Quantum Decoherence

In the last two posts (Quantum Measurement and Wavefunction Collapse), I’ve been exploring the notorious problem of measurement in quantum mechanics. This post picks up where I left off, so if you missed those first two, you should go read them now.

Here I’m going to venture into what we mean by quantum coherence and the Yin to its Yang, quantum decoherence. I’ll start by trying to explain what they are and then what the latter has to do with the measurement problem.

The punchline: Not very much. (But not exactly nothing, either.)

In order to understand decoherence it’s necessary to understand coherence, and to understand that we need to talk about phase. (Don’t be fazed, it’s not that bad.)

The starting point is the wave-like nature of matter. This is most apparent with light, where the wave/particle duality is ancient and relatively easily demonstrated. But ever since Louis deBroglie (pronounced “deh-broy”) we view all matter as having this duality. [For more, see What’s the Wavelength?].

As an aside, light, especially seen as photons, is matter. Light has energy but is a form of matter. Energy is a property all matter can have. All types of light, from radio waves to x-rays and gamma rays, are matter. What distinguishes them is their energy, which, in light, is expressed as frequency. Radio waves are low frequency (low energy) while x-rays and gamma rays are high frequency (high energy).

Frequency, casually speaking, is how often something happens. In physics, frequency refers to regularly occurring phenomena, such as swinging pendulums, vibrating springs, sine waves, or spinning objects. Here we’re interested in the last two, in particular the relationship between them:

Generating a sine (and cosine) wave

Figure 1. Generating waves with a wheel (red=sine wave, blue=cosine wave). [from Wikipedia]

There are three things to note in Figure 1.

Firstly, the direct relationship between the generated waves (red and blue) and the spinning wheel (green). This correlation is fundamental. Electrical power from the power companies waves in this manner because it’s generated by the spinning wheels of electrical generators.

Secondly, note that the peaks and valleys of the two waves don’t line up. This is because the red sine wave is a horizontal projection of the rotation while the blue cosine wave is a vertical projection. The vertical axis is 90° relative to the horizontal axis, so the blue wave is 90° out of phase from the red wave.

If the peaks and valleys lined up, we’d say the waves are exactly in phase or have a phase angle of 0° relative to each other. On the other hand, if the peaks of one matched the valleys of the other, we’d say their phase angle was 180°, which is as out of phase as it’s possible to get. They would be exactly out of phase, the opposite of being exactly in phase.

Figure 2. Waves that are exactly (180°) out of phase.

Obviously, projecting both waves onto the horizontal axis in the same direction generates identical in phase waves. Projecting one onto the horizontal axis in the opposite direction — 180° difference — generates a wave that is 180° out of phase. By varying the projection angles, we vary the phase angle.

As an aside, note that only the in 90° case can we say the blue wave is the cosine wave relative to the red sine wave. (In general, a wave generated by circular motion is called a sine wave.)

Thirdly, because both waves are generated by the same spinning circle, the distance between their respective peaks is the same for both. That distance is their wavelength and how frequently those peaks pass a fixed point is their frequency.


Phase is the angular distance between two waves relative to specific matching points on each, for instance their respective peaks or valleys or any other spot we identify.

Because of the relationship between sine waves and spinning wheels, phase is also the angular distance between two points on the wheel. If we think of a wheel as a clock, then, relative to 12 o’clock, 3 o’clock is 90° out of phase and 6 o’clock is 180° degrees out of phase. Above I said that 180° is as out of phase as possible — 6 being on the opposite side from 12. While 9 o’clock might seem like it’s 270° out of phase and, thus, more, it’s also -90° degrees, which is the same phase difference (in reverse) as 3 o’clock.

To segue into the notion of coherence, consider generating these waves with two spinning wheels, both projected left on the horizontal axis (like the red wave is). If the clocks were in phase — both striking 12 o’clock at the same moment — then the generated waves would also be in phase. To generate the above animation, the blue wave’s clock would have to strike 3 o’clock at the same moment the red wave’s clock struck 12 — the clocks would have to be 90° out of phase.

As an aside, we can calculate angular distance in degrees, which are arbitrary units (like inches or pounds), or we can use the natural units of radians where 180° degrees equals pi radians, and a full 360° is 2*pi radians. In physics we generally use radians. (One radian is about 57°. It’s the angle we get by moving along a circle the same distance as its radius.)

There is one more very important way we could control the phase between the two waves. Imagine generating two waves from one clock using the same projection. The waves would be exactly in phase but if one wave took a longer path to some comparison point, at that point they would be out of phase. How much out of phase depends on the relative lengths of the two paths.

Figure 3. Two waves (red and blue) from two clocks with different frequencies. They are in phase only momentarily at 0.0, +2.0, and +4.0 pi radians. They are (again momentarily) exactly out of phase at +1.0 and +3.0 pi radians. The green line traces the phase difference from 0° to 180° (aka 1.0 pi radians). (Note the scale difference there.)

Lastly, if the two clocks ran at different speeds, even if the generated waves were both from the same projection, then the phase angle constantly changes. How fast it changes depends on the clock speeds relative to each other. In the extreme case, think of how a stopped clock is right twice a day compared to a normally running clock.

§ §

Coherence is a broad topic (see its Wiki disambiguation page), but as the Yang to the Yin of decoherence, we can think of it as the property of a wave-like system that allows consistent interference effects. A crucial point is that the phase of a coherent system remains controlled and stable. In a decoherent system the phase is unstable, usually due to the effects of some other uncontrolled system.

One example of a coherent system is laser light. It has amazing and useful properties because it is a coherent form of light. Its photons march in lockstep and have consistent phase.

