Measurements and The Copenhagen Interpretation

[jambaugh]

I actually have a slight mistake in my characterization of Hurkyl's example. We shouldn't use a density operator to express our knowledge of the system, post measurement but pre-acknowledgment of measured value. Rather we should use a classical discrete pdf of the two "collapsed" modes. The reason being an entropy calculation on the density operator would yield non-zero entropy whereas having access to a record of the measurement outcome implies zero entropy. It should be even more in parallel with the die case. The case where we actually use a density operator we should be sure that the classical probabilities stem from correlation (entanglement) of the observable with variables in the entropy dump which are thereby inaccessible. The classical probabilities in the density operator stem from tracing over the inaccessible parts of the environment.

I believe this is just a technical qualification of the notation we could choose to utilize the same mathematical object, the density operator to represent both kinds of "classical" probabilities but with the type distinction made explicit. One is dealing with the same kind of entropy issues which arise in interpreting probabilities in a purely classical setting. Maxwell's demon is starting to rear his little pointy head.

I'll have to consider what this means in terms of my understanding of entropy. Hmmm...
I think essentially the CI must extend also in the classical realm. We should understand the meaning of entropy as expressing not a property of the system but a property of our knowledge about the system... said knowledge being of course linked via the empirical epistemology of physics to the physical constraints we impose in defining (physically setting up and example of) the system.

Hmmm... this would make the second law of thermodynamics an abstract law about information. I'll meditate on this.

 

[ThomasT]

I think essentially the CI must extend also in the classical realm. We should understand the meaning of entropy as expressing not a property of the system but a property of our knowledge about the system...

Hmmm... this would make the second law of thermodynamics an abstract law about information. I'll meditate on this.

Ok, so the 2nd LOT is an abstraction of and generalization based on the information available to us. What's there to meditate about?

Regarding your OP. I've learned to think of the meaning of wavefunction collapse in the CI in the same way that your posts indicate that you do. Wrt the CI wavefunction collapse isn't a problem. Given definite qualitative instrumental results (ie., measurements), info is updated. That's all it means.

Measurements (ie., information) are distinguished from models of the measurement process. Measurements refer to recorded instrumental events or output. These records are amenable to our senses and can be described using classical concepts. This provides the objective basis for the science of physics.

The CI, as I've learned it, says that definitive, objective answers to questions about deep reality are prohibited by the quantum theory.

 

[Hurkyl]

Here's an important thought experiment, I think.


We're all familiar with the scourge of thermodynamics: Maxwell's demon. Given a container with a small sliding window, this is the little imp that is capable of creating a vacuum simply by opening the window to let inside air molecules exit, and closing it before outside air molecules would enter.

We're all familiar with the half-silvered mirror experiment: we fire a beam of light through the mirror and it splits. We guide the split beams to another mirror, and they recombine.

We also know that if we add something along one of the beam paths that would interact with the light, then the two beams do not recombine at the end.

We also know the quantum eraser experiment: in some cases, it is possible to add a second interaction that "undoes" the first, and the two beams again recombine at the end.



Now, what happens if we combine Maxwell's demon with the idea of a quantum eraser? We have a setup like the half-silvered mirror experiment; let's make it an incredibly huge experiment, to give the demon ample time to do his stuff.

We pass one of the beams through a closed room of the demon's design. In the room, we place some sort of measuring device along the beams path that will print results onto a sheet of paper. We give the demon free reign to do whatever he wants inside his room, as long as he doesn't disable/disconnect/whatever our measuring device.

Now, the $1,000,000 question¹: under these restrictions, can the demon build a quantum eraser?

If so, then what about if we put Wigner's friend in the room, and he's allowed to read the sheet of paper?



1: If you believe in hidden variables, then also consider putting another restriction on the demon that he's only allowed to use quantum mechanics, and isn't allowed to peek at the hidden variables.

 

[jambaugh]

Here's an important thought experiment, I think.
...
Now, what happens if we combine Maxwell's demon with the idea of a quantum eraser?
...

This is a horribly ill posed thought experiment. Maxwell's demon is an inherently classical entity having deific knowledge of the objective state of an inherently classical system.

To set up the details of your experiment you must necessarily invoke counter-factuals. Maxwell's demon is supposed to simultaneously know values for all observables including mutually incompatible ones.

Make your demon a "quantum Maxwell's demon" and he ceases to be demonic. On of the great triumphs of QM was its resolution of Maxwell's demonic paradox. You can't invoke the demon without invalidating QM.

 

[jambaugh]

Ok, so the 2nd LOT is an abstraction of and generalization based on the information available to us. What's there to meditate about?

I have the same predilection toward objectifying systems as the next fellow. It is easy to fall into the trap of thinking of entropy as a system variable instead of a description variable. I see the 2nd law in action every time I eat and excrete waste. Just as it is hard to fight off the notion that $\Psi$ is the system's "quantum state" it is hard to fight off the notion that S is a concrete system observable. These notions, though consciously treated correctly need to be incorporated into our intuition or we tend waste time re-educating ourselves with what we already know consciously.

