CHSH inequality and the rejection of realism

I will start now a series on Bell's theorem and its importance to quantum foundations. Today I will talk about the Clauser, Horne, Shimony, and Holt inequality and its implication. 

Quantum mechanics is a probabilistic theory which does not make predictions of the outcome of individual experiments, but makes statistical predictions instead. This opens the door to consider "subquantic" or "hidden variable" theories which would be able to restore full determinism. However, even with the statistical nature of quantum mechanics predictions there is something more which can be investigated: correlations. What distinguishes quantum from classical mechanics is how the observables of a composite system are related to the observables of the individual systems. In the quantum case there is an additional term related to the generators of the Lie algebras of each individual systems and this in turn prevents the neat factorization of the Hermitean observables of the composite system. It is this lack of factorization which prevents in general the factorization of the quantum states. In literature this goes under the (bad) name nonlocality.

Now suppose we have two spatially separated laboratories which receive from a common source pairs of photons. The "left lab" L chooses to measure the polarization of the photons on two directions \(\alpha\) an \(\gamma\), while the "right lab" R chooses to measure the polarization of the photons on two directions \(\beta\) an \(\delta\), Let's call the outcome of the experiments: \(a, b, c, d\) for the directions of measurement \(\alpha, \beta\, \gamma, \delta\), respectively. The values \(a, b, c, d\) can take are +1 or -1.

Now let us compute the following expression:

\(C=(a+c)b-(a-c)d\)

Suppose \((a+c) = 0\), then \((a-c)=\pm  2\) and so \(C=\pm 2\)

Similarly if \((a-c) = 0\), then \((a+c)=\pm  2\) and again \(C=\pm 2\)

Either way \(C=\pm 2\)

Now suppose we have many runs of the experiment and for each run \(i\)we get:

\(a_i b_i + b_i c_i + c_i d_i - d_i a_i = \pm 2\)

from which we deduce on average that:

\(|\langle ab\rangle + \langle bc\rangle + \langle cd\rangle - \langle da\rangle|\leq 2\)

This is the CHSH famous inequality. Now under appropriate circumstances nature violates this inequality:

in an experiment with photons the average correlation between measurements on two distinct directions \(\alpha, \beta\) is: \(\cos 2(\alpha - \beta)\) and the inequality to be obeyed is:

\(| \cos 2(\alpha - \beta) + \cos 2(\beta - \gamma) + \cos 2(\gamma - \delta) - \cos 2(\delta - \alpha)| \leq 2\)

but if the angle differences are at 22.5 degrees we get that \(2\sqrt{2} \leq 2\) so what is going on here?

A natural first objection is that not all 4 measurements can be simultaneously be performed and so we are reasoning counterfactually. But in N runs of the experiment we get 2N experimental results and there is a finite number of ways we can fill in the missing 2N data and in each counterfactual way of filling in the unmeasured data the CHSH inequality is still obeyed.

A second potential objection is that there is no free will and there is a conspiracy going on which prevents an unbiased choice of the 4 directions. There is no counteragument for this objection except that I know I have free will. If free will does not exists then mankind has much deeper troubles than explaining quantum mechanics: try to explain morality and justify the existence of the judicial system. 

The introduction of bias can affect correlations and if the detection rate depends on the angle, then for appropriate dependencies one can obtain the quantum correlations. This is the so-called detection loophole, However, if such a dependency exists, it can be tested in additional experiments and the introduction of angle dependency only for Bell test experiments is indefensible. Loophole free Bell experiments while important to push the boundary of experimental technology have no scientific importance and they count only towards experimentalist's bragging rights.

Another way to obtain correlations above 2 is by appealing to contextuality: for example the value of \(a\) when measured by lab L when lab R measures \(b\) may not be the same when lab R measures \(d\). While quantum mechanics is contextual, in this case such an argument means that the the choice lab R makes influences the result of measurement at lab L which is spatially separated!!!

Last, if the values of \(a, b, c, d\) do not exist prior to measurement, this decouples again the value of \(a\) when lab R measures \(b\) from the value of \(a\) when lab R measures \(d\). 

Assuming free will is true, we have only two choices at out disposal to be able to obtain correlations above 2: 

• measurement in a spatially separated lab affects the outcome on the remote lab
• the outcome of measurement does not exist before measurement.

The first choice is taken by dBB theory because the quantum potential changes instantaneously and the second option is advocated by the Copenhagen camp. (I am excluding the MWI proposal because in it there is no valid derivation of Born rule. I am also excluding collapse models because they are a departure from quantum mechanics and experiments will soon be able to reject them).

Now here is the catch: the two labs need not be spatially separated and one experiment in lab L can unambiguously happen before the experiment in lab R. When the R lab measurement takes place it cannot affect the outcome in the L lab because that is in the past and already happened! 