In the quantum realm, phase and interference are aspects of the complex numbers used to describe quantum systems. These act like tiny clocks (with the caveat that, in some cases, the clocks are stopped, and we care about where they are stopped). This is why quantum math has so many constructs that look like this:


This complex exponential describes a “wheel” with frequency θ (theta). The x can stand for a rich variety of things depending on the system we’re describing. The t, if included, is the time component. In particular, note the i, the imaginary unit. Here again, complex numbers are important in quantum math. [See Circular Math and its many links (especially Beautiful Math) for more.]

For now, suffice to say constructs like this describe waves (which correspond to clocks or spinning wheels). We won’t delve further into the mathematics of it.


Quantum coherence is important in two regards: Firstly, it allows interference effects such as seen in two-slit and interferometry experiments. Secondly, as a “stopped clock”, it’s a critical part of quantum states (such as used in quantum computing).

Both of these are worthy of (multiple!) posts on their own. Here I’ll only summarize them so we can move on to decoherence and what (little) it has to do with measurement.

In two-slit experiments, a matter wave passes through two slits and interferes with itself to produce distinctive interference bands. While there is a classical wave description involving constructive and destructive interference, the quantum version is puzzling because, since we don’t really know what a matter wave is, we don’t really know what is interfering.

The presence of the bands tells us something is, but the phenomena isn’t like the classic version because what we see is based on probabilities, which remember, are the squares of the wavefunction at each point on the screen or detector. That wavefunction, by the way, contains complex exponentials similar to the equation above, and it’s these spinning wheels that interfere. The dark bands, unlike in the classical description, are not cancelations of wave energy but cancelations of probability.

Interferometry experiments, typically some form of the Mach–Zehnder interferometer, are more sophisticated and scientifically useful versions of the two-slit experiment. (The latter are mostly useful for just demonstrating the wave-like nature of matter.)

In quantum computing, the complex exponential describes the state of a qubit, and here we typically want a “stopped clock” — a fixed state. Recall that a (pure) quantum state is described by a vector with a length of one. The state is defined by where the vector points (see previous post) as well as by its phase.

In the Bloch sphere (image at top of this post), phase is represented as rotation around the sphere. (The basis vectors are the vertical axes labeled |0〉 and |1〉. See QM 101: Bloch Sphere for more.)

§ §

This brings us to decoherence. As the name implies, it’s the loss of a system’s coherence.

Two-slit and interferometry experiments depend on the phase of the waves being unperturbed. They require constant coherence. If environmental effects perturb the system enough, the interference effects are lost.

In quantum computing, if the phase of the qubit is perturbed, then the state of that qubit is no longer what it should be. This is similar to randomly flipping bits in classical computing — the computation becomes corrupted. Much of the engineering of quantum computers involves preserving the coherence of its qubits.

Note that, with regard to individual “particles” in a two-slit or interferometry experiment, altering the phase of their wave doesn’t mean it no longer interferes with itself. It means the interference shifts randomly due to the perturbation. If that happens to all the “particles”, the random nature of environmental influence smears the banding, destroying the overall effect.

So, quantum decoherence is the corruption of a quantum system’s phase by mixing it with the random phases from the environment. In turn, the quantum system’s phase is dispersed into the environment (imagine a tiny drop of ink in five gallons of water).

If coherence is a property a system can have, decoherence is a process it can experience.


Finally, what does decoherence have to do with measurement? Not much. At least, not much with the measurement itself. But I think it has a lot to do with the divide between a quantum system and the instrument that measures it.

In any large system, the phases of the individuals are dispersed, not just into the environment, but among the myriad individuals of the system. (Recall the 10²⁷ singers.) A large system has decohered and, thus, acts not as quantum system but as a classical one. In particular, quantum behaviors such as superposition and interference are no longer possible. (This is why cats are always either alive or not.)

Bottom line, when a classical system measures a quantum system, the effect is as if one more singer joined the crowd — that singer’s song is utterly and completely swamped by the “white noise” of the crowd.

§ §

Next time I’ll talk about measurement in some specific contexts, such as the infamous Schrödinger’s Cat experiment.

Stay coherent, my friends! Go forth and spread beauty and light.

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

51 responses to “Quantum Decoherence

  • Peter Morgan

    Your comment that “Electrical power from the power companies waves in this manner because it’s generated by the spinning wheels of electrical generators”, although usually true, put me into curiosity mode. I arrived at https://en.wikipedia.org/wiki/Power_inverter by way of https://en.wikipedia.org/wiki/Electricity_generation#Photovoltaic_effect. Solar energy is generated as DC, which is usually converted to AC without mechanically moving parts, although the physics of a power inverter somehow still has to provide the same mathematical effect.

    • Wyrd Smythe

      Indeed. AC is necessary for transmitting the current through the many power transformers along the way. Transformers only work with AC and usually are “tuned” to work most efficiently with AC of the expected frequency. That’s not the only reason we need AC from DC sources. Many home devices require it. Fan and other motors depend on it and so do many clocks (which use it as a timing source). Devices such as TVs require AC power to create the low-voltage DC that runs the device. What’s more, if you have a UPS, you have a system that stores (DC) power in a battery, and, if you lose the incoming AC, the UPS kicks in, inverting that DC into the AC your home devices expect.

      The concept is fairly old. Search for [inverters for cars] or [inverters for campers] and you’ll get many pages of hits. It has long been possible to buy inverters that connect to your car’s 12VDC battery and produce enough wattage to power a few “household” devices that depend on the voltage (and in such cases as mentioned above, the AC as well).

      Speaking of which, modern technology often shifts from mechanical to non-mechanical. Early inverters often used a tiny spinning or vibrating mechanism to create the AC, but modern ones are usually solid state.

    • Wyrd Smythe

      As a further thought to your last bit about the physics of an inverter and my last bit about modern technology replacing mechanical devices with solid state ones, often the circuitry of a modern inverter has something along the lines of a resonant circuit that acts like a harmonic oscillator — effectively an electronic wheel that generates a sine wave.

