Quantum teleportation

One of the most iconic quantum phenomena is quantum teleportation. But the reason why it is so iconic has nothing to do with the idea behind “beam me up, Scotty.”  In quantum teleportation, it is only the state of matter that is being transferred and not the matter itself. Usually, it is the state of light (a photon) that is being teleported. Quantum teleportation is iconic because it involves a mechanism that reveals a truly quantum nature.

How does it work? The state to be teleported is represented by photons that are specially prepared for the purpose. You can think of some light source that produces photons having specific properties that represent their state. We shall label one such photon as A. The resource that will mediate the teleportation process is a different bunch of photons representing an entangled state. This entangled state consists of a pair of entangle photons, which we label as B and C, respectively. To perform the process of teleportation, all we need to do is to make a joint measurement of photons A and B. It is the nature of this joint measurement that makes the process of quantum teleportation possible. The information that we obtain from this measurement tells us what transformation to perform on C to reproduce the state of A. Sometimes, we would not need to make any transformation. The state of C would already be that of A.

So, let’s look a little more carefully at the nature of the joint measurement. What do we mean by a joint measurement? To understand what it means, we need to discuss the state of photon A . There are many different possible states that this photon can have. All such states are collected into a set that we call a Hilbert space. Any of the states in this set can be represented as a superposition of a small set of states that we call a basis. One way to determine the state of a photon is the measure how much of each of these basis elements are required to make up the state of the photon. Such measurements are called projective measurements.

To understand joint measurements we just need to generalize our understanding of projective measurements a bit. What the measurement instrument in a teleportation experiment sees is not just A, but A and B together. The Hilbert space for the combination of the states of these two photons consists of all the combinations of all the states from their respective Hilbert spaces. One can produce a basis for the combined Hilbert space by combining the elements of the respective bases. There are different ways to do that, including some that would cause the elements of the combined basis to be entangled states. That is the key for quantum teleportation. One needs to make projective measurements of the combined state in a basis where the elements are themselves entangled.

Why would projective measurements in terms of an entangled basis cause teleportation? This mechanism is what makes teleportation an amazing process. It involves the multiple-reality nature of the quantum world. The entangled resource state can be interpreted in terms of such multiple realities. What joint measurements are doing to knit these multiple realities together with those presented by the input state A. But the latter is just one state (one reality), therefore, in the ideal case, only one of the realities of the resource state will survive the measurement process, the one where C has the same state as A. In a less ideal case, bits and pieces of A will be distributed over different realities. In that case, one can reconfigure the different realities with the aid of a unitary transformation on C, such that A becomes associated with just one reality in which C would then have the same state as A. The outcome of the joint measurement would tell us which unitary transformation to perform to achieve the necessary reconfiguration.

How does one make projective measurements in terms of an entangled basis? That is challenging, but people have identified at least two ways to do that. The first process and the one most often used is the Hong-Ou-Mandel effect. It is accomplished with the aid of a beamsplitter, causing a quantum interference effect. If two photons are observed simultaneously from the two output ports, then it signals the detection of a special entangled state called a Bell state, which implies a successful teleportation. The benefit of this method is that it does not require any unitary transformation of C.

Another way to perform a joint measurement is with the aid of the inverse of a process that would produce entangled states. In quantum optics, most entangled photon states are produces with the aid of a nonlinear optical process called parametric down-conversion. The inverse process is parametric up-conversion (also called sum frequency generation). While down-conversion converts a single incoming photon into two photons that are entangled to maintain energy and momentum conservation, the up-conversion process takes two incoming photons and combine them into one photon. A successful up-conversion implies a projection unto an entangled state to maintain energy and momentum conservation. Therefore, it can also be used for quantum teleportation. However, the process of up-conversion is very inefficient.

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A mechanism for quantum collapse

It is argued that the current impasse in fundamental physics is at least partially caused by the fact that there does not exist a credible understanding of the process of quantum collapse. This argument begs the question of those interpretations of quantum mechanics that incorporate quantum collapse. Therein lies a dilemma: interpretations of quantum mechanics are generally non-scientific – except for a few special cases – interpretations of quantum mechanics are not falsifiable. Therefore, even if we were able to come up with a mechanism for quantum collapse, it would not form part of our scientific understanding because it would not allow falsification.

