## The pictorial formalism of quantum mechanics

Today I want to talk about a nice formalism of quantum mechanics developed by Samson Abramsky and Bob Coecke. The underlying mathematical formalism is that of category theory

The pictorial formalism describes systems and processes. The best analogy (with very good reason) is with computer science. There FORTRAN was one of the earlier languages of functional programming. In Fortran one writes functions which take an input, perform some transformation (does a computation) and generate an output. One can formally represent such a program with a box connected by two wires: the input and the output. From a high level perspective it does not matter what goes on inside the box. More important than the inner workings is that those functions can be executed one after another and can be combined like Lego to generate complexity.

Similarly in the pictorial formalism one encounters linear transformations of the wavefunction and those transformations can be combined for complexity. What makes it all interesting is that one can operate simultaneously on several inputs (like on two particles in a singlet state), or on parts of composite quantum systems.

The pictorial rules are extracted from the usual Hilbert space formulation to guarantee agreement with quantum mechanics standard computations. The Choy-Jamilkowsky isomorphism is baked in from the beginning in the approach. Here are some primitive concepts. A state is a process with no input and one output, and a test (measurement) is a process with an input and no output:

The combination of a state followed by a test gives you the probability.

Other primitive notions are tensor product $$\otimes$$ and composition $$\circ$$:

$$f \otimes g = f~~while~~g$$
$$f \circ g = f~~after~~g$$

A fundamental relationship is this:

$$(g_1 \otimes g_2) \circ (f_1 \otimes f_2) = (g_1 \circ f_1)\otimes (g_2 \circ f_2)$$

which can be proven pictorially by inspecting this diagram:

(g1 while g2) after (f1 while f2) = (g1 after f1) while (g2 after f2)

Quantum information is trivially represented in this approach. For example here is how teleportation protocol is drawn:

Alice shares with Bob a Bell pair (the bottom triangle which represent a state) and Alice performs a bipartite measurement on the qubit to teleport (the leftmost line) and one of the Bell particle (the upper triangle). Then she transfers a classical bit to Bob who can use it on his half Bell pair particle to recover Alice's original qbit. The information flow can be continuously traced as in the line on the right. Sometimes the information seem to flow from the future to the past, but the line can be deformed by pulling the ends to straighten it and restore the causal order.

My poor drawings in Paint do not do justice to this very powerful method to represent the information flow. In explaining this approach I have a major weaknesses: I do not know the latest standardization of symbols. However I understand that a book on the pictorial formalism will soon become available and this will clarify the notation.

In the meantime I want to encourage the reader to look up for themselves this amazing approach. Here is an old but good reference:

In the pictorial approach complex quantum computations becomes child's play. One can even compare this method with Feynman's diagrams in terms of simplification of computation.

I said in prior posts that I was not aware a few years ago that my research area of reconstruction of quantum mechanics from physical principles is categorical in nature. But I can now take it a step further and create a pictorial proof of my results (it looks like a tangled mess so I won't draw it here). For this I need to introduce the concept of products and coproducts. In general any product (e.g. complex number multiplication, group operation, etc.) can be understood as a machine which takes two elements and generate another element. Flip the machine around and you get the related concept of a coproduct:

Because quantum mechanics is universal, it applies just as well to single physical systems and to composite systems which are represented using the tensor product ["while", $$\otimes$$]. It is the interplay between products, coproducts and the tensor product which completely determines the algebraic structure of quantum mechanics. All I need is a basic starting point: a product which appears naturally. And this is the Leibniz identity which comes out of the fact that the laws of nature are stable and unaffected by the passage of time. In the infinitesimal case this generates the Leibniz identity which is nothing but the good old fashion product rule of differentiation. Two mathematical representations of this product are the commutator and the Poisson bracket and they correspond to the quantum mechanics Hilbert space and phase space formulations. But I will talk more about this in future posts.

UPDATE:

"Has use of this pictorial/categorical formalism led to any new results? (as Feynman diagrams certainly did)"

I do not want to reveal without permission the identity of the person who asked this question, but I want to give here an extended reply which was not possible under Twitter's insane character limit.

My answer is that the pictorial formalism did not led to new results (as far as I know) as this formalism is strictly a reformulation of the Hilbert space formalism and what you can do in one you can do in the other one. The categorical approach proved its usefulness in quantum mechanics reconstruction in explaining why the results were the way they were, but this was hindsight, an "aha moment". The original motivation came from a very different direction: the attempt to find a common axiomatization for classical and quantum mechanics.

On Feynman diagrams I am not sure what were the new results which came from it. And here is why I say this. When I was in grad school studying QFT from Mandl and Shaw, the professor did not follow the book for the first half of the semester but instead he forced us to use non-relativistic pre-Feynman diagrams, just to appreciate what Mr. Feynman actually achieved. The non-relativistic diagrams were painful to compute, and you have like 16 non-relativistic diagrams for one relativistic Feynman diagram, but you can actually do exactly what Feynman diagrams could, just with a lot more work.

## Joy Christian's program of achieving quantum correlations with Clifford algebras

In the last post I explained how the algebra of the projector operators cannot always be Boolean, otherwise the Hilbert space formalism of quantum mechanics is invalid. Today I will stay in the classical-quantum divide area and I'll talk about an invalid proposal by Joy Christian which generated a lot of debate (and acrimony). When I attended the Vaxjo conference people looking up my archive record saw that I have argued against this proposal and I was asked to explain why it is invalid.

