# 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,

I understand now your question.
I think that the two interpretations you proposed are essentially equivalent, just as in thermodynamics. Anyway, this quantum-statistical concept is essentially for use in statistical mechanics and in thermodynamics.

I don't think "ignorance" point of view is really subjective. I think it is always presented as subjective but it is not. This is exactly similar to the conceptual problem in thermodynamics, the second law. Theoretically, you can describe a very complex system by a pure state also. But most often it makes no meaning to ask for detailled predictions on a complex system. Within the precision of usual experiments a statistical description is excellent. And it is not only reated to designing and analysing experiences. It is clear that the interactions of macroscopic bodies are not determined by minute details of their internl states. In equilibrium thermodynamics, temperature and pressure and eventually a few other state variable are enough. And these state variables determine the statistical properties on the microscopic level.

Another situation where statistics arise is related to the measurement postulate. After interaction of a quantum system with a measurement device, the quantum state of the system is undefined. Then the state is represent as a quantum-statistical state, with a density operator. Again, this description is not really subjective. In particular the role of the "measurement device" is exagerated in the postulate. The evolution of the system + "the measurement device" can be described -theoretically- without recourse to a density operator. Practically, the details that would result from the description of the entangled sytem would be useless, usually. Practically, describing this process with a final density operator catches the main feature: subsequently everything happens as if ... the postulate. This leads more to phylosophy than to physics, unless we are able to develop this point of view in a more formal way.

Michel

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