# The meaning of purity

1. Oct 16, 2006

### MaverickMenzies

I have been recently thinking about the nature of the purity of a mixed state. As far as I understand there are two equivalent physical interpretations of mixed states:

1) Ignorance Interpretation: Mixed states arise because of incomplete information concerning the preparation procedure of the state. This interpretation appears to be subjective since the "incomplete information" is to do with the observer and not the state itself.

2) Ensemble Interpretation: Mixed states can represent an ensemble of quantum systems perpared in different pure states.

However, mixed states appear in another context: entanglement. That is, by tracing out a subsystem from a pure entangled state of a composite system, one will yield a mixed state for the remaining subsystem. What is the interpretation of this mixed state? It seems to me that in this case, the Ignorance interpretation doesn't really apply since the loss of purity is objective i.e. the subsystem's cannot exist in pure states if the composite system is entangled.

In other words, does the purification theorem of mixed states require another physical interpretation?

2. Oct 16, 2006

### lalbatros

Dear All,
Dear MaverickMenzies,

I have a related question that troubles me since a long time.

Consider two basis states |a> and |b>.
I guess that the state |y> = norm * (|a> + |b>) is considered as "highly mixed".
While, for eps small, the state |y'> = norm' * (eps*|a> + |b>) is only "weakly" mixed.
Nice formulas may probably be written for a measure of mixing (entanglement) for such states.

My problem is:
Take these new basis states:
|A> = normA * (|a> + |b>) and |B> = normB * (|a> - |b>)
In this basis, the state |y> is pure state, obviously.

My question is then:
How is purity (conversly entanglement) defined.
Is there a preferred basis to define purity (or entanglement)
What's the real physics behind.
Purity and entanglement are certainly related, but I am more interrested in entanglement.

Michel

Last edited: Oct 16, 2006
3. Oct 17, 2006

### MaverickMenzies

Dear lalbatros,

This isn't what I mean when I talk about the purity of a quantum state. Any quantum state can be described by a density operator rho. A system is in a pure state if rho^2 = rho and can therefore be expressed as a vector. A mixed state (i.e. a state with purity less than one) is a state where rho^2 doesn't equal rho.

You seem to be talking about different superposition of pure states. This, however, is a different phenemenon.

4. Oct 17, 2006

### lalbatros

Dear MaverickMenzies,