The story begins with the EPR-B experiment and the derivation of the correlation

$$-a \cdot b$$

between Alice and Bob when Alice orients her detection device on direction a and Bob orients his on direction b. So the corelation curve is minus the cosine of the detection angles (the blue line below):

In this experiment the two spin 1/2 particles are in a singlet state:

$$|\Psi\rangle = \frac{1}{\sqrt{2}}( |up \rangle_{left} |down \rangle_{right} - |down \rangle_{left} |up \rangle_{right} )$$

and because the observables are $$a \cdot \sigma$$ and $$b \cdot \sigma$$ the correlation is:

$$\langle \Psi | (a \cdot \sigma)\otimes(b \cdot \sigma) | \Psi \rangle$$

So how can we compute this? We use an identity:

$$(a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma$$

which yields the final answer because $$\langle \Psi | \sigma| \Psi \rangle = 0$$ as the mean value for both Alice and Bob are zero for any direction because we started with a total spin zero state (a singlet state).

Now Joy noticed this identity and thought that it would be nice if he could use it in a classical setting to recover the $$-a\cdot b$$ correlation. There is a "little" problem: how to make the pesky $$i (a\times b) \sigma$$ term disappear?

So Joy came up with the following proposal: half the particle pairs obey:

$$(a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma$$

and the other half obey this:

$$(a \cdot \sigma)(b \cdot \sigma) = -a\cdot b -i (a\times b) \sigma$$

and so when averaged you get to the quantum correlation: $$-a \cdot b$$

But how can this be possible? It is all in the sign of $$\sqrt{-1}$$ Joy claimed. When complex numbers are represented in a plane, the imaginary unit corresponds to the vertical axis. So for half of the particle pairs we draw the imaginary axis bottom up, and for the other half up bottom. But do we really get the cancellation? Nope because $$a\times b$$ is a pseudo-vector which upon this reflection against the horizontal axis changes signs as well and the identity remains:

$$(a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma$$

and all of Joy's ill fated proposal is based on a "forgotten" -1 sign.

But if Joy would have presented his proposal like this, it would not have gotten very far. Instead Joy explained it all using the language of Clifford algebra which is not at all familiar to physicists. Also there was an associative faulty narrative about "topologically complete reasoning".

The main discovery of Joy was a no man's land at the intersection of math, physics, and philosophy: the mathematicians understanding Clifford algebra knew nothing of Bell, the physicists did not know how to counter Joy's philosophical narrative, and the philosophers had no clue of Clifford algebra. Add to this Joy's aggressive and patronizing defense of his proposal and you get a perfect storm of controversy.

The full story of debunking this nonsense would make for a nice soap opera. I was not the first who noticed the mathematical errors in Joy's proposal, I was the third out of four. Also I was not the first who wrote a paper about it, I was the second one out of three, but I was the first who uploaded it on the archive. There were other archive replies to Joy before me but nobody actually bothered to double check his math. The first reply by Marcin Pawlowski came very close to point out the problem but Joy's reply managed to discourage his critics into challenging his math:

"More specifically, the critics culminate their charge by declaring that, within my local realistic framework, it would be impossible to derive “a scalar in the RHS of the CHSH inequality. QED.” If this were true, then it would certainly be a genuine worry. With hindsight, however, it would have been perhaps better had I not left out as an exercise an explicit derivation of the CHSH inequality in Ref.[1]. Let me, therefore, try to rectify this pedagogical deficiency here."

And so people thought at that time that Joy is wrong, his physics and philosophical arguments were nonsense, but his math was correct and it was not a good idea to challenge him at that. But as it turned out all his math was only smoke and mirrors with more and more mathematical mistakes to cover up the prior ones, and I can write up an entire book about it.

All this controversy has hopefully came to an end with Joy resigning his FQXi membership, but he actually never accepted he was wrong and continues to this day to call his critics arguments: "strawmen arguments".

There was only one person who had more energy and spent more time than me debunking Joy's claims and this is Richard Gill

and the physics community owes him a debt of gratitude for putting this nonsense to rest. There were also two good things coming out of this challenge to Bell's theorem.

First, Sascha Vongehr came with what he called a Quantum Randi Challenge: show you beat Bell's theorem on a computer or shut up. With programming help from Cristi Stoica I came up with this simple Java Script program which runs in any web browser which anyone can use to try to disprove Bell's theorem until they really understand why it is an impossible task.

Second, James Owen Weatherall actually manage to fulfill Joy's hope to eliminate the extra term in a mathematically valid model which was not using Clifford algebras. But then would this count as a "disproof" of Bell's theorem? NO because the actual experimental outcomes are +1 and -1 and the correlations must be computed using them and not in a space of make-believe statistics.

So even if Joy's math were valid, it would not represent in any way a "disproof" of Bell's theorem.

Now what I found completely amazing was that after resigning his FQXi membership Joy received encouragements to continue the fight. It is not clear what fight. The fight to prove +1 = -1? Didn't I say soap opera?