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  • Wyrd Smythe

    Here’s a pretty good video about decoherence from Sabine Hossenfelder:

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  • Aditya Prasad

    It turns out that “decoherence” has many uses in the literature, and they aren’t as closely-related as one might think. You’ve captured some of its common usage (in particular, for NMR), but in QIS it’s taken on a different (and I think simpler) usage. If you’ve learned about density matrices, I’d be happy to explain it.

    • Wyrd Smythe

      Hello and welcome. Yes, I’ve encountered how “decoherence” almost seems a catchword. I’m especially askance at how “decoherence” supposedly explains how, under the MWI, matter dodges the Pauli Exclusion Principle. I have a rough understanding of density matrices, so by all means, I’d love to hear what you have to say.

      (As an aside, “entanglement” is another concept that seems used in different ways. Sometimes as what Roger Penrose calls “quantanglement” — a wavefunction describing two particles where a measurement on one affects the other — and also, as far as I can tell, just to mean quantum information being dispersed into the environment. In the latter, a measurement on the particle (or the environment) would not affect the so-called “entangled” parts. Confusing!)

      • Aditya Prasad

        Regarding density matrices, the book “The Structure and Interpretation of Quantum Mechanics” by RIG Hughes is what made it all click for me. It turns out to be really simple, though I can’t do it justice in a comment.

        Here’s an equivalent formulation that doesn’t use density matrices:

        Take the qubit |0>+|1>. Measure it in the basis {|0>+|1>, |0>-|1>}. Discover you get the first element with 100% chance. It is as though the two “components” of your qubit interfere: https://www.scottaaronson.com/democritus/lec9.html

        Now entangle it: |00> + |11>. If you measure either individual qubit in the preceding basis, you’ll now get 50-50 results, just as you would for a classical bit: https://www.scottaaronson.com/democritus/lec11.html. The interference is gone!

        The point is that any *sub*system appears to be classical, even though the overall state is still pure. Therefore, if you can’t access even a single one of the entangled qubits, you can’t exhibit quantum behavior any longer.

        I would work through that example carefully, because that’s what finally made things clear to me. You’ll find that it’s a bit of a pain, which is why density matrices were invented. They’re a clever way to encapsulate the same information.

        Never heard about this bit about the Pauli Exclusion Principle. Have a reference?

        Entanglement should mean the same thing in all contexts. The only confusing part is the details of the entangled systems. Entanglement with single qubits is easier to understand, but is in principle not different from entangling with fields (such as the gravitational field, which doesn’t have a force carrier in the standard model — though IANAP, so I have limited understanding here).

      • Aditya Prasad

        Basically, you have to understand how the individual systems are modeled. Are you familiar with the sense in which both qubits and wavefunctions are vectors in Hilbert spaces? The former is 2-dimensional, and the latter is (uncountably) infinite-dimensional, and so they require different descriptions (since nobody likes writing down tuples with infinitely many elements).

      • Wyrd Smythe

        Keeping in mind I’m learning this as I go, so it’s highly likely I’ve misunderstood you…

        “Take the qubit |0⟩+|1⟩. Measure it in the basis {|0⟩+|1⟩, |0⟩-|1⟩}. Discover you get the first element with 100% chance.”

        The link to Aaronson’s text. The section below “Exercise 2 for the Non-Lazy Reader:” — is that the section in question?

        There aren’t coefficients for the |0⟩+|1⟩ state. I assume 1/√2(|0⟩+|1⟩) — the “positive” superposition of |0⟩ and |1⟩ — and certainly if measured in the {|0⟩+|1⟩, |0⟩-|1⟩} basis would produce |0⟩+|1⟩. It has to because Ψ is already in that state. Same as having Ψ=|0⟩ and measuring in the {|0⟩,|1⟩} basis always returning |0⟩.

        What Aaronson is speaking of in that section is, as I understand it, slightly different. Given a starting state Ψ=|0⟩, applying a 90° rotation by using what looks like a Hadamard gate (except the minus would be in the lower right) does create the state 1/√2(|0⟩+|1⟩) and a further rotation by the same gate does result in a final state of |1⟩.

        Which, yes, I can see is a demonstration of quantum interference, and that requires a coherent system, but I’m not connecting the dots to how decoherence enters the picture. Except that if the qubit were to decohere, its state would be randomized. No doubt I’m just not getting it.

        “Now entangle it: |00⟩ + |11⟩. […] The interference is gone!”

        I feel I’m totally misunderstanding, because this is a very different situation to me. Now we have two qubits that are maximally entangled. Measuring either indeed gives 50/50 odds of |0⟩ or |1⟩. And immediately forces the other qubit to be the same.

        But I don’t see the connection or how this applies to decoherence. Bell pairs do have to remain coherent for the entanglement to survive. If the system decoheres, the entanglement vanishes.

        This linked page has material about the MWI and decoherence that I want to read carefully when I have a chance. Looks interesting!

        As far as density matrices goes, my understanding is that phase information is carried in the off-diaogonal members (usually in the form of complex exponentials). Decoherence is when those vanish or become real numbers.

        “Never heard about this bit about the Pauli Exclusion Principle. Have a reference?”

        Here’s the Wiki page. It’s the idea that fermions can’t share the same state (but bosons can). It’s why the periodic table — the electrons of atoms must each have their own state so they form shells.

        But in the MWI matter overlaps. Infinite worlds overlap. How is that possible if fermions can’t share the same state?

        “Entanglement should mean the same thing in all contexts.”

        Perhaps, and again this may be my misunderstanding, but it seems as if there is what I mentioned Roger Penrose calls “quantanglement” — which involves, for instance, bell pairs that are described by a single inseperable quantum state — and the more common entanglement that seems to just mean “all mixed up.”

