We’ll get back to measurement, interference and the double-slit experiment just as soon as I can get my math program to produce pictures of the relevant wave functions reliably. I owe you some further discussion of why measurement (and even interactions without measurement) can partially or completely eliminate quantum interference.
But in the meantime, I’ve gotten some questions and some criticism for arguing that superposition is an OR, not an AND. It is time to look closely at this choice, and understand both its strengths and its limitations, and how we have to move beyond it to fully appreciate quantum physics. [I probably should have written this article earlier — and I suspect I’ll need to write it again someday, as it’s a tricky subject.]
The Question of Superposition
Just to remind you of my definitions (we’ll see examples in a moment): objects that interact with one another form a system, and a system is at any time in a certain quantum state, consisting of one or more possibilities combined in some way and described by what is often called a “wave function”. If the number of possibilities described by the wave function is more than one, then physicists say that the state of the quantum system is a superposition of two or more basic states. [Caution: as we’ll explore in later posts, the number of states in the superposition can depend on one’s choice of “basis”.]
As an example, suppose we have two boxes, L and R for left and right, and two atoms, one of hydrogen H and one of nitrogen N. Our physical system consists of the two atoms, and depending on which box each atom is in, the system can exist in four obvious possibilities, shown in Fig. 1:
- HL NL (i.e. both the hydrogen atom and the nitrogen atom are in the left box)
- HL NR
- HR NL
- HR NR
Before quantum physics, we would have thought those were the only options; each atom must be in one box or the other. But in quantum physics there are many more non-obvious possibilities.
In particular, we could put the system in a superposition of the form HL NL + HR NR, shown in Fig. 2. In the jargon of physics, “the system is in a superposition of HL NL and HR NR“. Note the use of the word “and” here. But don’t read too much into it; jargon often involves linguistic shorthand, and can be arbitrary and imprecise. The question I’m focused on here is not “what do physicists say?”, but “what does it actually mean?”
In particular, does it mean that “HL NL AND HR NR” are true? Or does it mean “HL NL OR HR NR” is true? Or does it mean something else?
The Problems with “AND”
First, let’s see why the “AND” option has a serious problem.
In ordinary language, if I say that “A AND B are true”, then I mean that one can check that A is true and also, separately, that B is true — i.e., both A and B are true. With this meaning in mind, it’s clear that experiments do not encourage us to view superposition as an AND. (There are theory interpretations of quantum physics that do encourage the use of “AND”, a point I’ll return to.)
Experiment Is Skeptical
Specifically, if a system is in a quantum superposition of two states A and B, no experiment will ever show that
- A is true
AND - B is true.
Instead, in any experiment explicitly designed to check whether A is true and whether B is true, the result will only reveal, at best, that
- A is true and B is not true
OR - B is true and A is not true.
The result might also be ambiguous, neither confirming nor denying that either one is true. But no measurement will ever show that both A AND B are definitively true. The two possibilities A and B are mutually exclusive in any actual measurement that is sensitive to the question.
In our case, if we go looking for our two atoms in the state HL NL + HR NR — if we do position measurements on both of them — we will either find both of them in the left box OR both of them in the right box. 1920’s quantum physics may be weird, but it does not allow measurements of an atom to find it in two places at the same time: an atom has a position, even if it is inherently uncertain, and if I make a serious attempt to locate it, I will find only one answer (within the precision of the measurement). [Measurement itself requires a long discussion, which I won’t attempt here; but see this post and the following one.]
And so, in this case, a measurement will find that one box has two atoms and the other has zero. Yet if we use “AND” in describing the superposition, we end up saying “both atoms are in the left box AND both atoms are in the right box”, which seems to imply that both atoms are in both boxes, contrary to any experiment. Again, certain theoretical approaches might argue that they are in both boxes, but we should obviously be very cautious when experiment disagrees with theoretical reasoning.
The Fortunate and/or Unfortunate Cat
The example of Schrodinger’s cat is another context in which some writers use “and” in describing what is going on.
A reminder of the cat experiment: We have an atom which may decay now or later, according to a quantum process whose timing we cannot predict. If the atom decays, it initiates a chain reaction which kills the cat. If the atom and the cat are placed inside a sealed box, isolating them completely from the rest of the universe, then the initial state, with an intact atom (Ai) and a Live cat (CL), will evolve to a state in a superposition roughly of the form Ai CL + Ad CD, where Ad refers to a decayed atom and CD refers to a Dead cat. (More precisely, the state will take the form c1 Ai CL + c2 Ad CD, where c1 and c2 are complex numbers with |c1|2 + |c2|2 = 1; but we can ignore these numbers for now.)
