In the framework of POVMs, commutativity still plays a role, but it is demoted to simply being a sufficient condition for compatibility. There exist measurements that do not commute, yet they can originate from a single observable. None the less, commutativity is still considered an important measure of incompatibility.

In our recent paper we show, quite surprisingly, that when considering the incompatibility of quantum testers, commutativity becomes completely irrelevant. Before we continue, let me explain what these testers are. Whilst we usually measure a state given to us, testers (as shown in their simplest form in the picture below) are designed to measure the devices responsible for changes to states: the channels. To this end, one does not only measure what is on the output, but one can also choose the initial state that is injected into the channel. Simply put, a tester could tell us what the channel does to states.

Let us consider a simple example of a measurement where the channel preserves the polarisation of light. We can prepare a horizontally polarised photon and measure whether, after leaving the channel, it is still horizontally polarised. We can also measure how the vertical polarisation is preserved in a similar way. But can we measure both of these features at the same time? As it turns out, one can not.

One can immediately see that in this case the incompatibility will not lie in the final measurement, as the measurement of vertical/horizontal polarisation is the same. The incompatibility in this case is contained solely in the preparation of the state – one cannot prepare a state whose polarisation is both horizontal and vertical. Intuitively one might be tempted to ask whether a superposition of the two polarisations would work, but unfortunately by preparing the superposition one loses the information about the action of the channel on the individual states that are superposed. In fact, the two testers of the polarisation-preserving channels are maximally incompatible, even though (in their mathematical form) they commute – it takes the most amount of work to make them compatible in some sense.

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But wait, does it even make sense to compare a property of collections of measurements to a property of states of a composite systems? Isn’t it as misleading as comparing Finnish vodka to Slovak borovicka?

The basic idea in our comparison is simple: a quantum channel can be seen alternatively in the Schrödinger picture or in the Heisenberg picture. Consider, for instance, the following kind of experimental setting.

Here the red thing is a preparator that produces bipartite systems, blue tubes are quantum channels and on each side there is a collection of three measurement devices. For each run, both observers choose which measurement devices they use. An experimental run goes like this:

Observers are collecting measurement data, and in the end they have a correlation table which contains the information about their chosen measurements and the results they obtained. To violate a Bell inequality, the bipartite state must be entangled and the measuremens A, B, C must be incompatible. (And the same for A’, B’, C’.)

Let’s ignore the right hand side part of the picture and think about the situation from the perspective of the observer on the left. Now we come to the fact mentioned earlier: the blue channel can be seen alternatively in the Schrödinger picture or in the Heisenberg picture:

Now, if the blue channel is very noisy, it certainly spoils our correlation experiment. Hence, we want to know how much noise our experimental setting tolerates. Looking at the channel in the Schrödinger picture, we are worried that the state becomes separable. Meanwhile, looking at the channel in the Heisenberg picture, we hope that the collection of measurements does not become compatible. (Note, however, that these are just necessary criteria which are not enough to violate Bell inequality.)

I hope you are now convinced that it can make sense to compare noise tolerances of incompatibility and entanglement. Please read more from our arxiv papers!

(Thanks to Daniel Nagaj for his comments.)

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- What do you do to imagine and how do you imagine or visualize n-dimensional space (especially 4-dimensional)?
- How do you imagine or visualize wave-particle duality? In particular, in what way do you visualize what happens in the double-slit experiment (with single photons or electrons)? Do you imagine a particle, a wave, or something else, or nothing?

I discuss the answers that were given below. Since I cannot expect all readers to understand every concept I quizzed my friends on (wave-particle duality being the most likely to be alien to the average reader), I shall briefly introduce them.

Physicists always talk about dimensions, and they rarely stop at three. Sometimes they say that space-time is 4-dimensional, and sometimes they say that the universe has 11 dimensions (or some other arbitrary number). But very often, on a more abstract level not connected to the “physical” space we’re used to, they use n-dimensional space (where n can easily be of order 10^{23}). Some even talk about infinite-dimensional spaces (I know some that will engage in lengthy discussions about the intricacies of infinite-dimensional spaces, yet are quite lost when the discussion turns into finite dimensions). Finally, the layman tend to just look at these scientists and think how strange we are when we imagine such nonsense. This is how we – scientists – are usually perceived.

Let me say that we are not strange (not all of us at least), and I hope I can make this clear in the next few lines. With my first question I knew what to expect for an answer and I was not disappointed. The answer is simple: you simply do not imagine larger dimensional spaces, because you cannot. We live in a three-dimensional space and this is what we experience on a daily basis; we do not know anything else. This is exactly why we know how to imagine 3 dimensions (and smaller integer-valued dimensions, which are easily embedded in our 3D space) but we do not know where to put the fourth one. Even though we cannot visualize the fourth dimension (although some respondents had tried) it is an easy task to replace our intuition with known concepts. One can easily think of the fourth dimension as time, or color. However, this seems to be, in general, as far as we can push this idea.

So, what do we do with higher dimensions? We use analogy: in the same way one goes from one dimension to two and from two dimensions to three by adding just one more “identifier” (spatial position in this case) one can add time to get fourth, color to get fifth. But what’s more you do not have to imagine it – once you know the rules, you just accept them. In other words we just need to look at the underlying mathematical formalism and keep in mind the rules that are applied. A nice analogy to understanding the way we imagine things was given to me by Andrej Gendiar. After using a formalism a large number of times one is not getting any better at visualizing things, but you do acquire an intuition. As Andrej wrote: ” Now, I can see no difference among 3D, 4D, and infinite-D, because I do not see any difference between two integer numbers +4 and -4 (the latter one being absolutely unrealistic).” There is one more thing scientists do when they work in higher dimensional spaces and need to get intuition. They project the problem either into 3D or 2D space, which our brains are accustomed to. In the same way you can assess the properties of a hanging object from its shadows on the wall, one assesses what happens in higher dimensions from what happens in the lower-dimensional case.

