One Quantum World or Two?

The Astounding Link Between the P≠NP Problem and the Quantum Nature of Universe


With some straightforward logic, one theorist has shown that macroscopic quantum objects cannot exist if P≠NP, which suddenly explains one of the greatest mysteries in physics

The paradox of Schrodinger’s cat is a thought experiment dreamed up to explore one of the great mysteries of quantum mechanics—why we don’t see its strange and puzzling behaviour in the macroscopic world.

The paradox is simple to state. It involves a cat, a flask of poison and a source of radiation; all contained within a sealed box. If a monitor in the box detects radioactivity, the flask is shattered, releasing the poison and killing the cat.

The paradox comes about because the radioactive decay is a quantum process and so in a superposition of states until observed. The radioactive atom is both decayed and undecayed at the same time.

But that means the cat must also be in a superposition of alive and dead states until the box is open and the system is observed. In other words, the cat must be both dead and alive at the same time.

Nobody knows why we don’t observe these kinds of strange superpositions in the macroscopic world. For some reason, quantum mechanics just doesn’t work on that scale. And therein lies the mystery, one of the greatest in science.

But that mystery may now be solved thanks to the extraordinary work of Arkady Bolotin at Ben-Gurion University in Israel. He says the key is to think of Schrodinger’s cat as a problem of computational complexity theory. When he does that, it melts away.

First some background. The equation that describes the behaviour of quantum particles is called Schrodinger’s equation. It is relatively straightforward to solve for simple systems such as a single quantum particle in a box and predicts that these systems exist in a quantum superposition of states.

In principle, it ought to be possible to use Schrödinger’s equation to describe any object regardless of its size, perhaps even the universe itself. This equation predicts that the system being modelled exists in a superposition of states, even though this is never experienced in our macroscopic world.

The problem is that the equation says nothing about how large an object needs to be before it obeys Newtonian mechanics rather than the quantum variety.

Now Bolotin thinks he knows why there is a limit and where it lies. He says there is an implicit assumption when physicists say that Schrödinger’s equation can describe macroscopic systems. This assumption is that the equations can be solved in a reasonable amount of time to produce an answer.

That’s certainly true of simple systems but physicists well know that calculating the quantum properties of more complex systems is hugely difficult. The world’s most powerful supercomputers cough and splutter when asked to handle systems consisting of more than a few thousand quantum particles.

That leads Bolotin to ask a perfectly reasonable question. What if there is no way to solve Schrödinger’s equation for macroscopic systems in a reasonable period of time? “If it were so, then quantum theoretical constructions like “a quantum state of a macroscopic object” or “the wave function of the universe” would be nothing more than nontestable empty abstractions,” he says.

He then goes on to prove that this is exactly the case, with one important proviso: that P ≠ NP. Here’s how he does it.

His first step is to restate Schrödinger’s equation as a problem of computational complexity. For a simple system, the equation can be solved by an ordinary computer in a reasonable time, so it falls into class of computational problems known as NP.

Bolotin then goes on to show that the problem of solving the Schrödinger equation is at least as hard or harder than any problem in the NP class. This makes it equivalent to many other head-scratchers such as the travelling salesman problem. Computational complexity theorists call these problems NP-hard.

What’s interesting about NP-hard problems is that they are mathematically equivalent. So a solution for one automatically implies a solution for them all. The biggest question in computational complexity theory (and perhaps in all of physics, if the computational complexity theorists are to be believed), is whether they can be solved in this way or not.

The class of problems that can be solved quickly and efficiently is called P. So the statement that NP-hard problems can also be solved quickly and efficiently is the famous P=NP equation.

But since nobody has found such a solution, the general belief is that they cannot be solved in this way. Or as computational complexity theorists put it: P ≠ NP. Nobody has yet proved this, but most theorists would bet their bottom dollar that it is true.

Schrödinger’s equation has a direct bearing on this. If the equation can be quickly and efficiently solved in all cases, including for vast macroscopic states, then it must be possible to solve all other NP-hard problems in the same way. That is equivalent to saying that P=NP.

But if P is not equal to NP, as most experts believe, then there is a limit to the size the quantum system can be. Indeed, that is exactly what physicists observe.

Bolotin goes on to flesh this out with some numbers. If P ≠ NP and there is no efficient algorithm for solving Schrödinger’s equation, then there is only one way of finding a solution, which is a brute force search.

In the travelling salesman problem of finding the shortest way of visiting a number of cities, the brute force solution involves measuring the length of all permutations of routes and then seeing which is shortest. That’s straightforward for a small number of cities but rapidly becomes difficult for large numbers of them.

Exactly the same is true of Schrödinger’s equation. It’s straightforward for a small number of quantum particles but for a macroscopic system, it becomes a monster.