Post Script: In case anyone has questions regarding any past or present mathematical, physical, or philosophical claims by Joy, feel free to ask here and I will answer.

## Boolean logic and quantum mechanics

In the last physics post I made the following remarks:

"In quantum mechanics sets and Boolean logic do not apply. When you measure something in quantum mechanics you project to a subspace of the Hilbert space and the Boolean logic changes to the logic of projections. When a system has a property like say spin this is not representable as a point in a set."

which prompted this reply from Lubos Motl:

"Even more obviously, it is complete nonsense - as you state - that quantum mechanics violates the laws of Boolean logic."

I was in the process to systematically rebut point for point all prior objections from Lubos, starting with Bertlmann's socks. However, the importance all of other points pale in comparison with this incorrect statement of Mr. Motl. It makes no sense to split hairs on finer points of disagreement when basic well known facts about quantum mechanics are misunderstood.

To appreciate how blatantly incorrect is Lubos' objection, the implication that in quantum mechanics Boolean logic always applies amounts to denying the applicability of Hilbert spaces!!! Why? Because one can reconstruct the Hilbert space from orthomodularity, completeness, atomicity, and the covering property. But what does all this mean? Let's proceed.

The first thing we need to understand is that Boolean logic is a well defined mathematical structure with sharp axioms. In particular it needs to satisfy the so-called distributive property:

$$A \wedge (B \vee C) = (A \wedge B)\vee (A \wedge C)$$

no ifs, ands, and buts. However this is not always true in quantum mechanics. A simple way to see this is by using the uncertainty principle. Let the three propositions A, B, and C be the following:

A = momentum of the particle is in between $$p_1$$ and $$p_2$$
B = position of the particle is in between $$x_1$$ and $$x_2$$
C = position of the particle is in between $$x_2$$ and $$x_3$$

"momentum of the particle is in between $$p_1$$ and $$p_2$$ and the position of the particle is in between $$x_1$$ and $$x_3$$"

and the right hand side reads:

"momentum of the particle is in between $$p_1$$ and $$p_2$$ and the position of the particle is in between $$x_1$$ and $$x_2$$
OR
momentum of the particle is in between $$p_1$$ and $$p_2$$ and the position of the particle is in between $$x_2$$ and $$x_3$$"

So this seems to be identical, but if the $$x_1, x_2, x_3$$ are close enough then they can be picked in such a way that left hand side obeys the uncertainty principle while the right hand side violates it. And therefore the distributive property can be violated by quantum mechanics.

 No Distributivity Ghost of Classical Physics Allowed

After proving that Boolean logic is not applicable in quantum mechanics we need to figure out a replacement. After all, quantum mechanics is not in any way illogical. To do that we start with yes/no questions we can ask a physical system and attempt to organize it in a consistent way. Some questions are more general than others and when a general question is true, so are the particular ones. This means that we can define a partial order structure. Because we can ask nature no questions, or the trivial question of the existence of the physical system (which is always true) we have in fact a bounded lattice.

More can be said. Because the quantum logic area is rather dry, it is helpful to visualize the concepts using Hilbert spaces. The end goal is to reconstruct the Hilbert space from lattice properties, but we can start in reverse: from Hilbert space we will extract the properties of the lattices of propositions.

By the projection postulate, a measurement collapses the wavefunction to a 1 dimensional subspace of the Hilbert space corresponding to the eigenvector. Then one can decompose the Hilbert space into this subspace and its orthogonal complement. In general the complement is an involution and in terms of lattice of propositions we have what it is called an orthocomplement.

A lattice is called modular if it satisfies a weaker distributive law:

if $$a \leq c$$  then $$a \vee (b \wedge c) = (a \vee b) \wedge c$$

Also a lattice is called orthomodular if the modular condition holds only for b = orthocomplement of a. Therefore we have the following hierarchy:

distributivity => modularity => orthomodularity

[Exercise 1: come up with 3 propositions a, b, c about a quantum system for which modularity is violated due to the uncertainty principle. Hint: adapt the position and momenta example from above]
[Exercise 2: what happens if in your example for exercise 1 you replace modularity with orthomodularity? Why is the uncertainty principle not violated in this case?]

We still need more properties, that of completeness, atomicity, and the covering property to recover the Hilbert space.

When for all collection of elements in a lattice we have a infimum and a supremum the lattice is called complete.

An element is called an atom if $$0 \leq a \leq p$$ implies either $$a=0$$ of $$a = p$$. A lattice is called atomic when every nonzero element majorizes at least one atom.

We say that a covers b if $$a > b$$ and $$a \geq c \geq b$$ implies either $$c=a$$ or $$c=b$$. An atomic lattice has the covering property if for every a  and an atom p, such that $$a\wedge p = 0$$, $$a \vee p$$ covers a.

Now we have all the ingredients to reconstruct the Hilbert space for quantum mechanics. This was done by Constantin Piron in 1964 and the construction goes in three steps:
1. Embed the orthomodular, complete, atomic, with the covering property lattice in a projective space
2. Define an isomorphism from the projective space into a vector space
3. Restrict the vector space to a subspace corresponding to the elements of the lattice.
The details are too technical, but in the end one obtains quantum mechanics over reals, complex, and quaternionic numbers.

So, what does all of this mean?

