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.

This image has an empty alt attribute; its file name is 1C7DB1746CFC72286DF097344AF23BD2.png

Entanglement

Demystifying quantum mechanics VIII

Everything is no more or less in place to discuss one of the most enigmatic phenomena found in quantum mechanics: entanglement. It is sometimes called the quintessential property of quantum mechanics. 

We have discussed the fact that quantum mechanics introduces the concept of discrete entities that carry full sets of degrees of freedom, and which I called partites. Then we learned about the paradox introduced by Einstein, Podolski and Rosen (EPR) and how it led to the understanding that nature does not have a unique reality. Although it also allows that interactions could be nonlocal, we saw that such non-locality is not in agreement with our understanding of special relativity. The final ingredient that we need to explain quantum entanglement is the concept of a superposition. We can deal with that here.

The term superposition is a fancy way of saying that we are adding up things. Superpositions are also found in classical optics. There, one can observe interference effects when two waves are superimposed (added on top of each other at the same location). What makes the situation in quantum mechanics different is that the things that are added up in a quantum superposition can consist of multiple partites (multiple combinations of discrete entities) and these partites (discrete entities) do not have to be at the same location. Since each entity carries unique properties, as described in terms of the full set of degrees of freedom, the different terms in the quantum superposition gives complete descriptions of the state in terms of the set of discrete entities that they contain. 

Each the terms in the superposition can now be seen as a unique reality. The fact there are more than one term in the superposition, implies that there are multiple realities, just like the EPR paradox showed us. One can use the many-world interpretation to try to understand what this means.

Entangled entities
Artist impression of entangled entities

There are now different effects that these superpositions can produce. In some cases one can factorize the superposition so that it becomes the product of separate superpositions for each of the individual partites. In such a case one would call the state described by the superposition as being separable. If such a state cannot be factorized in this way, the state is said to be entangled.

What is the effect of a state being entangled? It implies that there are quantum correlations among the different entities in the terms. These correlations will show up when we make measurements of the properties of the partites. Due to the superposition, a measurement of just one of these partites will give us a range of possible results depending on which term in the superposition ends up in our measurement. On the other hand, if we measure the properties of two or more of the partites, we find that their properties are always correlated. This correlation only shows up when the state is entangled.

Some people think that one can use this correlation the communicate instantaneously between such partites if they are placed at different locations that are far apart. However, as we explained before, such instantaneous communication is not possible.

This discussion may be rather abstract. So, let try to make it a bit simpler with a simple example. Say that we form a superposition where each term contains two partites (two discrete entities). In our superposition, we only have two terms and the properties of the partities can be one of only two configurations. So we can represent our state as A(1) B(2) + A(2) B(1). Here A and B represent the identities of the partites and (1) and (2) represent their properties. When I only measure A, I will get either (1) or (2) with equal probability. However, when I measure both A and B, I will either get (1) for A and (2) for B or (2) for A and (1) for B. In other words, in each set of measurements, the two partites will have the opposite properties, and this result is obtained regardless of how far apart these partites are located.

The phenomenon of quantum entanglement has been observed experimental many times. Even though it is counterintuitive, it is a fact of nature. So, this is just one of those things that we need to accept. At least, we can understand it in terms of all the concepts that we have learned so far. Therefore, it does not need to be mysterious

This image has an empty alt attribute; its file name is 1C7DB1746CFC72286DF097344AF23BD2.png

The non-locality mess

Demystifying quantum mechanics VII

The title comes from a section heading in a paper a recently saw. Due to a serious issue with some confusion that exists in the literature, the author advocates that the physics community abandon the notion of non-locality in favor of correlations that can be observed experimentally.

The problem with the community is that it consist of a very diverse collection of people with diverse perspectives. So, the chances are small that they’ll abandon the notion of non-locality. However, it is not unreasonable that one may be able to clarify the confusion so that the community will al least know what they are talking about.

The problem comes in because people mean different things when they use the term “non-local.” The traditional meaning is associated with “spooky action at a distance.” In other words, it refers to a non-local interaction. This meaning is best understood in the context of special relativity.