Leaving aside that the experiment is both unethical and impossible in practice, it raises an important point about the word “AND”. It includes a place where we must say “AND“; there’s no choice.
As we close the box to start the experiment, the atom is intact AND the cat is alive; both are simultaneously true, as measurement can verify. The state that we use to describe this, Ai CL, is a mathematical product: implicitly “Ai CL” means Ai x CL, where x is the “times” symbol.
Later, the state to which the system evolves is a sum of two products — a superposition (Ai x CL) + (Ad x CD) which includes two “AND” relationships
1) “the atom is intact AND the cat is alive” (Ai x CL)
2) “the atom has decayed AND the cat is dead” (Ad x CD)
In each of these two possibilities, the state of the atom and the state of the cat are perfectly correlated; if you know one, you know the other. To use language consistent with English (and all other languages with which I am familiar), we must use “AND” to describe this correlation. (Note: in this particular example, correlation does in fact imply causation — but that’s not a requirement here. Correlation is enough.)
It is then often said that, theoretically, that “before we open the box, the cat is both alive AND dead”. But again, if we open the box to find out, experimentally, we will find out either that “the cat is alive OR the cat is dead.” So we should think this through carefully.
We’ve established that “x” must mean “AND“, as in Fig. 4. So let’s try to understand the “+” that appears in the superposition (Ai x CL) + (Ad x CD). It is certainly the case that such a state doesn’t tell us whether CL is true or CD is true, or even that it is meaningful to say that only one is true.
But suppose we decide that “+” means “AND“, also. Then we end up saying
- “(the cat is alive AND the atom is intact) AND (the cat is dead AND the atom has decayed.)”
That’s very worrying. In ordinary English, if I’m referring to some possible facts A,B,C, and D, and I tell you that “(A AND B are true) AND (C AND D are true)”, the logic of the language implies that A AND B AND C AND D are all true. But that standard logic would leads to a falsehood. It is absolutely not the case, in the state (Ai x CL) + (Ad x CD), that CL is true and Ad is true — we will never find, in any experiment, that the cat is alive and yet the atom has decayed. That could only happen if the system were in a superposition that includes the possibility Ad x CL. Nor (unless we wait a few years and the cat dies of old age) can it be the case that CD is true and Ai is true.
And so, if “x” means “AND” and “+” means “AND“, it’s clear that these are two different meanings of “AND.”
“AND” and “AND”
Is that okay? Well, lots of words have multiple meanings. Still, we’re not used to the idea of “AND” being ambiguous in English. Nor are “x” and “+” usually described with the same word. So using “AND” is definitely problematic.
(That said, people who like to think in terms of parallel “universes” or “branches” in which all possibilities happen [the many-worlds interpretation] may actually prefer to have two meanings of “AND”, one for things that happen in two different branches, and one for things that happen in the same branch. But this has some additional problems too, as we’ll see later when we get to the subtleties of “OR”.)
These issues are why, in my personal view, “OR” is better when one first learns quantum physics. I think it makes it easier to explain how quantum physics is both related to standard probabilities and yet goes beyond it. For one thing, “or” is already ambiguous in English, so we’re used to the idea that it might have multiple meanings. For another, we definitely need “+” to be conceptually different from “x“, so it is confusing, pedagogically, to start right off by saying that both mathematical operators are “AND”.
But “OR” is not without its problems.
The Problems with “OR”
In normal English, saying “the atom is intact and the cat is alive” OR “the atom has decayed and the cat is dead” would tell us two possible facts about the current contents of the box, one of which is definitely true.
But in quantum physics, the use of “OR” in the Schrodinger cat superposition does not tell us what is currently happening inside the box. It does tell us the state of the system at the moment, but all that does is predict the possible outcomes that would be observed if the box were opened right now (and their probabilities.) That’s less information than telling us the properties of what is in the closed box.