There are several conclusions:
1. The questions we can ask a quantum system form an orthomodular, complete, atomic, and with the covering property lattice.
2. There is a description duality: [QM in Hilbert space] - [orthomodular lattice + additional properties] as each can be derived from the other.
3. The logic of QM is that of projection operators (or subspaces in a Hilbert space) and not the Boolean logic of ordinary sets and Venn diagrams. In fact it is easy to visualize quantum logic statements using ordinary 3D space and picturing intersections and unions of points, lines, and planes.
4. Quantum OR and Quantum NOT are distinct from Classical OR and Classical NOT. Quantum OR corresponds to the superposition principle.
5. Quantum AND is the same as Classical AND.
6. The logic of QM does not always satisfy the distributivity property of Boolean logic.
Let's illustrate the point of 3D visualization trick with the modularity condition. First, let's prove that distributivity implies modularity:

From distributivity we get that $$a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c) = (a \vee b) \wedge c$$  when $$a \leq c$$. The point of modularity is that this weaker distributivity holds only for $$a \leq c$$ and not in general.

Now the ordinary 3D Euclidean space is endowed with an inner product and it is in fact a Hilbert space. Sure, it is not a complex Hilbert space which ordinary quantum mechanics demands, but Piron's result is equally valid for the usual complex number formulation as well as for the unusual quaternionic and real number formulations of quantum mechanics. This visualization trick works only for 3 dimensional Hilbert spaces but for pedagogical purposes it is enough to quickly reject invalid quantum logic identities. We only use it as a visual tool to illustrate quantum logic statements and build a visual intuition instead of being confined in the abstract and dry land of logic.

Back to modularity, suppose a is a line and c is a plane which includes a. Hence  $$a \leq c$$. If b is a point outside the plane c, $$b \wedge c$$ is the null set and the left hand side evaluates to $$a \vee (b\wedge c) = a \vee null = a$$. For the right hand side, $$a \vee b$$ defines another plane which intersects the plane c precisely at a and so $$(a \vee b) \wedge c = d \wedge c = a$$ as well.

 This is way cooler than flatlander's Venn diagram reasoning

Please note what $$a \vee b$$ corresponds to: it is the geometric figure spanned by the elements a and b which in this case above is the yellow plane. Alternatively it is the smallest subspace containing both a and b.

You can also try to draw yourself the case when b is a line which intersects the plane c in a point and figure out what the modularity relation would correspond to in this case. [Exercise 3: do it. Hint: the answer is that both sides of the modularity equality are in this case.]

[Exercise 4: why is classical OR different than quantum OR? Hint: Quantum OR for a and b is the smallest subspace containing both a and b. How is this different for classical mechanics? The fact that in the quantum case there are points in between a and b for "a OR b" corresponds to the existence of continuous transformations between pure state. No such thing exists in classical mechanics.]

[Exercise 5: why is classical AND the same as quantum AND? Hint: compare Venn diagram intersections with k-dimensional objects intersections.]

[Exercise 6: why is classical NOT different than quantum NOT (the orthocomplement)? Hint: imagine a quantum proposition a which is a line and a system whose state is a point not in a and not in NOT a. Picture it using the 3D method from above. Can this happen in Boolean logic?]

Why are quantum properties like spin measurement outcomes not always representable as points in a Venn diagram (which was the lesson from d'Espagnat's pedagogical simplification of Bell's theorem)? Because the questions we ask a quantum system are sometimes represented as lines, planes, etc. which have a geometric structure richer than that of a point. And why is that? Because of superposition which is the novel physical property which exists in the quantum world and is not present in the classical case.

To understand quantum mechanics you have to free yourself from thinking classically in terms of points, sets, Boolean logic, and Venn diagrams (equivalently thinking that physical systems have sharply defined (dispersion free was an old term for this) properties before measurement). This is all classical baggage unfit to describe nature. The logic of nature is far richer and has a clear geometric representation. Don't let the unfamiliar nature of complex Hilbert spaces stand in the way of your visual intuition.

[Hard Exercise 7: after playing with drawings of points, lines, planes to visualize modularity and orthomodularity, attempt to describe probabilities in quantum mechanics in the geometric representation of quantum logic. Hint: probabilities correspond to a simple geometric concept. What is it?]

Coming back to the original statement: "it is complete nonsense [...] that quantum mechanics violates the laws of Boolean logic" we see now that it is incorrect and moreover this is known to be incorrect for more than 80 years. The complete understanding of what is going on is known for more than 50 years. The quantum logic field is still an active research area. Yet again, category theory plays a key role because it provides semantic-free mathematical objects for logic, meaning it provides uniqueness proofs for logic concepts.

What I presented today is well documented in the literature. For the interested reader who wants to go in depth, an excellent entry point resource is the Beltrametti and Casinelli book: The logic of quantum mechanics.

## Monday, July 6, 2015

I just got back from a trip to Alaska and as I was preparing today to write a new physics post I got roped into a swim team B-meet timing activity which killed my free time. Since I don't want to delay my weekly post any longer and my remaining time will not do justice to the physics topic I want to write about, I will present my vacation impressions instead. I will return to physics topics at the end of this week.

