## Wednesday, August 28, 2013

Bell's Theorem

### Charlatans beware! So you think you can beat Bell's theorem?

 Number of experiments: Number of directions: Legend: Direction index|Data index|Measurement Alice|Measurement Bob

Before I present the problems with all current quantum mechanics interpretation, let me wrap up the discussion on Bell's theorem with this pedagogical tool above.

This is based on the original idea of Sascha Vongehr's Quantum Randi Challenge: either prove you can beat Bell's theorem in a controlled environment or shut up (http://www.science20.com/alpha_meme/quantum_randi_challenge_help_perimeter_physicist_joy_christian_collect_nobel_prize-79614) and was created with coding help from Cristinel Stoica.

This is a simple Java Script HTML page which illustrates Bell's theorem for the singlet state in an EPR-B experiment and allows the user to try their hand on beating Bell's theorem. The script works fastest on Google's Chrome, and potentially on Internet Explorer 10, but the older IE versions 7,8,9 are very slow (blame Microsoft for that). The page can be also downloaded from: http://www.florinmoldoveanu.org/Math_Philosophy.html

The advantage of Java script is that it can be modified by anyone and can be tested on any web browser. Next week I will add explanations of how this page works and add a small tutorial on Java script to enable the reader to change the code. Until then, please play with this page. Enjoy!

Let me explain how this program works. The first two boxes allow the user to enter the number of experiments and the number of directions. The number of experiments is how many times the EPR-B experiment is supposed to be repeated (we are after obtaining correlations and they require repeated measurements). The number entered here has to match the number of rows in the first large box. In there we generate a shared randomness element in the form of a random unit vector. The information in each row is shared between Bob and Alice outcome generation program and can be used to enforce a generation strategy. The generate buttons can randomly populate the boxes, or the user can erase all data and enter whatever he/she wants.

In the original Aspect experiment, the detector orientation is randomized and it is arbitrarily changed while the photons are in mid-flight. The number of directions times two has to match the number of rows in the Alice-Bob measurement directions box. The format is 3D Cartesian: the x component, the y component, and the z component with the values being comma separated. The odd entries (1,3,5, etc rows) are for Alice and the even entries (2,4,6, etc rows) are for Bob.

In the next box, the experimental outcomes of +1 or -1 are generated. The first two numbers are control indexes to identify the experiment orientation and the experiment run. The user can play with low numbers in the first two boxes to see how the data is generated and displayed.

Last, the Plot Data button displays the correlations from the prior box in graphical form. The user can check the validity of the data display by manually computing the correlations and verifying that the point on the graph are in the proper place.

Bell theorem states (and this program illustrates) that any local hidden variable model respecting three requirements generate a correlation which is a straight line, while quantum mechanics and nature shows a minus cosine correlation. The three requirements are: parameter independence, outcome independence, and binary outputs. Parameter independence and outcome independence are called Bell locality and next time I will show how the code behind implementation respects them. The experimental outcomes have to be +1 or -1 detector results and the correlations have to be computed using actual outcomes and not in other mathematical representations of the hidden variables.

Next time I will present a brief Java script tutorial and a brief description of the script behind this web page and this hopefully will allow the user to modify the script to his own hidden variable model in an attempt to recover the minus cosine correlation.

## Is the wavefunction ontological or epistemological?

### Part 6: Is the Moon there if we are not looking at it?

What does objective reality mean? Intuitively this is very clear: the world is out there independent of me, it exists “objectively”. But is this in agreement with quantum mechanics and with experimental evidence? Quantum mechanics predicts only the probabilities of experimental outcomes, and Einstein thought this must mean quantum mechanics is incomplete. By now we know quantum mechanics is the entire story so how can we reconcile probabilities with realism? One possible quantum mechanics explanation is the Bohmian interpretation (http://plato.stanford.edu/entries/qm-bohm/). Here, particles exists objectively and they are guided by a “quantum potential” allowing them to move in such a way that they recover the predictions of standard quantum mechanics.

But wait a minute; didn’t Bell prove the impossibility of local realism? How can this guiding potential allow the particle to achieve super-classical correlations, especially when one particle can be here and the other one at the other end of the galaxy? Simple: the particles move faster than the speed of light but without being able to carry signals!! No, no, no, unicorns and Santa Claus you may say. What about an electron? If the electron is whisked away it should radiate and we should see this radiation all around us. Also what happens in an atom if the electrons have definite positions? Would this mean the atom is unstable?

Because in part of the radiation problem there are no known generalizations of Bohmian mechanics for relativistic quantum field theory, and very likely there is not possible to obtain one. Also in this interpretation, the atom consists of stationary electrons at a fixed distance. Not only the existence of this particular distance is very strange, but also in general the Bohmian trajectories are known to be “surreal”.