But can the first measurement affect the second one? In dBB this is possible as long as the first particle and its quantum potential is still around to "guide" the second particle. However, if after the first measurement the first particle is annihilated by its antiparticle then its quantum potential vanishes. The behavior of quantum potential after annihilation  is a reason why a relativistic second quantization dBB theory is not possible: either the quantum potential sticks around and messes up subsequent measurements, or vanishes and then the correlations cannot occur in the case above. (dBB supporters pin their hopes on a "future to be discovered" relativistic dBB quantum field theory which never materialized and cannot exists for several reasons.)

So from the two choices above only one remains valid:

the outcome of measurement does not exist before measurement

Realism is rejected by Bell's theorem. However in literature Bell's result is presented instead as a rejection of locality. But this is an abuse of language: locality=state factorization. Nature and quantum mechanics are incompatible with a state factorization. State factorization is just factorization, not locality. Rejection of realism is the only viable option left. 

Can you generate entangled particles which never interacted?

This post is the continuation of the last one because (1) it attracted a lot of attention and (2) no consensus was reached by the viewers. In particular, Lubos Motl was insisting on the fact that you cannot generate entangled particles which never interacted. So I am making one last attempt to convince him of the contrary.

The setting is from last time: entanglement swapping. This time I will not write Latex and instead I will explain the picture above (please excuse my poor MSPaint abilities). For the sake of argument, I am using photons and I show their worldliness in black going at 45 degree angles. Alice and Bob have the red resonant cavities which capture the 1 and 4 photons (feel free to replace the cavities with long enough optical fiber loops). The cavities have a release or absorption mechanism which is activated by Charlie upon obtaining the result of a projective measurement on photons 2 and 3. On average Charlie obtains his desired output 25% of the time in which case he sends the signal to release the 1 and 4 photons to the outside world. The other 75% of the time Charlie sends the signal to absorb the 1 and 4 photons.

From the outside Alice, Bob, Charlie, and 1 and 4 photons are inside of a green box. At random times out of the green box comes out two entangled photons which never interacted in the past.

Now here is the key Lubos statement:

"Excellent. Now, the filtering is caused by the decision in Charlie's brain which is in the intersection of the two past light cones of events -measurements of particles 1,4, right?

So you haven't found any counterexample to my statement, have you?"

Now here is why Lubos is wrong: See the picture below. He contends it is the purple area which is important and the fact that Charlies decision is made inside it  (again excuse my lack of precision drawing 45 degree lines).

Why is the purple area irrelevant to the discussion? Because when the photons 1 and 4 are in their respective red cavities they do not interact with each other! 

Viewed from outside of the green box at random times comes out two entangled photons which never interacted in the past and could not have interacted in the past because first they were spatially separated, and second they got trapped in an isolation cavity long enough for the signal from Charlie to reach their cavity and release them.

It does not matter that Charlie's brain is in the intersection of the two past light cones of events measurements of particles 1,4 at the exit of the green box (the purple area). It matters that  Charlie's brain is not in the yellow area of the intersection of the two past light cones of the events of particles 1,4 entering the holding red cavities.

It is hard to explain all this in words without pictures. I tried to have a crude drawing last time in the comment section but formatting mangled it. I was suspecting Lubos was only pretending not to understand my argument, but now a week after the exchange I tend think he had a genuine misunderstanding because while he was thinking of the purple area I was talking about the yellow one.

Talking about other comments, last time a statement from Andrei caught my attention:

""S1 does not imply nonlocality, but nonrealism"

It does not imply non-locality for the particle that is measured, but implies non-locality for the distant particle. You measure one particle here and you create the value of the spin for both entangled particles (including the one that is far away)."

I will address this in a future post because a quick reply does not make justice to the topic. Is is true that "You measure one particle here and you create the value of the spin for both entangled particles (including the one that is far away)"? This is deeply related with Einstein's realism criteria:

"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 physical reality corresponding lo this physical quantity."

This is a very natural criteria given a classical intuition, but it is however false. And showing why and how it is false is not a trivial matter. And if Einstein realism criteria is false then this is not true either: "you measure one particle here and you create the value of the spin for both entangled particles (including the one that is far away)."

Measurement does change something about the remote particle: it's state. But if the states are related to epistemology as opposed to ontology then there is no nonlocality problem. The easiest way to understand this is in the Bayesian paradigm where I change my degrees of belief: local measurement changes my local degree of belief about the remote particle.

Correlations and Entanglement swapping

take 2

Finally I am ready to discuss the topic I promised: entanglement swapping. This s not a hard topic, and in fact it is usually given as a homework problem but since Lubos insists on doubling and tripling down on the fact that quantum correlations can arise only due to prior interaction:

"Whenever there's some correlation in the world – in our quantum world – it's a consequence of the two subsystems' interactions (or common origin) in the past."

