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I was researching the motivation behind introducing quantum channels and this is essentially what I've gathered. Suppose we have two subsystems, the system we're interested in where states exist in the Hilbert space $\mathcal{H}_{0}$ and the environment where states exist in the Hilbert space $\mathcal{H}_{e}$ The overall system is therefore a state in the Hilbert space $$\mathcal{H}=\mathcal{H}_{0}\otimes\mathcal{H}_{e}$$ Any evolution of the overall system must be a unitary evolution, as per the fundamental postulates of quantum theory. However, the subsystems can evolve according to non unitary evolutions, which we describe by quantum channels.

Projective measurements on a state are non unitary evolutions. So my question is this. Are projective measurements the result of a unitary evolution of the overall system which includes the measurement apparatus? That is, the Hilbert space of the environment, $\mathcal{H}_{e}$, includes the quantum states of the measurement apparatus. Further, if this is the case, do we have any idea what form this unitary evolution will take?

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Yes, a channel can always be described as a unitary evolution on a larger space followed by ignoring some of the degrees of freedom. There is also a deep connection between channels and measurements. Namely, to any POVM $\{\mu_a\}$ you can associate a "measurement channel" like $$\Phi_\mu(\rho) = \sum_a \operatorname{tr}(\mu_a \rho) |a\rangle\!\langle a|,$$ and this will describe the state resulting from someone measuring $\rho$ with $\mu$, but not telling you the measurement outcome (here, the label $a$). Even more generally, you can consider different channels associated to a given measurement with different post-measurement outcomes. Some related discussion can be found on this other post of mine, and links therein.

On the other hand, if your question is "can projective measurements be described as arising from unitary evolution in a larger system?", the answer is a resounding no. What you cannot describe with a channel is the process of going from a mixture of different possible post-measurement states, to a specific one. You can use a channel to describe the loss of coherence on some input $\rho$ due to measuring in a given basis, but you cannot use a channel to "explain" why a specific measurement outcome was found. Note that I'm essentially talking about the collapse of the wavefunction here. There's no way to describe the collapse of a wavefunction using only a channel.

One simple way to see this is that a channel will always (trivially) give the same output $\Phi(\rho)$ for any given input $\rho$. But measuring $\rho$ might result in different measurement outcomes at different iterations of the experiment, so no channel can replicate this behaviour.

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  • $\begingroup$ One should stop using the term "collapse of the wave function" altogether. Nothing of the sort is happening and the term was never properly defined in the literature to begin with. A wave function is the description of a quantum mechanical ensemble. The ensemble contains an infinite number of members. Adding one more actual experimental outcome does not change its description at all. $\endgroup$
    – FlatterMann
    Apr 19 at 19:17
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    $\begingroup$ @FlatterMann I don't really know what you are saying. By "collapse" I here simply mean the process of going from a quantum state to one of the possible measurement outcomes $\endgroup$
    – glS
    Apr 19 at 19:43
  • $\begingroup$ An actual "measurement outcome" is simply an irreversible energy transfer. In optical spectroscopy it's a photomultiplier tube detecting a photon by using the photoelectric effect after the light was passing through a spectrometer grating. What does that have to do with the emission process in the atom under investigation? Absolutely nothing. The actual problem with the von Neumann approach is that it doesn't actually talk about the physics of the atom (which is QFT). It is limited to modeling the spectral properties of the receiver system (described by measurement operators). $\endgroup$
    – FlatterMann
    Apr 19 at 19:57
  • $\begingroup$ What people are forgetting about the history of quantum mechanics is that Heisenberg suggested an S-matrix approach just a few years after Copenhagen. People understood in the 1930s, already, that playing around with wave functions didn't describe physics well. What was needed was a relativistic theory which could describe actual observations (like positrons etc.), not a theory that kept toying around mathematically with "how a spectrometer can be idealized by a linear operator". What an atom is and does is QFT, not density matrix and von Neumann solution theory. $\endgroup$
    – FlatterMann
    Apr 19 at 20:01
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Consider a unitary $U$ interaction between a system $S$ and the environment $E$ on their joint state $\rho_{SE}$ on $\mathscr{H}_S\otimes\mathscr{H}_E$. To get the reduced density matrix $\rho_{S}$ for $S$ you take the partial trace of $\rho_{SE}$ over $\mathscr{H}_E$. And when the result of a measurement has been recorded the relative state of the system corresponds to one of the outcomes.

