Graduate Stephen Weinberg on Understanding Quantum Mechanics

[A. Neumaier]

Understanding is pointless?

Dr. Du said the question is pointless, not the understanding. Understanding must be based on asking and answering questions that can be meaningfully formulated in the framework in which the theory is described. If a theory contains no notion of pure states, asking questions involving the latter is not meaningful.

 

[stevendaryl]

You are thinking only in terms of type I representations (in the classification of von Neumann). For these, which adequately describe the quantum mechanics of finitely many degrees of freedom, your statement is correct. However, the real world is occupied by macroscopic bodies, which need quantum field theory and infinitely many degrees of freedom for their description. Already a laser, which generates the quantum states with which Bell-type experiments are performed, is such a system. Once the number of degrees of freedom is infinite, the other types in von Neumann's classification play a role. In particular, in relativistic QFTs one has always representations of type III_1 (see the paper by Yngvason cited in the link given above.

Type III_1 representations behave conceptually very differently, as no pure states exist in these representations. In these representations one cannot rigorously argue about states by considering partial traces in nonexistent pure states! This shows that pure states are the result of a major approximating simplification, and not something fundamental.

First of all, I don't agree that any of the conceptual problems with quantum mechanics are resolved by using density matrices.

Second, I'm not sure I understand the claim about the nonexistence of pure states. I thought that in QFT, you can still work with pure states. At least, in perturbation, you can think of the pure states as being of the form of a superposition of states with zero, one, two, etc. applications of creation operators on the vaccuum.

 

[DrDu]

I disagree, because we can understand mixed states in terms of pure states with uncertainty (or in terms of pure states in which some of the degrees of freedom have been traced out). There is nothing conceptually new about mixed states that changes anything, as far as I can see.

I just had a look at the nice article by Yngvarson A. Neumaier cited. There are examples of how these type III theories arise for infinite tensor products of spin 1/2 which I mentioned earlier. The point is that the concept of a state as used in this article is not simply the infinite product of single particle spin wavefunctions.

 

[stevendaryl]

The recent (this year) article by Weinberg seems to me to be claiming that he considers there to still be unresolved conceptual problems with quantum mechanics. So if people are citing Weinberg as evidence that the use of density matrices resolves everything, it seems to me that they are disagreeing with Weinberg.

 

[A. Neumaier]

First of all, I don't agree that any of the conceptual problems with quantum mechanics are resolved by using density matrices.

I am not talking about density matrices. In QFT, states are described by positive linear functionals, in the simplest case given by density operators, infinite-dimensional versions of what you like to play with. Moreover, I didn't claim that all conceptual problems are resolved when working with density operators. Only that working with pure states cannot solve the conceptual problems since pure states are themselves an approximation.

Second, I'm not sure I understand the claim about the nonexistence of pure states. I thought that in QFT, you can still work with pure states. At least, in perturbation, you can think of the pure states as being of the form of a superposition of states with zero, one, two, etc. applications of creation operators on the vacuum.

I am sure you don't understand. Please read Yngvason's paper. The use of pure states is restricted to free quantum field theory, which is described in Fock representations, which have type I. One can do perturbation theory about a free theory, but only approximately (which makes rigorous arguments impossible) and only after renormalization (which destroys the Fock structure and causes the change of type).

Pure states are from a fundamental point of view useful approximations, nothing more. If one runs into conceptual problems when using them, the problems may well be caused by the approximations involved, especially if the arguments used assume that everything holds exactly without error.

 

[RockyMarciano]

I don't think density matrices solve anything, but how does that prevent us from realizing that pure states are just fictional approximations?

 

[A. Neumaier]

if people are citing Weinberg as evidence that the use of density matrices resolves everything, it seems to me that they are disagreeing with Weinberg.

You are talking about the empty set. I neither claimed that the use of density matrices resolves everything, nor was I citing Weinberg - the material I referred to is in Yngvason's paper linked to in my discussion of Weinberg's paper. Weinberg does not refer to him and may well be unaware of these matters. Finally, disagreement with Weinberg is no argument against truth.

 

[stevendaryl]

I took a look at the paper by Yngvason here:

https://arxiv.org/abs/1401.2652

It's very interesting, but I'm not sure I understand the point about the Type III states for which there are no pure states. Yngvason is defining a "pure" state \omega as one that cannot be written in the form \omega = \frac{1}{2} \omega_1 + \frac{1}{2} \omega_2 with \omega_1 \neq \omega_2. I don't understand the motivation for this definition.

