Mixed Quantum States: Definition, Uses & Homework

[Tac-Tics]

I have a book on quantum computation that explains the concept of a mixed quantum state. The definition is pretty plain, you just have a boring probability distribution over a set of quantum states.

What I would like to know is why we need mixed states. How are they represented physically in nature. What kind of leverage do they give us over pure states in describing an experiment? What kinds of homework problems would you expect to solve using them ;-)

 

[vanesch]

I have a book on quantum computation that explains the concept of a mixed quantum state. The definition is pretty plain, you just have a boring probability distribution over a set of quantum states.

What I would like to know is why we need mixed states. How are they represented physically in nature. What kind of leverage do they give us over pure states in describing an experiment? What kinds of homework problems would you expect to solve using them ;-)

Mixed states are standard statistical ensembles of "pure" quantum states, and they appear whenever we don't have complete information of the quantum state, pretty much in the same way as in classical physics. For instance, if you have a beam of "unpolarized" light, then you represent this as a mixed state of "up" and "down" spins for the photons. This is entirely different from a superposition of "up" and "down", which would result in just another polarization state, and which would show "interference" effects.

Now, there can be a funny interplay between the statistical properties of mixed states, and the statistical properties of the measurement results of a pure quantum state.

In a way, you can see a mixed state as "a superposition without interference effects". In fact, quantum mechanics manifests itself as the possibility to come up with different results than that of a mixed state, in a true superposition (a pure state).
Decoherence is the phenomenon of turning superpositions in mixed states.

 

[Tac-Tics]

In a way, you can see a mixed state as "a superposition without interference effects". In fact, quantum mechanics manifests itself as the possibility to come up with different results than that of a mixed state, in a true superposition (a pure state).
Decoherence is the phenomenon of turning superpositions in mixed states.

How would you go about preparing a particle in a mixed state?

 

[vanesch]

How would you go about preparing a particle in a mixed state?

The same way you prepare a dice to be in a mixed state

Seriously, a mixed state is an ensemble description. In fact, one of the peculiar things about the interplay between mixed state statistics and quantum statistics is that considering particles in a "mixed state" is indistinguishable from considering them in a randomly drawn pure state if that random drawing gives a statistically equivalent description as the mixed state. Worse, there are *different* ensembles of *different* pure states which are all observationally indistinguishable from the "mixed state". What describes a mixed state, or all of these ensembles, is the density matrix rho.

Simple example:
the mixed state "unpolarized electron".

You can see it as 50% |x+> and 50% |x-> (an ensemble of pure states)
or you can see it as 50% |z+> and 50% |z-> (another ensemble of pure states)
or you can see it as 25% |x+>, 25%|x->, 25%|z+> and 25% |z-> (yet another ensemble of pure states)
etc...

they are all observationally indistinguishable. They are all described by one and the same density matrix rho:
1/2 0
0 1/2