Is the Decoherent Histories Approach Consistent?

One particular approach of interpreting quantum mechanics is Decoherent Histories. All major non-Copenhagen approaches have serious issues: 

- MWI has the issue of the very meaning of probability and without a non-circular derivation (impossible in my opinion) of Born rule does not qualify for anything but a "work in progress" status.

-GRW-type theories make different predictions than quantum mechanics which soon will be confirmed or rejected by ongoing experiments. (My bet is on rejection since later GRW versions tuned their free parameters to avoid collision with known experimental facts instead of making a falsifiable prediction) 

-Bohmian approach has issues with "surreal trajectories" which invalidates their only hard ontic claim: the position of the particle.

Now onto Decoherent Histories. I did not closely follow this approach and I cannot state for sure if there are genuine issues here, but I can present the debate. On one hand, Robert Griffiths states: 

"What is different is that by employing suitable families of histories one can show that measurement actually measure something that is there, rather than producing a mysterious collapse of the wave function"

On the other hand he states:

"Any description of the properties of an isolated physical system must consists of propositions belonging together to a common consistent logic" - in other words he introduces contextuality.

Critics of decoherent (or consistent) histories use examples which are locally consistent but globally inconsistent to criticize the interpretation.

Here is an example by Goldstein (other examples are known). The example can be found in Bricmont's recent book: Making Sense of Quantum Mechanics on page 231. Consider two particles and two basis for a two-dimensional spin base \((|e_1\rangle\, |e_2\rangle), (|f_1\rangle, |f_2\rangle))\) and consider the following state:

\(|\Psi\rangle = a |e_1\rangle|f_2\rangle + a|e_2\rangle|f_1\rangle - b |e_1\rangle|f_1\rangle\)

Then consider four measurements A, B, C, D corresponding to projectors on four vectors, respectively: \(|h\rangle, |g\rangle, |e_2\rangle, |f_2\rangle\) where:

\(|g\rangle = c|e_1\rangle + d|e_2\rangle\)

\(|h\rangle = c|f_1\rangle + d|f_2\rangle\)

Then we have the following properties:

(1) A and C can be measured simultaneously, and if A=1 then C=1 

(2) B and D can be measured simultaneously, and if B=1 then D=1

(3) C and D can be measured simultaneously, but we never get both C and D = 1

(4) A and B can be measured simultaneously, and sometimes we get both A and B = 1

However all 4 statements cannot be true at the same time: when A=B=1 as in (4) then by (1) and (2) C=D=1 and this contradicts (3).

So what is going on here? The mathematical formalism of decoherent histories is correct as they predict nothing different than standard quantum mechanics. The interpretation assigns probabilities to events weather we observe them or not, but does it only after taking into account contextuality. Is this a mortal sin of the approach? Nature is contextual and I don't get the point of the criticism. The interpretation would be incorrect if it does not take into account contextuality. Again, I am not an expert of this approach and I cannot offer a definite conclusion, but to state my bias I like the approach and my gut feeling is that the criticism is without merit.

 

PS: I'll be going on vacation soon and my next post will be delayed: I will skip a week.

Measurement = collapse + irreversibility

I got a lot of feedback from last 2 posts and I need to continue the discussion. Even Lubos with his closed mind unable to comprehend anything different than the textbooks from 50 years ago and his combative style said something worth discussing.

But first let me thank Cristi for the picture below which will help clarify what I am trying to state. Let me quickly explain it: the interferometer arms are like the two sides of Einstein's box and once the particle was launched -for the duration of the flight- you can close the input and output of the interferometer, open the exit just in time and still have the interference. So this seems to contradict my prediction. But does it?

This time I do need to dig a bit deeper into the mathematical formalism. First, the role of the observer is paramount: no observer=no measurement. Second, the observer is described by quantum mechanics as well: there is the wavefunction of the quantum system, and there is the wavefunction of the observer. Now here is the new part: while we can combine the quantum system and the observer by tensor product and do the usual discussion of how unitary evolution does not predict an unique outcome, we need to combine the quantum system and the observer using the Cartesian product. This is something new, not present in standard quantum mechanics textbooks. However this follows naturally from the category theory derivation of quantum mechanics from first principles. There are equivalent Cartesian products corresponding to potential measurement outcomes:

\((|collapsed ~1 \rangle, | observer~ see~1\rangle ) \equiv ( |collapsed~2\rangle, | observer~see~2\rangle)\) 

This equivalence exists in a precise mathematical sense and respects the three properties of the equivalence relationship: reflexivity, symmetry, and transitivity. Break the Cartesian pair equivalence by any mechanism possible and you get the collapse of the wavefunction. 

