I What does decoherence have to do with phases?

[PeterDonis]

The original edition, as far as I can find out, was in 2000. The term "decoherence" in the literature with the usage I described goes back, AFAIK, to 1970 in a paper by Zeh. IIRC Zurek's first comprehensive review was published in the early 1980s.

This review by Schlosshauer gives an excellent list of references:

https://arxiv.org/abs/quant-ph/0312059

 

[PeterDonis]

the following review
https://arxiv.org/abs/1904.06560

The references in this paper give the 10th Anniversary Edition of Nielsen & Chuang as being published in 2011, which would mean the original edition was published in 2001.

 

[PeterDonis]

what I would call mainstream physics

If by this you mean "quantum computing", I would agree that the usage of "decoherence" is not necessarily the same as the usage of Zurek et al.. But I don't think "mainstream physics" is limited to QC, and I don't think the term "decoherence" is used outside the QC community in the same way that the QC community uses it.

If there is a paper by someone in the QC community that gives a detailed argument for why the QC usage of "decoherence" makes more sense than the previous usage, I would be interested in reading it.

 

[PeterDonis]

loss of phase due to e.g. external noise

It's interesting that you describe it this way, because on this usage, the scenario described in the OP, where a controlled preparation is made of an entangled state of two qubits but one qubit is not measured, is not an instance of decoherence. But the OP claims it is.

So now I'm somewhat confused as to what the actual usage in the QC community is.

 

[Talisman]

It's interesting that you describe it this way, because on this usage, the scenario described in the OP, where a controlled preparation is made of an entangled state of two qubits but one qubit is not measured, is not an instance of decoherence. But the OP claims it is.

So now I'm somewhat confused as to what the actual usage in the QC community is.

Well, there are sub-communities within QC, based on specific implementation. Of course NMR-based labs will understand it to be T1 and T2 relaxation. Now that I've read up on those (both in N&C and Wikipedia), I see why they (T2 in particular) are described as loss of polarization information. At the same time, the example in the "mathematical details" section of Wikipedia (here) covers the more general case of taking a partial trace over the environment, which leads to the aforementioned effects regardless of whether its state is known or not (though as we've beaten to death here, of course it is generally not, given the number of DOF involved). In theoretical QC, this seems to be a common example given to students.

For me, it is useful to understand both models. I can see this may not be true for others, and I do not wish to press the issue.

 

[Talisman]

It's interesting that you describe it this way, because on this usage, the scenario described in the OP, where a controlled preparation is made of an entangled state of two qubits but one qubit is not measured, is not an instance of decoherence. But the OP claims it is.

Perhaps I should be more clear here. I agree that nobody in their right mind actually prepares an entangled state, only to "lose" one of the partners, so as to make the other look incoherent, and then cries "decoherence." But using a well-defined state (like the one given here) exemplifies the essential problem described in the "mathematical details" section cited above in the simplest way possible, for ease of understanding. It demonstrates why we lose coherence in the first particle if we ignore the second, whether that ignorance is fundamental (e.g., inability to measure the environmental state) or contrived (e.g., simply throwing it away to prove a point).

It is a pedagogical tool, and I will be more careful to call it out as such in the future, since it apparently leads to confusion.

 

[f95toli]

It's interesting that you describe it this way, because on this usage, the scenario described in the OP, where a controlled preparation is made of an entangled state of two qubits but one qubit is not measured, is not an instance of decoherence. But the OP claims it is.

So now I'm somewhat confused as to what the actual usage in the QC community is.

That was sort of my point, I don't think anyone who works on QC today would use the word in the way the OP used it. This is why I keep pointing out that -in the way the word is used in QC or say spin physics (NMR, ESR etc)- there is no direct link between decoherence and entanglement. since you can certainly have decoherence in a single system (coupled to suitable environment). Models that e.g. describe decoherence as being caused by entanglement with a measurement device obviously have their uses; but I am not sure they make much sense in situations where your T1 (and therefore T2) is limited by say dielectric losses (which is frequently .the case for solid-state qubit).
I also don't quite see how say decoherence due to the Purcell effect (another important "engineering" consideration for quantum systems) would be modelled?.

 

[f95toli]

The references in this paper give the 10th Anniversary Edition of Nielsen & Chuang as being published in 2011, which would mean the original edition was published in 2001.

Indeed, the book is ancient by QC standards; the introductory chapters covering the basic QM and math are still great, but the rest of the book is no longer very useful.

 

[PeterDonis]

the book is ancient by QC standards

Yes, but not by decoherence standards.

 

[Talisman]

For whatever it's worth, I went and revisited Zurek's original paper (or at least, one of the originals) here: https://arxiv.org/abs/quant-ph/0306072

His example is to start with a system-detector state:
$$|\uparrow\rangle|d_{\uparrow}\rangle + |\downarrow\rangle|d_{\downarrow}\rangle$$

And then entangle it with an environment state:
$$|\uparrow\rangle|d_{\uparrow}\rangle|E_0\rangle + |\downarrow\rangle|d_{\downarrow}\rangle|E_1\rangle$$

With the only requirement being that $\langle E_0|E_1\rangle = 0$ (i.e., the environment is able to distinguish the states). Wherever they came from, the key point is that those DOF are no longer accessible. Then, tracing over them, we lose the off-diagonal elements.

