Consequences of the new theorem in QM?

Question

It seems there is a new theorem that changes the rules of the game in the interpretational debate on QM:

http://www.nature.com/news/quantum-theorem-shakes-foundations-1.9392

Does this only leave Bohm,Everett and GRW as possible candidates?

Answer 1

I believe this theorem, although it is simple, is somewhat useful, because it does rule out a certain class of interpretations of quantum mechanics, although none of these are the standard ones you see listed in books. For people who believe the Copenhagen interpretation (or many worlds/many-minds/CCC or even Bohm), the wavefunction is the physical state of the system, and this result is proving something vacuous, namely that the parameter $\lambda$ which describes the secret true state of the system, contains at least the wavefunction. But it is the wavefunction! So this theorem has nothing to say to those people, and that's the end of the story. The first answer you got was expressing this point of view.

To the extent that I am comfortable with the standard interpretation, I also have the same reaction to the theorem--- there is nothing new here. But the interpretations which the theorem is adressing are not really interpretations, they are attempts at new theories, and to understand them, one must be aware of the exact way in which quantum mechanics conflicts with intuition. It is difficult to find this in the classical literature, because this literature was written before people were comfortable with the notion of computing.

I believe the ideas below about 40% of the time. Most of the time, I believe the many-worlds interpretation. I don't see any point in distinguishing between philosophical options that are positivistically equivalent, or equivalent modulo philosophical mumbo-jumbo, so I consider many-worlds to be equivalent (modulo mumbo-jumbo) to Copenhagen, to consciousness causes collapse, to decoherence, and to many minds, and I am equally happy to use any of these languages for describing standard quantum mechanics (although many-worlds is more intuitive). These methods produce the same algorithm for simulating a chunk of the universe.

Bohm is different, because there is extra data in the simulation, the particle positions (or the scalar field values) which acts as a real-valued oracle which determines the outcome of experiments, and this means that the randomness is coming from a different source, so I don't think it is idle philosophy to distinguish Bohm from others.

But none of these interpretations are philsophically satisfying from a pure computational point of view. There is no theory which is satisfying from the computational point of view as of yet, and perhaps such a theory is impossible. But it would be nice to find an alternate theory, or else to prove this is impossible with good assumptions. To find the assumptions, there are partial criteria which are not often explicitly stated, which proponents groping for a more satisfying theory often use. I will compile them below, and explain what class of theories the new result rules out at the end.

Quantum mechanics is too big to be physical

The basic problem people have with quantum mechanics was already pointed out by Einstein, and this is the Schrodinger cat. The wavefunction for N particles is approximately the same size as a probability distribution on the N particles, and such humongous dimension mathematical objects usually come up only when referring to quantities which measure our information about a large system, not for quantities which represent the state of a system. In classical mechanics, the number of state variables grow linearly in the number of particles, but the number of variables required to describe a probability distribution of the particles' position grows exponentially. The exponential growth is not paradoxical in probability, because you can imagine that there is a real position underneath the distribution, and do Monte-Carlo to find measurable quantities without solving for the full probability distribution, which might be too big to describe.

The quantum state determines the quantum state in the future in a purely linear way, so it is evolving in a way that suggests that it is a measure of our ignorance of some variables, like a projection of a very high dimensional probability distribution, not in a way that suggests that it is a true measure of the physics of the objects. But in quantum mechanics, the wavefunction is a physical entity--- it is a measure of the physics. A wavefunction is the best-possible full specification of the physical state of an isolated system.

Aside from the ground state of a bosonic system, there is no productive way to reduce the computation in quantum mechanics using Monte-Carlo. This is nearly a theorem now, because of Shor's algorithm. Assuming only that factoring integers in nature requires a search (and Shor's algorithm by itself does not provide any classical method of factoring which is better than a blind search). Even if there were an extremely clever way to avoid a blind search in factoring, nature would have to figure this method out while running a quantum computer, without any hints coming from the quantum algorithm, so the efficient method would have to be closely related to Shor's algorithm, and it is unlikely that such an algorithm exists, even if a very complicated unrelated good factoring algorithm exists.

So quantum mechanics is not like probability, the different probabilistic branches are able to transmit just enough information between the branches to do nontrivial computational tasks with exponential speedup. But quantum mechanics is not like an NP universe--- it doesn't have a fork() instruction with a halt on any fork leading to a global outcome. So it occupies a strange intermediate place between a truly parallel universe physics and a probabilistic computation.

While it is possible that this is the way the universe really is, it is important to search theoretically for alternatives, just so that we learn how big the space of possibilities is. An interesting alternative would at least have to obey one of the following two restrictions:

• It is truly parallel: so that you can solve NP complete problems in polynomial time
• It is truly classical: it craps out when doing Shor's algorithm.