Closing the interferometer, or cutting Einstein's box in half kills the equivalence and the wavefunction collapses. However while the particle is still in flight the process is reversible!!! Open the interferometer's exits in time and you restore the equivalence, undo the collapse still get the interference (Han Solo kills the stormtrooper 100% of the time). 

However there is a way to make the collapse permanent: just wait enough time with the ends closed such that the energy-time uncertainty relation allows you to reduce the energy uncertainty to the point that you can detect the particle inside by weighing the half-boxes or the arms of the interferometer. Suppose the ends of the interferometer is made out of perfect mirrors. If you wait long enough (even though you are not physically weighing anything) and then reopen the exits will result in loss of interference: this is my prediction.

But what happens if you only wait a little bit of time and you are in between full interference and no interference? You get a weak measurement.

Now let me discuss Lubos objection and then come back to the nonlocality point Bricmont was making.

First Lubos was stating: "If you place some objects (a wall) at places where a particle is certain not to be located, the effect on the particle's future behavior is obviously non-existent". The objection is vacuous. Obviously I don't disagree with the statement, but his entire line of argument is incorrect because collapse is not a dynamic process. If collapse would have had a dynamic origin then we would have had a unitary explanation for it we would have had to talk about the "propagation" of collapse. What the Cartesian pair mathematical framework does is first getting the rid of the consciousness factor, and second clarifying the precise mathematical framework of how the observer should be treated within the formalism. Contextuality is paramount in quantum mechanics and cutting the box changes the context of the experiment.

Now onto Bricmont argument. Andrei stated in his comments: " I still do not see the relevance of all this in regards to the locality dilemma.". It has deep relevance as I will explain. And by the way, the rest of Andrei's comments were simply not worth answering-nothing personal: I don't have the luxury of enough free time to answer each and every comment.

Bricmont's point on Einstein's boxes was this: "either there is action at a distance in nature (opening B1 changes the situation at B2), or the particle was in B2 all along and quantum mechanics is incomplete "

Let's discuss the two options:

1. opening B1 changes the situation at B2
2. or the particle was in B2 all along and quantum mechanics is incomplete

Option one is clearly not the case. Wait long enough and the interference will no longer happen. At that point the particle IS in either B1 or B2 and shipping one box far away changes nothing. But how about option 2? Is quantum mechanics incomplete? Bohmian supporters think so because they augment the wavefunction with a hidden variable: the particle's initial position. Do we actually need this initial condition to make predictions? Not at all. Last thing to consider: was the particle in say B2 all along? If yes, there is no interference because the which way information. What about weak measurements? This is a case of even more examples of "surrealistic trajectories": combine two interferometers and you can obtain disjoint paths!!! The only thing which makes sense is that the particle does not have a well defined trajectory.

My question to Bohmian interpretation supporters is as follows: In the above picture close the arms long enough. What happens to the quantum potential? Does it dissipate? If yes how? If no, do you always get interference after opening the stormtrooper end regardless of the wait time? 

Finally back to measurement. There is no (strong) measurement without collapse. Collapse happens when a particular equivalence relationship no longer holds. Mathematically it can be proven that the wavefunction is projected on a subspace of the original Hilbert space. Moreover uniqueness can be proven as well: that this is the only mathematically valid mechanism in which projection can occur. Interaction between the quantum system and the measurement device can break the equivalence, but changing the experimental context can achieve the same thing as well. A measurement does not happen however until irreversibility occurs: there could be amplification effects, or as above enough time passes such that the energy uncertainty is low enough and the "which way" information becomes available regardless if we seek this information or not.

