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In the state of quantum superposition, as most famously illustrated by Schrodinger's Cat, we have a well-defined set of probabilistic outcomes that is not determined until observed. The cat is then said to "really" be in both states at once, while unobserved.

How does this differ from any other probability or what we might simply call "the future" or the "unknown." Isn't this simply a matter of introducing a time-bound "observer" into the picture?

From the naiveté of the question it should be clear that I'm hoping for some relatively simple distinction or epistemological assumption. Does it all hinge on the assumption of an "unobserved reality"?

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    It is a property of the Schrodinger wave equation that if you have two different solutions, then the sum of those two solutions is also a solution. There are different interpretations of this property. In the case of Schrodinger's cat, one solution is that the cat is alive, another is that the cat is dead, and therefore the sum of these two solutions - namely that the cat is both alive and dead - is another solution. This is one interpretation of the formalism. (Of course, Schrodinger's cat is to be taken in the spirit of a thought experiment.)
    – Nick
    Aug 10 '20 at 16:46
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    I should have added to my comment above that Schrodinger formulated his though experiment in order to illustrate the apparent absurdities of the theory. A cat is not a quantum system.
    – Nick
    Aug 10 '20 at 16:51
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    Are you asking about how quantum superposition differs from ordinary classical probabilities which can be interpreted purely in terms of epistemic uncertainty about parts of the world we don't know about? For example if someone flips a coin and covers it with their hand, I may say that subjectively there's a 1/2 probability heads is facing up and and 1/2 probability tails is facing up, but I assume there's some definite truth about how the coin is oriented under the person's hand.
    – Hypnosifl
    Aug 10 '20 at 17:02
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    Once you have a "well-defined set of probabilistic outcomes", a.k.a. preferred basis, everything specifically quantum goes away, the Born rule reduces quantum prediction to a distribution on a classical probability space of those outcomes. You can then use your favorite interpretation of "uncertainty" on them, but it is the "observer" (measurement device, environment) that has to supply the preferred basis in the first place, the superposition itself is basis invariant. That the cat is "in both states at once" from pop-sci is just wrong, it confuses superpositions (sums) with conjunctions.
    – Conifold
    Aug 10 '20 at 19:11
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    As a general note, this is a distressingly common pattern in philosophy: if you want to incorporate scientific conclusions, and you don't have a deep understanding of the science behind those conclusions, it is almost guaranteed that you'll misinterpret the results, and your argument will therefore be based on false premises. There's a whole lot of bunk swirling around that started when philosophers who didn't understand science tried to make arguments based on science. Aug 11 '20 at 16:45
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I recently answered a similar question on physics.SE here. What is special about the probabilities of quantum mechanics is that the randomness cannot be explained by a theory of nature that is both local and realist, while classical probabilities can. Quoting myself:

A "local hidden variable" theory is basically the classical idea of how the world works - everything has a list of well-defined properties, like position or momentum, and there is a "true" precise value for each of these at each time, and the laws of physics in principle determine the precise value at each other time from those at one instant. "Randomness" in this classical world is incidental, arising from incomplete knowledge, imperfect measurement devices, etc. When you flip a classical coin in the exact same way, it will always yield the same result. The "randomness" is just because humans are extremely bad at the level of consistency required to flip it "in the same way" again. The belief that there is a definite value for each property at all times is also called realism.

The other ingredient for a local hidden variable theory is locality, the idea that things that happen at one point cannot instantaneously affect the state of the world elsewhere, but that changes have to propagate at finite speed (lower than or equal to the speed of light, if you know about relativity, but that specificity isn't really necessary here).

Bell's theorem now says that quantum mechanics is incompatible with local hidden variable theories. No such theory can ever predict the results that we do, in fact, observe. So the probabilities of quantum mechanics are different from those of classical physics because they cannot be explained by a theory that is both local and realist. Interestingly, the standard technical formulation of quantum mechanics is neither really non-local nor really non-realist, but has more or less successfully delegated the decision about which of these traits we give up into a new realm of metaphysics called quantum interpretations.

Beware of anyone trying to tell you that quantum mechanics implies a particular ontological claim about the world, e.g. the existence of "parallel universes", a peculiar power of "conscious observers" or the existence of "pilot waves". All these are interpretations, but all that Bell's theorem says is that no local and realist interpretation can be valid for quantum behaviour. It does not select one of these interpretations over the other, and the reason I've called these interpretations metaphysics above is that they usually make no different claims about the results of experiments - pure physics cannot distinguish them, or they are careful to make different predictions from "standard" QM only in areas where experiments are as of yet infeasible.