        For instance, as far as I know, photons that bounce off an object, and are which sometimes then said to be “entangled” with it, may carry information away from the object, but measuring those photons doesn’t affect the object. The photons aren’t linked to the object the way bell pairs are linked.

        I can’t help but think Penrose came up with the term to distinguish from the “all mixed up” kind of entanglement. (Like we might say a box of rubber bands is entangled.)

        “Basically, you have to understand how the individual systems are modeled.”

        Absolutely agree! I think it’s critical to consider actual physical systems. I’m a hard-core realist, and I want to know what’s really going on!

        “Are you familiar with the sense in which both qubits and wavefunctions are vectors in Hilbert spaces? The former is 2-dimensional, and the latter is (uncountably) infinite-dimensional,…”

        Yes, and yes. Finite summations version integrations. I’ve posted about the Bloch sphere and two-state spin systems (especially photon polarization).

        (My ultimate goal in studying QM is learning to solve the Schrödinger equation and write some software to create an animation of the two-slit experiment. But learning to actually solve partial differential equations turns out to be very hard (for me). Not sure I’ll reach that goal, but I’m enjoying the learning so far.)

      • Aditya Prasad

        Alas, this is a poor medium for communication!

        Yes, I know what the Pauli exclusion principle is, but I had never heard about a connection to MWI. The whole point of them being “different worlds” is that they don’t belong to a single world, where things like PEP apply. But this conversation may take us too far astray of our original goal.

        I’m leaving out normalization constants because they’re too hard to type. Sorry!

        The state |0> will measure as |0>+|1> half the time. So will the state |1>. Naively, one might think that the state |0> + |1> could be treated as a classical combination of those two states, in which case it would *also* measure as |0>+|1> half the time. Of course, it does not — and when you try to see why not, you will find yourself using negative and positive amplitudes that cancel. This can be considered the simplest example of interference.

        An environment might now entangle with your qubit, producing the Bell state. *Now* if you try to calculate the above odds, you get 50/50 again — just like for a classical particle. Therefore, entangling with an environment destroys interference (and the degree destroyed depends on the maximality of entanglement).

        If the photon in your example does carry away information about an object, then the photon is entangled with that object in exactly the usual way you think of “entanglement.” Knowing the state of the photon gives you corresponding information about the object in precisely the way a Bell state does (but again, depending on *how* entangled they are).

      • Wyrd Smythe

        “Yes, I know what the Pauli exclusion principle is, but I had never heard about a connection to MWI.”

        Oh, sorry, I misunderstood what you were asking. It’s not something I read, but a question that’s occurred to me. Under the MWI, the many worlds are supposed to be taken as physically real, but I don’t understand how an infinite number of roughly identical worlds can overlap physically.

        There may be a different world wavefunctions superposed, but how does a world wavefunction apply to individual electrons? Those have only a certain number of quantum properties, and the PEP applies to that set of properties. So, what extra quantum property gets around the PEP?

        The only answer I’ve ever found is “decoherence”, but I can’t see how that works. (Maybe Aaronson’s webpage will finally provide an answer.) If two physical systems have decohered, for instance myself and the chair I’m sitting in, the last thing they can do is coincide physically.

        But, yeah, this is a distraction from the discussion. Just something about the MWI that bugs me.

        “The state |0⟩ will measure as |0⟩+|1⟩ half the time. So will the state |1⟩.”

        If measured in the {|0⟩+|1⟩, |0⟩-|1⟩} basis, I agree. (It might be worth mentioning that it’s not possible to actually measure on that basis. We can’t measure a superposition. What’s typically done is applying a rotation operator such that a (|0⟩+|1⟩) state rotates to a |0⟩ state and a (|0⟩-|1⟩) state rotates to |1⟩. Those we can measure.)

        “Naively, one might think that the state |0⟩ + |1⟩ could be treated as a classical combination of those two states, in which case it would *also* measure as |0⟩+|1⟩ half the time.”

        The thing is, when treating the system in the {|0⟩+|1⟩, |0⟩-|1⟩} basis, the state (|0⟩+|1⟩) is an eigenstate of that basis, so it will always measure as that state. In the {|0⟩+|1⟩, |0⟩-|1⟩} basis, the (|0⟩+|1⟩) state isn’t a superposition. In that basis, the |0⟩ and |1⟩ states are the superpositions. Which is why you’ll get them half-and-half if measuring (|0⟩+|1⟩) or (|0⟩-|1⟩) states in the {|0⟩, |1⟩} basis.

        I was going to do the math for the example Aaronson has in Lecture 9 when I got a chance. I’ll put it in a comment once I do. It might help clarify things. Or at least show how I see it, for whatever that may be worth.

        “Knowing the state of the photon gives you corresponding information about the object in precisely the way a Bell state does.”

        But does measuring that photon and obtaining some information about the object change the object’s wavefunction? It does with Bell pairs.

      • Aditya Prasad

        Ah, but it IS possible to measure in that basis — in fact, this is precisely what is going on in the spin-1/2 case. |0> = |z+>, √2/2(|z+>+|z->)=|x+>, √2/2(|z+>-|z->) =|x->.

        You are of course correct about being an eigenstate of the basis in the second example. This points to a fact that’s often overlooked in undergrad QM: “superposition” and “interference” are subjective labels (as can be seen in the spin case, where |x+> can be seen as a superposition or not).

        Yes, measuring the photon changes the wavefunction of the object, just as with any entanglement. Of course, this is what gives rise to “psi-epistemic” interpretations: they say that it is only our *information* that changed, since obviously “real effects” cannot propagate FTL.

      • Wyrd Smythe

        Ha! Well, yes, with spin states one can move the entire Stern-Gerlach device or rotate the polarization filter. In which case, it’s still eigenstates, not superpositions. (In quantum computing, as I understand it, qubits can only be measured on the {|0⟩, |1⟩} basis, but I’ve only yet dipped my toe in QC.) I think we’re on the same page here.