When I run across a realization which makes me say to myself "Oh yea I should have known that, it should have been obvious!" I take this as a warning bell that I haven't fully integrated facts into my intuitive understanding and hence I "meditate" on the issue to remediate the deficit.

Regarding your OP...The CI, as I've learned it, says that definitive, objective answers to questions about deep reality are prohibited by the quantum theory.

Yes and indeed the questions themselves are ill posed. But we evolved our though processes over the last eon or so dealing with our environment at the classical level. We are fighting a great deal of instinct in trying to properly understand QM.

 

[Hurkyl]

This is a horribly ill posed thought experiment.

Quantum erasers aren't an ill-posed thought experiment -- they are things we actually build in the real world and observe in action.

Make your demon a "quantum Maxwell's demon" and he ceases to be demonic.

On of the great triumphs of QM was its resolution of Maxwell's demonic paradox. You can't invoke the demon without invalidating QM.

No, seriously,

 

[jambaugh]

Quantum erasers aren't an ill-posed thought experiment -- they are things we actually build in the real world and observe in action.

Right! But mixing such with a "Maxwell Demon" is ill-posed. To invoke M's D you must be in a purely classical description. The demon is able to divine the objective state of the system without interaction with it. It thus knows values for observables which in QM are not simultaneously measurable and thus not simultaneously well defined. He grossly violates the uncertainty principle.

If you disagree then formulate your experiment in more detail and show me where I err.

 

[Count Iblis]

Right! But mixing such with a "Maxwell Demon" is ill-posed. To invoke M's D you must be in a purely classical description. The demon is able to divine the objective state of the system without interaction with it. It thus knows values for observables which in QM are not simultaneously measurable and thus not simultaneously well defined. He grossly violates the uncertainty principle.

If you disagree then formulate your experiment in more detail and show me where I err.

I also like to see more details about Hurkyl's thought experiment, but I don't see how MD necessarily violates the uncertainty principle. In a previous posting you wrote:

On of the great triumphs of QM was its resolution of Maxwell's demonic paradox. You can't invoke the demon without invalidating QM.

which I don't think is true at all. Even if a hypothetical MD in some setting would have to violate the uncertainty principle to work 100% efficiently, there is already a problem. It's decisions don't need to be correct all the time, what matters is that its decisions are more often correct that wrong. A coarse grained measurement of the local density on both sides of the door is enough to make a decision that is statistically correct.

The total entropy will go down and the memory of the Demon will contain more and more random information about its past decisions. According to quantum mechanics, the whole process must be unitary, so no information can be lost. If the entropy of the gas is going down, then that means that specifying the exact microstate given the macrostate requires less information. Given the (pseudo) random nature of the initial state, this is impossible unless all of the information that you now don't need anymore to point out the exact microstate is present in the memory if the Demon.

The impossiblility follows from counting states. The process would have worked starting from an intitial state being any of the possible microstates that the system could have been in. But the gas ends up in a final state that is one out of a smaller set of microstates. Then for the process to be unitary requires the possible final memory states of the Demon to compensate for this, otherwise two different initial states would be mapped under time evolution to the same final state.


This is then also an answer to an earlier cricism about me invoking the 2nd law to argue in favor of unitary evolution. While it sounds contradictory, the above argument explains that if you can violate unitary evolution, you can actually make an effective Maxwell's Demon that can really lower the entropy of a system.

 

[Hurkyl]

Right! But mixing such with a "Maxwell Demon" is ill-posed. To invoke M's D you must be in a purely classical description. The demon is able to divine the objective state of the system without interaction with it. It thus knows values for observables which in QM are not simultaneously measurable and thus not simultaneously well defined. He grossly violates the uncertainty principle.

If something isn't well-defined, then it cannot be part of the objective state of the system.

The superpower of Maxwell's demon is that he is not limited by what is practical or feasible -- he is only limited by what is, in principle, possible.

 

[jambaugh]

If something isn't well-defined, then it cannot be part of the objective state of the system.

The superpower of Maxwell's demon is that he is not limited by what is practical or feasible -- he is only limited by what is, in principle, possible.

I see that I am thinking of a different version of Maxwell's demon, i.e. an entity with "god like" knowledge of the system's objective state and dynamics and thus able to predict the outcome of all acts of measurement. This version is not meaningful in the quantum setting.

However the definition you are giving is more robust and I think more in line with Maxwell's original definition. With this definition I retract my objections of invoking such in a quantum context.

I however do not see the point. In all our though experiments we do not seem to be limiting the experimenter beyond what is possible and thus they are demonesque in this sense.

 

[jambaugh]

If something isn't well-defined, then it cannot be part of the objective state of the system.

The point here is that in a quantum setting "the objective state of the system" is itself ill-defined. That is at least within CI.