The new relative state when the $k$th outcome is seen may sometimes be described by a projection valued measure: $$\frac{P_k\rho_{S}P_k}{tr(P_k\rho_{S})}.$$ But sometimes a measurement is described by a POVM (e.g. - indirect measurements are often described by POVMs): a set of operators $E_j$ such that $\sum_jE_jE_j^{\dagger}=I$ and the post measurement relative state for the $k$th outcome is given by: $$\frac{E_k\rho_{S}E_k}{tr(E_kE_k^{\dagger}\rho_{S})}.$$ See Section 2.2.6 of "Quantum Computation and Quantum Information" by Nielsen and Chuang.

To understand measurement more generally you might want to look at the literature on decoherence, such as:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/1911.06282

https://arxiv.org/abs/1212.3245

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This may garner a lot of "animosity", but let me express this one more time: the prescription for "measurement" in quantum mechanics is that of an irreversible energy exchange between the quantum system and the measurement system. This is NOT a matter of unitary vs. non-unitary evolution as far as I can tell. Let me explain.

Physically we are taking the energy out of the quantum system under measurement and we are NEVER giving it back, which is not what a "closed" H=H0⊗He Hilbert space under unitary evolution describes. What you are setting up there is with a naive assumption about the measurement system is basically Schroedinger's cat, which says that in a completely isolated system under perfectly unitary evolution the cat will, eventually, come back to life. That Schroedinger's cat doesn't work in reality is obvious because there are no such completely isolated systems and we couldn't wait that long anyway (not to mention that such a system would, intermittently, also produce two kittens, a medium size puppy, a smallish brain in a vat and an infinity of other states).

So what happens in reality? Take the simple example of an excited atom: initially the atom is in an excited state and the electromagnetic field is in the ground state (photon number is zero). Eventually the atom will de-excite and there will be a quantum of electromagnetic field energy (a photon) and an atom in the ground state left. The electromagnetic field energy spreads out at the speed of light. Unless we are surrounding the atom with a perfect spherical mirror, that photon worth of field energy can never return to the atom. In other words: "atom" plus "empty space" is an irreversible system but the description of the electromagnetic quantum field is still perfectly unitary. Unitary and irreversible are not opposites if you allow for an infinite phase space, hence the choice can not be unitary/non-unitary. It has to be unitary-reversible vs. unitary-irreversible. Maybe that sounds pedantic, but it covers experimental reality.

This irreversible behavior is exactly what you can see in experiments on the bench. There is nothing particularly exciting about it. Nature simply tells us that the idealization of perfectly isolated/closed systems is nonsensical if we are asking the question of "What is the final state?". A final state only exists in irreversible systems. We are living in the largest possible such system. We call it the universe. It can absorb as many quanta of electromagnetic field energy as we can afford to throw at it and it will never become reversible. That is why we are surrounded by "final states" and a nearly classical macroscopic world.

So basically your entire "problem" is that you don't want to take "Yes, all physical systems are open." and "Yes, all physical processes are irreversible." for answers, even though absolutely everything you will ever see in the lab points to exactly that reality. How did they solve this in 1927? With Copenhagen cutting the existence of the system off after a measurement. Apply the Born rule and the system is gone. It solves the irreversibility problem perfectly fine by simply cutting the knot.