        Superposition is certainly relative to the basis. I’m not sure how much interference is. As I understand it, it depends on relative phase between two superpositions. I’ve never been entirely happy with that description due to Aaronson, the first one you linked to. (FWIW, I did the math. See below.) It seems more a special case of interference than the usual cases with two-slit experiments and beam-splitters. Start with a known state and rotate it. Then rotate it again. I think I do see the point being made, and perhaps you’re right I just need to really think about it. (But I find that many times things that are mathematically identical don’t to me seem physically identical, and as I mentioned, I’m a hard-core realist.) I worry that QM sometimes gets too lost in the math.

        In what way does measuring the photon change the object? I don’t have any problem with the nonlocality of, for instance, bell pairs. Or even Einstein’s spooky wavefunction nonlocal collapse, although that does require explanation. I think time is fundamental, but I can see space as being emergent from something deeper. Three-dimensional distance may be an artifact of classical reality. So, the photon linking “instantly” back to the object, no problem, but in what way can measuring the photon affect the object?

      • Aditya Prasad

        The photon carrying away information *must* be entanglement, because that’s the very definition of “carrying information away” (at the quantum level)! It entangled with some particular observable of the object, and information about the photon gives information about that observable. Whether you call this “affecting” the object or not, it’s the same thing that happens in EPR experiments.

        > it’s still eigenstates, not superpositions.

        An eigenstate of the spin-x observable is a superposition of spin-z eigenstates.

        BTW, it’s not the rotation part that is interesting (to me). The way to look at it is: how does a classical mixture of |0> and |1> behave (wrt the Hadamard basis), and “why” does the superposition |0>+|1> behave differently? One answer is “duh, it’s an eigenstate” and the other is “ooh, magic, interference!” It depends on perspective. This is what I mean when I say that calling it interference is subjective.

      • Wyrd Smythe

        “The photon carrying away information *must* be entanglement, because that’s the very definition of “carrying information away” (at the quantum level)!”

        I agree the photon carries away information from the object. No question.

        “Whether you call this ‘affecting’ the object or not, it’s the same thing that happens in EPR experiments.”

        I don’t, and I’m not seeing how that’s true. In EPR experiments, two particles are described by a single wavefunction. Fully described by this wavefunction, as I understand it. A measurement of one changes the wavefunction, and this is reflected in both. My understanding also is that such a measurement destroys the entanglement. Now they have separate wavefunctions.

        A photon bouncing off an object seems asymmetrical to me and not described by the same wavefunction. The photon and the object are linked in having affected each other in the past, and therefore having information about each other, but I don’t seem them as linked after the interaction. So, I don’t see the situation being similar to a Bell test, but perhaps there are dots I’m not connecting (always possible!).

        “An eigenstate of the spin-x observable is a superposition of spin-z eigenstates.”

        Yes. I’m sure we’re on the same page here. This tangent was in response to my saying that we can’t measure superpositions, just eigenstates. I think you agree but are pointing out any eigenstate is a superposition when viewed from another basis. I quite agree. Same page?

        “…how does a classical mixture of |0⟩ and |1⟩ behave (wrt the Hadamard basis),…”

        I’ve been thrown by what you’re calling “classical” and how it applies to quantum states. Isn’t any quantum state a superposition of a variety of basis states?

        I have a sense this takes us back to the beginning with decoherence to a non-quantum state, and I’ve encountered density matrices as being a big part of that. I do understand it has to do with the off-diagonal matrix members becoming zero (or real numbers rather than complex?).

        One more hill to climb!

      • Aditya Prasad

        “Classical” just means that you have a coin that is either |0> or |1> with 50% probability. Imagine flipping a classical coin, getting one of those at random (without knowing which), and doing measurements on it. In the Hadamard basis, you will get 50-50.

      • Wyrd Smythe

        Forgive me for being obtuse, but are you talking about a system (such as a coin) that physically can only have two states (and no superpositions)? As opposed to a two-level qubit measured in the {|0⟩. |1⟩} basis, which can only have two outcomes (although it actually has R² degrees of freedom)?

      • Aditya Prasad

        Starting a new thread because it’s hard to respond to the nested thread above.

        I just mean a qubit that known to be in one of the definite states |0> or |1> with 50% probability. We can calculate expectations for measurements on this classically indeterminate quantum state (and they will agree for all observables with the results for one qubit of a Bell pair).

        Yes, I believe that macroscopic objects can be in superposition. I don’t know how much sense it makes for me to consider myself as being in superposition, however, since I am me. I don’t believe it is meaningful to talk about a “God’s-eye perspective” where that would make sense. In this sense, perhaps I am closest to QBism: QM is a tool for agents to predict things. I’ve heard the term “distributed solipsism” (in a different context) and I kind of like it.

      • Wyrd Smythe

        Yeah, feel free to start a new thread any time. WP isn’t good about Reply links once a thread gets to a certain depth.

        Forgive my obtuseness, but wouldn’t any random qubit, if measured, give |0⟩ or |1⟩ with 50% probability? I’m sure you mean something more specific. Is the qubit meant to be in the |0⟩ or |1⟩ eigenstate with 50% probability — and not in a superposition of them? Clearly, I’m confused!

        “Distributed solipsism”! In terms of putative other copies of yourself, or in terms of other people in your world?

      • Aditya Prasad

        Suppose I give you a qubit that is either |z+> or |z-> with 50% probability, and ask you for the probabilities of spin-x outcomes. What are those odds?

        Distributed solipsism in terms of all beings. (And no, I don’t know how to define a “being” :)) I think the correct metaphysics is one that can’t be fully pinned down. The interesting bit for me is that my own world is not definite before information reaches me. That’s what I think QM is telling us: that there is genuine openness, and you are at the epicenter of it. I am the point at which things become real in my world — and the same for everyone else.