However, there are interpretations of quantum mechanics that do not involve quantum collapse. They present a possible way out of this dilemma. Whatever reason one would have to introduce the notion of quantum collapse in the first place, must somehow be reproduced in those interpretations of quantum mechanics that do not explicitly contain it. In other words, the observable phenomena that suggest a collapse mechanism, must somehow be reproduced in interpretations without quantum collapse. The explanation of any mechanism that would reproduce such phenomena would therefore provide a kind of mechanism for quantum collapse without quantum collapse. In this case, the falsification takes on the lesser form of a retrodiction where the observed phenomena are explained in term of a successful theory. That successful theory is standard quantum mechanics. In other words, what we’ll do is to use standard quantum mechanics without introducing quantum collapse to explain those phenomena that seem to require quantum collapse.

If we do not want to introduce quantum collapse we inevitably employ an interpretation of quantum mechanics that does not involve quantum collapse. The one that we use is the so-called many worlds interpretation. However, we first need to review it to remove some misconceptions. In effect, we’ll use a modified version of the many worlds interpretation.

The many worlds interpretation is often associated with a multi-verse that is produced by the constant branching of a universe due to the quantum interactions that take place in that universe. This notion is misleading. Taking a good hard look at the quantum mechanics formalism, one should realize that it does not support the idea of a multi-verse produced by constant branching. Instead, there is just one universe, but with an infinite multiple of “realities” that are associated with the infinite number of elements in the basis of the Hilbert space of this universe. The number of these realities never change – there is no branching. Instead, the basis always consist of an infinite number of discrete elements. The only effect of quantum interactions is to change their relative probabilities. So, the many worlds (or the multiple realities) correspond to the terms in the superposition of all these basis elements forming the state of the universe. This state evolves in a unitary fashion that incorporates all the interactions that take place in the universe and thereby produces a constant variation over time in the coefficients of the terms in the superposition.

With this picture in mind, we can consider what it means to have quantum collapse, or to understand any physical process that seems to suggest quantum collapse. As a first example of such a physical process, we use a historically relevant example presented by Albert Einstein during the 5th Solvay conference, which was held in Brussels in 1927. [For a transcript of the discussions at the 5th Solvay conference, see G. Bacciagaluppi and A. Valentini, Quantum Theory at the Crossroads, Reconsidering the 1927 Solvay Conference, Cambridge University Press (2009); arXiv:0609184.] At this conference, which played a prominent role in the development of quantum mechanics and the Copenhagen interpretation, Einstein described a scenario where an electron propagates toward a screen on which it is registered as a single point of absorption on the screen. He then presented two possible ways to view the process that takes place, exemplifying the problem of maintaining energy conservation without introducing action at a distance. This example is an apt demonstration of one of the key problems with the notion of quantum collapse.

Attendants of the Fifth Solvay Conference
Attendants of the Fifth Solvay Conference

Before we deal with the understanding of this process in the context of multiple realities, we first need to remove an unfortunate tacit assumption that we find in the discussions of such experimental scenarios. It mentions a particle. What particle? Or perhaps one should ask what is meant by the term particle? Does is refer to a localized lump of matter (or a dimensionless point) traveling on a world line? Or is it a more abstract notion associated with a finite mass and a discrete charge, without localization? I suspect, that the former is implies in these discussions, because it would help to explain the localization of the observed absorption. However, what we see is a localized absorption and not a particle. The notion of something that is itself localized is not the only possible explanation for the localization of an absorption process. The latter can simply be the result of a localized mechanism for the absorption process. In the scenario, the screen is assumed to be a photographic material, presumably consisting of little silver crystals that can register the electron.

So, a slightly different picture from that which was put forward by Einstein emerges. The electron is now represented by a wave function (not a particle) that propagates toward the screen. The screen contains numerous little crystals that can register the electron. However, assuming that the wave function represents only one electron, one would find only one such absorption event (otherwise we’ll have the problem of violating conservation principles). In terms of multiple realities, many of these crystals could serve to perform the absorption process. Each reality would correspond to a different crystal receiving the electron.