The Bohmian alternative, computationally speaking, is even bigger than quantum mechanics, it includes extra data, so from a computational perspective it is not any better than QM. So I will ignore it. I will also ignore the first alternative, because it is also bigger than QM. I will only focus on truly classical theories, underneath it all.

Bell's inequality and holographic nonlocality

The Bell inequality killed theories of local hidden variables for good. To see this, I will quickly recap Bell's theorem in the most intuitive way I know: if you have three students A,B,C in a line taking a test, and they are cheating, so that they look at their neighbor and copy the answers, if you know that the student in the middle, B, is 99% correlated with A, and also 99% correlated with C, you know that A and C are at least 98% correlated, without saying anything more. The reason is that if there are 1000 questions on the test, 10 answers are different between B and A, and another 10 questions are different between B and C, so there are at most 20 questions different between A and C. This is so intuitive, I hope it does not need a detailed explanation.

The bell inequality is violated for local spin measurements on anticorrelated entangled pairs. If you measure the spin of two distant entangled particles in a singlet along three directions, A,B,C, the answers for the two particles are always anticorrelated with each other. The A answer for one particle is 99% correlated with the B answer for the other, the B answer is 99% correlated with the C answer for the other, but the C answer is only 96% correlated with the A answer for the other. This means that if the electrons made crib-sheets ahead of time, when they were close, to say what answer they were going to give for each measurement, the content of the crib-sheets for the three directions would define students A,B,C (using either particle-- the crib-sheets for each direction are always going to be exactly opposite between the two), and A,B,C would have impossible correlations. So there are no local crib-sheets.

The fundamental reason for the violation is that the amplitude for mismatch gets doubled, but it is squared to get the probability, and the shape of the squaring function is round at the bottom, and for Bell's inequality to be satisfied, you need a probability function with a corner, like the absolute value function. This is a central feature of quantum mechanics, the rounding off, and it is exactly the reason that it is difficult to mock up quantum mechanics using statistics.

So Bell's theorem establishes that there are no crib-sheets, or else the crib-sheets are modified nonlocally. Does this kill hidden variables?

Not really, because we now know that there is gravitational holography--- that the fundamental description of gravity is nonlocal and far away. This requires that the correct description of entangled pairs is distributed nonlocally on a far surface, and it is now not inconceivable that one can reproduce nonlocal correlations, just because it is distributed.

Further, it is also no longer inconceivable that you can get speedups of a square-root nature, like in Grover's search algorithm, from a distributed description of a quantum state. Perhaps the fundamental variables describing a quantum system are distributed in a large number of copies on the cosmological horizon, and when you are doing Grover's algorithm, different sections of the horizon corresponding to your localized quantum computer are looking at different places in the database, so you get a square-root speedup (at least for systems much smaller than the size of the cosmological horizon). Under such models, the parallel-universe aspect of quantum mechanics would be limited in scope--- the amount of parallel computation you can do in a superposition would be limited by the total computational size in bits of the cosmological horizon.

This kind of point of view was suggested by t'Hooft, immediately after he discovered the principle of gravitational holography. His models for replacing quantum mechanics have been preliminary, and I don't find them particularly persuasive, but the idea seems to be that nonlocality due to holography might be the same nonlocality which would be required for hidden variables to make sense.

To be needlessly pedantic regarding the source of this idea (I don't want to put words in t'Hooft's mouth), I have not found this holographic-hidden-variable principle explicitly stated in t'Hoofts writing. He did like to draw a big horizon sphere with 0s and 1s all over it, but his writing on Bell's theorem and his hidden variable theories has been less clear. But I do believe this is the motivation behind his attempts at hidden variables. I ran into him once at a conference about 10 years ago, and he privately told me that he believes that a quantum computer will fail due to unidentifiable decoherence sources at a few hundred qubits, not for trivial environmental coupling reasons--- for failure of quantum mechanics reasons. This prediction suggests that he was thinking this way, at least back then.

Statistical Reversibility and Duck Feet

The fundamental problem with reproducing quantum mechanics is that quantum evolution is a reversible sort of thing, while stochastic evolution is randomizing and irreversible. This mismatch makes it very difficult to even conceive of a stochastic or classical substructure beneath quantum mechanics.