      • Wyrd Smythe

        What’s confusing me isn’t the probabilities of spin-X measurements (which are 50/50) but of exactly what’s meant by “a qubit that is either |z+⟩ or |z-⟩ with 50% probability”. I see multiple meanings:

        The qubit is known to be ½|Z+⟩ and ½|Z-⟩ — specifically in one of those two eigenstates (say because of a previous measurement on spin-Z).

        The qubit is an unknown random superposition 1/√2(|a⟩±|b⟩). In which case, 1/√2(|Z+⟩±|Z-⟩) is a valid superposition and satisfies the constraint “either |z+⟩ or |z-⟩ with 50% probability”.

        In both cases, though, spin-X measurements are 50/50. (The two cases would vary for non-orthogonal tests in that, a |Z+⟩ state is more likely to produce a spin-up measurement the closer the angle of measurement is to the positive Z-axis.)

        Which is all in the scope of QM, so I’m struggling with how it can be labeled “classical”.

        Anti-realism, solipsism,… suffice to say our metaphysics are quite different! 😆

      • Aditya Prasad

        I mean the former: one that is known to be in eigenstate |z+> OR in |z-> but you don’t know which, so you assign 50% odds. In that case, measurement of spin-x yields 50% expectation for both |x+> and |x-> .

        On the other hand, if the qubit is in definite state |z+> + |z->, then it will measure as |x+> 100% of the time (because it IS |x+>). (You may wonder how one can KNOW that it begins in that definite state, and the easiest answer is that you *prepare* it in |x+>, but then the story gets less interesting. Forgetting about *how* you know, the point of this exercise is to show that the classical mixture yields different odds than the superposition, and the reason can be meaningfully described as interference: amplitudes cancelling when you compute using the z basis.)

      • Wyrd Smythe

        Okay, I’m clear on it now. I kind of thought that’s what you meant but calling it “classical” really threw me. I reserve the term for the classical world that emerges from QM. (But then, it occurs to me your metaphysics might not include a classical reality. No collapse, no classical world?) FWIW, I posted about this topic: QM 101: Quantum Spin.

        The other thing that got in my way, I think, was how, as you no doubt know perfectly well, the situation can be formulated with the X-axis as the “fundamental” basis and the other two as superpositions:


        So, then the particle in known Z-axis eigenstates is defined being one of:


        Which is hard for me to think of as “classical” — mental block! 🤷🏼‍♂️

      • Aditya Prasad

        Just noticed this comment on the post you linked: “It does boil down to exactly what causes a branch, doesn’t it. It’s one of many things I keep hoping an expert who has really studied this would clear up. I don’t find “dealer’s choice” very satisfying.”

        This was precisely what kept me learning for years, and I finally got an answer that satisfies me a couple years ago.

        Sean Carroll will tell you that branches split “when decoherence happens” (maybe not an exact quote, but close). That sent me down the rabbit hole of learning about decoherence. In this context it turns out to be a synonym for “entanglement that is so complicated that we can no longer practically track or reverse it.” There is no point at which decoherence objectively “happens,” because “practically” is a subjective choice!

        The reason Carroll et al are happy with it is because it becomes “practically” impossible to track/reverse at a very early stage (nanoseconds or less in a Schrodinger’s Cat setup?). The only other option for an MWI’er (or *any* non-collapser, which is the vast majority) is to let it propagate until it reaches you. At that point you have no choice except to say that you are somehow the point at which possibilities become “real.” This is profoundly distasteful to most physicists (though Ed Witten is a notable exception, and I’m glad to have him in my corner :).

      • Wyrd Smythe

        I’m happy to discuss the MWI if you want to but should say up front that I’m entirely unsympathetic to the view. I’ve posted about the MWI quite a few times. Most recently a three-part series: MWI: Questions, part1, part 2, and part 3.

        If you want to go down that particular rabbit hole, we should move to one of those posts. Or better yet, perhaps on this post also from last year: BB #74: Which MWI? It has the virtue of almost no comments on the post so far.

        (The key difference between our views here being that I don’t believe macro-objects exhibit quantum behavior. I don’t think wavefunctions are meaningful for large objects.)

  • Wyrd Smythe

    As an aside,…

    Assuming the link refers to the section just below Exercise 2 for the Non-Lazy Reader:, the text beginning with ‘This “2-norm bit” that we’ve defined …’, I decided to do, as they say, the math.

    In part because I was struck by his unitary matrix being almost, but not exactly, a Hadamard gate. Which is defined for two-level qubits as:


    But Aaronson’s unitary matrix in the text is:


    It was only when I sat down and did the math that I realized what the A gate does and why Aaronson used it. Here I’ll show both, starting with the Hadamard gate.

    If applied to a qubit in the |0⟩ state, we have:


    And if applied to a qubit in the |1⟩ state, we have:


    OTOH, if we use the gate Aaronson uses, we have in the first case:


    Which is the same as above. For the |1⟩ case, we have:


    And because global phase is irrelevant:


    So, effectively the same state as above. Aaronson moving the minus sign makes no difference in this case. However, they rotate other states differently. In fact, they rotate the |0⟩+|1⟩ and |0⟩-|1⟩ states differently, as shown below.

    Next step, apply the gates to the superpositions formed in the first step. Again, I’ll do both.

    Using the Hadamard gate on the |0⟩+|1⟩ superposition:


    And on the |0⟩-|1⟩ superposition:


    So, the Hadamard gate, applied twice, restores the original condition. The diagonal rotation axis of the Hadamard gate swings the |0⟩ and |1⟩ states to the |0⟩+|1⟩ axis and then back again.

    Things are different when applying the Aaronson gate because it’s rotating the Bloch sphere on a different axis (the Y axis, not a diagonal).