But is there a more formal way to disprove classical realism, even non-local one? We need to start by the definition of realism. The best place to start is from EPR’s reality criterion:

“If without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity”

So for example, if I measure the position of a particle and I find a definite value, measuring again in quick succession would yield the same value because “the particle is there”. The reverse implication is given by EPR realism: if I predict with certainty the position of the particle (and any measurement would confirm my prediction) then the particle must really be there.

We can now prove that EPR’s reality criterion is actually inconsistent with quantum mechanics, and to do this we will reason very similarly with Bell from his famous theorem.

The gist is as follows: we will consider a quantum system for which we can predict with certainty both an outcome and a correlation. But given one measurement, a subsequent measurement (in a different configuration) must respect a quantum mechanics rule and this can be shown to destroy the correlation. In other words, the law of subsequent measurements, the certainty of the outcome and the certainty of the correlations are incompatible.  So which one should be sacrificed? The law of subsequent measurements is iron-clad and validated by experiments. Quantum correlations are indisputable also. What remains is realism. Late Asher Peres use to say: “unperformed experiments have no results”. But this is stronger. We may say: “unperformed experiments for which the outcome can be predicted with certainty have no results”.

The technical description of the argument is presented in http://arxiv.org/abs/1211.4270

Basically the argument is as follows:

Start with a Bell singlet state: (|+>|-> - |->|+>) and have Alice and Bob measure this on the vertical axis: one will get spin up and the other one spin down. Supposing the spins do exists independent of measurement, then they must be oriented on vertical axis (we don’t know how for each person) to achieve perfect anti-correlation. [Suppose they are oriented in opposite directions, randomly distributed. Then the measurement correlation is no longer minus the cosine of the angle between the measurement axis-as predicted by quantum mechanics and validated by experiments, but minus 1/3 of the cosine of the angle between the measurement axis]. This may look conspiratorial (after all we can select any other axis to the same end), but it is an experimental fact.

Because we can predict with certainty the spin alignment direction, the alignment direction must really exist by EPR reality criterion.

But now we can ask if this direction is compatible with Bell correlation. When measurement directions for Alice and Bob are orthogonal, the total correlation is zero. Is the vertical axis correlation compatible with this correlation? The answer is no by an impossible inequality: ¼=0.25 < sin^2(π/8) ≈ 0.1464 (see http://arxiv.org/abs/1211.4270)

There is still a way out however: remember Bohmian mechanics and non-local realism. Suppose the axis were in a place compatible with the correlation, but the very act of the intermediate measurement made the axis instantaneously realign. Fortunately relativity comes to the rescue and proves this is impossible. Why? Because if Alice and Bob are spacelike separated in a reference frame Alice does her measurement first and realigns the axis on her measurement direction, and in another Bob does his measurement first and realigns the axis on his direction. And the spin direction cannot have two values at the same time. The only way out is to conclude EPR realism is false.

But what about the Moon then? The Moon is there even when we don’t look at it because of decoherence: the moon is “observed” by the solar wind, cosmic radiation, etc, so someone is constantly looking at it!

And what about Bohmian mechanics? Would this argument not disprove this interpretation as well? Nope, because ve…eery conveniently, spin is not (cannot be) treated classically in this interpretation.

## Is the wavefunction ontological or epistemological?

### Part 5: The PBR Result

A few years ago a Nature paper by Pusey, Barrett, and Rudolf (PBR for short) (http://arxiv.org/abs/1111.3328) was the talk of the town. The paper claims the quantum states cannot be interpreted statistically, but this is an overreach and the proper claim is that the wavefunction cannot be “psi-epistemic”. A very good review on this can be found on Matt Leifer’s blog: http://mattleifer.info/2011/11/20/can-the-quantum-state-be-interpreted-statistically/ and thought provoking posts were written by Lubos Motl: http://motls.blogspot.com/2011/11/nature-hypes-anti-qm-crackpot-paper-by.html and Scott Aaronson: http://www.scottaaronson.com/blog/?p=822

To clarify the intent of the paper, let me quote part of the archive abstract: "we show that any model in which a quantum state represents mere information about an underlying physical state of the system, and in which systems that are prepared independently have independent physical states, must make predictions which contradict those of quantum theory"

So what does a psi-epistemic wavefunction mean? It means that one can have a probability distribution function over ontic states and it is possible to have overlapping functions over the same ontic states. Using this definition, here is how the proof goes:

Start with two pure states: |0> and |1> and construct a superposition |+> = (|0> + |1>)/sqrt(2)

