# I Is information lost in wavefunction collapse?

#### Stephen Tashi

And after decoherence its in a mixed state so superposition isn't really applicable anyway.

That's basically whats going on - after decoherence each element of the mixed state is interpreted as a world.
In a non-MWI interpretation of QM, is being in a "mixed" state a meaningful property for a single particle or "system"? Or is "mixed state" only a property of a population of particles or systems? ( For example, in a non-QM setting, when we speak of "the probability that a person is over 6 ft tall", we have in mind picking individuals at random from a population and measuring their height once rather than picking an individual from a population and measuring his/her height at 100 randomly selected times during the day. )

#### stevendaryl

Staff Emeritus
In a non-MWI interpretation of QM, is being in a "mixed" state a meaningful property for a single particle or "system"? Or is "mixed state" only a property of a population of particles or systems? ( For example, in a non-QM setting, when we speak of "the probability that a person is over 6 ft tall", we have in mind picking individuals at random from a population and measuring their height once rather than picking an individual from a population and measuring his/her height at 100 randomly selected times during the day. )
Being in a mixed state is not (in my opinion) an objective fact about a system, but is a fact about our model of the system. You can describe an isolated system, such as a single hydrogen atom that is not interacting with anything else, as a pure state. But if a system interacts strongly with the rest of the universe, then you have really two options:
1. Go the MW route, and try to describe the entire universe using quantum mechanics.
2. Describe the system of interest as a mixed state.
I consider it a consequence of how we draw the boundary of what the system of interest is.

#### stevendaryl

Staff Emeritus
Being in a mixed state is not (in my opinion) an objective fact about a system, but is a fact about our model of the system. You can describe an isolated system, such as a single hydrogen atom that is not interacting with anything else, as a pure state. But if a system interacts strongly with the rest of the universe, then you have really two options:
1. Go the MW route, and try to describe the entire universe using quantum mechanics.
2. Describe the system of interest as a mixed state.
I consider it a consequence of how we draw the boundary of what the system of interest is.
It's sort of similar to the modeling choices in statistical mechanics. If a system is isolated, you can model it as having definite values of quantities such as pressure, volume, total energy, total number of particles. If the system is in contact with an environment, then those quantities are not constants, so you have to talk about average values for them. You can enlarge the system of interest to include the environment, as well, and then volume and energy and number of particles becomes constants again.

#### Stephen Tashi

It's sort of similar to the modeling choices in statistical mechanics. If a system is isolated, you can model it as having definite values of quantities such as pressure, volume, total energy, total number of particles. If the system is in contact with an environment, then those quantities are not constants, so you have to talk about average values for them.
Presentations of statistical mechanics are often unclear about what is meant by an "average" value. To define an expected value precisely, it must be an expectation of a specific random variable. There can be averages with respect to randomly selected times, averages with respectd to a randomly selected container of gas, averages with repsect to a randomly selected point of space, etc. What kind of "average" is involved in a mixed state?

#### stevendaryl

Staff Emeritus
Presentations of statistical mechanics are often unclear about what is meant by an "average" value. To define an expected value precisely, it must be an expectation of a specific random variable. There can be averages with respect to randomly selected times, averages with respectd to a randomly selected container of gas, averages with repsect to a randomly selected point of space, etc. What kind of "average" is involved in a mixed state?
Technically, if you know the wave function for a composite state (system of interest + environment), then you can get a corresponding mixed state by
• Forming the composite density matrix.
• "Tracing out" the degrees of freedom that you're not interested in.
It's a kind of average, in the sense that the resulting density matrix can be written in the form:

$\sum_j p_j |\psi_j\rangle \langle \psi|$

which can be sort of thought of as a weighted average of different pure state density matrices $|\psi_j\rangle \langle \psi_j|$.

"Is information lost in wavefunction collapse?"

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