    For the A|0⟩(|0⟩+|1⟩) superposition:


    And for the A|1⟩(|0⟩-|1⟩) superposition:


    So, two applications of the A gate reverse the |0⟩ and |1⟩ states. Which obviously is Aaronson’s point. Nice! I’m sure what I’ve called the A gate has a formal name in QC, but I didn’t find it in a list of gates. I think it’s a straight two-dimensional rotation matrix for 45° along the Y axis:

    \displaystyle{R}_{y}(45)=\begin{bmatrix}{\cos 45}&{-\sin 45}\\[0.75 em]{\sin 45}&{\cos 45}\end{bmatrix}=\begin{bmatrix}{\frac{1}{\sqrt{2}}}&{\frac{-1}{\sqrt{2}}}\\[0.75 em]{\frac{1}{\sqrt{2}}}&{\frac{1}{\sqrt{2}}}\end{bmatrix}

    Which sort of makes sense, as it would rotate the Bloch sphere on its real plane (the XZ plane). Two 90° rotations would reverse the |0⟩ and |1⟩ states, and since it’s common to see θ/2, I suspect that’s how 90° becomes 45°. The |0⟩ and |1⟩ states are orthogonal, so two 45° rotations conceptually make an orthogonal angle.

    I was curious if not applying the global phase to the (-|0⟩+|1⟩) superposition made much difference:


    But a touch of global phase makes it the same:


    This might be a lot of old hats to you, but it was a fun exercise for me to work through.

    • Wyrd Smythe

      BTW, it’s those ½+½ and ½-½ parts that demonstrate the “interference” of the superposition.

    • Aditya Prasad

      I had not worked that out before. Nice! At the end, you could also have just factored a -1 out of the whole expression to see quickly that A(-|0>+|1>) = -A(|0>-|1>) = -|0>. In any case, you have more facility manipulating single qubits and visualizing the Bloch sphere than I do 🙂

      I really think you owe to yourself to learn at least the basics of density matrices and taking partial traces. As you may have learned already, the density matrix for a pure state |x> is the outer product |x> and |y> by 0.5(|x>+|1>)(<0|++|11>)(<00|+<1|. The off-diagonal terms have gone. This means we should predict it behaves like a classical mixture, which is also what we saw when we tried to measure one qubit in the Hadamard basis.

  • Aditya Prasad

    Ugh, I can see my comment got messed up 😦

    • Wyrd Smythe

      It sucks that WP doesn’t allow editing one’s own comments. I’ve messed up plenty! It’s especially annoying when you notice your error just as you click the [Submit] button. Split second too late! ARG!

      Density matrices are definitely on the list!

      I do know one can do some interesting things with outer products, such as:


      And I’ve seen constructions like this:

      \displaystyle\rho={c_1}|{0}\rangle\langle{0}|+{c_2}|{1}\rangle\langle{1}|+ {c_3}|{0}\rangle\langle{1}|+{c_4}|{1}\rangle\langle{0}|

      Which I think is a density matrix?

      • Aditya Prasad

        That’s right. The interesting thing is to take the density matrix of the superposed state, then of a classical mixture (= Identity, as you’ve shown), and see how they differ. Then take the density matrix of the Bell state. Then trace over the second qubit, and watch it look exactly like a classical mixture. Since a density matrix captures all information needed to predict measurements in all bases, the fact that it looks the same as a classical mixture tells you that it IS one for all practical purposes. That’s the punchline of (this kind of) decoherence.

        BTW, FWIW I am a hardcore anti-realist, so it is only natural that we will see some things differently 🙂

      • Wyrd Smythe

        Ha, well that just makes things interesting. Do you have a favored interpretation? (FWIW, I’m sympathetic to some version of OR involving gravity. I think Penrose and others are on to something there.)

        p.s. That’s probably it for me tonight. Been very interesting! Perhaps more later or on another post. Have a nice evening!

      • Aditya Prasad

        I don’t have a favorite interpretation, but I don’t believe in collapse. Instead, definite results are attained in “my world” only when superpositions interact with *me*. The same is presumably true of others (though I am prevented from ever being sure of this), but I only interact with the versions of them in my world.

        Good night!

      • Wyrd Smythe

        Good morning!

        Ha, yeah, collapse versus no collapse. Pretty much on opposite sides on that point! 😄

        So, QM applies to all levels of reality, and wavefunctions are meaningful for things like trees, people, bridges? You, right now, are in superposition with other nearly identical versions of yourself (and ones much less identical)?

  • Aditya Prasad

    Should have said no point at which decoherence *objectively* happens, because …

    • Wyrd Smythe

      I inserted objectively into your comment so future readers don’t have to notice your later comment. Later today I’ll delete it and my reply here. I’ll leave it long enough for you to see my reply (and complain if you feel I’ve crossed a boundary).

      • Aditya Prasad

        Also, I’m sure my decoherence times are off by many orders of magnitude, but I couldn’t be bothered to look up real numbers so I just wrote a small-sounding number to be safe 🙂

      • Wyrd Smythe

        From what I’ve read, I think you’re safely in the ballpark. (Those rapid decoherence times are a big factor in why I don’t believe macro-reality is quantum.)

        (Hmmm. These two comments make me think I’ll not delete this little sub-thread. I’ve never been entirely comfortable deleting comments, anyway.)

      • Aditya Prasad

        This is why it was crucial for me to get a grasp on precisely what decoherence is saying. It says that we lose the ability to exploit superpositions almost instantly in practice, but that they are still there in principle. If you want to lose superpositions in principle, a collapse mechanism (like in GRW) is necessary.

        My belief is that theories like GRW were created because the straightforward results of unitary evolution are just too weird. Same with modern MWI: so long as Carroll can maintain that worlds split “when decoherence happens,” the observer can be taken out of the equation. I see all such attempts as avoiding the punchline that nature is trying to deliver straight to our faces — though I appreciate that you will disagree!