The advantage of “OR” is that it does tell us the two outcomes of opening the box, upon which we will find
- “The atom is intact AND the cat is alive”
OR - “The atom has decayed AND the cat is dead”
Similarly, for our box of atoms, it tells us that if we attempt to locate the atoms, we will find that
- “the hydrogen atom is in the left box AND the nitrogen atom is in the left box”
OR - “the hydrogen atom is in the right box AND the nitrogen atom is in the right box”
In other words, this use of AND and OR agrees with what experiments actually find. Better this than the alternative, it seems to me.
Nevertheless, just because it is better doesn’t mean it is unproblematic.
The Usual Or
The word “OR” is already ambiguous in usual English, in that it could mean
- either A is true or B is true
- A is true or B is true or both are true
Which of these two meanings is intended in an English sentence has to be determined by context, or explained by the speaker. Here I’m focused on the first meaning.
Returning to our first example of Figs. 1 and 2, suppose I hand the two atoms to you and ask you to put them in either box, whichever one you choose. You do so, but you don’t tell me what your choice was, and you head off on a long vacation.
While I wait for you to return, what can I say about the two atoms? Assuming you followed my instructions, I would say that
- “both atoms are in the left box OR both atoms are in the right box”
In doing so, I’m using “or” in its “either…or…” sense in ordinary English. I don’t know which box you chose, but I still know (Fig. 5) that the system is either definitely in the HL NL state OR definitely in the HR NR state of Fig. 1. I know this without doing any measurement, and I’m only uncertain about which is which because I’m missing information that you could have provided me. The information is knowable; I just don’t have it.
But this uncertainty about which box the atoms are in is completely different from the uncertainty that arises from putting the atoms in the superposition state HL NL + HR NR!
The Superposition OR
If the system is in the state HL NL + HR NR, i.e. what I’ve been calling (“HL NL OR HR NR“), it is in a state of inherent uncertainty of whether the two atoms are in the left box or in the right box. It is not that I happen not to know which box the atoms are in, but rather that this information is not knowable within the rules of quantum physics. Even if you yourself put the atoms into this superposition, you don’t know which box they’re in any more than I do.
The only thing we can try to do is perform an experiment and see what the answer is. The problem is that we cannot necessarily infer, if we find both atoms in the left box, that the two atoms were in that box prior to that measurement.
If we do try to make that assumption, we find ourselves in apparent contradiction with experiment. The issue is quantum interference. If we repeat the whole process, but instead of opening the boxes to see where the atoms are, we first bring the two boxes together and measure the atoms’ properties, we will observe quantum interference effects. As I have discussed in my recent series of five posts on interference (starting here), quantum interference can only occur when a system takes at least two paths to its current state; but if the two atoms were definitely in one box or definitely in the other, then there would be only one path in Fig. 6.
Prior to the measurement, the system had inherent uncertainty about the question, and while measurement removes the current uncertainty, it does not in general remove the past uncertainty. The act of measurement changes the state of the system — more precisely, it changes the state of the larger system that includes both atoms and the measurement device — and so establishing meaningfully that the two atoms are now in the left box is not sufficient to tell us meaningfully that the two atoms were previously and definitively in the left box.
So if this is “OR“, it is certainly not what it usually means in English!
This Superposition or That One?
And it gets worse, because we can take more complex examples. As I mentioned when discussing the poor cat, the superposition HL NL + HR NR is actually one in a large class of superpositions, of the form c1 HL NL + c2 HR NR , where c1 and c2 are complex numbers. A second simple example of such a superposition is HL NL – HR NR, with a minus sign instead of a plus sign.
So suppose I had asked you to put the two atoms in a superposition either of the form HL NL + HR NR or HL NL – HR NR, your choice; and suppose you did so without telling me which superposition you chose. What would I then know?
I would know that the system is either in the state (HL NL + HR NR) or in the state (HL NL – HR NR), depending on what you chose to do. In words, what I would know is that the system is represented by
- (HL NL OR HR NR) OR (HL NL OR HR NR)
Uh oh. Now we’re as badly off as we were with “AND“.
First, the “OR” in the center is a standard English “OR” — it means that the system is definitely in one superposition or the other, but I don’t know which one — which isn’t the same thing as the “OR“s in the parentheses, which are “OR“s of superposition that only tell us what the results of measurements might be.
Second, the two “OR“s in the parentheses are different, since one means “+” and the other means “–“. In some other superposition state, the OR might mean 3/5 + i 4/5, where i is the standard imaginary number equal to the square root of -1. In English, there’s obviously no room for all this complexity. [Note that I’d have the same problem if I used “AND” for superpositions instead.]
