Is the Quantum/Classical Boundary the most important question in Physics?

[DarMM]

In which sense can kets represent mixed states and what has it to do with superselection rules?

I'd have to go into the representation theory of the observable algebra. Basically in more advanced contexts pure states and kets aren't the same thing, superselection being one aspect of this.
However I might cover this in a separate thread or Insight.

 

[kith]

It's still a vector state, i.e. it is a sum in the Hilbert space of two pure states.

Yes, mathematically, this state may be possible. But if we consider what states can be actually prepared, this state seems unphysical to me. If we prepare an arbitrary initial state by measuring a complete set of commuting observables of our system, there neither exists a Hamiltonian which leads to your state by unitary time evolution nor do open system dynamics lead to it.

 

[DarMM]

Yes, mathematically, this state may be possible. But if we consider what states can be actually prepared, this state seems unphysical to me. If we prepare an arbitrary initial state by measuring a complete set of commuting observables of our system, there neither exists a Hamiltonian which leads to your state by unitary time evolution nor do open system dynamics lead to it.

Do you mean evolution from an initial pure state under unitary dynamics does not lead to this state? Of course pure states evolve into pure states under unitary dynamics and this state is not pure.

That's the point of superselection rules, that not every sum of two pure states is another pure state.

All of this is immediate from the claim that the state is mixed despite it being a ket. What aspect are disagreeing with or pointing out?

 

[A. Neumaier]

Just to be clear, this is definition of superselection rules given by Wightman in "Superselection Rules: Old and New":

In the same paper Wightman also refers to "Superselection rules induced by the environment". So it seems to be considered valid by both Streater and Wightman that there can be superselection rules induced by the environment. Can you perhaps say where they are wrong with this?

I need to reread the paper to see what they claim with which justification.

It's still a vector state, i.e. it is a sum in the Hilbert space of two pure states. It's just that sometimes kets are actually mixed states. That's the "interesting" part about superselection rules.

Can you explain how a vector state is a mixed state? I understand how a mixed state can be pure in a bigger Hilbert space constructed by the GNS construction, but what you say is the opposite...

I'd have to go into the representation theory of the observable algebra. Basically in more advanced contexts pure states and kets aren't the same thing, superselection being one aspect of this.
However I might cover this in a separate thread or Insight.

I am looking forward to this. For now, can you just give a reference?

 

[DarMM]

Can you explain how a vector state is a mixed state? I understand how a mixed state can be pure in a bigger Hilbert space constructed by the GNS construction, but what you say is the opposite...
...
For now, can you just give a reference?

I think you may have terminology confused. A mixed state is any state which, as an algebraic state, is the sum of another two states, i.e. for all observables:
$$\omega\left(A\right) = c_{1}\omega_{1}\left(A\right) + c_{2}\omega_{2}\left(A\right)$$
A pure being one for which this is not true.
This is equivalent to having vanishing (pure) or nonvanishing (mixed) Von Neumann Entropy.

The GNS construction then allows one to express any state, pure or mixed, as a ket/vector state.
See Robert M. Wald "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics" p.83:

Note that the GNS construction always expresses any state (pure or mixed) as a vector state

Thus a vector state can be pure or mixed. However a pure state (entropy is zero) is never mixed (entropy non-zero).

 

[vanhees71]

In the SGE you cannot locate the cut between the spin of the atom and everything else. The measurement of the spin component is based on a full entanglement between the atom's spin and its position by its dynamical evolution in presence of the magnetic field. Of course, here we use the semiclassical approximation in treating the magnetic field as classical. I'm not sure, whether a full QED description is feasible.

Even in the usual S-matrix formalism you'd describe the magnetic field as classical background field (it's an interesting question, whether there's a fully relativistic description of the SGE, which I'm not so sure about since the properties of spin are more complicated in the relativistic theory than in the non-relativistic one to begin with).

The other thing with the preparability is also easy: If there's by some principle (like a SSR) no Hamiltonian exists that can prepare the state, then this state is indeed simply unphysical, because it cannot be observed. Isn't this the deeper reason for an SSR (like the spin or charge superselection rule) to begin with?

Another example that comes to my mind are neutrinos. You can't prepare mass eigenstates because there's no Hamiltonian available to do this. Consequently there are no neutrinos that can be interpreted as free particles (asymptotic free single-particle Fock states).