Macroscopic systems are made up of a number of constituent particles about equal to Avogadro’s number, which is 10^24.

So the number of elementary operations needed to exactly solve this equation would be equal to 2^10^24. That’s a big number!

To put it in context, Bolotin imagines a computer capable of solving it over a reasonable running time of, say, a year. Such a computer would need to execute each elementary operation on a timescale of the order of 10^(-3x10^23) seconds.

This time scale is so short that it is difficult to imagine. But to put it in context, Bolotin says there would be little difference between running such a computer over one year and, say, one hundred billion years (10^18 seconds), which is several times longer than the age of the universe.

What’s more, this time scale is considerably shorter than the Planck timescale, which is roughly equal to 10^-43 seconds. It’s simply not possible to measure or detect change on a scale shorter than this. So even if there was a device capable of doing this kind of calculating, there would be no way of detecting that it had done anything.

“So, unless the laws of physics (as we understand them today) were wrong, no computer would ever be able to execute [this number of] operations in any reasonable amount time,” concludes Bolotin.

In other words, macroscopic systems cannot be quantum in nature. Or as Bolotin puts it: “For anyone living in the real physical world (of limited computational resources) the Schrodinger equation will turn out to be simply unsolvable for macroscopic objects.”

That’s a fascinating piece of logic in a remarkably clear and well written paper. It also raises an interesting avenue for experiment. Physicists have become increasingly skilled at creating conditions in which ever larger objects demonstrate quantum behaviour.

The largest quantum object so far—a vibrating silicon springboard —contained around 1 trillion atoms (10^12), significantly less than Avogadro’s number. But Bolotin’s work suggests a clear size limit.

So in theory, these kinds of experiments provide a way to probe the computational limits of the universe. What’s needed, of course, is a clear prediction from his theory that allows it to be tested experimentally.

There is also a puzzle. There are well known quantum states that do contain Avogadro’s number of particles: these include superfluids, supeconductors, lasers and so on. It would be interesting to see Bolotin’s treatment of these from the point of view of computational complexity.

In these situations, all the particles occupy the same ground state, which presumably significantly reduces the complexity. But by how much? Does his approach have anything to say about how big these states can become?

Beyond that, the questions come thick and fast. What of the transition between quantum and classical states—how does that happen in terms of computational complexity? What of the collapse of stars, which are definitely classical objects, into black holes, which may be quantum ones?

And how does the universe decide whether a system is going to be quantum or not? What is the mechanism by which computational complexity exerts its influence over nature? And so on…

The computational complexity theorist Scott Aaronson has long argued that the most interesting problems in physics are intricately linked with his discipline. And Bolotin’s new work shows why. It’s just possible that computational complexity theory could be quantum physics’ next big thing.


From Link.

 

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The cat in the box scenario is a bit nonsensical. Schrodinger intended it to show that some quantum level notions do not apply to the classical world of our senses. If he had said it instead of writing it, I would expect that sarcasm would be evident in his tone.

Many non-experts believe that knowledge gained by an intelligent observer is required to cause collapse of the wave function to a particular value. This is incorrect.

I think that expert opinion is that the wave function collapses when some quantum level event has an effect at the classical level.

In the cat scenario, the release or non-release of the poison is a classical level effect. Certainly, the life or death of the cat are classical level effects.

BTW: The term collapse of the wave function seems like very strange semantics.

The wave function provides probabilities for various quantum level events.

If one throws a pair of dice, the result is reported as The shooter rolled some specific number. Would you prefer to say The probability table collapsed to that number?

The use of that phrase suggests that the wave function is some process or object rather than an equation.

 

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So the wave function is not real? How do explain the bands on the detection screen that form a wave pattern?

 

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Photons and matter, are both particle and waves, with the wave functions only half the story. As you scale up, the wave aspects of matter combine and cancel, but the particles aspects do not (they can't occupy the same space), resulting in the particles increasingly defining reality.

For example, a moving electron will create a magnetic field. If we take an atom with eight electrons, with all the electrons in motion, each is creating a magnetic field and waves, but often there is no net magnetic field coming from the atom due to wave cancelation. The electron particles don't cancel. This become more dominant with scale up.

The bias of some traditions, assumes waves are it and particles are simply a special case of wave addition. This is not true, as proven by the observation that scale up changes properties away from waves.

 

Spellbound: From your Post #3

Correct.

Similarly, Newtonian & Einstein equations relating to gravity are not real.

The wave function & various other algorithms, equations, laws of physics, et cetera are not real.

They do a good job of describing the laws of physics & can be used for practical purposes. They provide useful predictions & data relating to reality. They provide us with knowledge relating to what we call reality. The ink & paper or display devices used to display equations are real.

