On Uncommon Descent, Barry Arrington asks:
Let’s clear up this law of noncontradiction issue between StephenB and eigenstate once and for all. StephenB asks eigenstate: “Can the planet Jupiter exist and not exist at the same time in the same sense? That’s a “yes or no” question eigenstate. How do you answer it?
For some reason, Eigenstate’s response has gone astray, so here it is, as cross-posted elsewhere:
Eigenstate:
Theoretically, yes. In practice, the probabilities are so vanishingly small it’s indistinguishable from no.
Scale matters; superposition is fragile with respect to other elements in the system that force a classical resolution. Recent experiments have provided experimental verification that macroscale objects can be put in superposition (see here):
Quote
But although the rules of quantum mechanics seem to apply at small scales, nobody has seen evidence of them on a large scale, where outside influences can more easily destroy fragile quantum states. “No one has shown to date that if you take a big object, with trillions of atoms in it, that quantum mechanics applies to its motion,” Cleland says.
The “paddle” object in this experiment was just 30 micrometers long. But that’s freaking HUGE compared to the Planck length. Jupiter is just so many orders of magnitude bigger than that, that the prospects for superposition in that case become ONLY theoretical. Too many resolving influences make it statistically impossible.
The linked article describing the experimental evidence for QM weirdness “scaling up” includes this comment from a physicist at the U of Oregon:
Quote
“It’s wonderful,” says Hailin Wang, a physicist at the University of Oregon in Eugene who has been working on a rival technique for putting an oscillator into the ground state. The work shows that the laws of quantum mechanics hold up as expected on a large scale. “It’s good for physics for sure,” Wang says.
So if trillions of atoms can be put into a quantum state, why don’t we see double-decker buses simultaneously stopping and going? Cleland says he believes size does matter: the larger an object, the easier it is for outside forces to disrupt its quantum state.
(emphasis mine)
Those wacky physicists, I tell ya.
On an LNC-related note, this from the same article:
Quote
Next, the researchers put the quantum circuit into a superposition of ‘push’ and ‘don’t push’, and connected it to the paddle. Through a series of careful measurements, they were able to show that the paddle was both vibrating and not vibrating simultaneously.
(emphasis mine)
Sound familiar, StephenB (and Barry, if you’ve been reading our exchange)? I will note here that champignon’s comment on this being best viewed as a Law of the Excluded Middle issue is a point well taken. But that notwithstanding, you have QM weirdness in the real world ostensibly misbehaving against our propositional logic. “Vibrating” and “Not Vibrating” in the same sense, for the same object at the same time.
Here’s another example from a similar experiment (link), where Dr. Anthony Leggett of U of Illinois, Champaign, Urbana weighs in on a solar system body — not Jupiter but the moon (the moon was the example Einstein initiated these questions with: “does the moon exist if no one is looking at it?” :
Quote
For Dr. Leggett, quantum mechanics at the macroscopic level is still uncertain — and troubling. “It may bother me even more now,” Dr. Leggett said. “I’m interested in the possibility that quantum mechanics, at some stage, may be wrong.”
A few physicists have devised so-called macrorealistic theories to resolve the ambiguities of quantum mechanics. “What you get in quantum mechanics is not what you see,” said Dr. Philip Pearle, a professor of physics at Hamilton College in Clinton, N.Y. “Schrödinger felt this acutely. He himself felt something with quantum mechanics was wrong.”
Dr. Pearle and colleagues in Italy propose to add a term to Schrödinger’s equation that, in effect, constantly jiggles the fabric of the universe. Atomic-scale objects only jiggle a little and thus remain a blur, which preserves the predictions of quantum mechanics. Larger objects, like people or the Moon, jiggle more and quickly fall into a definite here and there, which corresponds to everyday experience.
(emphasis mine)
Barry, if you’ve read my earlier responses to StephenB on this, you will recognize the same ideas quoted here in my answers. Jupiter has a virtually zero statistical basis for avoiding decoherence, hence it will ALWAYS be there in the full, classical (non-superposition) sense.