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    This seems to be the only answer that actually has the key point in it.
    – John Davis
    Aug 11 '20 at 16:12
  • The proof of Bell's theorem has a loophole in that it assumes that for every measurement, there is a unique outcome. If you include this in the definition of "realism" then it's true that QM is incompatible with local realism, but if on the other hand you say a theory can still be "local realist" if it involves systems at different locations splitting into multiple local copies, then Bell's theorem no longer rules out locality. The MWI has been argued to be local in this sense, see the discussion of the arguments of Deutsch and Rubin here.
    – Hypnosifl
    Aug 11 '20 at 16:25
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    @Hypnosifl Yes, if I define realism differently from the standard definition exactly such that the MWI is realist, then Bell's theorem does not exclude local "realist" theories anymore, but that's a tautology. As I say, the notion of realism used here is essentially "The belief that there is a definite value for each property at all times" - if you have multiple local copies with distinct values, then there is no "definite value", and your theory is not realist by this measure. We can always redefine words to mean different things, what's your point? Aug 11 '20 at 16:39
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    As far as I know there is no standard technical definition of "realism" that addresses this issue, I have only ever seen it defined in somewhat vague verbal terms which might be read as compatible with the MWI (which does give an objective picture of reality independent of any particular observer's knowledge). Because of this vagueness, some prominent physicists like Gilles Brassard have argued that the MWI should be seen as "local realist", see here
    – Hypnosifl
    Aug 11 '20 at 17:17
  • Great answer :-) I would like to add, that the actual weirdness of quantum mechanics is not so much the concept of superposition, but rather the concept of entanglement. The fact that performing a measurement on a particle can instantly influence an entangled particle many light years away. Superposition is a prerequisite for entanglement, but it's the entanglement that really sets quantum theory apart. The Bell theorem formalizes this "spooky action at a distance" so far that it can be tested, and the tests say: "Yes, this spooky action exists. Screw locality or realism, you don't get both." Aug 11 '20 at 22:02
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The "classical" form of quantum mechanics (no hidden variables or "pilot waves") maintains that a state exists as a superposition of all possibilities until the act of measuring that state causes the associated wavefunction to collapse. The collapse of the wavefunction then follows the probabilities for each possible state (for example, 40% spin-up and 60% spin down for a certain particle, etc.). And only when that measurement is repeated many, many times does the 40/60 split become apparent.

In Shrodinger's thought experiment, the possible states are "cat alive" and "cat dead" and he maintained that the wavefunction does not collapse until you open the box and measure the state of the cat. (By the way, this was intended as an illustration of a particle physics effect and not as a statement about how cats get poisoned. In reality, such effects are vanishingly small for all macroscopic objects like cats, which consist of billions upon billions of individual particles.)

This business hinges on the fundamental idea that (only!) in the world of particle physics, the trajectory of a particle into its future cannot in principle be known until measurements are made on it in its present. If this essential fact is omitted from the mathematical formalism used by quantum physicists to solve problems like this, you get the wrong answers when predicting the outcome of an experiment.

You can think of this as illustrating the difference between the unknown and the undefined.

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    "such effects are vanishingly small for all macroscopic objects like cats, which consist of billions upon billions of individual particles" What does # of particles have to do with it? The real problem with macro objects like cats is environmental decoherence, but if it's possible to isolate a large system sufficiently from its environment (easier for systems near absolute zero) it can evolve into a superposition of macroscopically distinct states (like the live and dead cat) according to QM, and see archive.is/ylWb for an example
    – Hypnosifl
    Aug 10 '20 at 17:38
  • Thanks, sounds clear but I will have to chew on it a bit, being unfamiliar with the technical underpinnings. It seems the distinction is more in the technical anomalies of QM compared to classical mechanics than in the way we think of probabilities generally, though these too seem to harbor weird paradoxes. Aug 10 '20 at 17:39
  • Much of what passes for startling new discoveries in physics, over time and with reflection turns out to be merely common sense recognitions of the interactions of bodies in space. Newtons description of the measurement of gravity, instead of being claimed as a 'Law', would have been better served as a 'recognizable process of magnetic attraction and repulsion due to the molten cores and masses of globular objects locked into a repeatable pattern of circumlocution around a massive star'. It's really just a common sense observation once the nature of objects in a solar system are understood.
    – user37981
    Aug 10 '20 at 18:33
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    @CharlesMSaunders - Are you using "magnetic" metaphorically? The gravitational force is different from magnetic attraction (and there is no repulsive aspect to gravity, unlike with magnetism). It doesn't seem particularly common sense that there's an attractive force between two arbitrary masses regardless of composition since the attraction between human-scale objects is too minute to affect everyday life, and common sense also doesn't suggest why gravity should obey an inverse square law as opposed to some other mathematical function.
    – Hypnosifl
    Aug 10 '20 at 18:38
  • The same can be said for Einstein's theory of relativity; once escaping the mutual magnetic attractions and repulsions within a given solar system, it becomes simple scientific common sense to recognize that the potency of: gravity, time and energy will begin to dissolve and to wrap and warp around one another. It is not a 'Law' but another common sense regularity. Apply this process to QM and reflect on the interactions among nano-particles it is not so amazing or difficult to account for their interactions without making outrageously occult claims about how their actions disrupt physics.
    – user37981
    Aug 10 '20 at 18:42
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The key characteristic of a quantum superposition is that all superposed states are equally real (or potentially real) at the physical level. This is quite different from a classical probability, which assumes that one state is real and the probability reflects our ignorance of the true state of affairs. That, basically, is it.

We know this to be true because quantum entanglement, via Bell's Theorem, allows us to compare the probabilities predicted by quantum theory with those predicted by classical statistics. This experiment has been carried out many times and classical statistics always gives the wrong answer: superpositions really are inherent in reality and not due to ignorance.

What follows is just a bit of background comment.

Different "interpretations" of the quantum equations express different opinions about the nature of the reality - actual, potential or merely likely - of the various superposed states.

Schroedinger is often assumed to have been claiming that his cat was both alive and dead at the same time. In fact he was doing the reverse, he was highlighting the ludicrous consequences of the idea that superposed states were in any way real. Out of this controversy arose the standard "Copenhagen interpretation", that there is no objectively meaningful reality to the superposed states; enquiry is futile and the good physicist should just "shut up and calculate".

