What justifies the use of braket notation to label "macrostates?" Is there a rigorous procedure or is it just a heuristic?

Question

I've had this question in mind for a while, but this recent post reminded me to ask. The post is interesting because it demonstrates how a very naive application of labeling states (without understanding unitary time evolution) with braket notation can lead to false conclusions.

My question is, however, what exactly justifies the use of "$|\text{alive}\rangle$" and "$|\text{dead}\rangle$" labels in the Schrodinger's cat thought experiment? Doesn't "$|\text{alive}\rangle$" represent a whole continuum of possible states?

Note my question is decidedly not about Schrodinger's cat (I don't want to get side-tracked into the details of that). We could alternatively consider labels of proton states, which are themselves made of quarks that can be in many different states.

What is confusing is that $|\text{proton state }1\rangle$ might not just correspond to a single definite quantum state, but possible multiple ones (just live how a "dead" cat macrostate has many possible microstates), but then it seems like then we are describing mixed states and then treating them as if they were pure.

How can we be sure our labeling of the entire proton is "sound" in the sense that our "macrostate" labels are faithful to the "microstates?" For example, how can we be sure that $|\text{proton state }1\rangle$ "doesn't leak out" into $|\text{proton state }2\rangle$ (much the same way we fallaciously evolved $|\text{alive}\rangle\rightarrow |\text{dead}\rangle$ and $|\text{dead}\rangle\rightarrow |\text{dead}\rangle$ in the above linked post)? Moreover, how do we know there is no ambiguity as to whether a microstate belongs to $|\text{proton state }1\rangle$ or $|\text{proton state }2\rangle$ if we do come up with a labeling scheme?

Answer 1

It's wrong to use $|\text{dead}\rangle$ to mean "any state that has the property of containing a dead cat". But it's okay to say that $|\text{dead}\rangle$ is a state that contains a dead cat, without going into more detail about which state it is, as long as you consistently use that name for the same state. The problem in the other question was that it used the same name for more than one state.

Note that this contradicts Quantumwhisp's answer, which I think is wrong. In quantum mechanics, you can't simply ignore degrees of freedom that you don't/can't measure. You can trace them out to get a mixed state, but you can't just leave them out of your labeling of a pure state (and in Dirac's notation, $|\cdots\rangle$ always denotes a pure state).

I think your question is how you can know the action of the time evolution operator on the huge-dimensional space of live/dead cat states well enough to apply any labels. For some purposes, perhaps you can't. For other purposes, it's enough to say that some live-cat state evolves into some dead-cat state, omitting the details. You can usually assume that states reached by different "macroscopic paths" are orthogonal, or as near to it as makes no difference.

In the case of a proton it's easier because all degrees of freedom are straightforwardly measurable. Even though it's technically a complex soup of quarks and gluons, its ground state is unique up to position/momentum and spin direction, and there is a discrete (again up to those properties) spectrum of excited states that can be distinguished by mass. So I think the cat is a better example.

Answer 2

Such a labelling is only justifiable if we can do quantum superpositions with them, and it is easier to experimentally, operationally, figure out what is allowed and what is not. I would consider such a thing as something that can stump us upon first encounter, but should be obvious upon hindsight.

But first, we should point out a few preliminaries. Most importantly, a macroscopically large thing is not necessarily classical/``macrostate"!

We should also consider how we usually ``abuse" such notation in practice. When we talk about quantum optics, say, for basic one-photon experiments, we have a laser source, and with beam splitters and polarisers, we split the beam into $\left|A,\updownarrow\right>$ or $\left|B,\circlearrowleft\right>$, and by these we mean that the beam is currently moving along paths $A$ or $B$, and the other label is for vertical polarisation and left circularly polarised, respectively. The thing to note here is that the paths are macroscopically large and well separated from each other. All is well as long as we can bring them back to destructively interfere again. It does not actually matter that $A$ and $B$ really refer to, say, Gaußian beams of certain size and shape, and thus are "macrostates" of many different possible similar forms. What matters here is that, in any single experiment, the beam profile for either pathway is the same, i.e. they are in some specific microstate inside that macrostate, so that interference is possible. Note that $A$ and $B$ can be pointing in the same direction, and thus be nonsense if we consider infinitely large plane waves.

Next, we consider how people talk about decoherence. The pointer states of a measurement apparatus is incredibly obviously a macrostate. Actually, if you think a bit more about what the analysis is doing, it is that we are really using the states of the observed system as the macrostate labels of the measurement apparatus. Luckily, the aim of studying this is to quickly see that the random phases from the environment makes it practically impossible to later interfere the separated states again, and in this way, the measurement apparatus and environment combination causes our observed system to decohere into separate branches, if you would pardon my invocation of language from Many Worlds Interpretation (which, I emphasise, I am not in favour of). The universe's wavefunction could still be singular and unitarily evolved in time, but for the experimenter, the resulting experimentally relevant state to consider, can no longer be expressed by one wavefunction, and must instead be written as a density operator using Born probabilities.