Lastly, this, regarding the LNC-problematic nature of this second expirement:
Quote
The experiment combines two possibilities, known as a quantum superposition, for the direction of the flow of electric current: clockwise around the loop or counterclockwise.
The researchers measured an energy difference between the two states of the loop, a sign the current was a quantum superposition and not simply flipping directions.
Just as the cat is neither alive nor dead but a ghostly mix of the two possibilities, the current flows neither clockwise nor counterclockwise, but is a mix of the two possibilities.
(my emphasis)
Note that per superposition, this is not simply a matter of a “bi-directional current”. This is two otherwise exclusive one-way directions happening at the same time, exclusive states superimposed:
Quote
A measurement always gives one of the two possible answers, clockwise or counterclockwise, never a zero cancellation.
Glad to have the opportunity to settle this once and for all! Statistically, it will never happen for Jupiter, but it remains a theoretical possibility. It’s the same as wondering if I could fairly shuffle and deal a 52 card deck and deal the cards out, producing the same exact card order as the first shuffle a billion times in row. In theory, it cannot be eliminated as a possibility. As a practical matter, the odds are insdistinguishable from zero.
Have at it.
My only request is that we discuss the OP, not internet dramaz.
Cheers guys.
Superconductivity is a macroscopic manifestation of quantum mechanics.
What is being overlooked here is that decoherence is inevitable in macroscopic systems comprised of different kinds of particles at a high enough temperature that there is no unique coherent state into which all of them can be placed.
Superconductivity, for example, relies on the fact that electrons, interacting with the phonons in the lattice of a solid, are paired off into what are call Cooper pairs. These Cooper pairs of electrons become quasi-particles that fall into the class of particles called bosons. This allows them to condense into a coherent state that has large scale macroscopic effects.
Similar condensation into a superfluid state occurs for liquid helium 4 at low temperatures (liquid helium 3 is a fermion and doesn’t condense). There are other examples of Bose-Einstein condensation in other relatively simple systems. The key is that the systems are comprised of simple particles or atoms that all have a single state into which they can condense; and the temperatures are low enough to suppress transitions to any other states.
But such a condensation is not going to happen with the mixed collections of ordinary matter that make up the macroscopic things we see around us. All macroscopic matter in the normal environment in which we move around is full of “impurities” consisting of other atoms that we don’t normally notice as part of the makeup of normal things. And these atoms are not all going to just happen to be bosons – or pair off into bosons – with a single state into which they can all condense.
Because of this nearly arbitrary mixture of lots of atoms, there is virtually no likelihood of all of them combining into some sort of “quasi-particle boson” that will condense into a coherent macroscopic state.
But quantum effects are studied routinely in the lab; and that requires preparing some relatively simple systems in order that the effects can be observed.
There is also an interesting area of research often referred to as “mesoscopic” physics. This is the scale at which classical and quantum effects interact strongly.
I have worked in low temperature physics, superconductivity, electron g-factor anomaly, solid state devices, and infrared CCD imaging devices. All of these areas are steeped in quantum mechanics; and all these areas are extremely dependent on scale and temperature.
I’m not sure why this is called a law. It’s more of a definition. At any rate it applies to formal statements, not to empirical tests.
Any Platonist should know that the shadow world doesn’t match the world of form. 🙂
Yes, precisely. It is a law of logic, not a law of physics.
Neil Rickert,
Part of the confusion may come from the fact that some statements are formally not decidable. One often sees this conflated with statements that are empirically not decidable.
The boundary between these gets blurred when one wants to measure a phenomenon but cannot yet conceive of a way to make the phenomenon objectively observable. This challenge crops up in research frequently.
Neil Rickert,
Could the confusion come from assuming that the laws of logic govern the physical laws of the universe?
Joel H,
Yes, that’s the basic problem. Laws of logic are not incumbent on the laws of physics.
And even putting it that way makes me twitch a bit, as it equivocates on what we mean by “laws”. It’s not as prosaic, but it’s better put as:
The tools of logic are not incumbent on the dynamics of nature.