In the case of density matrices for non-relativistic quantum mechanics, a density matrix \rho can always be written in the form:

\rho = \sum_j p_j |\phi_j\rangle \langle \phi_j |

where p_j \geq 0 and \sum_j p_j = 1.

Then we can define a pure state to be one that can be written using only one vector:

\rho = |\phi\rangle \langle \phi |

That has the interpretation in terms of classical probability that \rho represents the situation in which the system is in state |\psi_j\rangle with probability p_j.

Yngvason is using a different definition of "pure state" (which presumably reduces to the same thing in the case of NRQM), and is saying that

• There are types of systems for which there are no pure states.
• (Therefore) density matrices cannot be interpreted as classical probabilities for being in this or that pure state.

I don't understand the definition or the conclusion, so I need to think about it a little more.

 

[stevendaryl]

You are talking about the empty set. I neither claimed that the use of density matrices resolves everything, nor was I citing Weinberg - the material I referred to is in Yngvason's paper linked to in my discussion of Weinberg's paper. Weinberg does not refer to him and may well be unaware of these matters. Finally, disagreement with Weinberg is no argument against truth.

That's certainly true, but I thought you were citing Weinberg's paper on Quantum Mechanics Without State Vectors
as evidence that the issues were resolved by using density matrices instead of state vectors.

 

[stevendaryl]

I notice that in the paper by Yngvason , he concludes with a warning that this discussion does not resolve any of the foundational issues with QM:

On the other hand, the framework of LQP does not per se resolve all “riddles” of
quantum physics. Those who are puzzled by the violation of Bell’s inequalities in EPR
type experiments will not necessarily by enlightened by learning that local algebras are
type III. Moreover, the terminology has still an anthropocentric ring (“observables”,
“operations”) as usual in Quantum Mechanics. This is disturbing since physics is concerned
with more than designed experiments in laboratories. We use quantum (field)
theories to understand processes in the interior of stars, in remote galaxies billions
of years ago, or even the “quantum fluctuations” that are allegedly responsible for
fine irregularities in the 3K background radiation. In none of these cases “observers”
were/are around to “prepare states” or “reduce wave packets”! A fuller understanding
of the emergence of macroscopic “effects” from the microscopic realm, without
invoking “operations” or “observations”, and possibly a corresponding revision of the
vocabulary of quantum physics is still called for.

 

[DrDu]

I notice that in the paper by Yngvason , he concludes with a warning that this discussion does not resolve any of the foundational issues with QM:

Yngvarson sais :"On the other hand, the framework of LQP does not per se resolve all “riddles” of quantum physics. " The shift from "all" to "any" is yours.

 

[stevendaryl]

Yngvarson sais :"On the other hand, the framework of LQP does not per se resolve all “riddles” of quantum physics. " The shift from "all" to "any" is yours.

My mistake.

 

[A. Neumaier]

Yngvason is defining a "pure" state \omega as one that cannot be written in the form \omega = \frac{1}{2} \omega_1 + \frac{1}{2} \omega_2 with \omega_1 \neq \omega_2. I don't understand the motivation for this definition.

It says that the state cannot be a mixture of two different states with equal weight. One can easily see that this is true for states $\omega(A)=\psi^*A\psi$ given by a wave vector $\psi$. One can see with a little more work that it is false for any state $\omega(A)=$trace$(\rho A)$ where $\rho$ has rank greater than 1. Thus the two definitions are equivalent for states on algebras of bounded operators, as they are used in Bell-type experiments.

 

[A. Neumaier]

• There are types of systems for which there are no pure states.
• (Therefore) density matrices cannot be interpreted as classical probabilities for being in this or that pure state.

I don't understand the definition or the conclusion, so I need to think about it a little more.

Point 2 follows directly from point 1 since there are no pure states, so talking about them is meaningless.

Even in the case of quantum mechanics of a 2-level system, where operators are matrices, density matrices cannot be interpreted as classical probabilities for being in this or that pure state. Unpolarized light has as density matrix half the identity matrix. One has infinitely many essentially different decompositions of the kind you describe - which one gives the ''correct'' interpretation? None, since the state is symmetric under the helicity $SU(2)$ while none of the decompositions is. Picking out one of them is well-defined only at the moment of measurement. Thus one can only get probabilities for passing an experimental test measuring this or that polarization. Thus your ontology is defective already at the level of single photons in a fixed beam.