 

[kith]

All of this is immediate from the claim that the state is mixed despite it being a ket. What aspect are disagreeing with or pointing out?

I agree that your ket formally corresponds to a completely mixed state and that we can use it to calculate the correct probabilities.

In order to consider it physical, I'd like to see a description where this state ket emerges from a well-prepared pure state (which necessarily belongs to only one of the superselection sectors) by a time evolution law. I don't see how this could be possible except for the ad hoc way where the density matrix is used in the calculation and only certain special mixed state density matrices are identified with kets by hand later on.

 

[DarMM]

In order to consider it physical, I'd like to see this state emerge from a well-prepared pure state (which necessarily belongs to only one of the superselection sectors) by a time evolution law

Why are you requiring this? All mixed states don't result from unitary time evolution of a pure state.

 

[kith]

Why are you requiring this? All mixed states don't result from unitary time evolution of a pure state.

I don't require unitary time evolution.

/edit: I slightly edited the wording of my last post.

 

[A. Neumaier]

I think you may have terminology confused. A mixed state is any state which, as an algebraic state, is the sum of another two states, i.e. for all observables:
$$\omega\left(A\right) = c_{1}\omega_{1}\left(A\right) + c_{2}\omega_{2}\left(A\right)$$
A pure being one for which this is not true.
This is equivalent to having vanishing (pure) or nonvanishing (mixed) Von Neumann Entropy.

The GNS construction then allows one to express any state, pure or mixed, as a ket/vector state.
See Robert M. Wald "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics" p.83:Thus a vector state can be pure or mixed. However a pure state (entropy is zero) is never mixed (entropy non-zero).

I know all this. But you were referring to the addition of kets, not of states (linear functionals). A sum of kets is always a pure state (in the Hilbert space of the ket). My question was how it is possible that you refer to it as a mixed state. Or was this only a slip of the pen?

 

[DarMM]

I know all this. But you were referring to the addition of kets, not of states (linear functionals). A sum of kets is always a pure state (in the Hilbert space of the ket). My question was how it is possible that you refer to it as a mixed state. Or was this only a slip of the pen?

First of all you said this:

I understand how a mixed state can be pure in a bigger Hilbert space constructed by the GNS construction

That's not true right? A mixed state may be a vector state in the GNS Hilbert space, but not a pure state.

Secondly a sum of kets is not a pure state precisely in the case where both belong to different superselection sectors. Unless you are have a certain precise definition of Hilbert space in mind.

 

[DarMM]

I don't require unitary time evolution.

/edit: I slightly edited the wording of my last post.

Well a very dumb example, but it could be fleshed out with a more realistic one, but imagine a "trap" of some kind that can only hold a single particle. Either spin-1/2 or spin-1. It's in contact with a reservoir of spin-1/2 and spin-1 particles with which it can exchange particles. The trap starts off with a single spin-1/2 particle. Its state after a finite time would be described by such a state.

 

[DarMM]

The cut is between what you as the user of quantum mechanics consider to be the quantum system and everything else (including the measurement apparatus and yourself).

Just to note the "cut" as such is present in any non-Kolomogorvian probability theory not just quantum mechanics. For instance PR boxes and the "Nearly quantum theory" of Barnum et al. This is because in all such theories you need some system external to the modeled one to select out the Boolean algebra of events.

So in a sense if one wishes to remove the cut, that is have a theory without the cut, you need to somehow restore classical probability.

 

[A. Neumaier]

A mixed state may be a vector state in the GNS Hilbert space, but not a pure state.

A vector state in the GNS Hilbert space is a pure state in the Hilbert space sense, but not in the $C^*$ algebra sense.

Secondly a sum of kets is not a pure state precisely in the case where both belong to different superselection sectors. Unless you are have a certain precise definition of Hilbert space in mind.

It is a pure state in the Hilbert space defined by the direct sum of the superselection sectors. What I don't understand is in which sense it can be a mixed state.

 

[kith]

Well a very dumb example, but it could be fleshed out with a more realistic one, but imagine a "trap" of some kind that can only hold a single particle. Either spin-1/2 or spin-1. It's in contact with a reservoir of spin-1/2 and spin-1 particles with which it can exchange particles. The trap starts off with a single spin-1/2 particle. Its state after a finite time would be described by such a state.