Many other speculative interpretations have been developed - hidden variables, pilot waves, parallel universes where every state has a home, and so on, but they are all either falsified in the lab (most hidden-variable theories predict classical results for entanglement experiments) or untestable metaphysics (which Schroedinger lampooned with his cat).

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    What does it mean exactly to say they are all "equally real"? There are for example hidden variables interpretations where there would be some definite truth about the positions of a system of unmeasured particles even though QM would treat as being in a superposition of different position states--it's true these interpretations involve untestable metaphysics, but depending on what you mean by "equally real" (and whether it's taken as conflicting with these interpretations), your statement might involve untestable metaphysics as well.
    – Hypnosifl
    Aug 10 '20 at 21:49
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    OK, what does it mean? I agree hidden variable theories seem "cooked up" and don't favor them myself, but either accepting them or rejecting them require untestable metaphysical assumptions, so if your comments about the "reality" of multiple paths are rejecting the assumption of hidden variables theories that say the particle has a single definite path it actually takes, then you are making such assumptions. As for the specific example you mention, it's been proven that Bohmian mechanics is identical to non-relativistic QM in all possible predictions, presumably including that experiment.
    – Hypnosifl
    Aug 11 '20 at 15:45
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    @Kevin - There is no conflict between relativity as a scientific theory and the purely metaphysical hypothesis of a "hidden" preferred definition of simultaneity that can't be determined experimentally, not even in principle, and "hidden" superluminal influences between quantum hidden variables which travel faster than light but never backwards in time with respect to this definition of simultaneity. Again, I don't favor such a view myself, but any arguments for or against this view also involve metaphysical claims rather than purely scientific ones.
    – Hypnosifl
    Aug 11 '20 at 16:12
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    @Kevin - You'll need to provide more detail on your proposed experiment, I don't know what you mean by "two simultaneous Bell experiment" (how do you know they are simultaneous if the truth about simultaneity is hidden?) or what it would mean to "force the information to propagate in both directions at once" (what are the directions you're referring to?) If it would require time to explain you might consider starting a new question here or in physics.se, something like "does this experiment show it's impossible to have a hidden-variables model with hidden simultaneity"?
    – Hypnosifl
    Aug 11 '20 at 16:32
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    @Kevin - It's trivial that you can have some form of metaphysical truth about simultaneity that doesn't conflict with relativity (simply pick some foliation of a relativistic spacetime into simultaneity surfaces, and declare this definition of simultaneity to be "metaphysically true" and all others "metaphysically false" with no changes to testable predictions). But you seemed to claim there was some specific thought-experiment that showed a problem with a hidden definition of simultaneity in connection with a hidden variables interp. of QM, I take it you don't have such a specific argument?
    – Hypnosifl
    Aug 11 '20 at 17:24
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This is essentially the same as ACuriousMind's answer but a little more practical, which I find helps in considering philosophy.

The key difference is that quantum mechanics includes, more or less, negative probabilities, such that you can superimpose situations that both allow for some phenomenon P and this combination disallows P. To quote Scott Aaronson,

Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let’s try to generalize it so that the numbers we used to call “probabilities” can be negative numbers. As such, the theory could have been invented by mathematicians in the 19th century without any input from experiment. It wasn’t, but it could have been.

As ACuriousMind has noted, when this happens you lose the ability to describe reality in a locally realist way, because if I locally change a + sign to a – sign, that can introduce correlations which no locally realist theory can explain. And it is worth you having an example in your mind. This example, I call “The Game of Betrayal.”

The rules of the game

The idea of this is that it is a coöperative game for a team of three players. If you are not familiar with coöperative games, the idea is that the players all win or lose together and are trying to defeat the rules of the game—some such games include Shadows over Camelot, Arkham Horror, Pandemic, Forbidden Island, and the first half of Betrayal at House on the Hill. The team tries to work together and The Game of Betrayal tries to set conditions where they have to work at cross purposes.

To that end, the game relativistically separates the players so that no known physical phenomenon can be used to communicate between them. Each player is in a room with a screen, a timer, and two buttons labeled 0 and 1. When the round starts, a goal will be flashed on the screen and the timer will start to tick down, and before the timer hits zero the player must press exactly one of the two buttons, which will be transmitted to a central location. Once the three button presses are logged, the sum of the three pressed numbers will be compared to the transmitted goals, and we’ll figure out whether the team collectively won the round. Hopefully that description is clear enough of the physical situation the players find themselves in.

One fourth of the time, we run a control round. In this round we ask everybody Make the sum of your numbers even and the team will win if it is, indeed, even—or will lose if the sum of their three chosen numbers is odd.

Otherwise we choose one of the three at random, who we will call here the “traitor” (but we will never tell them that, they will not find out until the round is over). To the traitor, we broadcast the identical message, Make the sum of your numbers even and we are lying to him/her. To the other two, we broadcast the true goal: Make the sum of your numbers odd. And the team will win if the number is odd, or will lose if the sum of their three chosen numbers is even. (You can flip the goals if you want to, have the control rounds be “odd” and the traitor rounds be “even”, that makes no substantial difference.)