      • Wyrd Smythe

        I’m working my way through Roger Penrose’s The Road to Reality (2004), and I was delighted when I read that he, too, questions unitarity. I’m not the only one who questions the supposed conservation of information. (In part, because our physical conservation laws are based on symmetries, and I’ve never heard of any symmetry that leads to conservation of information.) Unitarity comes, in part, from the linearity of QM, which is another thing I think is worth questioning. Reality seems decidedly nonlinear to me, and most physical laws are nonlinear.

        I’m not a fan of stochastic objective collapse theories, but I do like the Diósi-Penrose model that brings gravity into the equation. Everett himself allowed for the possibility of objective collapse but disdained and handwaved away the idea as being tied to some putative N (of particles). I think it’s likely more complicated than that. The Heisenberg Cut, I believe, will turn out to depend on number of particles, environmental conditions, gravity, and perhaps other factors.

        Though, obviously, I (and Penrose) could be completely wrong. There’s a space experiment proposed for the ESA that I’d really like see performed. It’s called MAQRO. From their website: The experiment involves “observing free quantum evolution and interference of massive dielectric test particles with radii of about 100nm and masses up to several 10^10 atomic mass units (amu).” It goes on to point out that the current record is about 10^4 AMU, so this would involve a big jump. It might even falsify the notion that quantum mechanics applies to macro-objects. Or not!

      • Wyrd Smythe

        “This is why it was crucial for me to get a grasp on precisely what decoherence is saying. It says that we lose the ability to exploit superpositions almost instantly in practice, but that they are still there in principle.”

        My analogy is a vast stadium filled with trillions of people singing the same song (in very very ragged unison). In come a group of hundreds singing a different song. They disperse into the crowd of trillions still singing their song. Some people around them might take up the song, but the trillions singing a different song utterly overwhelm them. Eventually the new group is absorbed into the trillions and their song is lost.

        The other picture is that, somehow, this group of hundreds gets the whole stadium of trillions singing their song or some blend of song. I’ve just never bought the idea that a crowd of hundreds would have much effect on a crowd of trillions. It’s throwing ping-pong balls at an ocean liner.

      • Aditya Prasad

        I think of amplification as quite distinct from decoherence. We know we can create a device that will kill a cat if a radioactive atom decays, and won’t if not — quite independently of the question of whether these two possibilities can exist in macroscopic superposition. That’s the amplification aspect.

        Decoherence says “well even if there IS a macroscopic superposition, there’s no way you can exploit it or even demonstrate it, so who cares?” And indeed for all practical purposes, nobody should care. But I still think it’s pointing us to something about the nature of reality.

        Of course, if we do get evidence for objective collapse, this perspective goes out the window! The more interesting case is if we don’t, since you can’t prove a negative and all that. I suspect we may end up wondering forever.

        BTW, I just put all my thoughts (on QM and tangentially-related topics) together in one (ugly) site, in case you find yourself bored and wanting to explore some bizarre ideas: https://github.com/monktastic/hackmd/blob/master/README.md

      • Wyrd Smythe

        “I think of amplification as quite distinct from decoherence.”

        Absolutely! I don’t know if you realized this, but this post is #3 of 5. There are two before and two after, all related to this general topic. The next post Measurement Specifics gets into my view about what’s happening when we amplify quantum events to the classical level. In a word: mousetrap!

        The last post in the series, Objective Collapse gets into my views about objective WF collapse.

        (The first two posts, Quantum Measurement and Wavefunction Collapse, just set the stage. I assume this post caught your eye because of its title and topic.)

        “The more interesting case is if we don’t, since you can’t prove a negative and all that.”

        “Interesting”? I think you misspelled “infuriating”! 😁 Yeah, no black swans. Like those poor folks chasing SUSY or Dark Matter.

        I’m about to take off to visit a friend and won’t be back until late. This will be my last comment for the day. Have a good one! (I’ll check out your page when I get a chance. Kinda booked until Tuesday or Wednesday or so.)

      • Wyrd Smythe

        I’m getting a 404 Page Not Found error on your GitHub link…

      • Aditya Prasad

        Thanks for letting me know. I’ve fixed the permissions.

        I skimmed posts 4 and 5 just now. Will leave a few comments on 5.

        BTW, Penrose discusses the sense in which decoherence is only “for all practical purposes” on p802 of RTR: https://physics.stackexchange.com/a/386163/47309

      • Wyrd Smythe

        I can access your GitHub page now! Caveat: got a lot on my plate right now (including a leaking water pipe and a plumber coming but not until Thursday morning).

        That Penrose quote is from the beginning of the section FAPP philosophy of environmental decoherence, which is part of the chapter 29, The measurement problem. It’s one of my favorite parts of the book, and I skipped to it soon after I started reading. (I’m linearly in chapter 20 where he’s still talking classical mechanics.) The reason I skipped to 29 was that it seemed a prerequisite for chapter 30, Gravity’s role in quantum state reduction, that I really wanted to read.

        After the quoted part, Penrose continues:

        “It would seem to be a strange view of physical reality to regard it to be ‘really’ described by a density matrix. Accordingly, such descriptions are sometimes referred to as FAPP […] The density-matrix description may be thus regarded as a pragmatic convenience: something FAPP, rather than providing a ‘true’ picture of fundamental physical reality.

        “There might, however, be a level at which the detailed phase relations indeed actually get lost, because of some deep overriding basic principle. Ideas aimed in this direction often appeal to gravity as possibly leading us to such a principle.”

        He introduces a few ideas that he describes in detail in chapter 30. In particular, he questions unitarity. (The black hole information paradox would be easily solved if information could indeed be lost.) FWIW, Penrose is also a pretty staunch realist.

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