A slightly different approach is to use what happens in a smaller dimension to acquire intuition about what happens in the higher dimension, just like you can gain some intuition about cubes by studying squares. In the end no one visualizes higher dimensions and all one can do is build some intuition about it from considering what happens in smaller dimensions.

While the first question was just to confirm my suspicions, the second question was more interesting to me. There is not much one can do to imagine larger dimensions, but I was quite curious how others try to imagine the concept of wave-particle duality.

Before we consider the given answers, what do I mean by wave-particle duality? When one lets a small particle such as a photon, electron or atom (anything that could be considered “quantum” in nature) go through a double slit and then measures where it hits the wall located past the slits one finds that two quite contradictory explanations are at hand: One is that the particle will hit the wall at a point-like location (like a particle, right?), but on the other hand the probabilities describing possible positions the particle could hit the wall seem to follow the rules for a wave – the probability of hitting the wall at some location is proportional to the height of the wave that would arrive at that location. There is certainly more to be said on the topic (see e.g. Wikipedia) but this blog is not about the effect. All we need to remember is that before hitting the wall, our photon/electron/atom behaves like a wave and then after the collision acts as if it is a particle. This is a strange phenomenon that is certainly counter-intuitive and has been with us from the onset of quantum theory. In the (classical) world around us, the waves are always waves and particles are always particles, and so there is this desire to ask: how can you imagine/visualise this *wave-particle duality?*

As in the previous case, I was not really surprised by the answers, although I had secretly hoped that somebody would have some amazing way of imagining it. The answers I got were quite closely related to the fact that the effect is statistical, meaning that the supposed “wave” is just a theoretical prediction of the (amplitude of the) probability that the particle is located on a given place at a given time. What people did was they visualized a wave, sometimes composed of small balls, or a “ship on the wave” that represented the particle.

It turned out that all these visualizations were, however, only a crutch. As is the case with imagining higher dimensions, what’s more important is the underlying mathematical structure. Your intuition is very limited and comes only from the observation of the classical world around us, which has no respect for “quantum laws”. That is why at one point or another our inherent intuition fails. We now have a firm grasp of the laws according to which quantum theory works and while their consequences might be counter-intuitive, we have to accept them. In the case of wave-particle duality, we just have to accept that the wave – as a mathematical tool – determines the probability that the particle hits the wall at a given location, and there is not much more to it.

As you can see, there is nothing in these cases that scientists are better at imagining than anyone else. The things we talk about that cannot be imagined are always based on a solid mathematical background that we respect. There is, however one point worth mentioning: while it is true that we cannot imagine these things, when one works with them a lot, one does acquire some intuition about these things, and in the end it is not the intuition that leads to knowledge, but knowledge that leads to intuition. In the same way that a cook, through hours of testing and tasting, acquires an intuition about which ingredients work well together, scientists, by working through the mathematics, acquire some intuition about how things work.

[1] The experiment was conducted on 38 participants. They answered the questions willingly and were allowed to answer anonymously. No animals or scientists were harmed during the experiment.

Many thanks to Tom Bullock for having the patience to go through the text and for his corrections.

]]>Suppose a quantum channel AB takes two systems as an input and gives two out. We keep the second input state fixed at all times, and call this a* blank state*. The situation looks something like this:

If we ignore one of the output states, we obtain a channel that takes in one input state and gives one out. We can either ignore the first or the second output, hence there are the two possibilities:

If the starting channel AB is specified, then we can calculate channels A and B. But we can also consider the inverse problem :

Suppose two quantum channels C1 and C2 are given. Is there a bipartite quantum channel C (and a blank state) such that C1 = ‘part A of C’ and C2 = ‘part B of C’?

The answer is: not always. Two channels can be incompatible! An exemplary instance is the one where both C1 and C2 are the identity channels. This case would look as follows:

The fact that two identity channels are not compatible in the previously specified way is known as the No-cloning theorem, now a standard piece in any quantum theory course.

The intention of this post was simply to point out that the No-cloning theorem can be seen as an instance of quantum incompatibility. Looking at it from this perspective puts it into a bigger frame.

Teiko

]]>In this first blog post let us tell you something about the topic in some broader term, namely, what is compatibility? It is exactly what the name suggests: it is the idea that two or more things can “work together”. Take for example following shadows:

As the picture shows, the three shadows come from a single object, although it takes a bit of thinking to decide whether it is so when only the shadows are available.

In the same way we ask a multitude of questions about compatibility of other different devices: a flat-head screwdriver is incompatible with a hex socket screw; tea and milk can be put together and make a good drink, and as such they seem to be compatible (I (Daniel) tested it on my taste buds, but would argue otherwise); and Slovak institutions do not offer good services, and so are also incompatible. These are everyday compatibility problems that arise, but there are many more hidden in the quantum realm and those provide questions we are interested in.

In the following posts we will try to dig deeper into these quantum questions and provide answers where we know them. We hope you will enjoy reading this blog as much as we will writing it.

Daniel and Teiko

(Many thanks to Tom Bullock for proof-reading)

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