A measurement can be more than an observer learning the value of a physical observable

Last post created quite a stir and I want to expand on the ideas from it. This will also help me get out of an somewhat embarrassing situation. For months now Lubos Motl tried to get revenge on his bruised ego after a well deserved April Fool's joke and became a pest at this blog. The problem is that although I have yet to see a physics post at his blog that is 100% correct, we share roughly the same intuition about quantum mechanics: I agree more much more with his position than say with the Bohmian, GRW, or MWI approaches. The differences are on the finer points and I found his in depth knowledge rusty and outdated. For his purpose: to discredit the opposite points of view at all costs this is enough, but it does not work if you are a genuine seeker of truth.

So last time he commented here: "A measurement is a process when an observer actually learns the value of a physical observable" which from 10,000 feet is enough. However this is not precise enough, and now I do have a fundamental disagreement with Lubos which hopefully will put enough distance between him and me. 

More important than my little feud with Lubos, I can now propose an experiment which will either validate or reject my proposed solution to the measurement problem. I do have a novel proposal on how to solve the measurement problem and this is distinct from all other approaches. I was searching for months for a case of a novel experimental prediction, but when I applied it to many problems I was getting the same predictions as standard quantum mechanics. Here is however a case where my predictions are distinct. I will not work out the math and instead let me simply present the experiment and make my experimental claim.

Have a box with a single particle inside. The box has a middle separator and also two slits A and B which can be placed next to a two-slit screen. We can then carry two kinds of experiments:

1. open the two slits A and B without dropping the separator allowing the particle to escape the box and hit a detector screen after the two-slit screen.
2. drop the separator and then open the two slits A and B allowing the particle to escape the box and hit a detector screen after the two-slit screen.

Next we repeat the experiments 1 or 2 enough times to see the pattern emerge on the final screen. Which pattern would we observe?

For experiment 1 we already know the answer: if we repeat it many times we obtain the interference pattern, but what will we get in the case of experiment number 2?

If dropping the separator constitutes a measurement, the wavefunction would collapse and we get two spots on the detector screen corresponding to two single slit experiments. If however dropping the separator does not constitute a measurement, then we would get the same interference pattern as in experiment 1.

My prediction (distinct from textbook quantum mechanics) is that there will be no interference pattern.

Are Einstein's Boxes an argument for nonlocality?

(an experimental proposal)

Today I want to discuss a topic from an excellent book by Jean Bricmont: Making Sense of Quantum Mechanics which presents the best arguments for the Bohmian interpretation. Although I do not agree with this approach I appreciate the clarity of the arguments and I want to present my counter argument.

On page 112 there is the following statement: "... the conclusion of his [Bell] argument, combined with the EPR argument is rather that there are nonlocal physical effects (and not just correlations between distant events) in Nature". 

To simplify the argument to its bare essentials, a thought experiment is presented in section 4.2: Einstein's boxes. Here is how the argument goes: start with a box B and a particle in the box, then cut the box into two half-boxes B1 and B2. If the original state is \(|B\rangle\), after cutting the state it becomes:

\(\frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\) 

Then the two halves are spatially separated and one box is opened. Of course the expected thing happens: the particle is always found in one of the half-boxes. Now suppose we find the particle in B2. Here is the dilemma: either there is action at a distance in nature (opening B1 changes the situation at B2), or the particle was in B2 all along and quantum mechanics is incomplete because \(\frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\) does not describe what is going on. My take on this is that the dilemma is incorrect. Splitting the box amounts to a measurement regardless if you look inside the boxes or not and the particle will be in either B1 or B2.  

Here is an experimental proposal to prove that after cutting the box the state is not \(\frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\):

split the box and connect the two halves to two arms of a Mach-Zehnder interferometer (bypassing the first beam splitter). Do you get interference or not? I say you will not get any interference because by weighing the boxes before releasing the particle inside the interferometer gives you the which way information.

If we do not physically split the box, then indeed \(|B\rangle = \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\), but if we do physically split it \(|B\rangle \neq \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\). There is a hidden assumption in Einstein's boxes argument: realism which demands non-contextuality. Nature and quantum mechanics is contextual: when we do introduce the divider the experimental context changes. 