That hardly seems controversial, put that way. But it is controversial in some quarters. If you think your mind is fashioned in the imago dei by the same dei that created all of nature, you might not see the tools as distinct from the ‘nature of nature’. The human mind, wired by God, to be pre-calibrated with the protocols needed for apprehending nature… all the way down.
On this view, the human mind may reach the frontiers of knowledge, and remain ignorant on this issue or that. But the protocol is defined by God such that it never becomes problematic and paradoxical.
Here’s the comment of mine that eigenstate refers to above:
champignon February 9, 2012 at 1:28 am
StephenB, eigenstate,
I confess to only having skimmed your discussion, so forgive me if I misrepresent either of your positions.
With that caveat, a few comments:
1. I think that sacrificing the law of non-contradiction is more of a problem than eigenstate acknowledges, because once you allow for contradictions, the truth of any proposition (and its negation) follows from the principle of explosion. Thus any model that allows contradiction fails to conform to our observed reality, in which some propositions are true and others are false.
2. This is okay, because I don’t think that quantum mechanics forces us to abandon the LNC anyway. However, it does force us to abandon the law of the excluded middle (LEM). Let me explain.
In the Cleland experiment, the paddle can be in three distinct states:
1) moving,
2) not moving, and
3) in a superposition of moving and not moving.
I maintain that a paddle in state 3 is not in state 1 and not in state 2. In other words, there is a difference between
3) saying that the paddle is in a superposition of moving and not moving, and
4) saying that the paddle is both moving and not moving.
#4 would be a contradiction, but it’s not true. #3 is not a contradiction and it is true. Thus there is no need to abandon the LNC in order to accommodate superposition.
On the other hand you cannot say that the paddle is either moving or it’s not moving; there is a third possibility: that the paddle is in a superposition of moving and not moving. Thus we do have to sacrifice the LEM in order to accommodate superposition.
3. I absolutely disagree with StephenB about causality being derivative of the LNC. Causality is not a principle of reason. It is an empirical observation at best, subject to all the limitations of inductive inference. Prior to QM, we seemed to be able to find a cause for every event, so we inferred that every event has a cause. Now we have observed phenomena that falsify this inference. The status of causality was, and is, a fact about the world, not a self-evident axiom of reason.
If causality is derivative of the LNC, as StephenB claims, then assuming acausality should lead to contradictions. I don’t think Stephen has demonstrated this, and I don’t think he can.
Causality is not a self-evident axiom of “right reason”.
And here is a response from olegt at AtBC:
Chomsky claimed to be a rationalist, whatever that means. I’d love to have a rational discussion of Chomsky, but it isn’t going to happen at UD.
You gotta love a man who embodies the failure of non-contradiction. Everyone loves and hates him simultaneously for various reasons. IDiots have loved him because he expressed doubts about evolution, but hated him for his dislike of Israel and condemnation of American foreign policy.
My own field is special education, which leans Skinnerian. My teachers hated Chomsky the way UD hates Dawkins. Personally I think Chomsky was right about humans having a language faulty, but wrong to doubt it evolved incrementally.
Trying to steer back to the topic of this thread, I think language fails us at this kind of discussion. There’s the temptation to Reify the terms of debate. The reason I bring up Chomsky is I think his analysis of language ignored the penumbras of expression and tended to treat the elements of sentences as formal objects with formal meanings. This never seems to work in real life. It’s why we continually talk past each other.
Yes, this is very much the problem. Some philosophers, and most conservative Christian theologians seem to take logic as metaphysical, as an aspect of the physical world. They tend to take John 1:1 an inerrant assertion that logic is part of their imagined creation.
The proper way to look at it, is that we make logical models of reality. And the conclusions of logic only apply to reality to the extent that the model fits. And that seems to be how eigenstate has been looking at it.
Yes, that is very close to the language and approach I used, and advocate.
I might offer a slight caveat here.
Most of the physicists who have thought deeply about it are still boggled by quantum mechanics; and this still applies even though most of us can “turn the crank” on quantum calculations and churn out the probabilities.