I would like to show how reversible things can happen in probability anyway, using an illustrative puzzle about thermodynamic reversibility:

Suppose I have two buckets of water, one at 100 degrees, the other at zero degrees. I am allowed to subdivide the buckets into little volumes in an arbitrary way, and touch any two volumes to each other in an arbitrary way, and let the heat flow to equilibrium between the two volumes. But I always remember which container came from which bucket, and at the end of the process, I pour all the water from the first bucket back into the first bucket, and all the water from the second bucket back into the second bucket.

So the hot bucket is now colder, and the cold bucket is now hotter. What is the maximum amount of heat that I can transfer from the hot bucket to the cold bucket using only heat flow?

It is a fun exercize to solve this problem, and if you want to do that, don't read any further.

The answer is that it is impossible to transfer the total amount of heat from the hot bucket to the cold bucket. Actually, no, this is not the answer, but I worried that someone might read too far by accident and I would ruin the problem, which is a lot of fun. The answer is that it is possible to transfer nearly all the heat from the hot bucket to the cold bucket by building a heat-exchanger.

This is a system of little thimbles which run in temperature from 100 degrees to 0 degrees in steps of (say) 1 degree. So you have a little 99 degree thimble, a little 98 degree thimble, etc, from each of the hot and cold buckets. You then touch the n-degree thimble from the cold bucket to the n+2 degree thimble from the hot bucket to make two n+1-degree thimbles.

The hot thimbles move down in temperature by 1 degree in this process, and the cold thimbles move up by 1 degree, and you get two oppositely moving conveyor belts, which end with all the hot-bucket thimbles reaching 1 degree, and all the cold-bucket thimbles at 99 degrees. By adjusting the difference in temperature to be small, you reduce the gap.

The lesson here is simple--- if you want to make reversible heat flow, you have to make the heat flow between things that are at nearly exactly the same temperature.

Now I will give a superficially different puzzle:

Suppose I have two rooms, with equal volumes of air. In one of the rooms, there is a single atom of Chlorine floating around, diffusing. I am allowed to subdivide the air in the room into little volumes (without testing for Chlorine), and touch the volumes together to allow the chlorine to diffuse between the volumes (if it is there). At the end, I restore the air in each volume to its corresponding room. What is the maximum probability with which I can transfer the chlorine to the other room?

In this formulation, the answer is almost blindingly obvious: the best you can do is end up with a 50/50 chance of finding the Chlorine in either room. Notice that if you replace temperature with "probability of finding the clorine", this problem is completely equivalent to the previous one, since heat and probability both diffuse. So the heat-exchanger solution to the previous problem gives a double surprise--- by doing a complicated choreography of touching thimbles of air moving along conveyor belts, you can transfer the Chlorine molecule between the two rooms with nearly certain probability! This violates intuition: consider that most of the time you are touching together empty thimbles, for the sole purpose of getting the (tiny) probability of having the clorine atom in the thimble sufficiently close to the (also tiny) probability of having the chlorine atom in another thimble, so that they may be safely touched to one another without causing irreversible entropy gain. This complicated dance of empty thimbles is absolutely necessary if you want reversible transfer.

This was a big surprise for me, because this is reversible linear dynamics arising from pure probability. If you use the _exact_same_process with a cholorine somewhere in one room and a flourine somewhere in the other room, with near certain probability, they will switch positions. So if you repeat this process, the chlorine will oscillate between the two rooms. Probabilistic systems are not supposed to oscillate like this, only quantum systems do that.

So the probability of the chlorine being in one of the two rooms is then evolving in a reversible complex-eigenvalue looking way, although the fundamental description is fully probabilistic and purely diffusive (and depending on the detailed model, the actual time dynamics is not quantum looking). The key requirement is that if a particle can go from region A to region B, the probability of the particle being found in region A must always be nearly equal to the probability for it being found in region B (this requirement is also stated by t'Hooft, which made me wonder why I can't recognize duck-feet quantum mechanics in his papers). This is natural to assume in a holographic type model, where all you know about the system is gross macroscopic variables which are the values of local observables for some system.

So the requirement on a classical theory which can replace quantum mechanics underneath, for me, is that it should be a classical probability theory which has the property that its pure-probability dynamics is described approximately at large scales by a reversible Hamiltonian quantum dynamics, because all the diffusion is between little regions where the probability of occupying the regions are nearly equal. I believe such a theory doesn't exist 60 percent of the time, I believe it does the other 40 percent.

I would call such a theory, if it exists, duck-feet quantum mechanics. Duck feet are nearly perfect heat exchangers.

The paper

The paper is refuting interpretations of quantum mechanics which assume that there is a secret state of the system $\lambda$, which is different from the wavefunction, and which does not determine the wavefunction. This includes all duck-feet quantum mechanics models, and t'Hooft's ideas, and nearly all other modern attempts at replacing quantum mechanics, because the wavefunction is just too darn big to be included in the state description of any reasonable classical-underneath-it-all model.