Analysis

There exists no three-person strategy in classical probability which beats this game more than 75% of the time. The easy way to see this is, classical probability allows you to construct the exact same situation by asking both questions of each individual person and then you choose randomly which sort of round it was afterwards. Either way you can describe any such strategy as a joint probability distribution over six random variables: say our people are Alice, Bob, and Carol, then the six variables are Aodd, Aeven, Bodd, Beven, Codd, and Ceven, and the four constraints we are placing upon this,

Aodd + Bodd + Ceven ≡ 1 (mod 2),
Aodd + Beven + Codd ≡ 1 (mod 2),
Aeven + Bodd + Codd ≡ 1 (mod 2), and
Aeven + Beven + Ceven ≡ 0 (mod 2),

suffice to construct an impossible equation «2 x ≡ 1 (mod 2)» and therefore classically, you must choose at least one of these constraints to violate.

Quantum players—players sharing an entangled quantum state over three qubits—can win each round 100% of the time, in theory. (In practice quantum states lose “coherence” as they interact with their environments and so even with very nice systems we might want to give some possibility of error, say 5%.) With enough rounds we can draw a clear distinction between these both and then issue a large cash prize and a small fee, to entice teams to build quantum computers and endure relativistic separation for several rounds and play this game. So for example if we require people to pass 38 out of 40 rounds, that accommodates a 5% error rate for the quantum players but if you can only pass each round with 75% probability then you only pass 38 out of 40 rounds something like 0.1% of the time.

The description of the simplest quantum strategy involves the single-qubit states |+> = √½ |0> + √½ |1> and |–> = |+> = √½ |0> — √½ |1>. By the normal rules of quantum mechanics the superposition

√½ |+++> + √½ |———> = ½ |000> + ½ |011> + ½ |101> + ½ |110>

only involves states which make the sum of the bits even, so measuring this “GHZ state” in the computational basis will be sufficient to pass control rounds.

What about in the traitor rounds? Well, if any two individuals perform the unitary transformation which takes

|+> ⇒ |+>,
|–> ⇒ i |–>

where «i = √(-1)», then the overall state must flip sign to the state

√½ |+++> – √½ |———> = ½ |001> + ½ |010> + ½ |100> + ½ |111>

at which point the sum of the three numbers is odd. So any one person unilaterally can change whether the sum will be even or odd, and two people can distribute this decision halfway between each other with no real problems.

Some properties to note

I want to observe some particulars about the quantum strategy that tend to be relatively common properties.

(1) Quantum mechanics generally does not allow things that are individually completely absurd to happen. If you had instead set the goal for the control case as “make all of your numbers zero” then I would not have the quantum freedom to generate the 100% solution. It is precisely because both 0 and 1 are plausible answers in all cases, that individuals have the freedom to tweak the phases of those numbers and cancel out some probabilities.

(2) Nobody can determine that something remarkable has happened until all of the bits are collected back together. This allows quantum mechanics to still obey the constraints of relativity in practice, no “information” can be sent faster-than-light even though local theories fail to describe what is happening.

(3) Even then, it takes many trials to reveal the remarkable thing, as clear from (1). So a locally realist theory must allow some probability of random success on the correlation: and then a quantum theory, expanding on this, generates a strangely higher-than-expected success rate.

(4) This example is carefully chosen to make the success rate 100% but often quantum mechanics pushes back on you. Quantum mechanics does not always allow 100% success rates on every possible experiment; the algebra is actually somewhat constrained. So this is the subject of a “Bell’s inequality,” named for John Stewart Bell’s analysis of the Einstein-Podolsky-Rosen paradox, and in the original analysis the correlation of quantum theory is not 100% for the maximal violation where you put the one axis at 45° relative to the other axis.

(5) The quantum system could be analyzed by the approach given for the classical system, but this would correspond to a different fundamental experiment. If we tried to ask every quantum player for two bits (one for each response to the two questions) and reassembled their chance of winning after the fact, we would still construct that impossible equation which proves at least a 25% failure rate. The spooky correlation really requires us to ask four bits out of each player corresponding to each of the four round types. So we would say that “the choice of what to measure is influencing the outcomes” but while that is a valid way of looking at it, perhaps a better statement would say that classical probability is somehow recursively embeddable in a way that quantum probability is not, so that things which look like they are valid alternate approaches classically are invalid quantumly.

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  • Thanks for such a detailed offering, a bit above my innumerate skill sets, but I can chew on it slowly, and it does have a lot of interesting aspects, with the focus on probability itself, which was my interest, not the kitty. Aug 12 '20 at 17:08
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One important aspect of Quantum Mechanics that frequently gets lost in discussions avoiding the mathematics is that there is no absolute distinction between "pure" and "superposed" states — it is all so to say in the eye of the beholder, and that beholder is typically not some human experimenter but rather a physical observable (technically: self-adjoint operator on the quantum state). @GuyInchbald hints at this in his answer by pointing out that designating some states as pure is the same thing as choosing a basis for the space of quantum states, but I fear the point needs to be stressed much stronger when raised in a philosophical context.

The case of spin

A system providing a good example of this arbitrariness of what to label as "pure" is that of the spin of an electron. The coarse popular science description is that the electron has the spin states 'up' and 'down', mathematically denoted +1/2 and −1/2 (for technical reasons that make sense in calculations). The slightly more refined description, which one encounters when wishing to discuss quantum superposition, is that the electron can also be in arbitrary superpositions of these states. After this comes the more seldomly seen (but really illuminating) further refinement that spin is really tied to a direction in physical space – we speak about 'spin up' and 'spin down' because there is a convention to use the eigenstates for spin in the Z direction as the standard basis when describing spin, but one could just as well use any other direction to define the basis states (those designated as "pure").

