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

In summary, mixtures are hermitian operators or matrices that can be constructed by taking a convex sum of outer products of state vectors. This notation can be used for mixtures that are not pure states, such as when tracing over unobserved degrees of freedom.
  • #1
naima
Gold Member
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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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  • #2
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
 
  • #3
My question is about notations.
How to write correctly such a mixture?
 
  • #4
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]
 
  • #5
Ok
thank you
I did not realize that i could write it in this way.
 

1. What is a pure state?

A pure state is a quantum state that describes a system with certainty. In other words, it is a state in which all the properties of the system are known.

2. How is a pure state different from a mixed state?

A pure state is a state with a single probability distribution, while a mixed state is a combination of multiple pure states with different probabilities. In other words, a mixed state represents a lack of knowledge about the exact state of the system.

3. Can a pure state be converted into a mixed state?

No, a pure state cannot be converted into a mixed state. However, a pure state can become a mixed state if it interacts with another system or if we measure only a part of the system.

4. How can we determine if a system is in a pure state or a mixed state?

We can determine if a system is in a pure state or a mixed state by performing measurements on the system. If the measurement results are always the same, the system is in a pure state. If the measurement results vary, the system is in a mixed state.

5. What are some real-world examples of pure states and mixtures?

A pure state can be represented by a single photon, which has a definite polarization. A mixture can be represented by a light beam that contains a mix of photons with different polarizations. In chemistry, a pure substance such as water can be represented as a pure state, while a solution of water and salt would be considered a mixture.

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