Alaska is a beautiful and expensive place. The price of much anything is doubled because it has to be shipped in. The local economy is based mostly on oil and government employment. The tourist industry is big business too in the summer. The summers in Alaska are very rainy and cold. A temperature of 65 degrees Fahrenheit is considered a "heat wave" by the natives. The timing of the trip was rather poor as the annual salmon runs did not yet start. And without salmons, the wildlife was hard to be seen. In fact I see more wildlife in my backyard every day than I saw in one week in Alaska if I am not counting mosquitoes and bald eagles

which were very common.

There was one thing I did not know: upon entering freshwater to spawn, salmons undergo chemical transformations and their meat turns green inside making it unusable for human consumption. However grizzly bears do not mind that. The salmons in grocery stores are all caught in saltwater.

Back to the trip, I sailed there on a southbound Princess cruise and I visited Anchorage, Skagway, Juneau, and Ketchikan

In Skagway I hiked on the Chilkoot pass and I also panned for gold in frigid water. It turned out that the gold flakes (about 1 millimeter square) were bought from New York at the price of $2 each and mixed with the river mud for gullible tourists to have an "authentic gold rush" experience at the excursion price of$125 per person :)

The Klondike gold rush brought at the time an influx of 100,000 people out of which only about 500 became rich. The people who were able to suffer two or more consecutive winters in Alaska were called "sourdoughs" because the experience turned them bitter and angry (hey I know a sourdough blogger- wink wink Lubos).

In Juneau I visited the Mendenhall glacier

Glaciers are rivers of of blue ice which flow at a rate of about 5 feet per day. In the process they make a lot of noise like thunder and small chunks of ice break up into the sea about every 5 minutes. It is extremely rare to have large icebergs formed this way.

The cruise boats go very close to the glaciers, but I don't have pictures to show as I lost my phone with all the good pictures in Skagway (but I'll get it back in a week). The up-close glacier view is truly majestic and awe inspiring.

All in all the trip was outstanding and worth the money but I should have probably postponed it for about a month to experience the salmon runs. I did not mind seeing only two grizzly bears about a mile away, but besides the bald eagles, three birds and a squirrel was a bit too little in terms of wildlife for a week.

## The socks of Mr. Bertlmann

It seems that I created quite a stir with my prior post and despite knee jerk emotional rants to the contrary which were mostly absurd misunderstandings like I am secretly a believer in classical physics, what I said there is still completely correct (up to grammatical mistakes and typos). One point of genuine disagreement however were about the well known paper of Bell: "Bertlmann's socks and the nature of reality" which I discovered it is greatly misinterpreted and misunderstood. There were other genuine disagreements which I will get to in future posts but I can only address one issue at a time. Today I will try to explain the Bertlmann's socks paper in the larger context of Bell's results.

Let me first set the stage. From its discovery, quantum mechanics was a constant source of debates and disagreements. Einstein had a great dispute with Bohr, Schrodinger did not like quantum mechanics implications and he concocted his famous cat in the box example. Less known is the position of Karl Popper, the discoverer of the falsifiability criterion. In 1959 Popper was trashing Heisenberg's uncertainly relations. His point was that the uncertainty relations correspond to physical characteristics after measurement and in principle there is no precision limitation to defining the position and momenta of a particle and so in his opinion Mr. Heisenberg was unnecessarily jumping to conclusions in his positivist approach. Then he said the following (this is a translation from English to Romanian and back to English so the original quote may be sightly different, but the meaning is clear enough):

"Because any proof of this kind must use quantum theory considerations applied to individual particles, hence formal probability statements, this must be translated word for word in statistical language. If we do that, we'll see that there is no contradiction between the particular measurements assumed to be precise and quantum theory in its statistical interpretation."

Why is this important? After all Popper is not know today to be a quantum guy. However back in 59 he was quite influential developing his own interpretation of quantum mechanics and the fact that he is not known today is because he was wrong and naturally got forgotten. But people today sometimes state that Bell's inequalities were already old news and Bell did not do much. My point is simply that around that time people were not aware of of those inequalities and Bell's results came as a shock.

So Bell put Popper's nonsense to rest with his result and showed that there is a contradiction in statistical terms between any local realistic theory and quantum mechanics. How? By the use of his correlation inequality. Bell had several motivations and today I will present his ideas from one particular point of view skipping the usual discussion with von Neumann. Bell started the analysis with the Bohm and Aharonov variant of the EPR gedankenexperiment in which a source of electrons emits pairs of electrons in a total spin zero state:

$$|\Psi\rangle = \frac{1}{\sqrt{2}}( |up \rangle_{left} |down \rangle_{right} - |down \rangle_{left} |up \rangle_{right} )$$

Measuring the spins for the left particle on direction a and for the right particle on direction b yields the correlation $$-a \cdot b$$ or minus the cosine of the angle between the two measuring directions. Can this be explained if the spins had pre-existing values before measurement? If the measurement directions are perfectly aligned, anti-aligned, or orthogonal, from total spin conservation it is easy to predict that the measurement correlations would be -1, 1, and 0 no matter what. And what would happen if the two electrons would have the spins on opposite directions to preserve the total spin zero state, but their spins would be randomly distributed in space? After about a page of an integration exercise you can convince yourself that the correlation would be in this case $$-\frac{1}{3}a \cdot b$$, so case close, right? Bell arrived at this -1/3 result too but he did not like it enough to ask to be put to an experimental test and he looked further. He noticed that the slope of the correlation curve is zero when the directions are parallel and that looked strange.