What is nice about the spin is that it is quite straightforward to observe, as is done in the Stern–Gerlach experiment: when an electron travels through a suitable nonhomogeneous magnetic field (ideally constant in time and direction, but strength varying between different points in space), its path will be slightly diverted according to the direction of its spin, very much as classically a rotating charged particle would be (hence the name 'spin'). Practically the experiment is conducted with a beam of electrons: passing through the magnetic field splits the beam according to the spins of the individual electrons, and that is what in the more abstract discussion of QM is described as making a measurement of their spins. (IMHO there are problems with the way that the 'measurement' concept is used in discussions of QM, but more on that later.)

From the classical physics point of view, the outcome of the Stern–Gerlach is weird, because the beam is always split into two subbeams: each electron is either shifted a certain distance in one direction, or an equal distance in the opposite direction, corresponding to spins of +1/2 and −1/2 respectively. For a classical rotating charge the distance would depend on the speed of rotation, so an initial interpretation could be that all electrons seem to always rotate at the exact same speed, but it's actually a lot weirder, because the classical effect would also depend on how well the axis of rotation aligns with the direction of the magnetic field; getting two distinct beams out would classically only happen if all charges rotate at the same speed and all have axes of rotation parallel to the magnetic field. In the Stern–Gerlach experiment you get two beams no matter what direction the magnetic field has, so it seems like electrons have an axis of rotation parallel to every direction there is! But that was just because we were trying to make a strained classical interpretation; the modern consensus is that electrons aren't rotating as such, they just happen to have a property known as spin which (among other things) causes them to interact with magnetic fields in a way that resembles how rotating charged particles would interact. That spin is quantised is in itself no weirder than that electrons "orbiting" a nucleus only has a discrete set of possible orbits; we may gladly think of 'up' and 'down' as the two possible values of the physical quantity 'spin'. As long as we only measure spin in the Z direction.

Things get weirder if we repeat the experiment with one of the subbeams coming out of the Stern–Gerlach apparatus, for example that of 'up' electrons. Feeding that through another Stern–Gerlach apparatus with magnetic field in Z direction will not split the beam, because these electrons are all in the same spin state, and are thus shifted the exact same amount. Feeding the beam through another Stern–Gerlach apparatus with magnetic field in Y direction will however split the beam 50/50 into two subbeams, because even though measuring the spin of any particular electron always produces a value of either +1/2 or −1/2, measuring the spin in the Y direction is something different than measuring it in the Z direction; one does not determine the other. Yet the actual spin state of the electron is always just a superposition of two basis states, which may be chosen as the 'up' and 'down' eigenstates in Z direction, but equally well as the two distinct eigenstates of spin in the Y direction, or the again two distinct eigenstates of spin in any other direction. None is more fundamental than any other, so it is perfectly fine to consider 'up' and 'down' as superpositions of their Y direction counterparts |σ_y = +½⟩ and |σ_y = −½⟩ (curse the lack of formula rendering on this SE!).

What happens if you place several Stern–Gerlach apparati in series (and always use only one output beam as input to the next) is that the beam splits if the magnetic field directions are different (in proportions depending on the angle between them; for a straight angle the split is equal, whereas for a smaller angle there is a bias towards the spin closer to that of the input beam) whereas it doesn't split if the directions are the same. There is in particular no "memory" going back further than that, so if you split Z, Y, Z then 1/8 of the original beam will come out 'up' from the last Z and 1/8 will come out 'down', even though all electrons in those subbeams came out in (say) the 'up' subbeam from the first Z. The standard way of doing the math here is to say that when the beam gets to the Y splitter, the relevant basis to use is that of the Y direction spin eigenstates. The electrons coming in are all in the Z direction spin eigenstate of 'up', but since 'up' in the Y basis is an equal absolute value superposition of the two basis states, each electron will have an equal 50% probability of going into the +½ subbeam or the −½ subbeam, after which they will then be in either of those two spin states instead. Then coming to the last Z splitter, everything is the same but with Z and Y interchanged, so again each electron has an equal 50% chance of going into either subbeam. It's all very neat, but just not what a training in classical physics would have led you to expect.

In particular it has been seen as inconceivable that the electrons would randomly fall into one subbeam or the other, even though that is what the experiments suggest. The aim of the various "hidden variable" theories has often been to try and restore determinism by expanding the state of an electron to make it predetermined what it would do when subjected to e.g. a particular sequence of Stern–Gerlach experiments (1/8 of the original electrons being predetermined to go up–left–up when subject to a Z,Y,Z split, and another 1/8 being predetermined to go up–left–down in the same experiment, etc.), but those theories have failed to match experimental results (especially when entanglement and interferrence enters the picture, but that's another level of complications).

More general reflections

Popular descriptions of QM often paint a picture where physical systems normally reside in one or another classical pure state, even though on microscopic scales you can temporarily create these weird quantum states that are superpositions of several pure states, but luckily those superpositions quickly "collapse" back to pure states, even though it is nondeterministic which pure state that will be the result. This picture is seriously flawed.

First, the "pure" states are not classical at all. The states one designates as pure are typically chosen to be easy to comprehend (to the extent that is possible), and one approach for such choices is to require that some observable (or quantity, in the classical terminology) has a distinct value; mathematically this means picking the eigenstates of that observable as the states to designate as pure (basis) states. But merely letting one observable have a definite value does not make the state classical — from the point of view of another observable, the state is typically a superposition of some other set of states where that second observable has definite values. A classical state would be one where both position and momentum of all particles have definite values, and such states simply do not exist.