So even if “OR” is better, it’s still not up to the task. Superposition forces us to choose whether to have multiple meanings of “AND” or multiple meanings of “OR”, including meanings that don’t hold in ordinary language. In a sense, the “+” (or “-” or whatever) in a superposition is a bit more “AND” than standard English “OR”, but it’s also a bit more “OR” than a standard English “AND”. It’s something truly new and unfamiliar.
Experts in the foundational meaning of quantum physics argue over whether to use “OR” or “AND”. It’s not an argument I want to get into. My goal here is to help you understand how quantum physics works with the minimum of interpretation and the minimum of mathematics. This requires precise language, of course. But here we find we cannot avoid a small amount of math — that of simple numbers, sometimes even complex numbers — because ordinary language simply can’t capture the logic of what quantum physics can do.
I will continue, for consistency, to use “OR” for a superposition, but going forward we must admit and recognize its limitations, and become more sophisticated about what it does and doesn’t mean. One should understand my use of “OR“, and the “pre-quantum viewpoint” that I often employ, as pedagogical methodology, not a statement about nature. Specifically, I have been trying to clarify the crucial idea of the space of possibilities, and to show examples of how quantum physics goes beyond pre-quantum physics. I find the “pre-quantum viewpoint”, where it is absolutely required that we use “OR”, helps students get the basics straight. But it is true that the pre-quantum viewpoint obscures some of the full richness and complexity of quantum phenomena, much of which arises precisely because the quantum “OR” is not the standard “OR” [and similarly if you prefer “AND” instead.] So we have to start leaving it behind.
There are many more layers of subtlety yet to be uncovered [for instance, what if my system is in a state (A OR B), but I make a measurement that can’t directly tell me whether A is true or B is true?] but this is enough for today.
I’m grateful to Jacob Barandes for a discussion about some of these issues.
Conceptual Summary
- When we use “A AND B” in ordinary language, we mean “A is true and B is true”.
- When we use “A OR B” in ordinary language, we find “OR” is ambiguous even in English; it may mean
- “either A is true or B is true”, or
- “A is true or B is true or both are true.”
- In my recent posts, when I say a superposition c1 A + c2 B can be expressed as “A OR B”, I mean something that I cannot mean in English, because such a meaning would never normally occur to us:
- I mean that the result of an appropriate measurement carried out at this moment will give the result A or the result B (but not both).
- I do so without generally implying that the state of the system, if I don’t carry out the measurement, is definitely A or definitely B (though unknown).
- Instead the system could be viewed as being in an uncanny state of being that we’re not used to, for which neither ordinary “AND” nor ordinary “OR” applies.
- Note also that using either “AND” or “OR” is unable to capture the difference between superpositions that involve the same states but differ in the numbers c1, c2.
The third bullet point is open to different choices about “AND” and “OR“, and open to different interpretation about what superposition states imply about the systems that are in them. There are different consistent ways to combine the language and concepts, and the particular choice I’ve made is pragmatic, not dogmatic. For a single set of blog posts that tell a coherent story, I have to to pick a single consistent language; but it’s a choice. Once one’s understanding of quantum physics is strong, it’s both valuable and straightforward to consider other possible choices.
23 Responses
dear Matt,
below fig. 5 you write
“But this uncertainty about which box the atoms are in is completely different from the uncertainty that arises from putting the atoms in the superposition state HL NL + HR NR “.
I’m very confused by that (and I guess that that was with didactic intent).
I would naively expect to be able to tabulate the possible ‘situations’ and the possible outcomes of an experiment.
If I can only guess which superposition has been prepared (classical ‘OR’, someone knows it, mother nature knows it, but I don’t yet) then I lack 1 bit of ‘knowable info’.
As you said, I also have to guess at the outcome of the experiment (quantum ‘OR’, no-one knows it, mother nature hasn’t made up her mind yet) and that represents 1 bit of ‘instantaneous’ info. (???)
Two different types of ‘information’? Each with separate double-entry bookkeeping?
The key difference is
1) If I know that either (HL NL) is true or (HR NR) is true, I know this is the case now even if I don’t do a measurement. I know that my friend could tell me which it was if I can just get ahold of him or her. No measurement is necessary. The experimental consequence is that if I join the two boxes as in Fig. 6, no quantum interference will be observed, because the quantum state contained only one possibility (even though I didn’t know which it was) at all times.
