How do I express p1 |v1> + p2 |v2> for a mixture, not a pure state?

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Discussion Overview

The discussion revolves around how to express a mixture of quantum states, specifically in the context of representing a mixture rather than a pure state. The focus includes notational conventions and mathematical representations relevant to quantum mechanics.

Discussion Character

  • Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions how to express a mixture using the notation p1 |v1> + p2 |v2>.
  • Another participant suggests that if the mixture is a classical ensemble, the expression is not valid, but if it arises from tracing over unobserved degrees of freedom, it can be represented as a pure state in a larger space.
  • A further inquiry is made about the correct notation for expressing such a mixture.
  • It is proposed that mixtures can be represented as Hermitian operators or matrices, constructed from state vectors through a convex sum of their outer products.

Areas of Agreement / Disagreement

Participants express differing views on the representation of mixtures, with some asserting that classical ensembles cannot be expressed in the same way as quantum mixtures, while others provide alternative methods for notation and representation. The discussion remains unresolved regarding the best approach to express mixtures.

Contextual Notes

Limitations include the dependence on definitions of classical versus quantum mixtures and the mathematical steps involved in constructing the Hermitian operators or matrices from state vectors.

naima
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Is there a way to write p1 |v1> + p2 |v2> if it is not a pure state but a mixture?
thanks.
 
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If it is a true classical ensemble then no. However, if it's something that you got from tracing over unobserved degrees of freedom then yes. The representation is then the original pure state on the larger space.

Cheers,

Jazz
 
My question is about notations.
How to write correctly such a mixture?
 
naima said:
My question is about notations.
How to write correctly such a mixture?

Mixtures are hermitian operators or matrices. You can construct them from state vectors by taking a convex sum of their outer products. So for example

[itex] \rho = a^* a \left|\psi\rangle \langle\psi\right| + b^* b \left|\phi\rangle \langle\phi\right|[/itex]
 
Ok
thank you
I did not realize that i could write it in this way.
 

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