I Mixed states vs pure states - physical POV

[bhobba]

Can I give you a better insight into the physical meaning of the density operator? Not really, I'm afraid. It's a great mathematical tool and very important but I confess I don't have a really good way to describe what it is in a 'physical' sense because it's not really all that clear to me either

The rock bottom basis of the Born Rule and the existence of states is really non-contextuality (plus a few other things like the principle of strong superposition that most people pretty much assume without even being told its an assumption - and I think there may be others like that as well - non-contextuality is by far the main one) as shown by Gleason. For me states, including mixed ones, are simply a mathematical artifact of that assumption. Definitionally they are the equivalence classes of preparation procedures of all the different outcomes of the Born rule - a preparation procedure belongs to the same class if it gives exactly the same result from the Born Rule when all possible operators are applied using it. I don't know how physical you would call that - but it seems to be the modern view.

Thanks
Bill

 

[stevendaryl]

The rock bottom basis of the Born Rule and the existence of states is really non-contextuality (plus a few other things like the principle of strong superposition that most people pretty much assume without even being told its an assumption - and I think there may be others like that as well - non-contextuality is by far the main one) as shown by Gleason. For me states, including mixed ones, are simply a mathematical artifact of that assumption. Definitionally they are the equivalence classes of preparation procedures of all the different outcomes of the Born rule - a preparation procedure belongs to the same class if it gives exactly the same result from the Born Rule when all possible operators are applied using it. I don't know how physical you would call that - but it seems to be the modern view.

My bugaboo about standard quantum mechanics is the role that a macroscopic/microscopic partition of the universe seems built-in to the very framework. If you describe the Born rule in terms of preparation procedures and measurement results, the two ends--the starting point and the ending point of an experiment---are macroscopic, while the middle part--the transition from initial state to final state--is treated microscopically, using smooth evolution (whether you use density matrices or wave functions). My feeling is that if there is nothing additional going on at the two macroscopic ends, then there should be a description of the whole shebang in microscopic terms, and the macroscopic description (in terms of preparation procedures and measurements) should be derivable, in a similar way that thermodynamics can be derived from Newton's laws plus statistics.

I'm not really asking for any response, just commenting that I personally don't find clarifying QM in terms of preparation procedures to be particularly enlightening. It's mathematically an elegant thing to do, but it seems like it's elegant at the cost of shuffling the hard parts out of the picture. What's left can be made elegant.

 

[vanhees71]

What you want, of course, exists for decades: It's called quantum statistics aka many-body theory and is as old as quantum theory itself.

 

[stevendaryl]

In general a mixed state can indeed be thought of as a statistical mixture of pure states - but only because it's formally indistinguishable from that. For example, suppose I have the mixed state of a spin-1/2 particle in a mixed state described by $$ | \psi \rangle = \frac 1 2 |z+ \rangle \langle z+ | + \frac 1 2 |z- \rangle \langle z- | $$ then I can interpret that as having been prepared as the pure state $| z+ \rangle$ with probability 1/2 and the pure state $|z- \rangle$ with probability 1/2. So basically having been prepared as an 'up ' or 'down' eigenstate of spin-z with equal probability.

That's the miracle of quantum mechanics that both makes it hard to nail down an interpretation and practically unnecessary.

• Start with a pure state of some small system.
• Let it evolve according to Schrodinger's equation, and eventually it will start to interact with larger systems--measurement devices, the environment, observers, etc.
• At this point, the original system is no longer describable by pure state, since it's become entangled with other systems. The best you can do is to trace out the degrees of freedom due to the parts of the universe that are unobservable or too complicated to analyze in detail. The result is a mixed state for the original system.
• But a mixed state can be interpreted as a statistical mixture, which we all understand from classical probability.
• So we can pretend that the system is in this or that state with a certain probability.

So somewhere along the way, we went from talking about superpositions, which are quantum-mechanical and a little mysterious, to mixtures, which can be understood in classical terms. Exactly how that transition took place is irrelevant when it comes to making predictions about the results of experiments.

 

[stevendaryl]

What you want, of course, exists for decades: It's called quantum statistics aka many-body theory and is as old as quantum theory itself.

No. That's not true. Quantum statistical mechanics does not answer the problems. It doesn't solve the measurement problem, but instead simply applies statistics to the ad hoc interpretation.

Look at the way classical statistical mechanics works. You start with laws of motion for particles. Those laws are non-statistical. Then you consider the larger problem of many, many, maybe 10^20 or so, particles all obeying those laws, but with different initial conditions. This problem is completely intractable using the original laws of motion, but to get macroscopic properties, which is good enough for many purposes, you can treat the collection statistically, and it all becomes manageable again.

The problem with QM is that the "laws of motion" themselves are only understood in terms of macroscopic quantities--measurements and preparation procedures. So we don't have the microscopic form of the laws of physics. That's what's weird.

 

[stevendaryl]

No. That's not true. Quantum statistical mechanics does not answer the problems. It doesn't solve the measurement problem, but instead simply applies statistics to the ad hoc interpretation.