Then for a tensor product corresponding to the system being prepared in either |0> or the |+> states, consider a cleverly chosen basis of four orthogonal states:
|Blue> = (|0>|1> + |1>|0>)/sqrt(2)
|Red> = (|0>|-> + |1>|+>)/sqrt(2)
|Gold> = (|+>|1> + |->|0>)/sqrt(2)
|Green> = (|+>|-> + |->|+>)/sqrt(2)

where |->=(|0> - |1>)/sqrt(2)

Now we have the following properties:

|Blue> is orthogonal with |0>|0>
|Red> is orthogonal with |0>|+>
|Gold> is orthogonal with |+>|0>
|Green> is orthogonal with |+>|+>

which means that measuring in this  basis would yield either:
the system was not prepared as 00
the system was not prepared as 0+
the system was not prepared as +0
the system was not prepared as ++

Let us paint the picture of the statements above using psi-epistemic definition:

Here we see the red and green overlap of the hypothetical probability distribution over some hypothetical ontical state. The key in in the overlap. This allows to claim an epistemic interpretation.

Let us highlight the probability distribution support for the 4 cases:

Not in |0>|0>:

Not in |0>|+>:

Not in |+>|0>:

Not in |+>|+>:

Now PBR paper claims that the following black area:

generates a contradiction because the 4 vector basis forms a complete orthogonal base and no matter the outcome result, it will generate a contradiction with the assumption of understanding the wavefunction as a probability distribution over some ontic state.

Convinced?

Technically the paper is correct, but can we reason in this fashion in quantum mechanics? The contradiction is based on 2-dimensional Venn diagrams. Potential Danger!

Suppose the 4 compact support areas are not 2 dimensional objects, and we need to reason in space!

Here there is no intersection of the 4 areas, and the earlier intersection was an optical illusion stemming from looking at this case from a particular viewpoint (that of classical concepts, and 2-dimensional Venn diagram).

This does not correspond to any quantum system, and it is not a counterexample to PBR, but it is an illustration of the dangers of thinking 2-dimensionally in terms of classical sets.

To improve this 3D picture's agreement with the PBR argument one can add for example depth to the color rectangles and extending the depth until any 2 rectangles begin touching. Also picture this on a torus with the red slab touching and overlapping with both the green and gold slabs, the green slab touching and overlapping with both red and blue, etc. (drawing this goes beyond my ability of using Paint).

There is a key piece of information in favor of PBR however: quantum AND is the same as classical AND. However, in higher dimensions, should we demand that all 4 areas must intersect at the same time? This is one of Lubos' criticism:  measuring “not 00” only eliminates this possibility. Because “unperformed experiments have no results” as Asher Peres put it, this is certainly a valid criticism under appropriate conditions.

To complete the PBR argument and generate the contradiction you need to regard the ontic states as having definite properties and this in turn allows reasoning with sets and 2D Venn diagrams (as a caveat this does not necessarily mean definite observable values and we make take for example the wavefunction itself as the ontic object). Therefore agreeing with PBR hinges on the very definition of what we might mean by “ontic states”.

In conclusion, does PBR proved that quantum wavefunction is not epistemic? Yes and no: yes, it is not psi-epistemic, but this is not the whole story in terms of epistemic explanations (see Matt's blog post).

Next time I’ll show that quantum systems cannot have definite observable values before measurement and attempt to get a handle of what we might mean by “objective reality”.

UPDATE

The topic of PBR is subtle and the original post was mildly changed to avoid giving the wrong impression. PBR does not disprove Born's interpretation but the consensus is that it disproves the quantum Bayesian interpretation which contends that the collapse postulate is nothing but a manifestation of information update about a quantum system. If the wavefunction is nothing but subjective degree of knowledge, then before the collapse, the wavefunction has some overlap with the wavefunction after the collapse and is subject to PBR's assumptions. Hence PBR rejects the quantum Bayesian interpretation.

Both Chris Fuchs and myself disagree with this analysis (for different reasons), and I contend that the Bayesian interpretation is "successfully reasoning consistently about an inconsistent system" because the collapse postulate is inconsistent with quantum mechanics. As such the wavefunction in this interpretation does not satisfy the "ontic state requirement" of having definite properties since technically the wavefunction in this interpretation does not rigurously exist.

## Is the wavefunction ontological or epistemological?

### Part 4: Building a Hilbert Space Intuition

Hilbert spaces are the natural arena of quantum mechanics and their dimension can be either finite or infinite. As late Sidney Coleman will joke, I intended to draw a 72 dimensional Hilbert space, but due to budgetary constraints I can only draw the case of dimensionality three. However this will turn out enough to understand the peculiarities of quantum mechanics.