 

[PeterDonis]

Zurek's original paper (or at least, one of the originals)

This isn't one of his original papers; they were published in the early 1980s. This, as the initial text states, is a "revisit" of a review article he wrote for Physics Today in 1991, when he felt that the field had developed enough for him to write such a review.

His example

Which specific equations in the paper are you referring to?

 

[Talisman]

This isn't one of his original papers; they were published in the early 1980s. This, as the initial text states, is a "revisit" of a review article he wrote for Physics Today in 1991, when he felt that the field had developed enough for him to write such a review.Which specific equations in the paper are you referring to?

Ah. The equations are identical to his 1991 paper, but I did not realize he had (and cannot find) earlier ones.

Equation 13, p.10.

 

[PeterDonis]

Equation 13, p.10.

Ok, so if I try to apply this equation to the scenario in your OP, which things in the scenario correspond to which terms in the equation?

 

[PeterDonis]

I did not realize he had (and cannot find) earlier ones.

The earlier ones might not be easily findable online, since they were published in the primitive times before the Internet.

 

[Demystifier]

You can find the same explanation here (in the section titled "Decoherence") from Scott Aaronson.

Excellent!

 

[Demystifier]

Or, as Demystifier says here: https://www.physicsforums.com/threads/irreversibility-vs-reversibility.909761/
When it becomes "irreversible in practice" depends on the technology available to you.

Incidentally, the ocean analogy is used also by Aaronson to explain the idea of hidden variables. https://www.scottaaronson.com/democritus/lec11.html

 

[Talisman]

Ok, so if I try to apply this equation to the scenario in your OP, which things in the scenario correspond to which terms in the equation?

Start with a pure state (the system + detector in Zurek's example, or the single qubit in mine). Maximally entangle it (with E in Zurek's case, or a second qubit in mine). If you now ignore / discard the second system, the first must be modeled as a mixed state. The off-diagonal terms "vanished." If you somehow do manage to track down and measure all the environmental DOF thereafter, you will have enough information to reconstruct those terms, but in any real-world scenario this is impossible, so you are stuck treating it classically.

 

[PeterDonis]

Start with a pure state (the system + detector in Zurek's example, or the single qubit in mine).

A single qubit can't be a system + detector, because a detector, by definition, must be able to register a macroscopic result that a human can perceive. Zurek's description of a measurement has two stages: environment-induced decoherence is the second. The first is entanglement of the system to be measured (which could be a qubit) with the detector (which can't, for the reason given above).

I know you said that you intend your example as pedagogy, not as an actual description of a real measurement, but good pedagogy still has to include all of the essential features of the thing it's describing. That includes the macroscopic nature of the detector.

Maximally entangle it

As I understand Zurek, it is not necessary that the entanglement with the environment be maximal. All that is necessary is that the environment states that correspond to different measurement results are orthogonal.

 

[Talisman]

A single qubit can't be a system + detector, because a detector, by definition, must be able to register a macroscopic result that a human can perceive. Zurek's description of a measurement has two stages: environment-induced decoherence is the second. The first is entanglement of the system to be measured (which could be a qubit) with the detector (which can't, for the reason given above).

I know you said that you intend your example as pedagogy, not as an actual description of a real measurement, but good pedagogy still has to include all of the essential features of the thing it's describing. That includes the macroscopic nature of the detector.

Sure, that's fair. This is physicsforums, after all, and not computerscienceforums. CS people (like Aaronson) tend to look for the simplest example that captures the interesting mathematical details, even if it loses important physical details, and that can indeed be problematic.

As I understand Zurek, it is not necessary that the entanglement with the environment be maximal. All that is necessary is that the environment states that correspond to different measurement results are orthogonal.

I don't know if this is also a definitional thing, but the way it was taught to me, entanglement with orthogonal states implies maximality.

 

[PeterDonis]

the way it was taught to me, entanglement with orthogonal states implies maximality.

For entanglement of two qubits, I believe that follows from the definition of maximal entanglement in terms of Von Neumann entropy.

For entanglement of a system + detector with an environment containing a huge number of degrees of freedom, however, I don't think orthogonality of the environment states (which won't be single states but huge subspaces of the environment Hilbert space) implies maximality of entanglement.

 

[Talisman]

Here is Artur Ekert, a major figure in QIS, giving the single-qubit + environment example a year ago:

The textbook version is here https://qubit.guide/12.2-decoherence-and-interference.html

For better or worse, this usage does indeed seem to well-established in this particular field.

 

[kith]

As others have noted, "decoherence" is used differently in different communities. I've found the section "A Few Words on Nomenclature" in Klaus Hornberger's 2009 lecture notes Introduction to decoherence helpful in the past. It gave me an overview of the different usages of "decoherece" and how it relates to similar concepts like "dephasing".