 

[stevendaryl]

Point 2 follows directly from point 1 since there are no pure states, so talking about them is meaningless.

Even in the case of quantum mechanics of a 2-level system, where operators are matrices, density matrices cannot be interpreted as classical probabilities for being in this or that pure state. Unpolarized light has as density matrix half the identity matrix. One has infinitely many essentially different decompositions of the kind you describe - which one gives the ''correct'' interpretation? None, since the state is symmetric under the helicity $SU(2)$ while none of the decompositions is. Picking out one of them is well-defined only at the moment of measurement. Thus one can only get probabilities for passing an experimental test measuring this or that polarization. Thus your ontology is defective already at the level of single photons in a fixed beam.

I certainly recognize that. That was actually going to be my next point.

In my opinion, every interpretation of QM amounts to shuffling the defect around, rather than addressing it.

 

[A. Neumaier]

Thus one can only get probabilities for passing an experimental test measuring this or that polarization.

In the notation of Yngvason, testable statements are represented by Hermitian elements $P$ with only eigenvalues 0 and 1, corresponding to the two possible results of disproving or confirming the statement in the experiment. This is equivalent to saying that $P^2=P=P^*$, the condition for an orthogonal projector. The probability for testing a statement $P$ is given by $p=\omega(P)$. This is completely independent of the notion of a pure state. von Neumann's classification is essentially a classification of the possible orthogonal projectors. Pure states exist only when the algebra contains projectors of rank 1. Then $P=\psi\psi^*$ where $\psi$ is a normalized state in the range, and if $\omega$ is also a pure state we get Born's rule.

 

[A. Neumaier]

every interpretation of QM amounts to shuffling the defect around, rather than addressing it.

If one acknowledges that specifying the state (in Yngvason's sense) is having specified the system then at least the defect that I mentioned in my post is completely absent, without introducing another defect anywhere else. Thus it is a superior ontology.

 

[stevendaryl]

If one acknowledges that specifying the state (in Yngvason's sense) is having specified the system then at least the defect that I mentioned in my post is completely absent, without introducing another defect anywhere else. Thus it is a superior ontology.

I suppose, but the issues that Yngvason said are unresolved by using density matrices are the most vexing aspects of the foundations of quantum mechanics, in my opinion.

 

[A. Neumaier]

the issues that Yngvason said are unresolved by using density matrices are the most vexing aspects of the foundations of quantum mechanics

Maybe, but they are more likely to be solved based on a physically correct basis (general states) rather than based on one (pure states) that is already known to be only of limited validity.

 

[rubi]

Let me clarify this issue.

An (algebraic) state on a $C^*$-algebra $\mathfrak A$ is a linear functional $\omega:\mathfrak A\rightarrow\mathbb C$ with $\omega(a^*a)\geq 0$ and $\omega(1)=1$. One example of a $C^*$-algebra is the set of bounded operators $\mathcal B(\mathcal H)$ on a Hilbert space $\mathcal H$ and an example of a state is the functional $\omega(A)=\left<\Omega,A\Omega\right>$, where $\Omega\in\mathcal H$ is any vector with $\lVert\Omega\rVert=1$. Another example would be $\omega(A)=\mathrm{Tr}(\rho A)$, where $\rho:\mathcal H\rightarrow\mathcal H$ is any positive trace-class operator with $\mathrm{Tr}(\rho)=1$. An algebraic state $\omega$ on $\mathfrak A$ is said to be mixed, if there exist $\omega_1,\omega_2 \neq \omega$ and $\lambda\in(0,1)$ such that $\omega=\lambda\omega_1+(1-\lambda)\omega_2$. Otherwise, it is said to be pure.

