Is a pure state a kind of mixed states?

  • #1
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Main Question or Discussion Point

Hi.
1. Does a pure state belong to mixed states

[tex]\hat{\rho}=\sum_k p_k|\psi_k><\psi_k|[/tex] where ##p_k=1## for k=i and otherwise 0 ?
2. Does quantum jump by observation work for both mixed and pure states ?
Your teachings will be appreciated.
 

Answers and Replies

  • #2
atyy
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If we use density operators to represent states, both pure states and mixed states can be represented in the same formalism. In the usual terminology, pure states are not mixed states.

The quantum jump by observation works for both pure and mixed states.

https://arxiv.org/abs/1110.6815
The modern tools of quantum mechanics (A tutorial on quantum states, measurements, and operations)
Matteo G. A. Paris
See postulates II.4 and II.5 on p9

https://arxiv.org/abs/0706.3526
"No Information Without Disturbance": Quantum Limitations of Measurement
Paul Busch
See Eq 3 and 4
 
  • #3
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I will read them fully. Thanks a lot.
See postulates II.4 and II.5 on p9
I thought pure states always take place after observation of both pure and mixed states. II5 tells us that mixed states appear if we do not record observed values. It is very interesting that recording or memory matters physics.
 
  • #4
vanhees71
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You prepare a pure state, e.g., by performing a simultaneous von-Neumann-filter measurement of a complete set of observables, and indeed states are most conveniently described by statistical operators, which are of the form as you wrote. They are self-adjoint positive semi-definite operators with trace 1. The pure states are exactly the projection operators, where exactly one of the ##p_k## is 1 and thus all others 0.

One cannot overstress the importance of the fact that pure states are NOT represented by unit vectors in Hilbert space but by the corresponding projection operators or, equivalently, unit rays in Hilbert space. In other words overall phase factors are not physical, and this has a lot of important consequences. One of the most important is that you can have half-integer spin and fermions. Our entire existence as living beings rests on the existence of fermions!
 

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