"Of course I stand by the statement. Causality/locality implies that any entanglement - or any correlation - between objects in two places has to result from their contact or interaction in the intersection of their past light cones."

I want to work out the problem in detail for anyone interested to see under what conditions two particles can become entangled even though they never interacted.

Let me start with the usual Bell states:

\(|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle)+|11\rangle)\)
\(|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle)-|11\rangle)\)
\(|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle)+|10\rangle)\)
\(|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle)-|10\rangle)\)

and have two pairs of Bell particles (which do originate as Lubos demands from a common origin in the past). It does not matter which Bell states we start with and for the sake of the example let's pick the following 4 particles in the state below:

\(|1234\rangle = |\Psi^-\rangle_{12}|\Psi^-\rangle_{34}= \frac{1}{2}(|0101\rangle-|0110\rangle-|1001\rangle+|1010\rangle)\)

Now suppose that the first pair  \(|\Psi^-\rangle_{12}\) is split between Alice and Charlie, and the second pair \(|\Psi^-\rangle_{34}\) is split between Charlie and Bob such that Alice has particle 1, Charlie has particles 2 and 3, and Bob has particle 4. Particles 1 and 2 share a common origin, and the same is true for particles 3 and 4, but particles 1 and 4 NEVER interacted in the past.

Now Charlie does a projective  measurement on say \(\Phi^+\) on particles 2 and 3:

\(P_{\Phi^+} = \frac{1}{2}(|00\rangle+|11\rangle)(\langle00|+\langle11|)\)
\(P_{\Phi^+} = \frac{1}{2}(|00\rangle\langle00| + |00\rangle\langle11| + |11\rangle\langle00| + |11\rangle\langle11|)\)

This means that we apply the projector:

\(I_1 \otimes P_{\Phi^{+}_{23}}\otimes I_4\) on \(|1234\rangle \):

Ignoring the overall normalization factors for simplicity sake this means we need to compute:

\(|00\rangle_{23}\langle00| + |00\rangle_{23}\langle11| + |11\rangle_{23}\langle00| + |11\rangle_{23}\langle11|)\)
\((|0101\rangle-|0110\rangle-|1001\rangle+|1010\rangle)=\)

\(-|1001\rangle - |0000\rangle - |1111\rangle - |0110\rangle= \)

\(-(|11\rangle+|00\rangle)_{14}|00\rangle_{23} -  (|11\rangle+|00\rangle)_{14}|11\rangle_{23} = \)

\(-(|11\rangle+|00\rangle)_{14} (|11\rangle+|00\rangle)_{23} = - |\Phi^+\rangle_{14}|\Phi^+\rangle_{23}\)

and now particles 1 and 4 are entangled despite never interacting in the past.

So how does Lubos explain this?

"In quantum information, this is a part of the "LOCC" ("local operations and classical computation" do not create or increase entanglement) principle"

True but irrelevant. LOCC cannot increase entanglement overall, but now two particles which never interacted became entangled.

"Entanglement swapping is surely not a counterexample of LOCC, there can't be any counterexample. It's just swapping."

Again true but irrelevant.

"It's like the entanglement is riding on a train A and changes the trains to another train B that happens to meet A at some point."

This is fuzzy handwaving talk. It is true one can do correlation swapping in the classical world: suppose Alice has left and right gloves and she puts them in a bag. Bob does the same thing with his gloves and Charlie picks up a glove from Alice's bag and one from Bob's bag. Upon inspection of what he extracted he know how the gloves left in the two bags are correlated. However in the quantum case things are qualitatively different because of the active role of the observer and the fact that the result of measurement does not exist before measurement. I think I know where Lubos is coming from. I am speculating that he truly believes that quantum correlations are like Beltramann' socks, and if so his position makes perfect sense. However this is an unsustainable position and I plan to show why in subsequent posts. Lubos war on Bell is unsustainable as well. If you want to criticize Bell you have to do it on his genuine fault: the idea of beables. But this is a topic for another day, Let's continue:

"So the pair entangled afterwards is described differently after the trains are switched but the entanglement is preserved and links two places that change continuously and at most by the speed of light."

This is a crackpot statement. The introduction of the speed of light arguments illustrates a fundamental misunderstanding. Basically here Lubos attempts to come up with a handwaving argument of sliding the light cones of particles 1 and 4 until overlap for the purpose of generating correlation. Collapse in the classical and the quantum world is the result of information being revealed. There is no such thing as a propagation of collapse and/or correlation at the speed of light or slower. Any talk of propagation of collapse/correlation is nonsense.