I like the example. It will take me probably a few days to answer, though.

 

[DarMM]

A vector state in the GNS Hilbert space is a pure state in the Hilbert space sense, but not in the $C^*$ algebra sense.

Pure and Mixed refer to a state's entropy though. Thus a state is pure if it has vanishing entropy.

In many (most) cases of course we represent pure states with vector states. Thus the two are often, but not always, the same.

It is a pure state in the Hilbert space defined by the direct sum of the superselection sectors. What I don't understand is in which sense it can be a mixed state.

It's a vector state, not a pure state as its entropy does not vanish.

 

[A. Neumaier]

Pure and Mixed refer to a state's entropy though. Thus a state is pure if it has vanishing entropy.

In many (most) cases of course we represent pure states with vector states. Thus the two are often, but not always, the same.It's a vector state, not a pure state as its entropy does not vanish.

With which definition of entropy? Where is this discussed?

 

[DarMM]

With which definition of entropy? Where is this discussed?

So first of all one cannot use the usual formula:
$$S\left[\rho\right] = -Tr\left(\rho\ln\rho\right)$$
to compute the entropy in the most general cases. This will give the wrong answer in many cases, or be impossible to apply in some cases. An example being the case of superselection sectors where indeed a state such as:
$$|\psi\rangle = \frac{1}{\sqrt{2}}\left(|a\rangle + |b\rangle\right)$$
with $|a\rangle$ and $|b\rangle$ being elements of different superselection sectors. This is actually a mixed state, but of course the usual formula will give $S\left[\rho\right] = 0$ which is incorrect.

It is for this (and other cases) that Araki developed the more general method of computing entropy in the following papers:
H. Araki,Relative entropy of states of von Neumann algebras I, Publ. RIMS Kyoto Univ. 11, 809-833 (1976)
H. Araki,Relative entropy of states of von Neumann algebras II, Publ. RIMS Kyoto Univ. 13, 173-192 (1977)

Although I should say these papers use modular theory and other laborious methods from the theory of C*-algebras. A good introduction to modular theory (in my opinion) is:
Bratteli, O., Robinson, D.W. (1979): Operator algebras and quantum statistical mechanics I
(Springer, New York, Berlin, Heidelberg)
Bratteli, O., Robinson, D.W. (1981): Operator algebras and quantum statistical mechanics
II, (Springer, New York, Berlin, Heidelberg)

The original papers by Takesaki are harder to read.

Another good guide to all of this is:
Ohya M, Petz D. 1993 Quantum entropy and its use. Berlin, Germany: Springer.

You have the ultimate development of entropy here as related to Connes cocycles.

 

[vanhees71]

I'm really very puzzled.

It thought in a model with superselection rules the above superposition is simply not describing a physical preparable state. If it were preparable mathematically it'd by definition represent a pure state (or rather in the usual sense of $\hat{\rho}=|\psi \rangle \langle \psi|$ representing this pure state to be pedantic).

Whether von Neumann entropy describes the entropy measure for a given physical situation is of course a question to be decided for the specific case, but at least it's the standard definition of the missing information, given the statistical operator relative to complete information, defined as given by preparation in (any) pure state.

 

[DarMM]

It thought in a model with superselection rules the above superposition is simply not describing a physical preparable state. If it were preparable mathematically it'd by definition represent a pure state (or rather in the usual sense of $\hat{\rho}=|\psi \rangle \langle \psi|$ representing this pure state to be pedantic).

It is preparable, it's just that it describes a mixed state not a pure state. That's the whole point, just because a state can be represented by a vector state does not mean it is a pure state.

You don't typically see this in QM in practice, but it is the case.

 

[DarMM]

If it helps the actual connection between pure states and vector states is that in an irreducible representation of the observable algebra the vector states are pure. What's happening with superselection is that one is dealing with vector states in a reducible representation. Thus they are vector states, but can be mixed.

 

[Morbert]

It thought in a model with superselection rules the above superposition is simply not describing a physical preparable state. If it were preparable mathematically it'd by definition represent a pure state (or rather in the usual sense of $\hat{\rho}=|\psi \rangle \langle \psi|$ representing this pure state to be pedantic).