But how does the situation within a particular reality manages the localize the electron wave function without causing action at a distance? Remember that the different realities are associated with the different basis elements in the Hilbert space. The basis of the Hilbert space is not unique. One can define infinitely many different basis, each of which is related to any other basis by a unitary transformation. In general, there is nothing special about any particular basis. In other words, the separation of the wave function of the universe into a superposition of different realities can be changed via unitary transformation into another set of realities, each of which is a combination of the previous set of realities. However, when it comes to a specific interaction process, such as the absorption of an electron by a specific crystal, then there is a special basis that would clarify the process. This basis corresponds to the measurement basis of the crystal.

To understand what this measurement basis is, one can determine what electron wave function would have been radiated by the crystal if the absorption process is inverted as the adjoint process. At the fundamental level, all processes are invertible. It is just that the probability for the required initial conditions to produce a specific inverted process may be vanishingly small. We can nevertheless assume that these required conditions exist so that it would produce the radiated wave function. The conjugate of the radiated wave function would describe the ideal measurement basis element for the absorption of an electron by this crystal. In effect, nature transforms the basis (the realities) into the measurement basis for the absorption of electrons by the crystals. All the different realities now correspond to different basis elements, each of which is associated by the absorption of the electron by a different crystal.

Obviously all these basis elements are localized at the their associated crystals. This localization is accomplished purely through a unitary transformation without any funny action at a distance. The way that the unitary transformation accomplishes this localization is through constructive and destructive interference among the elements of any other basis that may initially have represented the multiple realities.

The different realities in the localized measurement basis have different coefficients associated with them in the superposition. Some of these coefficients would be larger than others, indicating which crystal is most likely to absorb the electron. It may be that one specific reality has a coefficient that completely dominates. In that case, although there are multiple realities, one specific reality would dominate. This dominant reality may be the one that we perceive as the single reality in our experience.

Hopefully, this understanding gives a feasible picture of the process whereby quantum collapse seems to take place. More can be said about this topic, but since this discussion has already been rather long, I’ll postpone further discussions for another day.


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Testable proposals for the measurement problem

In a previous post, I made the statement: “Currently, there are no known experimental conditions that can distinguish between different interpretations of quantum mechanics.” Well, that is not exactly true. Perhaps one can argue that no experiment has yet been performed that conclusively ruled out or confirmed any of the interpretations of quantum mechanics. But, the fact is that recently, there has been some experimentally testable proposals. Still, I’m not holding my breath.

Recently, seeing one such proposal, I remembered that I also knew about another testable proposal made by Lajos Diósi and Roger Penrose. The reason I forgot about that is probably because it seems to have some serious problems. At some point, during a conversation I had with Lajos, I told him I have a stupid question to ask him: does quantum collapse travel at the speed of light? His response was: that is not a stupid question. So, then I concluded that it is not something that any of the existing collapse models can handle correctly. In fact, I don’t think any such model will ever be able to handle it in a satisfactory manner.

Thinking back to those discussions and the other bits and pieces I’ve read about the measurement problem, I tend to reconfirm my conviction that the simplest interpretation of quantum mechanics (and therefore the one most likely to be correct) is the so-called Many Worlds interpretation of quantum mechanics. However, the more I think about it, the more I believe that “many worlds” is a misnomer. It is not about many worlds or many universes that are constantly branching off to become disjoint universes.

Perhaps one can instead call it the “multiple reality” interpretation. But how would multiple realities be different from “many worlds”? That fact is that these realities are not disjoint, but form part of the unitary whole of a single universe. These realities can be combined into arbitrary superpositions. What more, these realities are not branching of to produce more realities. The number of realities remains the same for all time. (There are actually an infinite number of them, but the cardinality of the set remains the same.) The interactions merely change the relative complex probability amplitudes of all the realities. 

Anyway, just thought I should clear this up. I don’t see myself ever writing publication on this topic.