Can he arrive at this kind of correlation curve $$P(a, b)$$ while assuming that the outcomes A for Alice and B for Bob depend only on the local measurement direction (no superluminal signaling), on some hidden variable $$\lambda$$ and (very important) respecting the factorization condition below?

$$P(a,b) = <A(a, \lambda) B(b, \lambda) >$$

where the angle brackets mean average over $$\lambda$$. This factorization is the famous Bell locality condition in which the outcomes depend only on the local physics (the directions a and b in the local laboratories) and on a shared randomness "hidden variable" $$\lambda$$ assumed to be generated at the moment of the emission of the two electrons.

So Mr. Bell discovered that for any theory obeying the factorization condition from above he would not get a zero slope correlation curve but a "kink". See the picture below from another Bell paper entitled: "Einstein-Podolsky-Rosen experiments"

Also from the factorization (Bell's locality) condition from above it is not hard to obtain Bell's original inequality:

$$1+ P(b, c) \geq |P(a, b) - P(a,c)|$$

But what does this mean and why is the correlation slope flat for quantum mechanics and is a straight line for classical physics (which does obey Bell's locality condition). The key is in the factorization or lack of. Take a look at the singlet state wavefunction from above. You cannot factorize it between the left and right particles and you do not get the straight line correlation curve. The existence of the flat curve of quantum mechanics requires a different explanation. Enter the Bertlmann's socks paper now.

There are several Bell inequalities, and quantum mechanics and Nature does violate them. But why? The key pedagogical simplification came from Bernard d'Espagnat which came with this silly but true statement:

"The number of young women is less then or equal to the number of women smokers plus the number of young non-smokers"

Let's explain this better with Venn diagrams:

and let us call Women the set A, Non-smokers the set B, and Old the set C. Then the statement reads:

A and not C <= A and not B + B and not C

Is this true? Let's check:

A and not C = areas 1+6
A and not B = areas 1+2
B and not C = areas 5+6

A and not B B and not C = areas 1,6,2, 5 which is larger or equal with the areas 1 + 6 (equal when the areas 2 and 5 contain no elements).

So far so good, but what does this have to do with quantum mechanics and Nature? Mr.Berltmann enters now the stage:

Dr. Bertlmann was an eccentric person who was always wearing socks of different colors. As soon as you see the color of one of his sock you know the other one is not the same. Now in this case the socks have definite colors before you look at them which is different than the spin direction in the electron case which does not exist before measurement and this is the key difference. Can we put this in an exact mathematical statement and more important, can we test this in an actual experiment to show electrons are not like the socks of Dr. Bertlmann?

Now back to d'Espagnat, thank you Dr. Bertlmann for providing humor to a serious physics, mathematical, and philosophical problem.

When a characteristic (be it color of socks, gender, smoker status, color of eyes, etc) exists independent of measurement then the natural way to describe it is using the concept of a set because you can perform the simple test of belonging to your set or not and the result in unambiguous: you are either inside the set, or you are outside. You are either a smoker or you are not, you are male or a female, etc.

Sticking with socks for now, Mr. Bell considered 3 sets, A, B, and C as follows:

A=the number of socks which survive 1000 washes at 0 degrees Celsius
B=the number of socks which survive 1000 washes at 45 degrees Celsius
C=the number of socks which survive 1000 washes at 90 degrees Celsius

Then following the Venn diagram from above he considered if :

A and not B + B and not C >= A and not C

which would be true. But does this inequality hold for electrons as well? You cannot "wash 1000 times an electron at 45 degrees Celsius", but you can detect if the spin records up when measuring it with a Stern-Gerlach device oriented at 45 degree angle. So if the spin orientation of the electron exists independent of measurement we can have the following 3 sets:

A=the electron records spin up when passing through a Stern-Gerlach device oriented at 0 degrees
B=the electron records spin up when passing through a Stern-Gerlach device oriented at 45 degrees
C=the electron records spin up when passing through a Stern-Gerlach device oriented at 90 degrees

Sure, but what to do about this business of "A and not B". You cannot pass at the same time through two detectors! But here is the trick: you have two electrons in the singlet state. Moreover you know that no matter what direction you chose for the left detector, if the right detector is opposite aligned, both detectors will record the same answer because of the total spin conservation. Therefore "A and not B" means now that the left particle clicks up when measured at 0 degrees, and the right particle clicks up (which from spin conservation is equivalent with the left particle clicking down or the left particle not clicking up) when measured at 45 degrees. Sure, there is a bit of counterfactual reasoning, but it works.

So now we have another genuine Bell inequality:

the number of [left electrons clicking up when measured on 0 degrees and right electrons clicking up when measured on 45 degrees]
+
the number of [left electrons clicking up when measured on 45 degrees and right electrons clicking up when measured on 90 degrees]
>=
the number of [left electrons clicking up when measured on 0 degrees and right electrons clicking up when measured on 90 degrees]

And those 3 numbers can be easily computed using quantum mechanics and the answer is...

$$\frac{1}{2} \sin^2(22.5) + \frac{1}{2} \sin^2 (22.5) \geq \frac{1}{2} \sin^2(45)$$

or

0.1464 >= 0.2500 !!!!!!!!!!