The term collapse of the wave function suggests that the function is real rather than some description of certain aspects of reality.

To paraphrase from my previous post.

I wish I knew who originated the collapse phrase & why he/she used it.

BTW: The significance of knowledge obtained by an intelligent entity has been over rated, probably due to the interpretation of some experts’s remarks by a tech writer composing an article for the benefit of non-experts.

 

So what does the term "wave-particle duality" refer to if not a wave?

 

You approach this from an interesting perspective Dinosaur. It seems you are saying that the Quantum world and the Classical world aren't really split or separate. I.e. That particles don't really take on different characteristics and behaviors at the Quantum level? Or do they? And why?

 

Reality is defined by particles alone at the macroscopic level, but as waves as well as particles on the microscopic level? Is that what you're saying?

 

Dinosaur,

Does not the photon particles form a wave pattern on the detection slit when fired even once at a time?

 

SpellBound : From your Post #6

What remark are you addressing by the above?

 

SpellBound : From your Post #9

The above is an interesting phenomenon. Before an experiment could be performed sending one photon through a slit at a time & observing the results, the pattern generated was interpreted as due to constructive & destructive interference, which is a well understood effect.

The single photon at a time belies that interpretation. I do not know the current explanation for this experiment. It is interesting that the pattern is built up in a very random fashion. The pattern is not apparent until many photons have been sent though the apparatus.

 

Do you think superposition is a valid description of reality?

 

SpellBound: from your Post #12

I am not sure what the term means. If it means that certain quantum entities can be partially in more than one of two or more mutually exclusive states, it seems unreasonable.

I accept the Copenhagen interpretation which essentially claims there is no deep reality at the quantum level in the absence of some quantum level phenomenon having an effect at the classical level. He considers the classical world of our senses as real, but built on a quantum world which in some sense is not as real.

 

The particle and wave duality of matter/energy results in two distinct interactions. Particles can't overlap, but their waves can. For example, if we have an electron and proton, defining a hydrogen atom; wave functions, there is no net charge relative to wave addition. Yet both the negative and positive charges will continue to exist as the electron and proton. The wave addition cancels the net charge to zero, but the particle addition remains as two particles.

As we scale up, the forces of nature overlaps the waves so the waves cancel, but the number of charges and particles is conserve. As we scale down to individual particles, the waves remain more true and isolated, without the macro-level scale-up canceling. These wave better reflect the particle and are often used to define it.

 

I have done this example, under other circumstances. It is appropriate to this discussion. Picture a wave tank with two wave generators, one at each end of the tank. Both generate the same waves, but each wave generator is out of phase by 180 degrees. The result is the two approaching waves will cancel and there will be stillness in the center of the tank. Although there is no wave, we are sill pumping in energy into the tank. This is hidden energy in the sense it is not visible via the waves, but we know it is there, via the work from each generator.

We can get the "hidden" energy back if we create a physical partition in the middle. The partition will cause the waves rise on each side of the partition and appear to materialize out of the stillness. The particle is the partition, with the wave energy often hidden within addition of wave functions.

 

SpellBound: I think some of our differences are due to misunderstood semantics. I believe in the reality of waves & particles, but do not believe in the reality of the wave function. This is analogous to believing in the reality of gravitation while not attributing reality to either the Newtonian or the Einstein equations.

From your Post#7

I only said that the Wave function is not real. I did not deny the existence of waves. The semantics of collapse of the wave function seems to imply that it a real entity rather than a formula, equation, or algorithm.

I tried to indicate that semantics of the phrase collapse of the wave function implies that the function is a real entity rather than a description of some aspects of reality. It is usually important to distinguish between reality & the equations/laws describing reality. This distinction is especially important for Quantum Theory.

As mentioned in a previous Post

I did not intend to imply that there is no split between the quantum & classical levels of reality. To me they are definitely distinct although there are some problems close to the boundary between them and/or problems determining the precise boundary.

The wave/particle duality is a complex issue. I am not sure about the opinion of the real experts. While I took a pertinent 1-semester course circa 65 years ago, my knowledge of this & other aspects of Quantum Theory comes from SciAm articles & 5-6 books for laymen (owned by me) which I have read in the past 5-10 years.

An interesting comment is the following.

I forget who said it. Experiments designed to detect waves indicate that certain quantum entities are waves while particle experiments show that they are particles.

The quantum level of reality confuses some very knowledgeable experts. An excellent book for laymen is Quantum Reality by Nick Herbert. Nick took graduate level courses. He later taught at some university & did serious research relating to Quantum Theory.

In his book he describes a disturbance model of the Uncertainty Principle which he adopted while a student & continued to believe while teaching & doing research. In his book he admits that the disturbance model is an erroneous POV & no longer accepts it.