There is little question that the mathematics of superposition (in terms of the Schrödinger equation, i.e., the wave function, for example) describes what nature does. Yet it doesn’t seem logical in the conventional sense. Many of the predictions are simply counterintuitive (as, say, in the double slit example).
There are various ways to frame the problem, some of which suggest that superposition is simply a mathematical way of stating the uncertainty in our knowledge prior to “collapsing the wave function” with some kind of detecting apparatus that finally selects a particular state.
In a sense, this is not unlike the uncertainties in other types of classically described systems. We see similar issues with detecting objects with waves of any type.
If we could only observe the things around us by detecting BBs bouncing off them, we would begin to see uncertainties in detecting things approaching the size of BBs.
But bosons and fermions are a reality even though we don’t have adequate classical analogues that “make us comfortable.” Yet there is no doubt about the dramatic consequences of Fermi-Dirac and Bose-Einstein statistics.
The mathematical models fit superbly out to more than 15 decimal places; yet they don’t seem “logical.”
The principles of classical logic were inspired by human observations of the everyday world of classical objects. The quantum of microscopic objects turned out to be quite a bit different.
A classical bit is definitely in the state 0 or in the state 1, not in both. A quantum bit can be in a state that is technically known as a quantum superposition of the two, |0> + |1>. In a very real sense, it is both 0 and 1, at the same time. This sounds like an impossibility, but that is what the quantum theory posits an experiments confirm.
The enormous powers expected of a quantum computer are precisely tied to this ability of a qubit to be in both states at once. For example, the reason why a quantum computer could crack passwords with ease is that it can run computations on many strings at once. Instead of doing computations on many input strings (e.g., 0000, 0001, 0010, 0011, etc., 16 in total for 4 bits) and checking the outcomes until one is found to match, it works with a string whose qubits are all in states |0> + |1>, so that all 16 calculations are done at once. Hence the speedup.
Classic logic is not very helpful with qubits.
Hi Oleg,
I agree that we can only retrieve one bit of information from a two-level system. However, that is because only two states are possible after the measurement has taken place. Before measurement, all of the points on the Bloch sphere really are distinct.
Think of it this way. I can place a qubit into a superposition of 25% 0 and 75% 1. If I then ask you to measure it, I can’t tell you whether you’ll measure a 0 or a 1, but I can tell you that if we repeat the experiment many times, you’ll get 0s 25% of the time and 1s 75% of the time. Thus, the state before measurement really is distinct from the states where we always measure a 1 or a 0.
Mike Elzinga,
Thanks for a lucid description of QM.
David Utidjian,
🙂
Hi David. Long time no see.
Welcome to all new posters, and some info on the somewhat idiosyncratic game rules on this board.
I just moved a post to Guano, not because it offended my delicate sensibilities or because I disapprove, but because the idea of this site is to try to drill down to the issues beneath the personalities, and I try to be ruthless with anything likely to derail into personalities, fascinating though those are (hey, I’m a psychologist). So if a decision seems arbitrary, console yourself with knowledge that the move does not reflect on you. Moved posts will not be hidden, but as there are lots of internet sites where people can have fun dissecting the personalities of those they disagree with, and few where the issues are rigorously (well, fairly rigorously) adhered to, I’m trying to make this such a site.
Lovely to see you all 🙂
All this talk of abandoning either the Law of Non-Contradiction or the Law of the Excluded Middle seems to stem from mashing together our macroscopic classical intuitions and the quantum formalism. If you look at the formalism alone, I do not see any motivation for a revision of classical logic.
Consider a qubit. Classically, a register storing a single bit can have only two states, 0 and 1. So from the classical perspective it makes sense to think of 0 as not-1, since it is the only other possible state. In quantum mechanics, a qubit can be in many other states, superpositions of 0 and 1. All this tells us is that the classical belief that 0 and 1 are exhaustive does not apply. For a qubit, not-1 is not the same as 0. Not-1 corresponds to “either 0 or some superposition of 0 and 1.” With this revised notion, it is still true that a qubit’s state must be either 1 or not-1. The LEM holds. Also, it is impossible for a qubit to simultaneously be in the states 1 and not-1. So the LNC holds.