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The state of physics

One of the quirky things about me is that I don’t believe things I don’t understand. As a result of that, I’ve had a long turbulent relationship with the notion of black holes. See the thing is, for the longest time, I couldn’t understand how an event horizon can form if the time becomes frozen when the infalling matter approaches the point where the event horizon should form.

While I was grappling with this existential aspect of black holes, the rest of the world happily proceeded to invent wormholes, Hawking radiation, singularities, and eventually the information paradox. Together with event horizons, none of these ideas have entered the realm of establish scientific fact, which requires observational confirmation.

Eventually, I read somewhere that the reason an event horizon can form even though the time becomes frozen is because the location for the event horizon with and without this additional matter implies that the matter would past the point where a new event horizon would form in finite time. So, now I understand it and I believe that event horizons can form. But we are not done yet. What about the interior beyond the event horizon? It is still frozen in time. Where does the singularity come from? I still don’t believe that part, perhaps because I still don’t fully understand it.

In all this, the importance of the scientific method should be emphasized. Even if I don’t understand something, I would believe it if it has been observed. While event horizons may be difficult to observe directly, the singularity inside the black hole is completely impossible to observe. For that reason, it can never be part of our scientific understanding.

This year, the Nobel committee announced that the Nobel prize is award for work on black holes. Half of it goes to two people that inferred the existence of a massive black hole at the centre of the milky way galaxy based on the orbits of stars close to the centre. This work is based on scientific observation and therefore satisfies the requirements imposed by the scientific method.

The other half of the Nobel prize is awarded to Sir Roger Penrose “for the discovery that black hole formation is a robust prediction of the general theory of relativity.” If I understand correctly, the award is based on the Penrose–Hawking singularity theorems. (Hawking did not share the Nobel prize because he passed away.) So what is meant by “a robust prediction” here?

Sir Roger Penrose

Sir Roger Penrose is a formidable person. During his lifetime, he has produced a remarkable collection of ideas that range over diverse fields. The originality and complexity of these ideas give evidence to Penrose’s uniquely creative intellect. However, these ideas are of a mathematical nature and they show very clearly that Penrose is primarily a mathematician. Many of these ideas have never been confirmed by scientific observations. This lack of scientific confirmation includes the work on singularities in black holes for obvious reasons explained above.

It now brings me to the question, why would the Nobel committee decide to award a prize for “a robust prediction,” instead of something that has been confirmed in a scientific manner? The answer is probably related to the current state of physics. If we look at the work that was awarded recent Nobel prizes in physics, one can see that there must a problem. The problem is that progress in fundamental physics is slowing down or has come to a complete stop. There simply is nothing else to be awarded a physics Nobel prize anymore.

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The problem with the measurement problem

It is not really a topic I want to discuss. In fact, I don’t think it is worthy of including under my Demystifying Quantum Mechanics series. However, since even physicists don’t seem to get it, it is necessary to clarify a few things.

So the argument seems to go that even of one were to consider a completely mixed quantum states with equal probabilities for different outcomes then a measure would convert this mixed state into one with only one outcome and zero for all other outcomes. This transformation is then interpreted as a quantum collapse and the fact that this process is not understood is called the measurement problem.

The problem with this interpretation of the situation is just that: it is an interpretation. So it falls under the general topic of interpretations of quantum mechanics. Currently, there are no known experimental conditions that can distinguish between different interpretations of quantum mechanics. As such it is not physics, because it is not science. It falls under philosophy. As a result, it would not be possible to solve the so-called measurement problem.

Just in case you are wondering whether this measurement scenario can be interpreted in any other way that does not involve collapse, the answer is yes. The obvious alternative is the Many World interpretations. In terms of that interpretation the mixed quantum state describes the different probabilities for all the different world in which measurement are to be performed. If one would restrict the quantum state to any one of these worlds (or realities) then it would have 100% probability for a specific outcome even before the measurement is performed. Hence, not collapse and no measurement problem.

So, yes indeed, the measurement problem is a pseudo-problem. It is not one that can (or need to be) solved in physics.

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