Theorem (Gelfand, Naimark, Segal): Given a $C^*$-algebra $\mathfrak A$ and a state $\omega$ on $\mathfrak A$, there exists a Hilbert space $\mathcal H$, a representation $\pi:\mathfrak A\rightarrow\mathcal B(\mathcal H)$ of $\mathfrak A$ and a vector $\Omega\in\mathcal H$, such that for all $A\in\mathfrak A$, we have
$$\omega(A)=\left<\Omega,\pi(A)\Omega\right> \text{.}$$

Now, what is the relevance to quantum theory?
1. The algebraic terminology of pure and mixed states matches the QM terminology in the following sense: If $\omega_1(A)=\left<\psi,A\psi\right>$ and $\omega_2(A)=\left<\phi,A\phi\right>$ are vector states on $\mathcal B(\mathcal H)$, then $\omega=p_1\omega_1+p_2\omega_2$ with probabilities $p_1$, $p_2$ is given by $\omega(A)=\mathrm{Tr}(\rho A)$ with $\rho=p_1\left|\psi\right>\left<\psi\right|+p_2\left|\phi\right>\left<\phi\right|$. In that sense, the algebraic terminology agrees with the standard terminology.
2. Every algebraic state $\omega$ can be realized as a vector state $\omega(A)=\left<\Omega,\pi(A)\Omega\right>$ by the GNS construction, even a state $\omega(A)=\mathrm{Tr}(\rho\pi(A))$ defined by a density matrix. The confusion between Arnold and stevendaryl is due to a disagreement about terminology. While Arnold refers to states as "pure", if they are pure algebraic states (which is what AQFT people do), stevendaryl refers to states are pure, if they are defined by vectors in a Hilbert space (which is common in QM). Of course, Arnold's terminology is more restrictive, because every state can be realized as a vector state in some Hilbert space by the GNS theorem and thus the distinction between vector states ("pure states" in stevendaryl's terminology) and non-vector states is completely unphysical. Nevertheless, in standard QM, it has practical relevance, because in that case we have the Stone-von-Neumann theorem, which (pretty much) singles out the standard Schrödinger representation and we usually like to represent our states in that particular representation. Of course, in the case of QFT, no such uniqueness result is available and hence, it is unreasonable to distinguish between vector states and non-vector states.

 

[Auto-Didact]

After carefully reading Yngvason's paper, one can conclude: its a cogent summary of mathematical objects in contemporary AQFT and gives possible distinctions w.r.t. similar objects in standard non-relativistic QM. It is however not concerned per se with interpretational issues, i.e. what Weinberg et al. are directly concerned with. However, Yngvason does state in a footnote on p.18

24 See [12] for important steps in this direction and [49] for a thorough analysis of foundational issues of QM

giving 2 references w.r.t interpretations, specifically:

[12] P. Blanchard, R. Olkiewicz: “Decoherence induced transition from quantum to classical
dynamics”, Rev. Math. Phys. 15, 217–244 (2003).

[49] J. Fröhlich, B. Schubnel: “Quantum Probability Theory and the Foundations of Quantum mechanics”, arXiv:1310.1484v1 [quant-ph].

The latter is a comprehensive review of the field, while the former a more specific environmentally induced decoherence proposal for open quantum systems. It seems that this environmental decoherence proposal is Yngvason's preferred interpretation given his AQFT understanding. In other words, taking density operators (or matrices) as more primary ontologic concepts implies embracing environmental decoherence, which is after all a purely FAPP pragmatic (as opposed to fundamental) philosophy as John Bell said.

 

[DrDu]

rubi, I think at least in ordinary QM vector representations of mixed states are reducible, while those of pure states are irreducible. Is this true for all C* algebras?

 

[rubi]

rubi, I think at least in ordinary QM vector representations of mixed states are reducible, while those of pure states are irreducible. Is this true for all C* algebras?

This is a little bit subtle. What is true is the following: The vector representation $(\mathcal H_\omega,\pi_\omega,\Omega_\omega)$ given by the GNS construction of a state $\omega$ is irreducible if and only if $\omega$ is pure. However, that doesn't mean that all vector representations are irreducible. For example, we can take the irreducible GNS representation $(\mathcal H_\omega,\pi_\omega,\Omega_\omega)$ of a pure state $\omega$ and specify the following data: $\mathcal H=\mathcal H_\omega\oplus\mathcal H_\omega$, $\pi=\pi_\omega\oplus\pi_\omega$ and $\Omega=\frac{1}{\sqrt{2}}\Omega_\omega\oplus\Omega_\omega$. Define a state $\xi(A)=\left<\Omega,\pi(A),\Omega\right>_{\mathcal H}$. We then find $$\xi(A) = \frac{1}{2}\left<\Omega_\omega\oplus\Omega_\omega,\pi_\omega(A)\Omega_\omega\oplus\pi_\omega(A)\Omega_\omega\right>_{\mathcal H}=\frac{1}{2}\left<\Omega_\omega,\pi_\omega(A)\Omega_\omega\right>_{\mathcal H_\omega}+\frac{1}{2}\left<\Omega_\omega,\pi_\omega(A)\Omega_\omega\right>_{\mathcal H_\omega}=\frac{1}{2}\omega(A)+\frac{1}{2}\omega(A)=\omega(A) \text{.}$$ Thus $\xi=\omega$. However, $(\mathcal H,\pi)$ is clearly reducible, because $\mathcal H\oplus 0$ is an invariant subspace. So we have found a reducible vector representation of a pure state $\omega$. This is of course possible, because only the GNS representation of $\omega$ needs to be irreducible. Our representation $(\mathcal H,\pi,\Omega)$ is not the GNS representation of $\omega$. So even in QM, vector representations needn't be irreducible.