All QM does is expand the number of possible states. If we restrict ourselves to the classical notion of states, then it will seem as if QM is violating classical logic. But if you look at the actual Hilbert space structure of QM in order to pick out states, there is no violation of these laws.
keiths,
“Really distinct” is a loaded expression. As I wrote before, this is a subtle issue and we will have to carefully define what each of us means.
I will start by contrasting points on the Bloch sphere, which parametrize quantum states of a qubit, with states of a classical particle located on a sphere. In classical physics, two states of a particle with different positions on a sphere are truly distinct. Be they diametrically opposite, one degree, or just an arc second apart, they are distinct physical states. They are not the same unless the two positions have exactly the same coordinates. Thus there are infinitely many physical states of a particle on a sphere.
In quantum mechanics, the situation is different. States corresponding to diametrically opposite points on the Bloch sphere are distinct in the same sense. In the mathematical formalism of QM, the scalar product of these two states is zero. The states are orthogonal as they are eigenstates of the same physical variable, with different eigenvalues. (Hi eigenstate!) States that are not diametrically opposite are not orthogonal. States one degree apart on the sphere are almost identical. One can continuously go from fully distinct to fully identical as the two points on the sphere come closer. This sort of fuzziness has no analog in classical physics.
Characterizing the stationary state of the paddle as “not moving” is ambiguous. It is true that a paddle that is stationary is not moving, but it is not true (in the Cleland experiment) that being stationary is the only way for a paddle to be in a not-moving state. If the paddle is in a superposition of the moving and stationary states, then it is not in the moving state.
So it is true that the paddle is either moving or not moving. The second disjunct covers both the stationary and the superposition states.
Sotto Voce,
There is a problem with your (re)definition of NOT in quantum mechanics. I will explain that later.
olegt,
Why is your standard for states being “really distinct” that their inner product must vanish? I don’t see the motivation for this definition. It seems pretty clear to me that an electron in the z-spin up state is really distinct from an electron in the x-spin up state, even though these states are not orthogonal.
I should mention that even though I do not think QM requires (or even encourages) a revision of classical logic, I am not a StephenB-eque believer in the transcendent power of human reason. I think it is entirely possible (though unlikely) that our empirical investigation of the world will eventually lead us to adopt a different system of logic. I just don’t think we have encountered this circumstance yet.
Incidentally, it was mentioned upthread that abandoning the LNC would have disastrous consequences, allowing us to prove anything. This is only true if we abandon LNC while maintaining the rest of classical logic. There are well worked out alternate logical systems available in which the LNC does not hold, but they are not explosive (i.e. one cannot prove anything from their axioms). Look up “paraconsistent logic”.
olegt,
Echoing Sotto Voce (sotto voce echolalia?), I think your dot product criterion for distinctness is too restrictive. Two nearby states on the Bloch sphere lead to distinct measurement results if the experiment is repeated. If the results are distinct, how can you argue that the states are not?
Sotto Voce,
Regarding paraconsistent logics, my impression is that they are not powerful enough to substitute for classical logic in scientific reasoning. Am I wrong about that?
keiths,
It’s true that paraconsistent logics have fewer inferential resources than classical logic. It is only by restricting possible inferences that they prevent contradictions from entailing everything. Are you saying that there are inferences that are crucial to scientific reasoning that cannot be made in any paraconsistent logic? That’s possible, I suppose. I don’t know if it’s true or not, and I’m not entirely sure how one would go about establishing its truth. Do you recall the source of this claim?
The flip side of this is also worth noting though: a logic which is inferentially weaker will have more models. So there are more models of paraconsistent logic than of classical logic. There is a sense in which paraconsistent logics apply in more “worlds”. If our universe is a model of classical logic, it will also be a model of paraconsistent logic, but the converse is not true. It is possible that our universe (or the best scientific description of it) is a model of paraconsistent logic and not a model of classical logic.
