?

 

[Oxvillian]

I think you more or less answered the question, such as it is.

You should probably look up the "Bloch sphere" (or Bloch ball) - it's a neat way of visualizing the mixed states of a 2-state system.

 

[Oxvillian]

at best we get a n-1 dimensional plane of solutions (for n pure states).

The space of the mixed states is actually bigger than this - for example with a 2-state system we can build a density matrix using the 4 objects |0><0|, |0><1|, |1><0| and |1><1|, subject to the constraint that the trace has to be 1. So the mixed states of a qubit live in a 3-dimensional space.

(not sure if that's what you meant)