2) If instead (HL NL + HR NR) is true, then I only know what the results of measurements might be. No one can tell me which is true. The experimental consequence is that if I join the two boxes as in Fig. 6, quantum interference will be observed, because the quantum state contained two possibilities that could interfere when the boxes were joined.
It’s hard to get the language right, or to agree on what it should be. But the fact is that the experimental consequences are observable. Moreover, they’re the same as in the double slit experiment: there is a measurable difference between saying “the object definitely went through one slit or it definitely went through the other slit, but I don’t know which of the two it was” and saying “at a certain point the object was in a superposition of going through one slit OR going through the other slit,” which has nothing to do with my personal knowledge about anything.
Sorry, I’ve probably missed your ideas about superpositions of superpositions. If a system may be in state A or B, then I can prepare superposition (A+B) or (A-B). If then somebody prepares superposition of these superpositions (A+B)+(A-B), he prepares pure A state. All the plusses (and consequently all the ors) have exactly the same meaning.
Exactly: there is no such thing as a superposition of superpositions, and at no point did I suggest that one can create such a thing.
I said “(A+B) OR (A-B)”, which is a situation of not knowing which superposition is in effect.
Technically, this is known as a “mixed state”, built from two pure states which happen (in this basis) to be superpositions.
This very confusion is *why* the + and – in the superpositions can only be called “OR” with caution. The meaning of the OR in the middle is not the same as the meaning of “OR” inside the parentheses.
Your mistake is thinking the “OR” in the middle between the (A+B) and the (A-B) is a + (or a -). It’s not. It’s a statement of uncertainty that is the same sort as when I say that I don’t know if my friend’s child is 25 years old OR 30 years old; it is not an addition or subtraction symbol and has nothing to do with superposition.
However the superposition itself means, that I do not know, in which state the system is. So I still do not see any difference in superposition A+B and superposition of A+B OR A-B.
Are you mean that the difference is whether the unknown is caused by the limitations of quantum mechanics or whether just my friend preparing the superposition hide it, but the information is generally available?
To your final question: yes. There is a difference between information being unavailable for basic physics reasons (the superposition) and information being unavailable to me personally even though someone else could have it (usual situations about probability.)
Indeed, if I try to measure whether A is true or whether B is true, I won’t see much difference between A+B, A-B, or a situation that involves my friend having picked one or the other without my knowledge. However, suppose I can measure whether the quantum state of the system is symmetric or antisymmetric under exchange of A and B. [In our two atom example, this is like asking whether the system is left-right symmetric or left-right antisymmetric. Such questions can often be answered experimentally; it’s called a measurement of “parity”] If the system is in the state (A+B), then my measurement of its “parity” will say it is symmetric. If it is in the state (A-B), then I will find it is antisymmetric. If it is in an state that might be one or the other, then, of course, I might measure one or the other. Finally, if it is in the state A = (A+B) + (A-B), then I will also measure one or the other.
So now you ask: wait, haven’t we just gotten back to where we started, where I can’t tell the difference between A and (A+B)OR(A-B)? No, because now we consider both types of measurements.
(A+B) is ambiguous about A vs B, but unambiguously symmetric
(A-B) is ambiguous about A vs B, but unambiguously antisymmetric
A is unambiguous about A being true (and not B), but has ambiguous symmetry
(A+B)OR(A-B) is ambiguous about both of them; it is ambiguous about A vs B due to quantum superposition, and ambiguous about symmetry because I happen not to know which superposition was chosen.
Similarly (A)OR(B) is ambiguous about both them, but the reasons are reversed: A and B are both quantum superpositions of the (A+B) and (A-B) states, and I happen not to know whether A is true or B is true even though I know that one of them is true.
Both your and my computer do not like “basis” as an adjective and autocorrects it. If you CTRL/CMD + F for “basic”, you will find three that should be “basis”.
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You mention basis states, so now I understand more of your reasoning.
(Note that, in what follows, I know Matt doesn’t need me to be so verbose. I write with the lay reader in mind.)
Mathematicians probably cringe at the way we use technical terms. Usually, we use “basis” to mean an “orthogonal basis” (or “orthonormal basis”), and that seems to be how you mean it by discussing mutually exclusive possibilities.
