Quantum Superposition Paradox

Superposition is a weird thing, and there's definitely disagreement over how to think about it. Mostly, there is disagreement over whether or not it literally represents multiple realities. That is to say, when Schrödinger's cat is in a superposition, is it both alive and dead at the same time?

It's hard to talk about that without getting into metaphysical debates (see my entry on the measurement problem). While metaphysics is good to get into, it'd be nice to avoid that, at least initially.

So maybe a good starting place for talking about the interpretation of superpositions are logical sentences about them.

Multiple Realities

For example, if it's true that 'Schrödinger's cat is in a superposition state', does that logically imply the statement 'the cat is alive' as well as the statement 'the cat is dead'. More symbolically, is it the case that

'Superposition' -> ('alive' & 'dead') ?

I'd wager that this view of superpositions is widely held. Most will probably think of superpositions as implying multiple realities in the very meaning of the term. This would be an a priori assertion, and it would agree with the phrasing above.

Opposed to this view would be the belief that there is no true logical relation that looks like the one above. One could argue, for example, that superpositions could possibly imply multiple realities, but that it would have to be decided empirically. This would mean that the above logical statement wouldn't always be true, and that it would depend on the circumstances.

See how much cleaner that is than normal? I don't have to talk about metaphysical things, but still get to articulate something providing some insight.

Contradictory Realities

The second half of this is what relationship is between the states of life and death. Do they contradict? If so, the statement

~('alive' & 'dead')

is directly contradictory with the superpositions statement above. In other words, one of these two statements must be false.

This second statement certainly seems natural to accept as well. We certainly think of life and death as being mutually exclusive states. But what could show that they really are mutually exclusive states, and not just compatible states that appear incompatible to our mistaken intuition?

More formally, maybe we can say that mutually exclusive pairs of physical states will create two conflicting set of predictions, both of which can't be true. For example, a macroscopic object can be both falling and electrically charged. There is no reason why these two physical states should make conflicting predictions, so a physical object can be in both states at the same time.

(I think the full story of why this is the case looks like this: since a system is said to be in a physical state only when it fits a physical model, and physical models are what we need to make predictions, having a contradiction in the predictions of two models applied to the same system suggests an error somewhere further up in reasoning. Either a model was applied that should not have, or there was an error in how one or more of them were applied. Feynman describes using this very line of thought to call out the existence of errors in other physicist's reasoning and impress others without working through all of the math. It's in Surely You Must Be Joking Mr. Feynman p.244-5. The fact that the statement could be wrong either because an incorrect model was applied or a correct model was incorrectly applied means that this second statement has an a priori sense to it and an empirical sense to it.)

Two mutually exclusive states would give predictions that contradict. For example, having a macroscopic object (not currently under quantum effects) be both charged and not charged would create contradictions. There would be two sets of predictions created by the two possible situations: when the object is charged and when it is not, and those predictions could differ significantly. Therefore, only one of them can be true, and a well designed experiment will typically decide between the two.

So are quantum states different? Do quantum effects somehow make states that are normally mutually exclusive so that they are not in the right circumstances?

An interpretation that would argue that this statement is false is the many-worlds interpretation. While classical states like 'alive' and 'dead' are mutually exclusive, in the quantum realm they are not. The simple fact for this being that they exist in separate worlds. It's not a contradiction to say that the cat is both dead and alive because the dead cat exists separately from the alive cat, and it is this world-splitting aspect of quantum mechanics that keeps these two states from producing contradictory predictions.

So yeah, I dunno how helpful this approach will be, or if there is any literature on breaking it down this way. I'm calling it the quantum superposition paradox, which seems appropriate, but I don't know if there would be a better name for it.

Schrödiner's Cat: Explain Live I'm Five

Recently I answered a thread on Reddit. Someone asked for the Schrödinger's cat thought experiment to be explained to them "like they were five" (basically, just use simple language and simple terms to explain something complicated). I was happy with the result, so I thought I'd share it.

******

Schrödinger's cat is really strange, and that is the way it is meant to be. Schrödinger thought about this to try and make quantum mechanics seem just too weird to accept. He was wrong, though: quantum mechanics is true. So now we have a story that sounds really weird but actually reveals something hard to understand.

Matter is a Wave

The easy understand part: if you shrink down really tiny, all of matter looks like waves. These aren't sound waves or water waves, though they sometimes act similarly.

Imagine making a splash in a pool and looking at it from the side. You will see a little wave go to the left and one will go to the right. Matter looks kinda like that sometimes: a little bit of the splash goes to the left and little goes to the right. This state is called a superposition.

Now here's the weird part: even when there is only one particle, it still behaves this way. "Splash" this one particle, and it wiggles to the left, and wiggles to the right. One might ask: which way did it actually go? Well, it did not go to the left, and neither did it go to the right. It didn't literally go both directions, it just went the direction you go when you go left and right at the same time. It wiggles in both directions because it is in a superposition.