I am pretty sure Lubos will counter this and  attempt to clean the handwaving but if he is precise he will have to pick between two unacceptable options: (1) introducing considerations of propagation of correlations or (2) explain the 1 and 4 particle entanglement similar with my gloves example above and contradicting the role of the observer. If I were to make a prediction he is going to pick option 2 and argue that quantum correlations are precisely like Beltramann' socks.   

"Why teleportation isn't a counterexample - or a source of nonlocality - was also discussed in detail in Susskind's recent arXiv:1604.02589."

Here I talk about entanglement swapping, not teleportation. True they are closely related, and true, entanglement is not created by LOCC. Again the contention is on this statement: "Causality/locality implies that any entanglement - or any correlation - between objects in two places has to result from their contact or interaction in the intersection of their past light cones"
which is proven false by the computation above.

It is always dangerous to make grandiose statements using "all", "any", etc.

I won't get into the ER=EPR argument, but at some point in the future I'll explain why it is irrelevant in the quantum foundation/interpretation area.

"Feel free to write another completely wrong blog post - you have already written dozens of those - this URL is one giant pseudointellectual dumping ground."

On quantum interpretation, both me and Lubos are in the Copenhagen camp and I (civilly and without burning bridges) disagree more with the Bohmian or the MWI positions than with Lubos. From 10,000 feet if you do not care about subtle points and excluding his vitriol Lubos is basically correct in his quantum mechanics intuition.

However there is a fundamental difference between us: I am (neo) Copenhagen because my work proved to me this is the correct point of view while Lubos is a self-appointed defender of quantum orthodoxy. Lubos shoots from the hip, is not aware of subtle points in quantum mechanics, and hides his ignorance by ad hominem attacks. I do not care about him being a jerk any more than I care about being criticized for not wearing pink slippers at work. I do care very much however about being right and if I am proven wrong I do acknowledge my mistakes (nobody is perfect). Like Trump, Lubos never acknowledges mistakes and doubles and triples down. And I just could not resist calling the emperor naked when this happens.  

Impressions from California

I just got back from a one week vacation in California, and although I wanted to complete last week post, I could not concentrate on physics, so I will postpone that post one more week and today I want to share my impressions from the trip.

I visited San Francisco, Lake Tahoe, and Yosemite National Park. I have been to San Francisco before but this time I had more time to explore the city and surroundings. Besides the regular attractions like Fisherman's Wharf, Ghirardelli square, the cable car rides, and the sea lions on pier 39, I went to see the giant redwoods in Muir forest. The view is nice if you can find a parking space: there were cars lined the side of the road for 4 miles next to the entrance. Also coming back to San Francisco I had to cross back on the Golden Gate bridge which just happens to be a toll road in this direction. An ominous sign scare you with a $500 fine for not paying the toll, but when you get to the toll booth there is nobody there.  At the toll plaza they take a picture of your license plate and they chase you after. When you google the toll information, you get to the official site which lets you pay the toll within 48 hours. However the idiotic site first asks you to enter the time you pass the toll and then it informs you that from that time forward up to the end of the day you were allowed one crossing. So if you missed the crossing time by one minute you wasted the payment and you are still a toll violator. Also on the negative side, the weather in San Francisco is incredibly cold. The sun is shining brightly causing quick sunburns, and yet the air temperature requires a decent coat in the middle of summer. 

Next stop was Lake Tahoe-a beautiful four seasons resort. There you can either spend time at the beaches enjoying an incredibly clean water which goes knee deep for about half a mile, 

Emerald Bay Lake Tahoe

 

or you can take a gondola up the mountain for zip lines, high ropes climbing, mountain coaster rides, and rock climbing activities.   

The pinnacle of the trip was the Yosemite National Park. This is a big place of breathtaking beauty choke full of tourists (about 50% were Europeans). The traffic and tourist density surpasses that of downtown Washington DC at the Smithsonian museums. The chipmunks are almost domesticated and they beg you for food every time you start eating anything. 

The main attraction is the valley from where you can hike to Vernal and Nevada Falls

and also if you apply for a permit some time in advanced to Half Dome peak (this is a 2 day hike)

The picture above is from the Glacier Point and you see the half dome and the two waterfalls on the center right.  The hike to the Nevada fall is a strenuous 6 hours hike (2000 feet elevation difference) and you need to carry 2 liter of water per person to do it. 

The places to stay inside the park require one year booking in advance, and the first hotel outside the park is about 30 miles away from the center of the valley. The first day was dedicated exploring the valley and hiking to the waterfalls, the second day I went to the glacier point to see the entire thing, and the last day I drove on the north road of the park to some beautiful views of the high mountains, lakes, and meadows. Everywhere there were long lines: 1 hour to enter the park, 90 minutes to take the free bus to glacier point, two hour drive to the north of the park which is in the far mountains in the picture above.  All in all I was deeply impressed by the beauty of the Yosemite which surpasses everything else I saw in the US.