Measurement Specifics

In the last three posts (Quantum Measurement, Wavefunction Collapse, and Quantum Decoherence), I’ve explored one of the key conundrums of quantum mechanics, the problem of measurement. If you haven’t read those posts, I recommend doing so now.

I’ve found that, when trying to understand something, it’s very useful to think about concrete real-world examples. Much of my puzzling over measurement involves trying to figure out specific situations and here I’d like to explore some of those.

Starting with Mr. Schrödinger’s infamous cat.

If you have any interest in quantum physics, you’ve heard of this strange, and I hasten to add, imaginary, example of animal cruelty. It’s yet another physics thought experiment — an imagined, but concrete, experiment designed to illustrate or illuminate an idea. In some cases, thought experiments aren’t practical, maybe not even possible, but even then, they invoke the dynamics of the real physical world.

We begin with Schrödinger’s own words:

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer that shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.

As an aside, Schrödinger reportedly loved animals and often used them in his thought experiments. More important is his first sentence; he considered the idea of a cat in a superposition of dead and alive as “ridiculous” (and I quite agree).

Yet, if we take quantum mechanics as a valid description of reality at all levels, we’re left trying to explain exactly why a cat superposition is ridiculous. Why can photons and other quantum systems have a superposition of states, but larger (classical) objects do not?

We can appeal to measurement (or observation), but a conundrum arises regarding what constitutes such a thing. Observation gives us a classical world. What, precisely, divides quantum behavior from classical behavior? What, and where, is the so-called Heisenberg Cut?

If we appeal to human observation, the cat is in a superposition of states until the researcher opens the box to find, either a cat that died at some point during the hour, or a (very unhappy) living one.

This leads to the idea that, if famous mathematician Eugene Wigner is outside the lab, then from his point of view, his friend the researcher is superposed between sorrow and cat-scratched until Wigner enters the lab and asks about the result.

Looking outward, Wigner’s associates see Wigner in superposition of knowing, and not knowing, the result until they hear it from him. The “collapse” continues outwards on an expanding wave of communicated information.

One aspect of this is Wigner asking his friend how it felt to be in superposition — a question that would likely seem quite strange. It seems sensible to accept that, at the very least, the researcher “collapsed” the cat’s wavefunction upon opening the box, but the only change of state they could report would involve their new knowledge.

We can’t ask the cat the same question, but unless we think human consciousness is special (and some do), then it seems also sensible to think the cat “collapses” the wavefunction, too. A cat observes its environment and surely would react in a definite manner to poison in it.

Looking inwards, does the Geiger counter observe itself? Does it observe the radioactive sample? What about the sample? Does it observe itself? How big does something have to be for classical dynamics to emerge?

Note this doesn’t deny that large objects, at root, are quantum systems, but that the classical behavior that emerges utterly swamps the quantum nature. (Just as 10²⁷ singers utterly swamp the song of any one singer or any group of singers.)

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Let’s go through this piece by piece. We’ll start with the radioactive sample.

Radioactive elements have a property called a half-life period, which varies from tiny fractions of a second to millions of years (or more). During the half-life period, on average, half the atoms of that element release one or more particles (as some form of radiation) and decay into another less massive kind of atom.

An important aspect of half-life is that there is no way to predict which atoms decay, only that approximately half of them will. The larger the sample, the more the approximation approaches exactly 50%. Single atoms have a 50% chance of decaying by the half-life period. Which means that for any group, by the half-life period, half of them will have decayed.

Wait… radioactivity and poison?

Schrödinger specified a radioactive substance with a 50% chance of releasing one particle (of radiation) in one hour. This depends on both the half-life and the sample size. For a given decay rate, the larger the sample, the more particles it emits. There is also the consideration that the emitted particle might miss the detector. Or if it doesn’t miss, the detector might fail to detect it.

The sample size seems important to the question of quantum versus classical. A single radioactive atom with the right decay rate (50% after one hour) seems likely to act like a quantum system. We’ve successfully demonstrated quantum wave behavior in objects with 2000 atoms, so it seems one atom certainly should have quantum dynamics.

We can wonder whether a sample comprised of billions of trillions of atoms might not observe itself. (One gram of uranium-235 has about 2.56×10²¹ atoms.) Somehow any such sample “knows” to decay according to its half-life. Is this quantum or classical behavior? Does our search for the Heisenberg Cut end at the radioactive sample? How much does sample size matter?

But even if we say the sample can be treated classically, the emitted “particle” almost certainly is a quantum system. Most forms of radiation are comprised of single particles such as electrons or photons.

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Schrödinger also specified a Geiger counter, and here, even if we consider the radioactive sample a quantum system, I think the Geiger counter — the actual measuring device — is clearly not.

Geiger-Müller tube
[from Wikipedia]

The key is the Geiger-Müller tube, which is the heart of the device. The tube contains a low-pressure gas and two high-voltage electrodes. The gas insulates the electrodes so no current flows.

Ionizing radiation from radioactivity creates pairs of positive ions and free electrons. The voltage across the electrodes accelerates the electrons towards the anode (and the ions towards the cathode). As electrons near the anode, the electrical field increases accelerating the electrons enough to create new pairs. This creates a multiple cascade effect that amplifies the effect of the radiation and creates a strong signal.