Consider two separate events, which one can think of as points in space at certain moments in time. These events are separated in different ways. Let’s call them A and B. If we start from A and can reach B by traveling at a speed smaller than the speed of light, then we say that these events have a time-like separation. In such a case, B could be caused by A. The effect caused by A would then have travelled to B where a local interaction has caused it. If we need to travel at the speed of light to reach B, starting from A, the separation is called light-like and then B could only be caused by A as a result of something traveling at the speed of light. If the separation is such that we cannot reach B from A even if we travel at the speed of light, then we call the separation space-like. In such a case B could not have been caused by A unless there are some non-local interactions possible. There is a general consensus that non-local interactions are not possible. One of the problems that such interactions would have is that one cannot say which event happened first when they have a space-like separation. Simply by changing the reference frame, one can chance the order in which they happen.

As a result of this understanding, the notion of non-local interactions is not considered to be part of the physical universe we live in. That is why some people feel that we should not even mention “non-locality” in polite conversation.

However, there is a different meaning that is sometimes attached to the term “non-locality.” To understand this, we need three events: A, B and C. In this case, A happens first. Furthermore, A and B have a time-like separation and A and C also have a time-like separation, but B and C have a space-like separation. As a result, A can cause both B and C, but B and C cannot be caused by each other.

Imagine now that B and C represent measurements. It would correspond to what one may call “simultaneous” measurements, keeping in mind that such a description depends on the reference frame. Imagine now that we observe a correlation in these measurements. Without thinking about this carefully, a person may erroneously conclude that one event must have caused the other event, which would imply a non-local interaction. However, based on the existence of event A, we know that the cause for the correlation is not due to a non-local interaction, but rather because they have a common cause. In this context, the term “non-local” simply refers to the fact that the observations correspond to events with a space-like separation. It does not have anything to do with an interaction.

When it comes to an understanding of entanglement, which we’ll address later in more detail, it is important to understand the difference between these two notions of non-locality. Under no circumstances are the correlations that one would observe between measurements at space-like separated events B and C to be interpreted as an indication of non-local interactions. The preparation of an entangled state always require local interactions at A so that the correlated observations of such a state at B and C have A as their common cause. The nature of the correlations would tell us whether these correlations are associated with a classical state or a quantum state.

This image has an empty alt attribute; its file name is 1C7DB1746CFC72286DF097344AF23BD2.png

Einstein, Podolski, Rosen

Demystifying quantum mechanics VI

When one says that one wants to demystify quantum mechanics, then it may create the false impression that there is nothing strange about quantum mechanics. Well, that would be a misleading notion. Quantum mechanics does have a counterintuitive aspect (perhaps even more than one). However, that does not mean that quantum mechanics need to be mysterious. We can still understand this aspect, and accept its counterintuitive aspect as part of nature, even though we don’t experience it in everyday life.

The counterintuitive aspect of quantum mechanics is perhaps best revealed by the phenomenon of quantum entanglement. But before I discuss quantum entanglement, it may be helpful to discuss some of the historical development of this concept. Therefore, I’ll focus on an apparent paradox that Einstein, Podolski and Rosen (EPR) presented.

They proposed a simple experiment to challenge the idea that one cannot measure position and momentum of a particle with arbitrary accuracy, due to the Heisenberg uncertainty. In the experiment, an unstable particle would be allowed to decay into two particles. Then, one would measure the momentum of one of the particles and the position of the other particle. Due to the conservation momentum, one can then relate the momentum of the one particle to that of the other. The idea is now that one should be able to make the respective measurements as accurately as possible so that the combined information would then give one the position and momentum of one particle more accurately than what Heisenberg uncertainty should allow.

Previously, I explained that the Heisenberg uncertainty principle has a perfectly understandable foundation, which has nothing to do with quantum mechanics apart from the de Broglie relationship, which links momentum to the wave number. However, what the EPR trio revealed in their hypothetical experiment is a concept which, at the time, was quite shocking, even for those people that thought they understood quantum mechanics. This concept eventually led to the notion of quantum entanglement. But, I’m getting ahead of myself.

John Bell

The next development came from John Bell, who also did not quite buy into all this quantum mechanics. So, to try and understand what would happen in the EPR experiment, he made a derivation of the statistics that one can expect to observe in such an experiment. The result was an inequality, which shows that, under some apparently innocuous assumptions, the measurement results when combine in a particular way must always give a value smaller than a certain maximum value. These “innocuous” assumptions were: (a) that there is a unique reality, (b) that there are no nonlocal interactions (“spooky action at a distance”) .

It took a while before an actual experiment that tested the EPR paradox could be perform. However, eventually such experiments were performed, notably by Alain Aspect in 1982. He used polarization of light instead of position and momentum, but the same principle applies. And guess what? When he combined the measurement result as proposed for the Bell inequality, he found that it violated the Bell inequality!