Non physicist, non-philosopher ploughing in regardless:
It seems to me that it’s not so much that “the law of non-contradiction” doesn’t work at quantum and astronomical scales but that ontology doesn’t work at quantum and astronomical scales. We call things “things” if they go on for a bit, and “events” if the don’t. And we say that “things” exist and “events” happen. And we say “at the same time” if we are talking about things fairly close together.
But at some space and scales that’s just not going to work properly – events and things will smoosh together, and time – the dimension along which we distinguish things from events, and define simultaneity – itself is confounded by distance and speed.
Ask yourself this question. What is the meaning of time if matter and motion did not exist?
Then ask the following: How do we know that “time” passes? What is a clock, really?
Existence and events are intricately intertwined. In relativity, an event is a point in a four-dimensional continuum called space-time. But it is not necessary to think about special or general relativity in order to begin to recognize that time can only be defined if there is matter and motion at the very least.
Some phenomenon has to be singled out as a “clock” whose positions in space-time can be paired off with the positions of other phenomena in space-time. Then there are the technical details of how one chooses a “clock.”
The passage of time can only be recognized if there is some material medium that has a hierarchy of memory that can record and compare events after they have occurred.
There is a problem with this your NOT operation. It has trouble saying no. 🙂
This definition works fine for classical points on a sphere that I mentioned in an earlier comment. Any two points are fully distinct unless they coincide. So the negation of a single point is the entire sphere except for that point. Applying NOT twice yields that point again.
Not so with the states of a qubit. The classically inspired NOT (cNOT for short) fails to do its job. If the area cNOT-1 contains every state on the Bloch sphere, excluding |1>, then it contains the entire sphere, including |1>. This is easy to see.
Picture |1> and |0> as the north and south poles of the Bloch sphere. Any other point on the sphere is a superposition of |1> and |0> with various coefficients. States on the equator are equal-parts superpositions of |1> and |0>. We can take two diametrically opposite states on the equator, subtract one from the other to obtain |0>, or add them to obtain |1>. So the supposedly excluded state |1> can be easily reconstituted from states in cNOT-1. This sort of negation does not work.
The standard NOT operation for qubits has no such problem. NOT-1 is 0. You can’t reconstitute the state |1> from |0>, because these two states are linearly independent.
I think the problems with the operation cNOT go rather deep. It seems incompatible with the very mathematical structure of quantum mechanics, the linear algebra of Hilbert spaces. Every physical observable and every physical transformation is represented by a linear operator (a matrix) in the Hilbert space. Knowing how it acts on some basis states provides information about its action on any state. We know well how the standard NOT operator acts on the basis states of a qubit: NOT |0> = |1>, NOT |1> = |0>. Its action on any other point on the Bloch sphere can be deduced from that. (It maps any point onto its diametrically opposite.) What is the action of cNOT on |0> and |1>? It has got to be a superposition of |0> and |1> and any such superposition is a single point on the Bloch sphere. And if it is a single point, it had better be the opposite point, right?
Hello! I’ve been following the debate over at Uncommon Descent, and I thought I’d come over here and over a few cents that might help.
If one takes the time to look at the arguments that Aristotle uses for “the law of non-contradiction,” it becomes quite clear that he’s talking about *objects* and *properties*. So he’s already working within a specific ontological framework when he’s talking about what it makes sense to say. Conversely, if we change the ontological framework, then we might well need different rules about what it makes sense to say. So it’s not that logic is a “mere tool,” nor to say that it’s one of “reason’s rules”, but to say that the logic and the ontology go hand-in-hand.
For that matter, I don’t know if this has been mentioned here, but there was a short-lived project in the 30s and 40s, “quantum logic,” which aimed at eliminating paradoxes precisely by rejecting the law of the excluded middle. (Not the same as the law of non-contradiction, though close enough.)
An addendum to my previous comment.
One can also try to define projection operators that define areas 1 (the north pole) and cNOT-1 (the rest of the Bloch sphere). Such a projector operator should yield one when it acts on state 1 and zero when it acts on a state from cNOT-1. Such an operator simply does not exist.