The theorem states that if the wavefunction is not determined by $\lambda$, there are two different non-orthogonal wavefunctions $|+>$ and $|0>$, preparing either of which could lead to the same $\lambda$. The wavefunction cannot be determined by $\lambda$ in a reasonable model. It is determined by $\lambda$ in Bohmian mechanics (because the wavefuncion is part of the force description between the particles), and this is why Bohmian mechanics is no good as a substitute for quantum mechanics--- it mismatches with classical computation just as much, and so it is just as fundamentally incomrehensible as quantum mechanics.

The paper then assumes that distant systems are independent, so that two systems are described by $\lambda$ and $\lambda'$. Using a particular skewed wavefunction basis, mixing up $|+>|0>$ and $|+>|+>$ etc, they show that there are measurements which give a result in quantum mechanics with certainty that cannot give a certain result using the hidden state $\lambda$ and $\lambda'$.

I believe that this is only a mildly informative result, because the state $\lambda$ is assumed to be like a classical state, so that the description of a two-particle system is by independent variables. It is not clear that you can't violate this theorem by having correlated probability distributions for the hidden variables of the two seemingly unrelated systems, and identifying classical state information with information about the correlated probability density, so that the description of the independent systems is not just concatenating the two descriptions of either system.

but the theorem does rule out a naive mental model where each electron has a secret crib-sheet inside, which tells you the outcome of all experiments, so that the number of variables on the crib sheets does not grow exponentially in the number of electrons. Perhaps this is a straw man position, but I have seen some inexperienced people who hold it.

Answer 2

The paper does not go into details about what interpretations would be disproved by their results. There's a good reason for this: There are no interpretations that would be disproved by their results. They are disproving a straw-man. Here is the central result proved by the paper, phrased in a less obscure way:

"If a system is in the pure state $|+_Z\rangle$ then it is definitely not in some other different pure state $|+_X\rangle$ or whatever."

If this seems obvious and uncontroversial, it is! Admittedly, in the conclusions section, they claim they are saying things that are not obvious...but they're wrong.

Let's start from the beginning. They define the debate by saying there are two pure quantum states, $|\phi_0\rangle$ and $|\phi_1\rangle$. There is one procedure to prepare $|\phi_0\rangle$ and a different procedure to prepare $|\phi_1\rangle$. They say there are two schools of thought. The first school of thought (the correct one) is that "the quantum state is a physical property of the system", so that "the quantum state is uniquely determined by [the physical situation]". That's the one that they will prove is correct. They say the alternative (the incorrect one) is that "the quantum state is statistical in nature", by which they mean "a full specification of [the physical situation] need not determine the quantum state uniquely".

Let's say you have a spin-1/2 system, in state $|+_Z\rangle$. Then...HOLD ON A MINUTE! I just committed myself to the first school of thought! I said the system was really in a certain quantum state!

In fact, everyone doing quantum mechanics is always in the first school of thought, because we say that a system has a quantum state and we do calculations on how the state evolves, etc., if the system is in a pure state. (Not necessarily true for mixed states, as discussed below.)

What would be the second school of thought? You would say, "I went through a procedure which supposedly prepares the system into the pure state $|+_Z\rangle$. But really the system doesn't just have one unique state. It has some probability somehow associated with it. This exact same procedure might well have prepared the state $|+_X\rangle$ or whatever.

Real physicists have a way to deal with this possibility: Mixed states, and the density matrix formalism. If you try to prepare a pure state but don't do a very good job, then you get a mixed state, for example the mixed state which has a 70% chance of being $|+_Z\rangle$ and a 30% chance of being $|+_X\rangle$.

So again, as I said at the start, they have proven the obvious fact: "If a system is in the pure state $|+_Z\rangle$ then it is definitely not in some other different pure state $|+_X\rangle$ or whatever."

With such an obvious and uncontroversial premise, how do they purport to conclude anything that is not totally obvious? Let's go to the conclusions section. They conclude that the "quantum process of instantaneous wave function collapse [is different from] the (entirely nonmysterious) classical procedure of updating a probability distribution when new information is acquired." Indeed, if a spin is in the state $|+_Z\rangle$, then acquiring new information will never put it in the state $|+_X\rangle$. You have to actually do something to the system to change a spin from $|+_Z\rangle$ to $|+_X\rangle$!! For example, you could measure it, apply a magnetic field, etc.