Unless, of course, like in modern quantum computation experiments, we find some way to control the quantum noise, i.e. reverse the accumulated phases and make sure that the interference pattern continues to appear. This is where it gets interesting, so we need some mathematical treatment. One thing that seems to be needed for physics/maths to work for our universe is some form of reductionism, that we may always talk about complex systems by first considering simpler systems and combining them, even though the discovery of quantum entanglement itself forced us to abandon ``spatially separated and forces compensated/minimised" as a good enough isolation. By this I mean that we are still able to study, say, finite dimensional Hilbert spaces and combine them to talk about bigger systems. So, assuming that the observed system is simple enough to just be described by a simple wavefunction for simplicity, then the state of the universe can be written as $$\left|\psi\right>_s\otimes\left|\phi\right>_r\otimes\left|\chi\right>_i\tag1$$ where the subscripts mean the system, the relevant other degrees of freedom of the system, and irrelevant degrees of freedom, respectively.

The important part is to note that, when you design an experiment so that, after some manipulations, you want to interfere the system again, it is important to keep the relevant degrees of freedom pristine, whereas the irrelevant parts can go to hell. That is, the universe may unitarily evolve the state to $$\tag2 \left|\psi_1\right>\otimes\left|\phi_1\right>\otimes\left|\chi_1\right>\qquad\text{and}\qquad \left|\psi_2\right>\otimes\left|\phi_2\right>\otimes\left|\chi_2\right> $$ and if that is the case, the difference between $\left|\phi_1\right>$ and $\left|\phi_2\right>$ will cause the system to fail to interfere. Control of the quantum noise really means that we can take these systems, and ``reverse" the noise in these degrees of freedom, so that the actual state can be treated as $$\tag3 \left[e^{i\theta_1}\left|\psi_1\right>+e^{i\theta_2}\left|\psi_2\right>\right]\otimes\left|\phi_c\right>\otimes \left[e^{i\varphi_1}\left|\chi_1^\prime\right>+e^{i\varphi_2}\left|\chi_2^\prime\right>\right] $$ where the important part is that the relevant degrees of freedom have become a common state $\left|\phi_c\right>$ so that interference can happen by tweaking $\theta_2-\theta_1$ (always doable), even though the actually realised states are $$\tag4\left[ e^{i(\theta_1+\varphi_1)}\left|\psi_1\right>\otimes\left|\chi_1^\prime\right>+ e^{i(\theta_2+\varphi_2)}\left|\psi_2\right>\otimes\left|\chi_2^\prime\right> \right]\otimes\left|\phi_c\right>$$ that is not manifestly entangled if you trace out the irrelevant degrees of freedom.

Note carefully that there is a whole list of stuff that is being covered here. First of all, the control of the quantum noise converts both $\left|\phi_1\right>$ and $\left|\phi_2\right>$ into $\left|\phi_c\right>$ by converting $\left|\chi_1\right>$ to $\left|\chi_1^\prime\right>$ and $\left|\chi_2\right>$ to $\left|\chi_2^\prime\right>$ and phase factors. This is basically completely general and so this part should be tolerable. It is kind of like in thermodynamics, you have some heat reservoir to absorb the unwanted entropy.

It is also important to note that we have not actually identified what degrees of freedom are relevant and what are irrelevant, only assumed their existence and partition. This is a big assumption, but one that is operationally shown to exist: What we do is to just try things out experimentally, until some sufficiently good control of quantum noise restores the interference pattern, and if there is no such thing as irrelevant degrees of freedom, then one has to seriously wonder why there can ever be interference patterns at all, because this is really always happening if we want to do experiments, especially when you consider an experiment whereby the interference pattern is obtained by waiting and repeating data acquisition.

i.e. it is incredibly important that it is possible to do the bait and switch from the actual entangled wavefunction, to the one whereby the irrelevant degrees of freedom are separated out. This is because, once the partial trace is taken, or the random phases are averaged out, if the result needs to be written as a density operator, then there is no way back to a wavefunction description. If we are to stay with pure states, the Cartesian product state must appear to be factorisable, so that the partial tracing leaves us with a gigantic density operator that is a Cartesian product of the pure state part and the irrelevant part, and then we can pretend that we are dealing with a pure state, ignoring to even write down the relevant degrees of freedom.

Considering how much physical content is being packed into this, it should be unlikely that we can state this in a rigorous manner, but it is at least not total abject nonsense.

If you accept all of these, then you can see that there is actually quite a lot of ``macrostate" looking things that can be discussed as if they are much simpler than they actually are. Note, however, that if you assume that every stable nuclei has a unique ground state, then even though the proton is a complicated composite body, its ground state is what we usually label, and so it is not an example of the "macrostate" looking things that I am talking about. It just so happens to actually be a labelling of a specific microstate that is conveniently available without any need for nuance; excited states thereof will behave differently and labelled differently, and thus not need further help.

One should thus properly treat Schrödinger's cat states as seriously as we can, because it actually is trying to tell us something about the universe we live in. It is all too easy to find actually realised versions of these things; might as well thoroughly understand it once and for all, and not have to keep bashing head against walls over the many realisations.