I think it's the case that the state is not physical only insofar as there is no observable that is a projector onto the state. But it is still physical as a preparation.

 

[DarMM]

I think it's the case that the state not physical only insofar as there is no observable that is a projector onto the state. But it is still physical as a preparation.

Precisely. It's as physical as any other mixed state, i.e. describes a preparation.

 

[kith]

Just a quick question: are there kets which correspond to partially mixed states?

 

[DarMM]

Just a quick question: are there kets which correspond to partially mixed states?

In the GNS construction you can form a vector state corresponding to any mixed state.

More in line with the case here you can just change the coefficients in front of the kets

 

[vanhees71]

It is preparable, it's just that it describes a mixed state not a pure state. That's the whole point, just because a state can be represented by a vector state does not mean it is a pure state.

You don't typically see this in QM in practice, but it is the case.

Then we have different definitions of pure states.

 

[DarMM]

Then we have different definitions of pure states.

Possibly. The definition I'm using is typical in rigorous QFT, quantum foundations and quantum information. It reduces to your definition (i.e. a pure state is a ket) in the case when the kets are an irreducible representation of the observable algebra.

However due to things like the GNS construction your definition isn't very useful in the case of QFT in curved spacetimes and situations like that, since one can take any statistical operator and recast it as a vector state in another representation.

This is a problem because if we take a statistical operator then it has $S > 0$, however in the GNS representation, according to the usual definition of entropy, we would have $S = 0$. Even though it's really the same state. Thus we need a representation independent notion of purity and entropy.

 

[Elias1960]

Just to note the "cut" as such is present in any non-Kolomogorvian probability theory not just quantum mechanics. For instance PR boxes and the "Nearly quantum theory" of Barnum et al. This is because in all such theories you need some system external to the modeled one to select out the Boolean algebra of events.

So in a sense if one wishes to remove the cut, that is have a theory without the cut, you need to somehow restore classical probability.

This seems relevant to the discussion we had about generalizations of probability theory.

Because it suggests that in all those non-Kolomogorvian probability theories all one needs is to add some external system which selects the "Boolean algebra of events", and for the combined system we have a Kolmogorovian probability theory, with the elementary events defined as the external animal which defines the Boolean algebra of events together with the corresponding events. Thus, simply the straightforward generalization of the Kochen-Specker construction for quantum theory.

 

[DarMM]

This seems relevant to the discussion we had about generalizations of probability theory.

Because it suggests that in all those non-Kolomogorvian probability theories all one needs is to add some external system which selects the "Boolean algebra of events", and for the combined system we have a Kolmogorovian probability theory, with the elementary events defined as the external animal which defines the Boolean algebra of events together with the corresponding events. Thus, simply the straightforward generalization of the Kochen-Specker construction for quantum theory.

There is a significant difference though. One selects the Boolean subalgebra from a fundamentally non-Boolean structure, but that choice is not given a probability. The Kochen-Specker construction has the choice itself as an event with a probability in a larger Boolean algebra.

 

[Elias1960]

It's still a vector state, i.e. it is a sum in the Hilbert space of two pure states. It's just that sometimes kets are actually mixed states. That's the "interesting" part about superselection rules.

Sorry, but this makes no sense. Kets are pure states. Point. For a mathematician, the only reasonable reaction to an attempt to name them mixed states is a facepalm.

Usual superselection rules give only that they cannot be prepared. But that they cannot be prepared does not justify to give them false names.

The same holds for those weaker notions of superselection rules where one can prepare such states, like the state of a Schrödinger cat. Here we obviously have no operator to measure this pure state. All measurements we have give the same result as some large number of other pure as well as mixed states. But this also does not give a permission to name a pure state a mixed state. These are simply indistinguishable but nonetheless different states.

I would not object to phrases like "pure states become indistinguishable from mixed states", but to name a pure state a mixed state because of this is positivism going insane.

(Of course, the whole idea of considering $C^*$ algebras as something of some fundamental importance smacks of positivism. So, I would not really be surprised if this is indeed standard language in that community. In this case, I would characterize this as an example of an absurd consequence of the remaining influence of positivism in physics.)