Second, as mentioned above, what is "pure" or "superposition" is merely a matter of how we choose to describe the space of quantum states, not an aspect of the reality. Though that said, one should also be aware that some descriptions are more strained than others; there are states that would be labelled as superpositions in any humanly sensible description of the system.

Third, one should not make the mistake of believing that classical physics is always clear and intuitive whereas quantum mechanics is weird, because they both harbour plenty of things that are weird from the everyday layman perspective: shadows that are objectively less dark in the middle for one thing (by classical wave theory of light), or the finer points of celestial mechanics. But since the early writers on QM were classically trained physicists, they were biased in that these were familiar phenomena, not this new quantum weirdness. Even today the physics curriculum first deals with the classical material before addressing the quantum view, because the classical material is "easier". (Some parts surely are, but other parts I'm not so sure about; you can do a lot of QM with just linear algebra, whereas classical physics relies heavily on PDEs.)

Measurements

Another thing that frequently gets misrepresented is the matter of "measurements" in QM. From classical physics, we're used to the idea that a measurement reveals what is already there, a fact about the world that was true regardless of whether we knew it or not. Those in the business of actually making measurements know that it is not always quite that easy; a voltmeter has a large but finite impedance, so the mere act of connecting it to two points of an electrical circuit will slightly change the currents and therefore also the voltages in that circuit, however in that case the distortion is typically small enough that it may be ignored. For measuring other quantities we may be less lucky, but for the sake of philosophy it is common to disregard practical aspects such as imperfections of measurement devices (among other things, because it makes the discussion much messier).

Either way, measurements in QM aren't that easy. A common description (which IMHO is misleading) of what is going on is that the observable again has a definite value (as in the above classical model), because that is true in the pure states, but since the state at hand unfortunately is a superposition this definite value degrades into a random variable. By "collapsing the wavefunction", the act of measuring forces this random variable to pick a definite value and thus reveal its underlying truth. This description is correct insofar as one can use it to carry out the calculations, but it is not so useful for philosophical enquiries. There is even a strong version of this description according to which the superposed state is unknowable, unlike the pure states which can be known, but that strong version is simply wrong and the spin example gives a good explanation of why.

If in the repeated Stern–Gerlach experiment we pick the 'left' subbeam coming out of the Y direction splitter, then the spin states of the electrons in that beam are simply 'left'; we know this, because we just measured this to be the case — there are no ifs or buts. For then calculating the effect of the subsequent Z direction splitter, the above description would ask us to instead view this 'left' state as the equivalent superposition of 'up' and 'down'; according to QM, this is just the same thing. By measuring the spin in the Z direction, we then force each electron to pick one of those two possibilities, or at least that is one way to interpret the calculations. A seemingly similar, but according to QM (and experiments) incorrect, probabilistic interpretation is that half the electrons going in are predetermined to become 'up' electrons and the other half predetermined to become 'down' electrons; that erroneous conclusion is however easy to reach if one believes "a measurement reveals what is already there".

A better picture of measurements in QM is that a measurement of an observable of a quantum system is an interaction with that system which forces it into a state where said observable has a definite value, subject to certain rules relating the probabilities of the possible outcomes to the amplitudes of the corresponding pure components in the pre-measurement state. This may sound strange, but it matches what many measurement processes actually do: to measure the vertical/horizontal polarisation of a photon, one aims it at a polarisation filter (of e.g. vertical polarisation), and if the photon gets through it has been measured to be polarised vertically, whereas if it is reflected it has been measured to have the complementary polarisation of horizontal. Even if the photon was in fact known to be polarised at a 45° angle before encountering the polarisation filter, it will be either vertical or horizontal when leaving it. Interactions between individual particles and pieces of macroscopic experimental apparatus have a definite tendency to behave in this manner; something as simple as having a particle pass through a particular hole constitutes a measurement of the fact that the particle was at the position of that hole, i.e., a measurement of its position. Measurements can be rather subtle.

On the other hand, it is subjective whether a measurement has in fact occurred, because measurements happen when you extract classical information from a quantum system; when describing an experiment, there is at least in principle always an option to delay the point at which the measurement takes place, by expanding what you regard to be the quantum system (to include more of the experimental apparatus, particularly detectors and registration)! Concretely, if your quantum state only includes the spins of the electrons, then the Stern–Gerlach apparatus performs a measurement of the spins by forcing each electron to go either into one subbeam or the other. However if you expand the quantum state to include the position of the electron, then the apparatus simply performs the reversible state transformations

|↑,0⟩ ⟼ |↑,+d⟩  (spin up at position 0 goes to spin up at position +d)
|↓,0⟩ ⟼ |↓,−d⟩  (spin down at position 0 goes to spin down at position −d)

What happens when the spin is instead |σ_y=−½⟩ = ( |↑⟩ − i|↓⟩ )/√2 (the imaginary unit i here is significant – without it we would instead have the state |σ_x=−½⟩) is that the transformation acts independently on the spin up and spin down terms, mapping

( |↑,0⟩ − i|↓,0⟩ )/√2  ⟼ ( |↑,+d⟩ − i|↓,−d⟩ )/√2

The electron coming out of the apparatus is thus in a state that is a superposition of 'spin up at position +d' and 'spin down at position −d', rather than in just one of the two. Because this is an entangled state, we now have the extra option of deducing the spin from measuring the position, but the Stern–Gerlach apparatus itself has not measured the spin. It would (at least in theory; I'm not versed well enough in the experimental aspects to be sure of how practical it is to do with electrons) be quite possible to recombine the two beams using a second Stern–Gerlach appartus with the magnetic field in the opposite direction, and thereby recover the original state of the electron. The measurement does not happen until the electron hits a detector not part of the quantum system, and theoretically there is no problem of redefining that detector and its records as being part of an even larger quantum system, in which case the superposition persists until someone looks at those records to see what the detector registered.

