# Pure or mixed?

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1. Nov 30, 2014

### normvcr

The short question: For the current state of a given system, is there a way to determine whether the state is pure or mixed?

Of course, one can take as many ensembles as you need, do as many experiments as you need, and determine where the state sits in the entire state space. However, I find this brute force approach unappealing, as it requires knowledge of the entire state space.

The refined question: Is there a way to determine whether the state is pure or mixed, without knowing the structure of the entire state space? i.e. Is there some intrinsic property of a state that will tell you whether it is pure or mixed?

2. Nov 30, 2014

### Staff: Mentor

Not to my knowledge for a general system However, if you were to pick a particular system like a glass of water and a drop of ink. You could determine the pure statem if the water has no color and a mixed state of it does.

3. Nov 30, 2014

### normvcr

Interesting. I would say that this example uses our knowledge of the entire state space -- classical line segment with the two endpoints being the pure states.

4. Nov 30, 2014

### naima

if you measure the density matrix of your state the Von Neumann entropy tells you if it is pure.

5. Nov 30, 2014

### bhobba

Yes there is.

For a large number of similarly prepared systems you can, in principle, determine its state, pure or mixed, to a vanishingly small probability by observations on each of those systems. You may have to divide them into large sub groups and do different observations on each - but we are speaking about matters of principle.

Sure - by simple trial and error on a large number of similarly prepared systems - remember - conceptually it can be as large as you like.

Thanks
Bill

Last edited: Nov 30, 2014
6. Nov 30, 2014

### StevieTNZ

Do you have a reference as to how it would done? I'm curious now..., and want to add to my QM knowledge.

7. Nov 30, 2014

### bhobba

The book you often quote gives a way.

But its easy - by pure brute force.

Simply guess a state and observe a large number of systems for that state to see if you get all the same yes outcome. Then guess another and repeat. There is a vanishingly small probability it will give a false positive.

Thanks
Bill

8. Nov 30, 2014

### atyy

It's generally called something like "quantum state tomography", and is a brute force approach like bhobba mentioned, eg. http://arxiv.org/abs/1303.7154. I believe the OP was asking whether a non-brute force approach for an arbitrary unknown state exists (I think it doesn't, but am not sure).

9. Nov 30, 2014

### StevieTNZ

"Sneaking a Look at God's Cards"?

10. Nov 30, 2014

### bhobba

That's correct.

That's what the thought experiment you like does - it tells, in that case, the difference between a mixed and pure state.

Thanks
Bill

11. Dec 1, 2014

### normvcr

I appreciate the discussion and responses. Is there something in "Sneaking a Look at God's Cards" that I need to know about to know what you guys are talking about?

12. Dec 1, 2014

### Staff: Mentor

Last edited by a moderator: May 7, 2017
13. Dec 1, 2014

### StevieTNZ

Last edited: Dec 1, 2014