So let's look at type $III_1$ factors $\mathfrak A$ now. Can there be irreducible vector representations? I think what is going on is the following: At first, the fact that every state on $\mathfrak A$ must be mixed seems to invalidate this. However, what it really means is the following: Let's take the GNS representation $(\mathcal H_\omega,\pi_\omega,\Omega_\omega)$ induced by a state $\omega$ on $\mathfrak A$. Clearly, it must be reducible due to the theorem I stated in the beginning. However, that means that there is an invariant subspace of $\mathcal H_0\subseteq \mathcal H_\omega$ and I can take it to be minimal, i.e. $(\mathcal H_0,\pi_\omega\rvert_{\mathcal H_0})$ is irreducible and I can take any vector $\Psi\in\mathcal H_0$. Then I can define the state $\xi(A)=\left<\Psi,\pi_\omega(A)\Psi\right>_{\mathcal H_0}$ and it clearly defines a vector state in an irreducible representation on $\mathfrak A$. How can this be? The answer is that the data $(\mathcal H_0, \pi_\omega\rvert_{\mathcal H_0},\Psi)$ does not arise as the GNS data of some state on $\mathfrak A$. If I apply the GNS construction to the state $\xi$, I will end up with a reducible representation, because Yngvason tells us that $\xi$ must be a mixed state and our theorem tells us that mixed states produce reducible GNS representations.

(However, I still find it fishy. The states should form a convex set and as such, it should have extremal points, which should correspond to pure states. I need to think about this more.)

 

[A. Neumaier]

Every algebraic state can be realized as a vector state by the GNS construction, even a state defined by a density matrix.

But this is not the way stevendaryl uses the terms since for him, mixed states are composed of pure states, in a fixed Hilbert space, while the GNS construction produces a different Hilbert space for each (pure or mixed) state!

 

[A. Neumaier]

The states should form a convex set and as such, it should have extremal points

This does not follow. The plane is a convex set without extremal points.

 

[A. Neumaier]

it clearly defines a vector state in an irreducible representation

A vector state in an irreducible representation of the observable algebra $\cal A$ in a Hilbert space $\cal H$ can still be mixed. It is guaranteed to be pure only relative to the algebra of all bounded operators on $\cal H$. But this algebra is far bigger than the III_1 algebra $\cal A$, and contains lots of operators that have no interpretation as observables. This is the essential difference to the case of type I algebras.

 

[stevendaryl]

A request: If you feel you understand Yngvason's point about Type III systems without pure states, could you write up a post about it, aimed at people who understand the basics of Hilbert spaces, but not the technical points of C^* algebras, and all that? Someone mentioned a sort-of accessible example involving an infinite array of spin-1/2 particles, or something like that.

 

[A. Neumaier]

example involving an infinite array of spin-1/2 particles

This example is given in equations (27) and (29) in Yngvason's paper. The GNS construction is fully described in Wikipedia. It produces a large Hilbert space with a huge algebra of bounded operators, most of which are (typically) not in the algebra of observables one starts with. In particular, this happens when one starts with a mixed state of the algebra of bounded operators of a small Hilbert space. (For example, one obtains from an $n$-dimensional physical Hilbert space in most cases an $n^2$-dimensional nonphysical Hilbert space.)

 

[rubi]

But this is not the way stevendaryl uses the terms since for him, mixed states are composed of pure states, in a fixed Hilbert space, while the GNS construction produces a different Hilbert space for each state!

