Determine if System State is Pure or Mixed?

  • Context: Graduate 
  • Thread starter Thread starter normvcr
  • Start date Start date
  • Tags Tags
    mixed pure
Click For Summary

Discussion Overview

The discussion centers on determining whether a given quantum state is pure or mixed. Participants explore various methods and concepts related to this determination, including the use of density matrices, entropy, and experimental approaches. The conversation touches on theoretical implications and practical challenges without reaching a consensus on a non-brute force method applicable to arbitrary unknown states.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that measuring the density matrix and calculating the Von Neumann entropy can indicate whether a state is pure.
  • Others propose that for specific systems, such as a glass of water with ink, visual cues can help determine the state, although this may rely on knowledge of the state space.
  • A participant argues that a brute force approach involving many similarly prepared systems can, in principle, determine the state with high confidence, although this method may not be appealing to all.
  • There is mention of "quantum state tomography" as a brute force method to ascertain the state, with some uncertainty about the existence of a non-brute force approach for arbitrary unknown states.
  • Participants reference a thought experiment from the book "Sneaking a Look at God's Cards" as a means to differentiate between mixed and pure states.
  • Some express curiosity about references and methods to enhance their understanding of quantum mechanics related to the topic.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a non-brute force method for determining whether a state is pure or mixed. There are multiple competing views on the effectiveness and applicability of various approaches discussed.

Contextual Notes

The discussion highlights limitations in the proposed methods, including dependence on specific system knowledge and the challenges of applying brute force techniques universally. There is also an acknowledgment of the need for further exploration of references and theoretical frameworks.

normvcr
Messages
28
Reaction score
2
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?
 
Physics news on Phys.org
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.
 
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.
 
if you measure the density matrix of your state the Von Neumann entropy tells you if it is pure.
 
normvcr said:
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.

normvcr said:
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
 
Last edited:
bhobba said:
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.
 
StevieTNZ said:
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
 
StevieTNZ said:
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).
 
bhobba said:
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"?
 
  • #10
StevieTNZ said:
"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
 
  • #11
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?
 
  • #13
Last edited:
  • Like
Likes   Reactions: atyy

Similar threads

Undergrad Qbit pure vs mixed state space
  • · Replies 35 ·
2
Replies
35
Views
3K
Graduate Mixed versus Pure Quantum States for the Singlet
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad Mixed states from entangled state
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Can a Pure State be Interpreted as a Mixed State for a Single Particle?
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Decoherence Pure States Into Mixed States
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Exact symmetry, quantum states, and symmetric dynamics
  • · Replies 8 ·
Replies
8
Views
1K
Undergrad Mixed states vs pure states - physical POV
  • · Replies 65 ·
3
Replies
65
Views
10K
Undergrad Actual example of a mixed state not part of entangled state
  • · Replies 26 ·
Replies
26
Views
3K
Undergrad Difference between superposition and mixed state
  • · Replies 15 ·
Replies
15
Views
4K
Graduate Is it possible to distinguish between pure and mixed states?
  • · Replies 30 ·
2
Replies
30
Views
6K