If I understand you correctly and, in particular, assuming orthogonal basis states are essential, I don’t think your double-slit OR explanation of the previous post works. I question whether in the double-slit scenario there are left and right orthogonal basis states. To be clear, I am not concerned with the exact words used, but instead I am concerned with the implication that a measurement corresponds precisely to one of the two slits (or, in our language, the left or right ket).
Here’s my low-dimensional thinking:
Consider a two-dimensional diagram of the double-slit experiment. Let’s say the situation has a symmetry both in the slits and the amplitudes at the slits. A minuscule distance downstream of the barrier with the slits, imagine a line parallel to the barrier.
We will picture the wave function amplitudes along this line at one point in time. The wave function emerging from the slits to first-order can be imagined as a composition of two top-hat functions along this line, each of which is in front of a slit. A top-hat function will have a non-zero value in front of the slits and zero everywhere else. (I imagine the point in the wave function’s cycle when it is real.)
But first order is not good enough. We know discontinuities and zeroes almost everywhere are not allowed in wave functions. So the top-hat functions will have round corners and have tiny, non-zero values everywhere (except when the wiggles cross zero).
For the left and right kets (the top hats) to be orthogonal, the integral of their product must be zero. I don’t see how this integral can be guaranteed to be zero. It seems the integral could easily be finite. Maybe I am wrong, but I think the onus is on the one claiming they are orthogonal.
Moving to 3D, for left and right states to be always orthogonal, one would have to show that the wave functions of the basis states emerging from holes of any shape are orthogonal.
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I also have issues with Schrödinger’s cat explanations. Yes, an alive and dead cat are mutually exclusive. We see one or the other. But, thinking mathematically in terms of wave functions, for all the different configurations of ALIVE and DEAD that are possible, would any physicists claim that they know that the integral of ALIVE * DEAD is zero? Could they argue that the wave functions of any particular choice of ALIVE and DEAD are truly orthogonal and thus form basis states?
I don’t think there is the particular superposition as commonly described. I have never seen a thorough, rigorous justification (by Einstein, Schrödinger, or anyone else) of the first step in this thought experiment, and I think that’s where the mistake is made.
(If it matters, in the above para, ALIVE includes the undecayed atom and apparatus and DEAD includes the decayed atom and apparatus.)
Hi Matt, only my second post here so hopefully this is formatted correctly. I believe I found a typo.
Ai should be Ad
in “where Ai refers to a decayed atom and CD refers to a Dead cat.”
Thanks, fixed!
I find OR un troubling. I guess the A is true OR B is true OR both are true is something I should consider in the future but A OR B is conceptually helpful. Is it possible that OR is troubling because we have not yet brought Feynman into the picture?
If we were to look at OR from the Path Integral formulation with effectively infinite possible integrals, we would come to OR like thinking very quickly. The set of the possible would then expand beyond this binomial choice of superposition. If we think about it this way, it seems really easy to say there are a lot of possibilities but only one survives measurement.
Nope, the quantum OR is not like either meaning of English OR.
Meanwhile, the path integral formulation is simply a reformulation of the problem which neither clarifies the conceptual issues nor makes them worse. It’s the same. You can obtain one approach from the other with a set of mathematical steps, and in fact we show this explicitly in quantum physics classes.
This seems to imply the predicate logic isn’t the right paradigm. All the truth tables of predicate logic can be constructed via some finite combination of {nand, nor} or {and, or, not}. if this is not sufficient, then simple predicate logic is the wrong kind of logical calculus to use. What kind of logic do we need?
I think that’s probably correct, and probably on many levels. I know there have been long debates about this, but unfortunately I am not expert on them and cannot provide much insight. As you have seen, getting the logic correct in quantum physics seems to require even keeping track of the numbers that appear in a superposition. It’s not entirely clear to me that verbal logic can entirely do the job.
But it’s also true that we don’t know what the logic should act on. You have to choose the basis objects that are to be subjected to your analysis, and we aren’t in agreement as to what they should be.
It sounds like the additional “or” you really need for quantum physics is the “xor” (exclusive OR) of computer programming. This is the familiar English “OR” but with the difference that if A and B are both true then the operator returns FALSE.