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    $\begingroup$ You are simply insisting on your particular interpretation choice rather than seeing that there are a whole lot of possible interpretation choices. Also, a proper understanding of Schrödinger's cat would see that the Ammonia maser is a direct physical manifestation of it---the stupid wall does not matter. Instead, the correct transition from Schrödinger's cat type experimental results to the classical type experimental results, has to be seen from the example in Anderson's "More is Different" paper. Yet, we also know that careful experimentation can revert that to Schrödinger cat style too. $\endgroup$
    – naturallyInconsistent
    Apr 19 at 2:06
  • $\begingroup$ Nitpick: if you measure the energy of a superposition of different energy eigenstates, it is possible to give the quantum system energy, rather than only ever extracting. $\endgroup$
    – naturallyInconsistent
    Apr 19 at 2:08
  • $\begingroup$ @naturallyInconsistent That there are no closed systems is not an interpretation. It's a fact. It is also a fact that "final state" requires irreversibility. That's just the usual definitions applied to measurement. I don't think any of what I am saying is controversial. It's just not being expressed this clearly in the books that are dealing with quantum mechanical solution theory. $\endgroup$
    – FlatterMann
    Apr 19 at 2:10
  • $\begingroup$ @naturallyInconsistent Yes, it is. In an introductory textbook about atomic physics that is called "absorption spectroscopy" vs. "emission spectroscopy". I am just keeping it simple. I hope people can generalize what I am saying in their own mind. $\endgroup$
    – FlatterMann
    Apr 19 at 2:12
  • $\begingroup$ That is just not experimentally proved facts. It is typical to state whether you assume the entire observable universe as a closed or an open quantum system. i.e. one can be agnostic about that, and I am taking that viewpoint. $\endgroup$
    – naturallyInconsistent
    Apr 19 at 2:12
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I am not familiar with quantum channels but I know a little about this. You are definitely missing a bit more, namely, that you should also explicitly include the measurement apparatus, separately from the environment. Due to the fact that there is no such thing as a purely classical object, in other words, both the measurement apparatus and the environment are all quantum too, we have to consider that $$ \mathcal H_o \, \otimes \, \mathcal H_m $$ is an open quantum system, where $\mathcal H_o$ is the Hilbert space of the object, and $\mathcal H_m$ is the Hilbert space of the measurement apparatus. By the purification of quantum states, we know that all such open quantum systems can be seen as a part of a bigger closed quantum system, the missing part we can safely assume to be the environment. i.e. $$ \mathcal H = \mathcal H_o \, \otimes \, \mathcal H_m \, \otimes \, \mathcal H_e $$ is the relevant complete Hilbert space we are taking as the stage. Here is a good time to discuss a small detour: We usually think of the measurement apparatus as a macroscopic device, in which case we would need to separate the ``pointer states" as macrostates. That would be quite annoying to discuss. Instead, we might follow the original pioneers and consider microscopic systems as the measurement apparatus for simplicity.

Once we have the complete Hilbert space $\mathcal H$, basically everybody can accept that the correct next step is to unitarily evolve the entire object+measurement+environment in time. This causes all 3 things to be entangled with each other.

From here on, there are plenty of interpretation choices. The most natural is to take the decoherence choice (which is common to Many Worlds, Pilot Wave, and quite many others), that the unitary evolution led us to simple object+measurement states (each) entangled with complicated environment states. Because the environment quickly devolves into much more entangled states, this means that undoing the entangling happening in the environment is incredibly unlikely to happen, so that then each object+measurement state essentially will no longer interfere with other such object+measurement states. This has the attractive properties that

  1. the process of measurement is not mystical
  2. the process of measurement (including the ``collapse") occurs over finite time
  3. it has been simulated and results agree with experiments
  4. the theory is internally consistent
  5. it explains why modern experiments with tighter and tighter noise controls can see bigger and bigger stuff being put in quantum superpositions, as opposed to Anderson's interesting example in his ``More is Different" paper that after a certain size, the quantum superpositions get washed out.

The mathematically standard thing to do here, is to form the density operator of such a triply entangled pure state, and then do a partial trace over the environment. That then gives a impure state, a mixed density operator over object+measurement. Each pure object+measurement state composing the impure state would evolve independently.

It is when you want to extract the independent pure object+measurement state do you actually impose the projection. Of course, once you apply projection, it will be compatible with all accepted interpretations, since they all agree with the Copenhagen prescription.

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  • $\begingroup$ The problem with all of that is that entanglement is not a measurement. Collapse is not even a well defined technical term and all measurements are, by definition, infinite in duration. Whoever told you that a measurement takes a finite time, or worse, no time at all, simply didn't understand what a measurement is. No experiment has evert seen "superposition". Superposition is a mathematical property of Hilbert space. It doesn't exist in reality and it is most certainly not a noise issue. $\endgroup$
    – FlatterMann
    Apr 19 at 4:35
  • $\begingroup$ It is your beloved Copenhagen interpretation that is insisting that a measurement instantaneously collapses a quantum system that was initally in a superposition. But I suppose you are still awaiting your infinitely long measurement to tell you this. Please keep to standard physics. Superposition is incredibly supported by experimental evidence. $\endgroup$
    – naturallyInconsistent
    Apr 19 at 4:40
  • $\begingroup$ Copenhagen doesn't talk about collapse. It simply tells you how to calculate the probabilities for the ensemble. That's it. A quantum system is never in superposition. The ensemble of the system can be in superposition, but even that is physically meaningless because if I change the base then I can collapse a sum in the old base into a single function in the new one. This is all just paper pushing by the solution theory. It's not physics. A Fourier transform is not physics. It is something we do on paper to solve equations. $\endgroup$
    – FlatterMann
    Apr 19 at 4:46

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