And guess what? Not only quantum mechanics violates this inequality, Nature does it too just as quantum mechanics predicts it does.

So what happened? How can this be true? In quantum mechanics sets and Boolean logic do not apply. When you measure something in quantum mechanics you project to a subspace of the Hilbert space and the Boolean logic changes to the logic of projections. When a system has a property like say spin this is not representable as a point in a set. The Venn diagrams have to be generalized from flat circles in a plane to subspaces and their intersection is not as naive as in the picture above. Quantum OR and Quantum NOT are very different than classical OR and classical NOT. All this is because of the novel property of superposition which does not exist in classical physics. Superposition is what makes the Hilbert space a relevant mathematical description to what is going on.

And this is the business of Bertlmann's socks paper.

Now back to the misuse and misunderstandings of this paper. Last time I stated:

"[I cannot take Schack's Bertlmann comment at face value as this would imply he disagrees with Bell's mathematical statements from his famous Bertlmann's socks paper and that would be wrong]."

to which Lubos Motl objected. When you state that "quantum correlations are like Bertlmann socks" at face value you state that there are no differences between classical and quantum correlations and that the difference between the kink vs flat curve of correlations is not there. The big point of Bertlmann's socks paper is that quantum and classical correlations are fundamentally different. And this is not me stating it, it does not come from a faulty understanding of the paper, but it is stated by Bell himself in the very first sentences of the paper and you cannot get more explicit than that:

"The philosopher in the street, who has not suffered a course in quantum mechanics, is quite unimpressed by Einstein-Podolsky-Rosen correlations. He can point to many examples of similar correlations in everyday life. The case of Bertlmann's socks is often cited."

If the correlation curves are not fundamentally different, then you can create classical models of quantum effects, which in turn means that the spin has a definite orientation before measurement. But I know Schack does not believe that because he always emphasizes the importance of Kochen-Specker theorem. The right way to understand his statement was as I stated before:  quantum correlations are just correlations and no explanations are needed in general and I agree with this point of view because there is no way to explain them by reduction to hidden variables which is the content of Bell's theorem. [My position is a bit stronger than what QBism advocates. QBism appeals to the trip between Alice and Bob needed to be able to compute the correlations and this makes perfect sense in their approach. I however say respect nature for what it is and just stop whining about the lack of an explanation to appease your classical intuition which is the result of biological evolutionary pressures.]

But stating it like this: "quantum correlations are like Bertlmann's socks" invites protests and follow up clarification questions from the people who do understand very well the Bertlmann's socks paper. In other words, it adds spice to conversation and it is a provocation for reaction, a friendly poke aimed at the Bell experts who may also (but not always-I am a counterexample and I am not alone) believe in something more: beables. But beables, the unfinished research project of Bell, are a topic for another time.

Also, back to Bell's factorization condition. This is called Bell locality and next time I'll dig into it some more. Nature violates Bell locality precisely because nature is quantum mechanical and not classical mechanical. It does not mean you can send signals faster than the speed of light and violate relativity. If you have a problem with the name you are not alone, but you are in a minority, tough luck, this is a standard term now. If you want to change it, do something really important in the foundations of quantum mechanics on par with what Bell did and then rename it to whatever you like. Calling the foundations community idiots leads nowhere.

Side announcement: I will be going on vacation for a week tomorrow and I will not have internet access. Therefore I will not be able to read or reply to reactions about this post. My next post will also be a bit delayed.

Update: I just came back from a trip to Alaska and I'll need a couple of days to get up to speed and write the next post. You can expect it at the end of Monday.

## Is Nature is Local or Nonlocal?

In quantum mechanics there are two strong points of view. On one hand the philosophers of physics insist that Bell showed us that nature is nonlocal: "What Bell Did", and on the other hand qubists and practitioners of high energy physics stress that nature is purely local and there is no "tickle at a distance". Now  last time I called this debate sterile because both sides are right as they talk about different things. Also I have yet to meet supporters of a camp not agreeing with the mathematical points of the other camp, and so it is all purely a matter of perspective. Hidden behind this seeming disagreement are the epistemic and ontic points of view.

Let's try to disentangle the arguments and explain this local-nonlocal divide. Let's start with the case for nonlocality. This point of view starts with quantum correlations. In the words of Bell: "correlations cry out for explanations". Now only two kinds of explanations for correlations were ever found:
1. common causes from the past
2. an event causing the other one
and neither of them are valid explanations for quantum mechanics correlations. The first kind of explanation falls under local hidden variable approach and this was disproved by Bell, while the second kind is forbidden by the special theory of relativity because spatial separated experiments were performed where there was not enough time for the signal to propagate from Alice to Bob side. The absence of a third explanation is typically stated as nonlocality. Mathematically this is expressed as violation of Bell's locality condition:

$$p(s, t | a, b) = p^1 (s|a) p^2 (t|b)$$

which is equivalent with parameter and outcome independence.