Proof. We have P|1> = |1> (this state is in 1) and P|0> = 0 (this state is in cNOT-1). For any other point on the sphere, say, |x> = a|1> + b|0>, with nonzero coefficients a and b,
P|x> = aP|1> + bP|0> = a|0>.
This is a contradiction because we expect that P|x> = 0 because |x> is in cNOT-1.
This means that Sotto Voce’s NOT operation is cannot be define on the Bloch sphere.
Thanks for that explanation, olegt. This is the best articulation of this problem I’ve run across.
olegt,
Very interesting, Oleg! But I’m still not convinced, I’m afraid. Two points of contention for me, the first one minor and the second one major.
1. I’m not sure what’s so problematic about the fact that |1> can be reconstructed by adding elements of not-|1>. Consider directions on a plane represented as unit vectors in a 2-D space. There is a unique vector corresponding to the direction NORTH. There isn’t a unique vector corresponding to not-NORTH. All the other vectors are not NORTH. Now it turns out that you can take two non-NORTH vectors, add them and recover the NORTH vector. What does this show? I don’t think it shows that we were wrong to identify all the vectors except NORTH as not NORTH.
2. In your discussion you are conflating the logical operation of negation (I’ll call it L-NOT) with the computational operation NOT. When we are performing computations, we want our gates to transform states to other states, so it makes sense to think of NOT as a unitary operator acting on the Hilbert space of qubit states. In this case NOT |1> must correspond to one specific state, and you are right that it makes sense for this state to be |0>.
But L-NOT is not a map from states to states. It is a sentential operator. When we express the LNC as “~(A & ~A)”, the variable A here is supposed to stand for a proposition or a statement, not an object or a state. It makes no sense to ask “What is the negation of the state |1>?” where by negation we mean the operation L-NOT. Only statements can be negated by this operation. It maps statements to other statements with the opposite truth value.
L-NOT is not an operator on the Hilbert space, because L-NOT does not act on states. You have the statement “The qubit is in state |1>.” This statement is only true if the qubit is in one specific quantum state. Negating the statement, we get “The qubit is not in state |1>”, which means the same as “The qubit is either in state |0> or in some superposition of states |1> and |0>.”
Here’s an example that might make the fallacy clearer. Take the sentence “It is raining.” This describes a specific state of the weather. The negation of this sentence is “It is not raining.” Now someone might say, “What does this negation mean? What is the state of not-raining? Is it sunniness, is it fog?” But of course, this is misguided. When we say it is not raining, we are not attributing a particular state to the weather. We are ruling out a state. This is exactly what we are doing when we say “The qubit is not in state |1>”, where the “not” is interpreted in the logical (L-NOT) sense rather than the computational one.
Short version of my previous post:
It is absolutely true that the properties of classical systems have the structure of a Boolean lattice while the properties of quantum systems do not have this structure. This is what Oleg has elegantly pointed out. This is the consequence of moving from a commutative algebra of dynamical variables to a non-commutative algebra.
However, this does NOT mean that descriptions of quantum systems do not obey classical logic. One can have non-Boolean structure at the level of properties/states without having to abandon the LNC or the LEM at the level of statements or propositions. Non-Boolean lattices can appear in models of classical logic.
Sotto Voce,
Thanks for detailed explanations. It seems that L-NOT 1 boils down to “the qubit is not in pure state 1.” I agree that a qubit either is in pure state 1 or it isn’t. But that, to me, looks like a completely trivial statement that has nothing to do with quantum mechanics whatsoever. A system either fits our description or it doesn’t. Ho hum.
I also agree that the computational NOT for a qubit looks like an operator mapping a state onto a state. But that is only true for a two-dimensional Hilbert space. With three (or more) physical states |0>, |1>, |2>, NOT-|1> is any state that is a superposition of |0> and |2>. Hence my addendum discussing projectors.
Off to bed.
Thanks!
Yes, I get that, I think.
I think the point I was trying to make was something like the one Carl Sachs has made much more eloquently below.