Let's take a more interesting example, an EPR pair in the state $(|++\rangle+|--\rangle)/\sqrt{2}$. After preparing the state, it really truly is in this specific quantum state. If we carefully manipulate it while it's isolated, we can coherently change it into other states, etc. Now we separate the pair. Someone who wants to describe the first spin as completely as possible, but has no access to the second spin, would take a partial trace like usual to get a density matrix. He then gets an email that the second spin is in state +. He modifies his density matrix to the pure state +. You will notice that their example does not show that this so-called collapse violates any laws of quantum mechanics. Their disproof is specific to pure states, and would not work in this mixed state example. Therefore, they cannot conclude that "the quantum collapse must correspond to a real physical process" in the EPR case.

One more example: A spin in state $|+_Z\rangle$, and you measure it in the X-direction. The Schrodinger equation, interpreted with decoherence theory, says that the wavefunction of the universe will coherently evolve into a superposition of (macroscopic measurement of +) and (macroscopic measurement of -). In the paper, they say this in a different way: "each macroscopically different component has a direct counterpart in reality". This is just saying the same thing, but sounds more profound. I should hope that anyone who understands decoherence theory will agree that both of the macroscopic measurements are part of the universe's wavefunction, and that the universe really does have a unitarily-evolving wavefunction even if we cannot see most of it. We rarely care, however, about the wavefunction of the universe; we care only about the branch of the wavefunction that we find ourselves in. And in that branch it is quite reasonable for us to collapse our wavefunctions and to say that the other branches are "not reality" (in a more narrow sense).

-- UPDATE --

I tried to reread the paper in the most charitable way that I can. Now, I think I was a bit too harsh above. Here is what the paper proves:

CENTRAL CLAIM: Say you have a hidden-variables theory, so when you "prepare a pure state $|\psi\rangle$", you actually prepare the state $\{|\psi\rangle,A\}$, where A is the hidden variable which randomly varies each time you prepare the state. It is impossible to have $\{|\psi_0\rangle,A\}=\{|\psi_1\rangle,A'\}$, if $|\psi_0\rangle\neq|\psi_1\rangle$. In other words, the hidden-variable-ensembles of different pure states do not overlap.

They are disproving a straw-man because there is no interpretation of quantum mechanics that asserts that the hidden-variable-ensembles of different pure states must overlap with each other. Even hidden-variable theories do not assert this. There is a so-called "statistical interpretation" in the literature (advocated by L. Ballentine), which also does not assert this.

So this is a straw-man, because nobody ever argued that hidden-variable ensembles of different states ought to overlap. But, it's not a manifestly ridiculous straw-man. At least, I can't think of any much simpler way to prove that claim. (Admittedly, I do not waste my time thinking about hidden-variables theories.) I can imagine that someone who was constructing a new nonlocal-hidden-variables quantum theory might like to know that the hidden-variable-ensembles should not overlap.

Answer 3

IMHO, the PRB paper overlooks the fact that probabilities come from decoherent coarse-graining. At the fine-grained level, a pure state is a pure state. However, for all practical purposes a complete fine-grained description of a system which is not simple like a molecule can't be made. What coarse-graining does is to turn a pure state into a mixed state, and then, their theorem does not apply. Even when observing a simple system like a molecule, the measurement apparatus has to be described in a coarse-grained manner, and so, the probabilities come into play at the level of the measurement apparatus.

Answer 4

It is not possible to specify the state $\lambda$ of a system without measuring it from the outside by interacting with it dynamically and changing it and entangling with it. To specify the state is to change it so that it is no longer the same as before.

Anyhow, PBR assumes in practice, we can measure a system in any orthonormal basis we choose, no matter how convoluted. This may be true in principle, but not in practice. Decoherence has a theory of einselection into so-called pointer bases. Some bases are more natural than others, natural being more easily measured, and some bases are practically impossible to measure.

Answer 5

I really don't think this theorem is as groundshattering as it is made out to be. The main proof of their article is for a qubit prepared in a particular pure state. For realistic systems of interest, typically we have many internal degrees of freedom and the dimensionality of the Hilbert space is huge. Let's leave aside the difficulty of preparing a system with many internal d.o.f. in a particular pure state as opposed to a mixed state. When there are so many internal d.o.f., the construction of the authors require the same pure state to be prepared a really huge number of times. For systems of any significant size, the number of prepared clones will turn out to be impractical. I mean, even in classical probability theory, if two scientists assign different probability distributions to some experimental preparation, and this experiment is prepared in the same way a really really huge number of times, then statistically, with very high probability, at least one of the scientists will be shown to be overwhelmingly likely to be wrong.

The authors are really sneaking in an ensemble through the back door, and ensembles have statistical interpretations.