Final opinions

I believe the above is all established physics. What I haven't seen is anyone drawing the (IMHO) obvious conclusions of the above points with respect to more philosophical concepts such as determinism/nondeterminism, so the following rather counts more as my own opinions. It is however perfectly possible that this just happens to coincide with one of the standard interpretations of QM — a lot of them seem to have rather illogical names, so it is quite likely that I would fail to find this even if I spent a month looking for it.

There is a paradox present in the ordinary description of quantum mechanics, in that quantum systems are supposed to evolve unitarily — a property stronger than deterministically in that not only is the future completely determined by the present, but the past is also so determined, since everything is reversible (no information is ever created or destroyed, just rearranged) — until a measurement occurs, at which point the system makes a random transition that creates new information (result of measurement) and destroys old (actual state before measurement). This is a paradox because the physics laboratories in which such measurements happen are built up from matter that supposedly interacts in ways that obey the unitary laws of quantum mechanics — if at the micro scale everything is unitary, then how can it fail to be so also at the macro scale?!?

One solution is apparent in the idea that measurements happen when you extract classical information from a quantum system. The catch is that in a quantum mechanical universe, there is no such thing as classical information, although at macroscopic scales you can get darn good approximations of it (or at least: so it seems). Consequently measurements cannot exist either (which resolves the paradox), although in interactions between micro and macro systems there has to be something which manages to produce very good approximations of them — maybe the superpositions do not in fact collapse, but rather one of the outcomes get heavily suppressed (somewhat like in Grover's algorithm)? Or more likely, it's down to entanglement with the environment — certainly if you had a quantum system running the usual experiments to test whether the laws of quantum mechanics hold you would get an overwhelming probability for the conclusion 'yes', but low probabilities for any particular outcome in many intermediate stages which anyway aren't important. This is speculation, but speculation that could potentially be examined mathematically: would unitary interactions between macroscopic and microscopic quantum systems behave in ways that approximate the QM laws for "measurements" (for macro systems that approximate classical information processing)?

If they do, this also puts an interesting spin on the matter of QM randomness, since it philosophically comes out pretty much the same as the deterministic pseudorandomness used for cryptography: both rely on external entropy sources (in the quantum case: entangling with the environment) to produce results that come out as effectively random. It's only that in the quantum case this happens spontaneously, whereas in classical computers we need fancy hash algorithms to achieve similar effects.

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What is important here, is that you have already specified a 'basis'. In this case, the two states of the cat being alive or dead form such a basis. In quantum mechanics, a measurement is determined by specifying a basis (ignoring so-called POVM's here). The possible measurement outcomes that can be obtained are exactly the states in the basis, and the probabilities are determined by how `close' your initial state is to each of the basis states.

But there is now something funny we can do, we can choose a different basis to measure in. Our basis has two states in it, both equal superpositions of the cat being alive and dead. However, one of them has a plus sign in front of the dead state, and the other one a minus.

Let us now measure with this basis. Assuming our initial state is the equal superposition between alive and dead with a plus sign, we will always get that outcome. But this is quite peculiar, we started off with a state that had some uncertainty attached to it, but by measuring in a different way, we will get deterministic outcomes. Compare this with the classical scenario, where such a thing is not possible in a non-trivial way.

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Imagine for a moment that we sit down with a few friends to play a hand of poker. The cards get dealt, and in that moment before everyone picks up their hands we have a classical probabilistic situation. No one knows what their cards are; the best we can do is make an educated (probabilistic) guess about what kinds of cards we might get. Of course, the cards themselves are not probabilistic. Once the cards have been dealt, the hand we each have is determined. We just don't know what it is, so our knowledge is limited to probabilities. Incidentally, this is what card cheating leverages; people who cheat at cards do things that give them just a bit more knowledge about the actual state of the cards than the rest of us have — marking cards, dealing seconds, etc. — which makes their probabilistic assessments that much better.

Now, imagine that instead of using a normal (classical) deck of cards, I pull out my quantum deck and deal from that. In that moment before we pick up our hands we have the exact same probabilistic situation in terms of our knowledge, but the physical reality is different. Instead of having face-down hands of determined but unknown cards, the face-down cards have no explicit values. They are a 'superposition' of all the possible hands that could be dealt from that quantum deck. The moment I pick up my hand, the cards snap into focus as one particular hand, and every other card in the deck instantly re-evaluates its superposition to exclude the cards that appeared in my hand. In fact, this highlights the difference between classical and quantum probabilities. Once we've all picked up our cards, I don't know what's in your hand and you don't know what's in mine, but that's just lack-of-knowledge. Your cards and mine are both concretized (fixed in value), even though we both have to make probabilistic assessments of the other's hand. But the un-dealt cards that remain in the quantum deck are still in superposition, having no concrete value, and able to take any value except the ones that have already been concretized in player's hands.