But possibly I am just confused 🙂
Oh absolutely. Logical truths should be trivial statements with no (or very little) empirical content. Which is why StephenB’s insistence that the law of non-contradiction proves the existence of God or something is so ridiculous.
Sotto Voce,
But then the L-NOT approach entirely misses the point. The question raised by StephenB was Can the planet Jupiter exist and not exist at the same time in the same sense? For a qubit based on a single photon in a resonator cavity, the question boils down to “Can the photon exist an not exist at the same time in the same sense?” A photon exists = state |1>, a photon does not exist = state |0>, the standard negation of state |1>. Or if you take into account states with more than one photon, “Can photons be present and not be presnt in the cavity at the same time in the same sense?” That boils down to standard negation is the cavity in state |0> and also in states other than |0> (i.e., |1>, |2>, etc.)?
Waves white flag…
As long as I’m this far over my head I would like to complicate the question a bit with a question about the nature of time. From the math deficient layman’s point of view, physicists are discussing time as discrete (can I use that word here?) as opposed to continuous.
The metaphor would be the computer clock, which has no in-between states. Is there any way in the real world to measure or detect events simultaneously? It would seem that any observation requires clock ticks, and that states could change between ticks. Hence the observation of tunnelling?
I realize I’m invoking something like a Newtonian universal clock, but that’s just a symptom of my confusion.
StephenB and kairosfocus have served up a red herring.
Logic works on assertions of TRUE and not-TRUE.
Whatever we assert, is input to our logic processing system.
If we assert an input equal to TRUE, the LNC holds whether that input relates to Jupiter or a temperature alarm.
It is not the responsibility of logic to verify an assertion, that’s up to the layer doing the assertion.
“kairosfocus” can prove this by simply tying a pot to a generous mix of TTL level and CMOS level logic gates running at different VCC supplies.
By adjusting the pot, you should see the LNC appear to be invalidated which of course is not true at all.
This is typical ID/Creation bad science.
I seem to have merited a special banning thread at UD. For a while I thought they didn’t consider me worth the effort, so it’s a bit flattering to be singled out.
I expect to see DrDr Dembski ride back in on Palm Sunday, now that the place has been cleaned up.
LOL.
And they resent being called IDiots. For those interested the thread is Only Those Who Admit the Foundation of Argumentation Will Be Allowed To Argue at UD. The Arrington inquisition is well under way.
Neil Rickert, Petrushka,
Clearly, Barry does not see that both censorship and “open debate on the controversy” cannot exist at the same time! 🙂
Link to kairosfocus’s response to Eigenstate’s post as posted here.
This megaphone diplomacy is rather odd, but there’s no reason why it shouldn’t work as both communities seem to be reading both sites 🙂
And while I know it is a lot to ask in the circs, can I request that people stick rigorously to the actual arguments.
Think of it as levelling up.
Superposition is such a fundamental and well-understood concept that any of us who have worked with this stuff find kairosfocus’s garbage truck dumps nothing more than pretentious avoidance.
The experimental facts are now technologically ubiquitous and easy to understand. The double slit experiment can be done for photons or electrons in any undergraduate physics laboratory.
As I mentioned above, an increasingly complex system of atoms and molecules quickly reaches the point where there is not a single coherent state into which all particles can simultaneously condense at the same time and temperature. Thus one cannot represent such a complex system as a simple wave function that can interfere with itself and produce a superposition of states.
But that doesn’t invalidate superposition and the fundamental notion that simple systems can exist in a superposition of states.
NOT-(Moon AND NOT-Moon) = (NOT-Moon OR Moon) does not disprove superposition in quantum mechanics.
Elizabeth,
Elizabeth,
Odd is a mild way of putting it. You would think folks so certain of their rectitude would be willing to obey Jesus’ commandment of Matthew 28:19. But, so be it. I’ll go back to lurking.
ben h,
You have been counted….
http://www.antievolution.org/cgi-bin/ikonboard/ikonboard.cgi?act=ST;f=14;t=5141;st=720#entry201233
The case could be made that those who are banned from UD are not interested in an open debate.
Jupiter the planet can exist and at the same time Jupiter the god need not exist.