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

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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.

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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.

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Discreteness

Demystifying quantum mechanics V

Perhaps one of the most iconic “mysteries” of quantum mechanics is the particle-wave duality. Basically, it comes down to the fact that the interference effects one can observe implies that quantum entities behave like waves, but at the same time, these entities are observed as discrete lumps, which are interpreted as particles. Previously, I explained that one can relax the idea of localized lumps a bit to allow only the interactions, which are required for observations, to be localized. So instead of particles, we can think of these entities as partites that share all the properties of particles, accept that they are not localized lumps. So, they can behave like waves and thus give rise to all the wave phenomena that are observed. In this way, the mystery of the particle-wave duality is removed.

Now, it is important to understand that, just like particles, partites are discrete entities. The discreteness of these entities is an important aspect that plays a significant role in the phenomena that we observe in quantum physics. Richard Feynman even considered the idea that “all things are made of atoms” to be the single most important bit of scientific knowledge that we have.

Model of the atom

How then does it happen that some physicist would claim that quantum mechanics is not about discreteness? In her blog post, Hossenfelder goes on to make a number of statements that contradict much of our understanding of fundamental physics. For instance, she would claim that “quantizing a theory does not mean you make it discrete.”

Let’s just clarify. What does it mean to quantize a theory? It depends, whether we are talking about quantum mechanics or quantum field theory. In quantum mechanics, the processing of quantizing a theory implies that we replace observable quantities with operators for these quantities. These operators don’t always commute with each other, which then leads to the Heisenberg uncertainty relation. So the discreteness is not immediately apparent. On the other hand, in quantum field theory, the quantization process implies that fields are replaced by field operators. These field operators are expressed in terms of so-called ladder operators: creation and annihilation operators. What a ladder operator does is to change the excitation of a field in discrete lumps. Therefore, discreteness is clearly apparent in quantum field theory.

What Hossenfelder says, is that the Heisenberg uncertainty relationships is the key foundation for quantum mechanics. In one of her comments, she states: “The uncertainty principle is a quantum phenomenon. It is not a property of classical waves. If there’s no hbar in it, it’s not the uncertainty principle. People get confused by the fact that waves obey a property that looks similar to the uncertainty principle, but in this case it’s for the position and wave-number, not momentum. That’s not a quantum phenomenon. That’s just a mathematical identity.”

It seems that she forgot about Louise de Broglie’s equation, which relates the wave-number to the momentum. In a previous post, I have explained that the Heisenberg uncertain relationship is an inevitable consequence of the Planck and de Broglie equations, which relate the conjugate variables of the phase space with Fourier variables. It has nothing to do with classical physics. It is founded in the underlying mathematics associated with Fourier analysis. Let’s not allow us to be mislead by people that are more interested in sensationalism than knowledge and understanding.

The discreteness of partites allows the creation of superpositions of arbitrary combinations of such partites. The consequences for such scenarios include quantum interference that is observed in for instance the Hong-Ou-Mandel effect. It can also lead to quantum entanglement, which is an important property used in quantum information systems. The discreteness in quantum physics therefore allows it to go beyond what one can find in classical physics.

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Partiteness

Demystifying quantum mechanics IV

Yes I know, it is not a word, at least not yet. We tend to do that in physics sometimes. When one wants to introduce a new concept, one needs to give it a name. Often, that name would be a word that does not exist yet.

What does it mean? The word “partiteness” indicates the property of nature that it can be represented in terms of parties or partites. It is the intrinsic capability of a system to incorporate an arbitrary number of partites. In my previous post, I mentioned partites as a replacement for the notion of particles. The idea of partites is not new. People often consider quantum systems consisting of multiple partites.

What are these partites then? They represent an abstraction of the concept of a particle. Usually the concept is used rather vaguely, since it is not intended to carry more significance than what is necessary to describe the quantum system. I don’t think anybody has ever considered it to be a defining property that nature possesses at the fundamental level. However, I feel that we may need to consider the idea of partiteness more seriously.

Classical optics diffraction pattern

Let’s see if we can make the concept of a partite a little more precise. It is after all the key property that allows nature to transcend its classical nature. It is indeed an abstraction of the concept of a particle, retaining only those aspects of particles that we can confirm experimentally. Essentially, they can carry a full compliment of all the degrees of freedom associated with a certain type of particle. But, unlike particles, they are not dimensionless points traveling on world lines. In that sense, they are not localized. Usually, one can think of a single partite in the same way one would think of a single particle such as a photon, provided one does not think of it as a single point moving around in space. A single photon can have a wave function described by any complex function that satisfies the equations of motion. (See for instance the diffraction pattern in the figure above.) The same is true for a partite. As a result, a single partite behaves in the same way as a classical field. So, we can switch it around and say that a classical field represents just one partite.

The situation becomes more complicated with multiple partites. The wave function for such a system can become rather complex. It allows the possibility for quantum entanglement. We’ll postpone a better discussion of quantum entanglement for another time.

Multiple photons can behave in a coherent fashion so that they all essentially share the same state in terms of the degrees of freedom. All these photons can then be viewed collectively as just one partite. This situation is what a coherent classical optical field would represent. Once again we see that such a classical field behaves as just one partite.

The important difference between a particle and a partite is that the latter is not localized in the way a particle is localized. A partite is delocalized in a way that is described by its wave function. This wave function describes all the properties of the partite in terms of all the degrees of freedom associated with it, including the spatiotemporal degrees of freedom and the internal degrees of freedom such as spin.

The wave function must satisfy all the constraints imposed by the dynamics associated with the type of field. It includes interactions, either with itself (such as gluons in quantum chromodynamics) or with other types of fields (such as photons with charges particles).

All observations involve interactions of the field with whatever device is used for the observation. The notion of particles comes from the fact that these observations tend to be localized. However, on careful consideration, such a localization of an observation only tells us that the interactions are localized and not that the observed field must consist of localized particles. So, we will relax the idea that fields must be consisting of localized particle and only say that, for some reason that we perhaps don’t understand yet, the interaction among fields are localized. That leaves us free to consider the field as consisting of nonlocal partites (thus avoiding all sort of conceptual pitfalls such as the particle-wave duality).

Hopefully I have succeeded to convey the idea that I have in my mind of the concept of a partite. If not, please let me know. I would love to discuss it.

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