Using XOR and regular OR would then make it clear that which combinations are exclusive vs. inclusive. In fact, it is precisely the observable or “pure” states that are modeled with ‘XOR’ whereas all the superpositions are modeled with plain ‘OR’. For your superposition of superpositions example, it becomes:
(HL NL XOR HR NR) OR (HL NL XOR HR NR)
This is still less info or precision than using the actual “+” and “-” (or complex #s) to distinguish the 2 superposition possibilities. But it captures the fact that if you ultimately measure/observe the system, you’re going to get either “HL NL” or “HR NR” and nothing else. It also captures the qualitative fact that no matter how complex one of these ‘superpositions of superpositions’ situations gets, the set of all possible observable states of the system is just all the distinct arguments to the various ‘XOR’ operators in its representation.
No, I’m afraid you’ve missed the point. The quantum OR is not XOR.
If we use XOR (which is not of computer programming originally, but dates back eons into the earliest ages of human thought,) then the state I mentioned (which is not a superposition of superpositions, but rather a lack of knowledge about two possible superpositions) is
(HL NL QOR HR NR) XOR (HL NL QOR HR NR)
where QOR is a quantum “or” not found in classical computing (but of course found in quantum computing in some way or other — I dont know the current lexicon), and XOR is the exclusive “or”. If you look carefully at what I wrote in the text of the post, I explicitly stated in words that the “OR” I was using in the middle was the “or” in “either…or…”, which of course is XOR.
So in fact you fell directly and deeply into a common conceptual trap, thinking that quantum physics is just computer science that quantum physicists are too uninformed to understand. 😉
I suggest you read the post again, carefully this time.
Maybe you could write SOR when you mean superposition-OR for short, especially in examples where you need to mix it with regular OR (for “I just don’t know which one”).
Very well written and clear. Thank you. Have you ever encountered the retrocausal interpretation of Huw Price (among others)? You say “It is not that I happen not to know which box the atoms are in, but rather that this information is not knowable within the rules of quantum physics.” But this doesn’t seem to be true, for you do know what which box the atoms are in *after* you make the measurement, and presumably the time after you make the measurement is still subject to the rules of quantum physics. The retrocausal interpretation says that if we assume an A-theory of time and that the fundamental laws are time-reversible, quantum uncertainty is exactly what we’d expect to see. Now this means you have to accept that the future in some sense causes the past, but as Price points out this isn’t really a difficulty if you accept the time-invariance of the underlying laws, there’s no difference in saying that the past causes the future or the future causes the past.
Sorry meant to say B-theory, tenseless time
There is no time-reversal-invariance here. The state of the atoms and the measuring device is different after the measurement than before, just as in my post on the double-slit experiment. Before the measurement, the device and the atoms are independent: the wave function is (HL NL + HR NR) D0, where D0 means the device is ready to measure. After the measurement the state is HL NL DL + HR NR DR, which says either the device says “left” and both atoms are in the left box, or the devicec says “right” and both atoms are in the right box. The measurement involves correlating the device with the state of the atoms — that’s what makes it a measurement — and although the device’s state after the measurement tells me which box the atoms are in after the measurement, nothing tells me the state of the atoms before the measurement. And since the states before and after are different, there’s no way to use time-reversal invariance to gain information; even if you use it, going backward in time undoes the measurement and leaves me with no information.
I believe that is a benefit of the retrocausal interpretation, since allowing information from the future to affect the past could lead to logical paradoxes (Grandfather paradox etc.).
I guess I’m losing the thread here. I thought the point was to establish that we know something about where the atoms were in the past based on a measurement that establishes where the atoms are now. That’s the thing that the quantum OR brings into question.
Is electron spin one of the things you will get onto. If an electron is spin up along the Z axis then maybe you could say it is spin up AND down along the X axis. But then if you are about to measure it along the X axis it becomes an OR ?
Spin (and many other cases) give us situations where a superposition from one basis (spin along the z axis) is not a superposition from another basis (spin along the x axis.) But that’s not really an and/or issue. You can still just say that spin in the +x direction is equivalent to spin +z OR spin -z.
The problem is that spin in the -x direction is also spin +z OR spin -z — just as for the two-atom superpositions I used in the final section on OR.
Actually you have the same issues with the two atoms in the two boxes. If instead of measuring the atoms’ positions, you measure the parity of their wavefunction [i.e. is it symmetric or antisymmetric with respect to flipping L and R] then for the state HL NL + HR NR you will get the answer “symmetric”, without ambiguity. Similarly for the state HL NL – HR NR you will get the answer “anti-symmetric”. In other words, these states are NOT superpositions with respect to the parity-state basis. The spin-x and spin-z is only slightly more complicated, because you also have spin-y to think about.