My Brain Hurts

This is weird. Lots of smart people did not like this. It made their brains hurt and they were unhappy with the idea. "How can something be in two places at once?" they would ask. Schrödinger asked this as well. He was a smart man, though, and came up with a story to help him argue.

He asked "If a particle in a superposition can move in two directions at once, then couldn't a cat in a superposition be both dead and alive at the same time?" Obviously, this made quantum mechanics sound silly.

Quantum Mechanics: Not Silly

But these people were wrong: quantum mechanics is not as silly as it sounds, it is just really really hard to understand. When a particle wiggles both left and right, it would be wrong to say that it "went to the left". It would also be wrong to say that it "went to the right". That is because it did not wiggle in only one of those directions: it wiggled in both directions because it was a wave.

Still Confused?

You might still wonder: how can something be in more than one state at once?

The answer is that it was not in several different states, it was only in one state, a superposition state. Remember, we know that the quantum wave did not go to the left AND to the right, it wiggled in both directions. It did not become two waves moving away from each other, it was still just one wave spreading apart.

You should know that no one has done this experiment with a cat. It turns out to be something that is very very hard to do. It is hard to visualize also. But the cat is made up of atoms that can wiggle in two directions. One of those directions might make the cat die, and the other direction will let the cat live. The cat is somehow in a superposition state of alive and dead. When we open the box, the cat stops wiggling and we see either an alive or a dead cat. How exactly it stops wiggling and how it becomes only one or the other is something very smart people argue about, so don't worry about that part yet.

If it seems really confusing, that is because it is.

Quantum Bayesianism Interpretation

Recently I picked up the June 2013 issue of Scientific American and read an interesting article about the quantum Bayesianism interpretation of quantum mechanics. I have not studied this interpretation at all, and the article piqued my interest, so I decided to look into it. Here are some of my initial reactions to the interpretation after looking into the commentary N. David Mermin's commentary about the topic.

Resources:
Commentary by N. David Mermin

Sci Am Article

Bayesian Probability

Quantum Bayesianism is one of the latest interpretations that has taken a swing at making sense of quantum mechanics, a subject that is notoriously difficult to make sense of. It seems to be based around extending certain notions found in the subjective Bayesian interpretation of probability.

The subjective Bayesian interpretation of probability, as I understand it, is something like this: Probability is not something that is intrinsically inherent in any system; it is simply a judgment of how likely that thing is going to happen.

This is in contrast with the Frequentist interpretation of probability, which says that probability is some intrinsic property of a situation, which can be determined through many trials. For example, we know that the probability of flipping a coin and having it land on heads or tails is 50/50 because we are able to perform many trials. Flipping a coin many times will yield an equal number of heads as tails, if the coin is a fair one.

Both the Bayesian and the Frequentist interpretations will agree that flipping a fair coin a great number of times will result in an even distribution: this is uncontroversial. The disagreement is that the Frequentist argues that this is what defines the probability of something happening in any given situation. A statement of probability has meaning because if we were to perform many trials in the same situation we would see a ratio develop. In this way, the Frequentist interpretation of probability claims that probability ascribes a value that is inherent to some situation.

In contrast, the subjective Bayesian interpretation of probability says that we should not think of probabilities as something inherent to an object or system. Instead, they are aspects of our belief about the object or system.

Relevance to Quantum Mechanics

I am not interested in delving much further into this debate in the philosophy of mathematics since it is not relevant to our discussion on quantum interpretations. The important thing to pull from the above is the last bit: Bayesianism considers probability to be an aspect, not of some object or system, but of our belief about that object or system. In this way, Bayesianism claims that probability is inherently subjective.

This is relevant to quantum mechanics because one of the first attempts to make sense of quantum states has stuck around. This is the generalized Born interpretation, and it states that:

"If the state of the system is |α⟩, then the probability that a measurement finds the system in a state |β⟩ is |⟨β|α⟩|2" (Ohanian 1990 p.102)

This means that we make sense of quantum states by thinking of them as probabilities that the quantum system will be “measured” to be in some other state.

Quantum Bayesianism has connected this notion to the subjective Bayesian interpretation of probability. It says that because quantum states are defined using probabilities, this means that they are inherently subjective. That is to say, they are aspects of our beliefs about the world and not aspects of the world itself.

“Since quantum states determine probabilities, if probabilities are indeed assigned by an agent to express her degree of belief, then the quantum state of a physical system is not inherent in that system but assigned by an agent to encapsulate her beliefs about it. State assignments, like probabilities, are relative to an agent.” (Mermin Physics Today 2012, p.8)

The Quantum Bayesianism Interpretation

Quantum Bayesianism, or QBism for short, claims to solve different paradoxes of quantum mechanics. For example, Schrödinger’s cat does not describe a cat that is both dead and alive at the same time. The cat is described by a quantum state, which is nothing more than a subjective description; it doesn’t describe the actual cat. If we were to actually open the box the cat is clearly either alive or dead.