It’s not unlike a snowball starting an avalanche.

There are other classical examples. Consider a mousetrap. Or a gun with a hair trigger. In both cases, a tiny input — the slight pressure of a mouse or finger — triggers the massive release of stored energy. In these cases, mechanical leverage accounts for some cascade for trigger amplification, but the bulk of the output comes from the huge amount of stored energy in the mousetrap spring or burning gunpowder.

In general, when measuring quantum systems, we need to amplify something tiny and quantum into something large and classical we can measure. This often involves two important aspects: A system with lots of stored energy (like the high voltage in the Geiger-Müller tube); and an avalanche effect triggered by a tiny input.

Photomultiplier tubes are another example. They amplify low levels of light by factors as high as 100 million. They can also be configured in avalanche mode to detect single photons. Like Geiger-Müller tubes, they use high voltages to store energy and rely on a progressive cascade to create a usable signal. Now there are also single-photon avalanche diodes (SPADs) that use a high reverse-bias voltage and cascade effects to amplify the effect of a single photon.

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So, I think Schrödinger’s decidedly classical cat depends on a classical Geiger counter that uses self-supplied high voltage to act like a mousetrap for ionizing radiation. The first interactions of the radiation particle/waves with the Geiger-Müller tube trigger the mousetrap. The stored energy and cascades amplify it to a classical level.

A general physical rule links high resolution with high noise. In an efficient system, noise is small, so it’s more apparent at small scales. When detectors observe at single-particle scales, noise is a major factor. It causes false detections and can suppress valid events. The idea with Geiger counters is lots of clicks, so missing a few, or a few extra, aren’t a problem. In experiments depending on, for instance, single photons, false and missed detections need consideration.

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We can look at two kinds of experiment involving single-particle detection: beam splitters, such as interferometers; and entanglement tests, such as of Bell’s inequality. Many (perhaps even most) of both kinds use photons, and I’ll assume that in what follows.

Beam Splitter. A photon from the laser either reflects off or transmits through the mirror. The probability depends on the mirror.

Beam splitters use a probabilistic branch that splits the path of the photon into two possible end points, each of which has a detector. A typical example uses a half-silvered mirror that gives the photon a 50/50 chance of either passing through or being reflected. Both transmitted and reflected paths end at detectors. This makes a “quantum coin flip” — with odds controlled by how silvered the mirror is.

However, a problem arises because of the single-photon source. At this level there is uncertainty between time and energy, so it’s not possible to control or know the exact time the source releases a photon. That makes it hard to know if the two detectors miss or imagine a photon. In a beam splitter, either one or the other detector fires, so it’s hard to tell if a detection is noise, or silence is a missed photon.

Entanglement tests use two photons with entangled spin (polarization). Because there are two particles, and it’s often possible to detect both, coincidence counting helps distinguish valid events.

Either way, pairs of separate detectors watch for photons. These detectors, probably SPADs, like the Geiger-Müller tubes, use stored energy at an easy tipping point, to act like mousetraps for photons. The amplification from quantum to classical comes from the cascading avalanche driven by the stored energy.

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Beam splitter experiments share with the Schrödinger’s cat experiment clear either-or outcomes. The photon reflects or transmits; the cat is either dead or alive. With photons we must worry about noise; with the cat we must worry about scratches. At least we can actually perform the first kind of experiment.

And the beam-splitter helps highlight another aspect of what quantum mechanics suggests when applied at the detector level. If we assume a quantum view, then we have a superposition of:

Where the two qubits represent the two detectors firing (1) or not (0). Ideally, the outcomes can only be |01〉 and |10〉. We assume |11〉 (both firing) is impossible, because it’s noise; and we ignore |00〉 (neither firing), because we can’t distinguish a missed photon from no photon.

But the detectors aren’t related to each other, they’re separate devices. For them, the superposition is between firing and not firing. Between detecting something and detecting nothing. That doesn’t seem like a superposition of states to me. I don’t think we can treat a detector as a quantum object, so we also can’t so treat the pair.

There is a view that each of an entangled pair entangles with the respective detector. But entanglement is a quantum behavior we don’t see in classical objects. (Under the MWI, we still don’t see it because we become entangled with the detectors and become superposed ourselves. Which would allow answering Wigner’s weird question, except that he branches into worlds where it’s still a weird question.)

Regardless, we’re still stuck with “spooky action at a distance” because the photon wave seems to at least approach both detectors but only interacts with one. Somehow the other detector instantly knows to not interact. Alternately, the wavefunction instantly vanishes everywhere so there is no chance for interaction anywhere else.

Entanglement tests highlight this instant connection. It can’t be used to transmit information, so it doesn’t seem to violate special relativity. Quantum non-locality seems to be a feature of our reality. I don’t have a problem with that; I already think space could be emergent from something more fundamental that supports what seems non-local from our point of view. Therefore, I’m not as concerned about the wavefunction vanishing as I am exactly what it is.

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Which is all for another day. For now, I’m going to leave it here. No doubt I’ll pick it up somewhere down the road.

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

∇