Determine if System State is Pure or Mixed?

[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?

 

[jedishrfu]

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.

 

[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.

 

[naima]

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

 

[bhobba]

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

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.

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?

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

 

[StevieTNZ]

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.

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

 

[bhobba]

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

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

 

[atyy]

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

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).

 

[StevieTNZ]

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

"Sneaking a Look at God's Cards"?

 

[bhobba]

"Sneaking a Look at God's Cards"?

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

 

[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?

 

[jedishrfu]

Its a book reference:

https://www.amazon.com/dp/069113037X/?tag=pfamazon01-20

 

[StevieTNZ]

Specifically, there is a thought experiment on pages 373-376.

EDIT: Similar discussions can be found in David Albert's 'Quantum Mechanics and Experience' - https://www.amazon.com/dp/0674741137/?tag=pfamazon01-20