As a reference for this post, I recommend this paper: http://arxiv.org/abs/quant-ph/0007041v1 by Diederik Aerts and collaborators and I will attempt to explain why the logical operators “OR” and “NOT” are different in quantum mechanics.

As a physical theory subject to experimental verification, quantum mechanics generates sets of propositions about a (quantum) physical system, and those propositions are formally identified with projection operators in a Hilbert space. Technically, those propositions generate an order structure: a family of complete, separable, orthomodular, atomistic with the covering property lattices (isn’t this a mouthful?).  The reverse property stands as well, and Piron was able to reconstruct quantum mechanics’ Hilbert space starting from those properties. Diederik Aerts studied under Piron and his main research interest is in composite systems: http://www.vub.ac.be/CLEA/aerts/

In general, a Hilbert space is a generalization of the usual Euclidean space and its main property is the existence of an inner product. By definition, a Hilbert space is defined over complex numbers, but it can be generalized over any number system in which case it is called a Hilbert module. Since it is well known that quantum mechanics can be expressed over reals, complex, or quaternionic numbers, we can use the usual 3D space as an example of a three dimensional Hilbert space (over real numbers) and try to understand the quantum mechanics postulates in this case (the inner product would be the ordinary dot product). The obvious advantage is that we can visualize right away what is going on.

In ordinary three dimensional Euclidean space, a point is specified by three numbers: x,y,z, but in quantum mechanics, a point would represent a particular quantum state. Statements (propositions) about the quantum system correspond to projections on points, lines, and the entire space. We can understand points, lines, and planes as quantum states corresponding to either experimental preparation procedures, or to experimental outcomes (collapsed subspaces).

So now let’s take a look at the picture below:

Suppose “State A” it is a subspace of the original Hilbert space corresponding to a projection P_a and suppose we can represent this as the vertical line in the picture. The projection P_a would mean that measuring the observable “A” would yield the outcome “a” (“A” could be the position, momentum, spin operators, etc).

The orthogonal complement of State A will be the horizontal plane called “Not A”, itself a Hilbert subspace corresponding to another projection.

Let us say that the QM system is the green dot and measuring “A” on it would yield the “a” value. Since the green dot is not on the “Not A” plane, this means that testing for the P_{not a} projection would mean that the answer is false. Hence, if a is true, not a is false.

Now suppose the quantum state is prepared in the Hilbert subspace Not A

The same game can be played in reverse and nothing out of the ordinary happens.

But how about the case below?

Now the state is neither on “State A” nor on “Not A”. Hence, we can state this: “the quantum entity is in a state such that a is not true, without not a being true”. And this is not the case in Boolean logic. Quantum NOT has a nonclassical behavior.

A similar game can be played with quantum OR:

In this case the green dot does not belong to either subspaces A or B but it belongs to the closure of the union of the two subspaces (the entire 3D space). Therefore: “a b can be true without a ‘or’ b being true.” It is easy to see that if the green dot belongs to either A or B subspaces, it belongs to AB. This means that if a ‘or’ b is true it follows that a b is true.

Now what really happens in a Hilbert space is that in QM the ordinary Venn diagrams which illustrate the boundary of finite sets are no longer confined on a plane!!! In fact, the logic of quantum mechanics is the logic of projective spaces and the Venn diagrams acquire an n-dimensional border.

The lack of distributivity property (in QM we have orthomodular lattices and not Boolean algebras) can be explained  geometrically by the fact that Venn diagrams for QM are n-dimensional objects no longer confined to a 2D plane.

In several quantum mechanics theorem (like Gleason’s theorem http://en.wikipedia.org/wiki/Gleason's_theorem), one basic assumption is that the Hilbert space dimensionality is higher than two. The reason is that in two dimensional Hilbert spaces, Venn diagrams can be confined in a plane and classical explanation of quantum effects can be easily obtained.

Recalling the three circles in the prior Bell Theorem post, it is clear now how the Bell inequality is violated: the inequality was based on the classical physics requirement that Venn diagrams must be represented on a 2D plane (which is not the case in QM)

We can now also “visualize” the famous double slit experiment (http://en.wikipedia.org/wiki/Double-slit_experiment ) question: which slit did the particle go through? This is a direct physical illustration of the nonclassical OR behavior pictured above: one slit corresponds to subspace A, the other to subspace B, and the quantum state is located on neither one. Finding out “which way information” amounts to projecting the state to either A or B and hence destroying the interference.

So QM is still strange, but at least now we can create a mental picture of the Hilbert space geometry and we can visualize the meaning of various famous QM results.

Next time I am going to use this technique on a recent result called PBR which claims to show that the wavefunction is not epistemological. Did PBR actually succeed in proving this very strong claim? Yes and no. Please stay tuned.