Yes, stevendaryl uses a different terminology. There are even algebraic states that can't even be realized as density matrix states, if the Hilbert space is fixed (although the density matrix states are in some sense dense in the space of algebraic states). The concept of algebraic states is much better for theory building if the Hilbert space isn't known yet.

This does not follow. The plane is a convex set without extremal points.

Of course, you're right. I'm stupid. The state space may be non-compact.

The point is that a vector state in an irreducible representation of the observable algebra $\cal A$ in a Hilbert space $\cal H$ can still be mixed. It is guaranteed to be pure only relative to the algebra of all bounded operators on $\cal H$. But this algebra is far bigger than the III_1 algebra $\cal A$, and contains lots of operators that have no interpretation as observables. This is the essential difference to the case of type I algebras.

Yes, I acknowledged this in my post. Even an irreducible vector state needn't be a pure algebraic state. $\mathfrak A$ was a type $III_1$ algebra. But the fact that you're dealing with type $III_1$ algebras doesn't mean that you need to drop the standard QM formalism of vector states in irreducible representations. In fact, the pure/mixed decomposition isn't the most interesting one. People are more interested in theories with unique vacuum states and in order to get that, one should rather be looking at the ergodic decomposition with respect to the time evolution.

 

[A. Neumaier]

could you write up a post about it, aimed at people who understand the basics of Hilbert spaces, but not the technical points of C^* algebras, and all that?

The GNS construction is fully described in Wikipedia. It produces a large Hilbert space with a huge algebra of bounded operators, most of which are (typically) not in the algebra of observables one starts with.

Let me illustrate what happens in a C^*-algebra-free way (hence without GNS):

We consider an explicit representation of mixed states by a state vector in the simplest case of an $n$ level system. Here the Hilbert space $\cal H$ consists of all complex vectors of size $n$, and the associated observable algebra $\cal A$ consists of all $n\times n$ matrices. (This is the simplest example of a $C^*$-algebra, but we do not need that.) $\cal A$ has a unitary representation on another Hilbert space $\cal\hat H$, consisting of all $n\times n$ matrices with inner product $<\Phi|\Psi>:=$tr $\Phi^*\Psi$. The action of $A\in\cal A$ on $\Psi$ is simply given by multiplication $A\Psi$.

Every state of $\cal A$ is representable as a vector state on $\cal\hat H$. Indeed, the linearity of the state $\omega$ implies that we can write $\omega(A)=$tr$\rho A$ with some $\rho\in\cal A$, and the state properties then imply that $\rho$ is Hermitian positive semidefinite with trace 1. Thus it is a (typically mixed) state in the traditional sense. It has a Hermitian positive semidefinite square root $\hat\rho\in\cal\hat H.$ By construction, $\hat\rho\hat\rho^*=\hat\rho^2=\rho$. In the inner product of $\cal\hat H$, we have for any $A\in\cal A$ the relation $<\hat\rho|A|\hat\rho>=$tr $\hat\rho^* A\hat\rho=$tr$\hat\rho \hat\rho^*A=$tr $\rho A=\omega(A)$. Since this implies for $A=1$ that $\hat\rho$ is a vector of norm 1 in $\cal\hat H,$ we have represented the state $\omega$ as a vector state on $\cal\hat H$.

Thus $\omega$ is seemingly represented as a pure state on $\cal\hat H$. But of course it is still the same mixed state that it was initially! This seeming paradox finds its explanation in the fact that there is a subtle difference between the state $\omega$ (defined on $\cal A$) and the state $\hat\omega$ on $\cal\hat H$ defined on the algebra $\cal \hat A$ of linear operators $\hat A$ on $\cal\hat H$ by $\hat\omega(\hat A):=<\hat\rho|\hat A|\hat\rho>$. The precise mathematical relation between these two states is that the state $\omega$ is the restriction of $\hat\omega$ to the observable algebra $\cal A$. That the state $\omega$ can be written as a mixture of pure states applies only to observables from $\cal A$. On the other hand, $\cal \hat A$ contains many linear operators (e.g., the generator of the Lindblad equations considered in Weinberg's paper) that do not belong to $\cal A$, hence have no physical meaning as observables. The decomposition of $\omega$ as a mixture does not extend to all these other linear operators. Thus there is no conflict with $\hat\omega$ being pure.