Furthermore, QBism says that looking in the box does not cause the cat to become either dead or alive. Other interpretations, such as the multi-worlds interpretation, would claim that measuring the cat would cause its wave function to collapse. QBism argues that collapse is not a change of state in the system, but in our information about the system; therefore there is no need for multiple universes or other metaphysical postulations.

“Collapse” is still a meaningful word to QBism, but it signifies the change in a quantum state and not a change in the system itself.

Metaphysical Claims

QBism may appear initially to be claims about mathematical probabilities, but it is truly a metaphysical claim. QBism relies on the distinction between mind and matter. That is to say, we can identify things which are part of the world outside of our minds, and we can also identify things which are part of our mind and do not exist in the world. For example, a hallucination is an example of something that is in our minds but obviously does not exist in the world.

The central claim of QBism is, of course, that quantum states fall into that second category: they are part of minds and not part of the world.

This is a problem, however. QBism will have trouble answering the question “does quantum mechanics describe the world outside of our minds?” We are compelled to answer: “it describes some part of it, yes”, but QBism would lack the ability to make that claim. Remember, if quantum states do not describe the world, and are really part of our minds, it is not clear how quantum mechanics describes the real world that is outside of our heads.

Proponents of QBism could claim that quantum mechanics describes the world outside of our minds; they simply would have no basis for this claim. They would have to claim that quantum mechanics describes the world in some objective way that we simply do not understand yet. This is counter to our intuition that we have good reason to believe that quantum mechanics describes the world, and not simply an aspect of our minds.

The obvious alternative is that quantum mechanics describes nothing more than some part of our minds. All of this business of wave functions and evolution of the state vector are nothing more than inventions of our minds that have no bearing on external reality. This would lead to a kind of quarantined idealism for specifically quantum states. I feel as if this is a rather counterintuitive and controversial claim, so I guess that most proponents of QBism would advocate the former realism.

A Return to Schrödinger’s Cat

Perhaps an example is in order? Imagine the case of Schrödinger’s cat again. QBism has good reason to argue that a cat is not both dead and alive at the same time. They could say that it is in a superposed state, but they would have to qualify this and say that this only means that we are able to ascribe certain probabilities to it and that it is not a state that the cat is in.

QBism would then have difficulty saying anything about the cat. On the one hand, I have seen it written that QBism says that the cat can be considered either dead or alive; we simply do not know which.

“[Quantum Bayesianism] says that of course the cat is either dead or alive (and not both). Sure, its wave function represents a superposition of alive and dead, but a wave function is just a description of the observer’s beliefs.” (Baeyer SciAm June 2013, p.49)

The problem with this is that the statement “the cat is actually either dead or alive when it is described by quantum mechanics as being in a superposition” is not true! Superposition accounts for interference effects, whereas classical states cannot.

For this reason, QBism fails to explain how quantum states seem to describe the behavior of the system while failing to actually describe reality outside of our minds.

Strengths of the Interpretation

Let me end on a positive note about the interpretation. Though it fails to give an adequate account of some basic questions we might have about quantum mechanics (such as “how does quantum mechanics describe the world?” and “what state is Schrödinger’s cat actually in, if not a superposition?”), it is a surprisingly sophisticated interpretation.

I think that it zeroes in on good questions that should be asked in examining quantum interpretations. We should be taking a second look at the probabilistic nature of quantum mechanics and asking questions about the generalized Born interpretation.

In addition, N. David Mermin’s commentary of QBism outlines the problem of interpreting quantum mechanics as one of a “split” between classical and quantum mechanics. This is a step towards phrasing the debate as one about the inability to translate between two contradictory theories. This is a more advanced way to tackle the problem as compared to some others (such as worrying about the role of the “mind” and “observer”). I am not sure if this is unique to Mermin’s commentary or if it is a feature of literature about QBism as a whole. Regardless, I think that QBism seems to be an especially strong theory when viewed under the light of this particular way of phrasing the debate.

In addition, QBism holds a rather robust objection to the claim that choice of measurement basis affects reality and that consequently the mind of the observer is necessarily part of quantum mechanics. It denies that superpositions represent multiple realities, and that therefore collapsing superpositions constitute a change in possible reality. This, I think, is the right objection to the claim.

So on the whole, I found quantum Bayesianism to be a much more sophisticated interpretation than I initially anticipated. I do think that it fails to account for the reality that quantum mechanics describes, as I detailed above. I will continue to look into the interpretation.