If a person with the knowledge & credentials of Herbert can misunderstand a fundamental concept like the Uncertainty Principle, it indicates the Quantum level of reality is weird & difficult to comprehend.

From Nick Herbert’s Book

The last of the above is my paraphrase of a lengthier paragraph.

There is much disagreement at the interpretation level. The following are some of the interpretations described by Herbert

Variations on the Copenhagen POV developed by Bohr & Heisenberg, which is my personal view. This interpretation maintains that a quantum entity (an electron for example) has no properties prior to it having an effect at the classical level.

The observer-created reality concept seems erroneous. However, I wonder if the problem is due to a tech writer dumbing down some statements by an expert. If this semantics implies that a conscious observer is required, it is surely erroneous. If it merely implies that reality of some properties at the quantum level do not exist until they have an effect at the classical level, the notion is equivalent to the Copenhagen interpretation.

The Many Worlds interpretation is the easiest to understand & the silliest when its implications are considered. It is accepted by many scientists with serious credentials. I think that the believers are seduced by a desire to have an easily understood explanation. Note that it requires a universe corresponding to each possible outcome of a quantum level event. That is billions of new universes per second, with each spawning new universes in the next second. That seems to be a high price to pay when the only benefit is ease of understanding.​

Nick Herbert’s book includes several more interpretations, but this Post is already too lengthy.

 

A microscopic entity can enjoy brief or even extended periods of isolation. Whereas macroscopic objects are composed of so many particles / atoms -- that are constantly interacting with themselves and the environment -- that it's usually impossible for such "group organizations" to escape quantum decoherence. For every undisturbed "building block" in a body there are a legion more which are being disturbed. That, and the unharmonious mix of individual states, eliminate superpositions being encountered for the household pet or the backyard tree in our everyday world of large complexities and non-systemic aggregates.

Ironically, however, in Everett's work the biggest "object" of all (the universe) received a wave function. Or does it really count anymore than claiming that Fido or the Larch have [unconstructed] wave functions, when the total information subsumed under / implied by the concept cannot literally be expressed for such macrophysical things?

Peter Byrne: ...Well, the device he used to do that—and I don't want to get too technical here, but it's hard not to use this term—is a universal wave function. A wave function is basically just a mathematical list of every possible configuration of a quantum object, like a hydrogen atom. A universal wave function lists every possible configuration of every single elementary particle in the universe. And there are a lot, so you can't actually write it down! The way you symbolize a wave function is with the Greek letter psi. It's kind of like a U with a stake going down through the middle. The first time he ever saw this symbol, Everett's son Mark said, "What's that little devil's pitchfork?" It does look like a little devil's pitchfork.

So Everett came up with this universal wave function, which is just [Austrian physicist Erwin] Schrödinger's equation for describing how elementary particles move around writ large—that is, applied to the whole universe! And it makes beautiful mathematical and logical sense. Actually, it's very much in use in physics today. However, it has consequences to it that people were and remain uneasy with, which basically is that everything that is possible happens. This assertion, which Everett backed up mathematically, solves, according to him and his supporters, the so-called measurement problem, which is kind of the "dirty little secret" that has been afflicting quantum mechanics since it was invented and then formalized in 1920s... http://www.pbs.org/wgbh/nova/physics/many-worlds-theory-today.html ​

 

CC,

Reality is all-encompassing?

 

Not mine.

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I just shop here...

Seriously, this is the pointless argument I see all over the place.

Particles are excitations (or standing waves) of the field that describes them. There is no "wave/particle duality" to debate.

Unless everyone disagrees with me, of course.

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Hi Spellbound. Thanks for the PM. I do have an opinion, but mine is not any better than Everett's, or Byrne's, except mine makes more sense TO ME, lol. The superposition that they describe along with a description of the measurement problem, means to me that the particle always exists with its states in tact in a precise location somewhere within the wave function, but only in one place at a time, not everywhere within the possible space at all times. For that to be true, reality in regard to that particle is not all-encompassing, if by that you are asking CC if reality is the entire wave-function instead of the particle itself that is located at a specific point in the available space, in my view. I have to qualify that answer by saying that I am an advocate of the Hidden Variables interpretation of QM, meaning that there could be a foundational level of order smaller than the level that we call the fundamental level. The Fundamental level is where we observe, describe, and theorize the characteristics and nature of particles in the Standard Particle Model.

Fundamental particles are said to have no internal composition, while in my view, at the Foundational level, they are complex standing wave patterns, with two wave components, directionally inflowing gravitational wave energy, and spherically out flowing gravitational wave energy. The particle moves in the direction of the net highest inflowing wave energy density, or so I hypothesize.

I hope that answers your PM, and I don't mean to jump in here and derail any ongoing discussion.