Now no qbist is denying that quantum mechanics violates parameter and outcome independence because this is a solid mathematical and experimental fact. But the local point of view starts with no-signaling, or the inability of Alice to influence the outcomes for Bob (and unsurprisingly no nonlocality supporter is denying this either). In the QBist point of view, each measurement is local and quantum mechanics is a tool which updates my personal degree of belief in order to make sense of what I observe. The Alice-Bob correlations can only be determined when the two sides come in contact and for this to happen travel at speeds lower than the speed of light is required.

To better understand this debate I encourage you to watch this meeting moderated by Brian Greene.

At 1:22:00 Rudiger Schack makes a provocative statement: quantum correlations are like Bertlmann socks. I think this is just an extravagant way of saying that quantum correlations are just correlations and no explanations are needed in general. [I cannot take Schack's Bertlmann comment at face value as this would imply he disagrees with Bell's mathematical statements from his famous Bertlmann's socks paper and that would be wrong].

Now since both sides agree on the mathematics and on experiments, but disagree on interpretation maybe there is a middle ground. Abner Shimony introduced the expression: "passion at a distance" but in the charged atmosphere of today in quantum foundations this is not a popular point of view.

Behind the local-nonlocal debate there is a fracture of interpretation: is quantum mechanics ontic or epistemic? Jean Bricmont expresses best the ontic point of view around 4: 35 in the interview below:

"you need a theory about the world whose fundamental concepts are not expressed, the meaning is not expressed in terms of measurment".

The opposing epistemic point of view was best expressed by late Asher Peres: "quantum mechanics while correct it is not universal, some things must remain unanalyzed".

For now the supporters of each camp do not agree at all with the opposite point of view and seems that nothing can change their minds as each position is perfectly self-consistent. But what is my position because I am neither in the epistemic nor in the ontic camp?

First, Asher Peres position is wrong because his argument is pure handwaving inspired by Godel's incompletness theorem. In Godel's proof there is this key step of arithmetization of syntax without which the proof falls apart, and this is missing from Peres' musings. More important, quantum mechanics can be reconstructed from the assumption of its universality. I believe the epistemic point of view is essentially correct, but I disagree that the Bayesian point of view gives you the complete story. In fact I predict that quantum collapse happens in nature by itself (similar with spontaneous symmetry breaking) and that there is a boundary between quantum and classical due to dynamically generated superselection rules. This implies a testable extension of the quantum formalism and I'll talk about this in future posts. The same approach which allowed me to reconstruct quantum mechanics from physical principles predicts a unique extension of quantum formalism using Grothendieck group construction. Let experiments decide if I am right or wrong.

I also think the basic demand expressed by Jean Bricmont is perfectly valid, but I disagree that the Bohmian interpretation is the way to go. The main fault of Bohmian's approach is distinguishing the complex number formalism of quantum mechanics and splitting the wavefunction into the real and imaginary parts. The quantum harmonic oscillator can be successfully solved in phase space or in the quaternionic formalisms and one obtains the same predictions. However the actual representations are very different in mathematical terms, and who says complex wavefunctions deserves ontic status and quaternionic wavefunctions do not?

Finally, is nature local or nonlocal? Local or nonlocal are bad words lacking a precise enough meaning. Nature is pure quantum mechanical, quantum mechanics is universal, locality-independent and no-signaling.

## Impressions from Vaxjo

I just came back from the QTFT conference in Vaxjo which was excellently organized by Professor Andrei Khrennikov.

I have a ton of interesting information to report from there but for today, still suffering the jet lag and organizing my notes, I will only paint an impressionistic view of the conference experience.

I have never been to Sweden before and I was pleasantly surprised to see how well Sweden is connected to the world. Everyone I met spoke English without any accent, the small Vaxjo town was cozy, and the hotel had excellent service on par with three times as expensive hotels in US. A strange experience was the short dark hours, due to the Nordic latitude, and I can only imagine how winter would look like. Also it was rather cold, like a nice November day but it got warm as the week progressed. If you walk from town to the university you go around two beautiful lakes and the surroundings provided a very nice setting for quantum mechanics private discussions.

The conference featured a lecture from Theodor Hänsch, the recipient of the 2005 Physics Nobel Prize. Then the conference placed the focused on several interesting and essential in my opinion areas: experiment and interpretation, qubism, categorical quantum mechanics, quantum-like models outside physics.

I was able to learn that we may be about two years away from experimental confirmation or rejection of the current GRW-type collapse models, I understood the finer points of distinction between Copenhagen and qbism interpretation, I experienced the amazing depth of the category theory usage in quantum mechanics (and I think the time to launch a journal dedicated to this is fast approaching), and I got delighted by quantum-like effects in psychology.

The discussions happened on four levels: during the formal presentations, during the coffee breaks, during walks around the lakes on the trips back to town, and in the welcoming arms of the Bishop: the local pub where many fine points of quantum interpretations were very seriously debated until the closing hours.

I found it surprising to see the passion that Bell's theorem still elicits as well as the debate between locality vs. nonlocality in quantum mechanics. The funny part is that both sides agree that quantum mechanics violates Bell's locality condition which is the essential part, and as a neutral observer (since I have my own interpretation) the fight looks to me completely sterile and useless.

I also discovered that I am not the only one bitten by the hope to solve Hilbert's sixth problem one day.

Overall, it was a very pleasant and extremely productive time for me and I wish I will be able to return to this conference every year.