What this underlying reality actually is is difficult to imagine; that's why there's so many different hypotheses about it. Generally I find it easier to think about this in terms of systems than objects. That's why I talked about a quantum deck rather than a set of quantum cards, because the individual cards have to be entangled with each other in the context of the deck for this to work (otherwise two of them might be observed to be the King of Spades). A system of this sort is (perhaps) like a big ball of tangled yarn: pull on one thread and the whole ball tightens or loosens or unravels. But analogies are all going to fail at some point, because the quantum world is fundamentally different than the classical world we are adapted to.

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  • Thanks, this is a nice illustration for someone like me without the formal training. I know you can't really get it without the math, but you can get a kind of useful picture from imaginative descriptions like this. Math is the power tool, but metaphor isn't shabby. Aug 12 '20 at 17:12
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Your answer is related to Consciousness view of superposition.

H. D. Zeh, The Problem of Conscious Observation in Quantum Mechanical Description

The problems of formulating a process of observation within quantum theory arise because of quantum nonlocality (quantum correlations or \entanglement" as part of the generic state), which in turn may be derived as a consequence of the superposition principle. For dynamical reasons, this non-locality does not even approximately allow the physical state of a local system (such as the brain or parts thereof) to exist . Hence, no state of the mind can exist parallel" to it (that is, correspond to it one-to-one or determine it).

The problem does not only concern the philosophical issue of matter and mind. It has immediate bearing on quantum physics itself, as the state vector seems to suer the well known reaction upon observation: its \collapse". For this reason Schr¨odinger even argued that the wave function might not represent a physical state (neither of the system itself, nor of a system carrying information about it), but should rather have a fundamental psychic meaning".

A dynamical collapse of the wave function would require nonlinear and nonunitary terms in the Schr¨odinger equation . They may be extremely small, and thus become eective only through practically irreversible ampli-cation processes occurring during measurement-like events. The superposition principle would then be valid only in a linearized version of the theory perhaps related to wave function renormalization. While this suggestion may in principle explain quantum measurements, it would not be able to describe definite states of concsiousness unless the parallelism were articially restricted to quasi-classical variables in the brain. Since nonlinear terms in the Schr¨odinger equation must lead to observable deviations from conventional quantum theory, they should at present be disregarded for similar reasons as hidden variables. Any suggested violation of the superposition principle must be viewed with great suspicion because of the latter's great and general success. For example, even superpositions of dierent vacua have proven heuristic (that is, to possess predictive power) in quantum field theory.

The quantum world (described by a wave function) would correspond to one superposition of myriads of components representing classically different worlds. They are all dynamically coupled (hence \actual"), and they may in principle (re)combine as well as branch. It is not the quantum world that branches in this picture, but consciousness (or rather the state of its physical carrier), and with it the observed (apparent) world .

https://arxiv.org/abs/quant-ph/9908084

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    The idea that consciousness plays an essential role in quantum physics is a minority view that is only held by certain variants of the "Copenhagen Interpretation" of quantum physics, see the last paragraph of the "Divergent Views" section of the Copenhagen Interpretation SEP article. The idea that superposition differs in some important sense from classical uncertainty is much more broadly accepted. Also note that environmental decoherence seemingly produces the same changes to the statistics as measurement by conscious beings.
    – Hypnosifl
    Aug 10 '20 at 17:31
  • Thank you. I have some trouble with your terminology here, but I'll try to follow up on this "consciousness" interpretation. It is true that a complete physical interpretation of "observation" was part of my curiosity. In this case, however, I was more interested in the distinction between "superposition" and any state we call "probabilistic" or "future," with "observer" remaing constant. Aug 10 '20 at 17:49
  • See my comment on the original question. This is an example of the trend that I was talking about. Aug 11 '20 at 21:18
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I think the Schrodinger's Cat thought experiment is actually a confounding factor for your understanding. Its meaning and purpose is not what most people think. As Schrödinger wrote:

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber ...

The point that this idea was 'ridiculous' has somehow been lost in the popular interpretation of the thought experiment and leads people to a simplistic and wrong idea about quantum mechanics (QM). Essentially, it leads people to believe that a superposition of states is equivalent to not knowing the true state.

When I'm playing poker with my friends. I don't know what cards are going to appear. That knowledge is unknown to me until the cards are revealed. There's nothing astounding or interesting about this. It's definitely not something that required a large team of extremely brilliant scientists to figure out.

What is astounding about QM is that we can actually demonstrate in repeatable experiments that real objects are actually (i.e. in reality) in multiple mutually exclusive states at once. If that sentence isn't strange to you, read it again. It's absolutely self-contradictory but proven to be true. Proven beyond a doubt. That's what is so 'shocking' about QM. It contradicts our experience of reality and the logic that we have built to understand it.

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  • Can you provide references to such experiments? From my limited QM training, this just means that the system is a linear combination of its eigenvalues/eigenstates. It's nothing strange, just that the linear combination of eigenstates is also an eigenstate. And these eigenstates are just the wave function, which has a probabilistic interpretation. So it's not really in each of these states at once, it's in a state corresponding to a linear combination of eigenstates, which provide a complete basis for all possible states. Of course you could be talking about something totally different... Aug 18 '20 at 21:17
  • @user5728491 A wave function is a mathematical construct and nothing more. It only exists in our minds. You are essentially describing the Copenhagen interpretation that the waveform collapses (i.e. 'something happens') when observation occurs. For examples look no further than the double-slit experiment. Another option is to read about the various approaches to quantum computing in use. Here's an article about various interpretations of QM.
    – JimmyJames
    Aug 19 '20 at 15:11

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