So, what does this imply? It means that at least one of the assumption made by Bell must be wrong. Either, the physical universe does not have a unique reality, or there are nonlocal interactions allowed. The problem with the latter is that it would then also contradict special relativity. So, then we have to conclude that there is no unique reality.

It is this lack of a unique reality that lies at the heart of an understand of the concept of quantum entanglement. More about that later.

This image has an empty alt attribute; its file name is 1C7DB1746CFC72286DF097344AF23BD2.png

Transcending the impasse, part II

Classical vs quantum

It is a strange thing. Why the obsession with something that in the end comes down to a rather artificial distinction. Nature is the way it is. There is no dualism in nature. The distinction we make between classical and quantum is just an artifact of the theoretical model we build to understand nature. Or is it?

Well there is a history. It started with Einstein’s skepticism about quantum mechanics. Together with some co-workers, he eventually came up with a very good argument to justify the idea that quantum mechanics must be incomplete. At least, it seemed like a good argument until it was eventually shown to be wrong. It was found that the idea that quantum mechanics is incomplete and needs some extra hidden variables does not agree with experimental observations. The obsession with the distinction between what is classical and what is quantum is a remnant of this debate that originated with Einstein.

Today, we have a very successful formalism, which is simply called quantum mechanics, and can be used to model quantum phenomena. Strictly speaking, there are different versions of the quantum mechanics formalism, but they are all equivalent. The choice of specific formalism is usually based on convenience and personal taste.

Though Einstein’s issues with quantum mechanics may have been resolved, the mystery of what it really means remains. Therefore, many people are trying to probe deeper to find out why quantum mechanics works the way it does. However, despite all the probing, nothing seems to be discovered that disagrees with the quantum mechanics formalism, which is by now almost a hundred years old. The strange concepts, such as entanglement, discord, and contextuality, that have been distilled from quantum physics, turn out to be aspects that are already built into the quantum mechanics formalism. So, in effect all the probing merely comes down to an attempt to understand the implications of the formalism. We do not uncover any new physics.

But now a new understanding is rearing it ugly head. It turns out that the quantum mechanics formalism is not only successful for situation where we are clearly dealing with quantum physics. It is equally successful in situations where the physical phenomena are clearly classical. The consequence is that many of the so-called quintessential quantum properties, are actually properties of the formalism and are for that reason also present in cases where one can apply the formalism to classical scenarios.

I’ll give two examples. The one is the celebrated concept of entanglement. It has been shown now that the non-separability, which signals entanglement, is also present in classical optical fields. The difference is, in classical field it is restricted to local properties and cannot be separated over a distance as in the quantum case. This classical non-separability display many of the features that were traditionally associated with quantum entanglement. Many people now impose a dogmatic restriction on the use of the term entanglement, reserving it for those cases where it is clearly associated with quantum phenomena.

It does not serve the scientific community well to be dogmatic. It reminds us of the dogmatism that prevailed shortly after the advent of quantum mechanics. For a long while, any questioning of this dogma was simply not tolerated. It has led to a stagnation in progress in the understanding of quantum physics. Eventually, through the work of dissidents such as J. S. Bell, this stagnation was overthrown.

The other example is where certain properties of quasi-probability distributions are used as an indication of the quantum nature of a state. For instance, in the case of the Wigner distribution, any presence of negative values in the function is used as such an indication of it quantum nature. Nothing prevents one from using the Wigner distribution for classical fields. One can for instance consider the mode profiles of classical optical beams. Some of these mode profiles produce Wigner distributions that take on negative values at certain points. Obviously, it would be misleading to use this as a indication of a quantum nature. So, to avoid this situation, one needs to impose the dogmatic restriction that one can only used this indication in those cases where the Wigner distribution is computed for quantum state. But then the indication becomes somewhat circular, doesn’t it?

It occurs to me that the fact that we can use the quantum mechanics formalism in classical scenarios provides us with an opportunity to question our understanding of what it truly means to be quantum. What are the fundamental properties of nature that indicates scenarios that can be unambiguously identified as quantum phenomena? Through a process of elimination we may be able to arrive at such unambiguous properties. That may help us to see that the difference between the quantum nature of things and the classical nature of things is perhaps not as big as we thought.

This image has an empty alt attribute; its file name is 1C7DB1746CFC72286DF097344AF23BD2.png