Bohmian's supporters will argue that always \(|B\rangle = \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\). There is a simple way to convince me I am wrong: do the experiment above and show you can tune the M-Z interferometer in such a way that there is destructive interference preventing the particle to exit at one detector.

Gleason's Theorem

It feels good to be back to physics, and as a side note going forward I will do the weekly posts on Sunday. Today I want to talk about Gleason's theorem. But what is Gleason's theorem?

If you want to assign a non-negative real valued function \(p(v)\) to every vector v of a Hilbert space H of dimension greater than two, then subject to some natural conditions the only possible choice is \(p(v) = {|\langle v|w \rangle |}^{2}\) for all vectors v and an arbitrary but fixed vector w.

Therefore there is no alternative in quantum mechanics to compute the average value of an observable A the standard way by using:

\(\langle A \rangle = Tr (\rho A)\)

where \(\rho\) is the density matrix which depends only on the preparation process. 

Gleason's theorem is rather abstract and we need to unpack its physical intuition and the mathematical gist of the argument. Physically, Gleason's theorem comes from three axioms:

• Projectors are interpreted as quantum propositions
• Compatible experiments correspond to commuting projectors
• KEY REQUIREMENT: For any two orthogonal projectors P, Q, the sum of their expectation values is the expectation value of P+Q: \(\langle P \rangle + \langle Q\rangle = \langle P+Q\rangle\)

In an earlier post I showed how violating the last axiom (which is the nontrivial one), in the case of spin one particles, can be used to send signals faster than the speed of light and violate causality. But how does Gleason arrives at his result?

Let's return at the original problem: to obtain a real non-negative function p. Now add the key requirement and demand that for any complete orthonormal basis \(e_m\) we have:

\(\sum_m p(e_m) = 1\)

For example in two dimensions on a unit circle we must have:

\(p (\theta) + p(\theta + \pi/2) = 1\)

which constrain the Fourier expansion of \(p (\theta)\) such that only components 2, 6, 10, etc can be non zero. In three dimensions the constraints are much more severe and this involves rotations under SO(3) and spherical harmonics. I'll skip the tedious math, but it is not terribly difficult to show that the only allowed spherical harmonics must be of order 0 and 2 which yields: \(p(v) = {|\langle v|w \rangle |}^{2}\).

The real math heavy lifting is on dimensions larger than three and to prove it Gleason first generalizes  \(\sum_m p(e_m) = 1\) to \(\sum_m f(e_m) = k\) where k is any positive value. He names this "f" a "frame function". Then he proceeds to show that dimensions larger than three do not add anything new.

If you are satisfied with the Hilbert spaces of dimension 3, the proof of the theorem is not above undergrad level, and I hope it is clear what the argument is. But what about Many Worlds Interpretation? Can we use Gleason's theorem there to prove Born rule? Nope. The very notion of probabilities is undefined in MWI, and I am yet to see a non-circular derivation of Born rule in MWI. I contend it can't be done because it is a mathematical impossibility and I blogged about it in the past. 

Einstein's reality criterion

Sorry for the delay, this week I want to have a short post continuing to showcase the results of late Asher Peres which unfortunately are not well known and this is a shame.

In the famous EPR paper, Einstein introduced a reality criterion:

"If, without in any way disturbing a system, we can predict with certainty ... the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity"

Now it is generally accepted that the difference between classical and quantum mechanics is noncomutativity. While there are some subtle points to be made about this assertion (by the mathematical community), from the physics point of view the distinction is rock solid and we can build in confidence upon it.

Now consider again the  EPR-B experiment with its singlet state. Suppose the x and y components of the spin exists independent of measurement and let's call the measurement values: \(m_{1x}, m_{1y}, m_{2x}, m_{2y} \). From experimental results we know:

\(m_{1x} = - m_{2x}\)
and
\(m_{1y} = - m_{2y}\)

And now for the singlet state \(\psi\) let's compute:

\((\sigma_{1x} \sigma_{2y} + \sigma_{1y} \sigma_{2x})\psi\)

which turns out to be zero. The beauty of this is that \(\sigma_{1x} \sigma_{2y} \) commutes with \(\sigma_{1y} \sigma_{2x}\) and by Einstein's reality criterion extended to commuting operators it implies that \(m_{1x} m_{2y} = - m_{1y} m_{2x}\) which contradicts \(m_{1x} = - m_{2x}\) and \(m_{1y} = - m_{2y}\)

This contradiction is in the same vein as the GHZ result, but it is not well known. The catch of this result is that measuring S1xS2y cannot be done at the same time with a measurement of  S1yS2x and so we are reasoning counterfactually. However, counterfactual reasoning is allowed in a noncontextual setting (in classical mechanics and in quantum mechanics for commuting operators) and the result is valid. 

von Neumann  and Gleason vs. Bell

Returning to physics topics today I want to talk about an important contention point between von Neuman and Gleason on one had, and Bell on the other. I had a series of posts about Bell in which I discussed his major achievement. However I do not subscribe to his ontic point of view and today I will attempt to explain why and perhaps persuade the reader with what what I consider to be a solid argument. 

Before Bell wrote his famous paper he had another one in which he criticized von Neumann, Jauch and Piron, and Gleason. The key of the criticism was that additivity of orthogonal projection operators not necessarily implies the additivity of expectation values:

\(\langle P_u + P_v \rangle = \langle P_{u}\rangle + \langle P_{v}\rangle \)

The actual technical requirements in von Neumann and Gleason case were slightly different, but they can be reduced to the statement above and more importantly this requirement is the nontrivial one in a particular proof of Gleason's theorem. 

Andrew Gleason

To Bell, additivity of expectation values is a non-natural requirement because he was able to construct hidden variable models violating this requirement. And this was the basis for his criticism of von Neumann and his theorem of the impossibility of hidden variables. But is this additivity requirement unnatural? What can happen when it is violated? I will show that violation on additivity of expectation values can allow instantaneous communication at a distance.

The experimental setting is simple and involves spin 1 particles. The example which I will present is given in late Asher Peres book: Quantum Theory: Concepts and Methods at page 191. (This book is one of my main sources of inspiration for how we should understand and interpret quantum mechanics. )

The mathematical identity we need is:

\(J_{z}^{2} = {(J_{x}^{2} - J_{y}^{2})}^2\)

and the experiment is as follows: a beam of spin 1 particles is sent through a beam splitter which sends to the left particles of eigenvalue zero for \(J_{z}^{2}\) and to the right particles of eigenvalue one for \(J_{z}^{2}\).

Now a lab on the right decides to measure either if \(J_z = 1\) or if \(J_{x}^{2} - J_{y}^{2} = 1\)

For the laboratory on the right let's call the projectors in the first case \(P_u\) and \(P_v\) and in the second case \(P_x\) and \(P_y\)

For the lab on the left let's call the projectors in the first case \(P_{w1}\) and in the second case\(P_{w2}\). 

Because of the mathematical identity: \(P_u + P_v = P_x +P_y\) the issues becomes: should the expectation value requirement hold as well?

\(\langle P_{u}\rangle + \langle P_{v}\rangle  = \langle P_{x}\rangle + \langle P_{y}\rangle \)

For the punch line we have the following identities:

\(\langle P_{w1}\rangle = 1 - \langle P_{u}\rangle - \langle P_{v}\rangle\) 

and

\(\langle P_{w2}\rangle = 1 - \langle P_{x}\rangle - \langle P_{y}\rangle\) 

and as such if the additivity requirement is violated we have:

\(\langle P_{w1}\rangle \neq \langle P_{w2}\rangle\)

Therefore regardless of the actual spatial separation, the lab on the left can figure out which experiment the lab on the right decided to perform!!!

With this experimental setup, if additivity of expectation values is false, you can even violate causality!!!

Back to Bell: just because von Neumann and Gleason did not provide a justification for their requirements, this does not invalidate their arguments. The justification was found at a later time. 

But what about the Bohmian interpretation of quantum mechanics? Although there are superluminal speeds in the theory, superluminal signaling is not possible in it. This is because Bohmian interpretation respects Born rule which is a consequence of Gleason't theorem and it respects the additivity of  expectation values as well. Bohmian interpretation suffers from other issues however.