Quantum statistical mechanics is describing a different problem, which is that you have many, many identical systems. That is not what is going on in say an EPR experiment. In that case, it isn't that we have many, many twin pairs. We have a single twin-pair. What's macroscopic are:

• The device that produces the twin pairs
• The device that measures the spin (or polarization)
• The observers

 

[stevendaryl]

The problem with QM is that the "laws of motion" themselves are only understood in terms of macroscopic quantities--measurements and preparation procedures. So we don't have the microscopic form of the laws of physics. That's what's weird.

So the laws themselves are described in macroscopic terms (probabilities for measurement outcomes given certain preparation procedures). Then you propose to eliminate the dependency on macroscopic phenomena by describing those macroscopic phenomena as many, many microscopic objects obeying the same laws. But since the laws themselves refer to macroscopic phenomena, you haven't actually eliminated the reference to macroscopic phenemena.

It seems circular to me, but maybe it's actually some kind of bootstrapping, or recursive understanding of the universe. The microscopic is understood in terms of macroscopic, which is understood in terms of microscopic, etc. Maybe there is some mathematical sense in which this is a convergent process.

 

[vanhees71]

Well, can you prove that the measurement devices used to measure the polarization of photons or the spin of particles etc. etc. are not described by quantum statistics of macroscopic "very-many-body systems"? If so, then you'd be right in saying that there is a division of the world in microscopic and macroscopic behavior.

I don't see the necessity for such a schizophrenic worldview at all, and there's also nothing in QT which tells you where the microscopic world should end and the macroscopic should begin. The more refined our preparation methods become the more success quantum experimenters have in preparing larger and larger (partially even macroscopic) systems in specific quantum states and showing, as expected, all the expectations related with them (e.g., double-slit experiments with buckyball molecules, entangled phonon states of diamonds (even at room temperature!), Bose-Einstein condensates of trapped gases, etc. etc.).

 

[Mentz114]

Quantum statistical mechanics is describing a different problem, which is that you have many, many identical systems. That is not what is going on in say an EPR experiment. In that case, it isn't that we have many, many twin pairs. We have a single twin-pair. What's macroscopic are:

• The device that produces the twin pairs
• The device that measures the spin (or polarization)
• The observers

I don't think this is entirely accurate. In a typical QM experiment the apparatus can be described as a series of concentic spheres, with the coldest, most-shielded part in the centre surrounded by ever more 'classical' apparatus
There was a recent paper which describes an experiment that mimics 'Maxwells demon' in which the experiment can be driven from outside the central area by laser or MW pulses shining into the ultra-cold region where the QM laws of motion rule and information extracted by pulses coming out. So the apparatus has quantum parts.
Furthermore what is happening in there is predictable and well-understood.

The paper is here https://arxiv.org/abs/1702.05161

The figure on page 10 is interesting.

 

[stevendaryl]

I don't see the necessity for such a schizophrenic worldview at all

I'm calling your approach schizophrenic. If the only meaning of the quantum state is to give probabilities for measurement outcomes (or other macroscopic quantities such as mean values) in terms of preparation procedures, then that's necessarily schizophrenic. A non-schizophrenic theory would not mention macroscopic quantities such as measurements or preparation procedures in the fundamental laws, but would be able to derive whatever is being claimed about those things.

 

[stevendaryl]

Well, can you prove that the measurement devices used to measure the polarization of photons or the spin of particles etc. etc. are not described by quantum statistics of macroscopic "very-many-body systems"

Absolutely, I can't prove that, and I don't believe it. I believe that there should be a non-schizophrenic formulation of quantum theory, but the current formulation is schizophrenic.

If so, then you'd be right in saying that there is a division of the world in microscopic and macroscopic behavior.

I'm saying that the formalism has this division. I'm not saying that the division is inherent in nature. I believe it's not.

 

[vanhees71]

I'm calling your approach schizophrenic. If the only meaning of the quantum state is to give probabilities for measurement outcomes (or other macroscopic quantities such as mean values) in terms of preparation procedures, then that's necessarily schizophrenic. A non-schizophrenic theory would not mention macroscopic quantities such as measurements or preparation procedures in the fundamental laws, but would be able to derive whatever is being claimed about those things.

Physics is defined as describing preparation procedures, measurements, quantitative observations of Nature. All theories of physics are right about this way to investigate what's objectively going on. If you want something else, it's not physics!

 

[vanhees71]

Absolutely, I can't prove that, and I don't believe it. I believe that there should be a non-schizophrenic formulation of quantum theory, but the current formulation is schizophrenic.

This is your very individual idea about what the natural sciences should provide. Nature doesn't care about our feelings and expectations how she should behaves. She is just is she is.

 

[stevendaryl]

This is your very individual idea about what the natural sciences should provide. Nature doesn't care about our feelings and expectations how she should behaves. She is just is she is.

My complaint is not in how nature behaves, but the schizophrenic way that we describe it. I don't believe that nature is schizophrenic.

 

[stevendaryl]

Physics is defined as describing preparation procedures, measurements, quantitative observations of Nature. All theories of physics are right about this way to investigate what's objectively going on. If you want something else, it's not physics!

You're confusing your own philosophy of physics with physics itself.

 

[vanhees71]

It's not my philosophy of physics. It's what physicists do in all kinds of physics labs and theory institutes around the world.