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Measuring Entropy/Entanglement in a Trapped Ion Quantum Simulator

Quantum simulation with trapped ions is one of the most promising platforms for quantum technology so far, offering almost unrivalled control at the single particle level. Recent technological developments mean trapped ion quantum simulators are increasing in size and now routinely work with tens of trapped ions. As such, this technology is starting to move out of the regime where it can be classically simulated; the necessary computing power needed to perform the classical simulation is simply too large. It is therefore necessary to develop protocols and techniques which can characterise such systems, to verify that they are performing as expected and to extract quantities of interest such as the presence of entanglement.

Learning Objectives:
This talk aims to give an overview and understanding of how quantum simulation with trapped ions is practically implemented. It will explain the relationship between entropy and entanglement, culminating in an experimentally demonstrated protocol to measure entropy in chains of trapped ions

Speaker: 
Tiffany Brydges, Quantum Optics and Spectroscopy Group at University of Innsbruck, Austria

Bio:
Tiffany Brydges is a graduate student, currently finalising her thesis and working on experimental quantum simulation with trapped ions. The ultimate aim of the experiment is to perform 'quantum simulation' tasks which are too complex to be calculated on a classical computer. Her PhD work looked predominantly at generating interesting dynamics on chains of ~20 ions, which is at the edge of what can be simulated classically, and implemented techniques to characterise these dynamics without the need for comparison with classical simulations. These techniques can then be used in the near-future when the experiment works with larger numbers of ions, and so can no longer be classically simulated.

The first webinar is about measuring entropy and entanglement in a trapped-ion quantum simulator, which is presented by our speaker Tiffany Brydges from the University of Innsbruck in Austria. I would also, like to highlight that Tiffany's presentation will be followed by a brief talk given by Dr. Colin Coates, who is the product manager, Andor Technology. And at the end of both presentations, we will have a Q&A session.

Tiffany: Thanks, Ines, for the nice introduction. And yeah, I am Tiff Brydges from the group of Rainer Blatt in Innsbruck. And So,, the University of Innsbruck and IQOQI in Innsbruck as well. And this talk is about the work I did around my PhD, where we were looking at how to characterize entanglement through using entropy in a trapped-ion quantum simulator. And So,, I have tried to make this talk understandable to a broad group of people, So, it'll start a bit, the basics of quantum simulation and quantum information. So, if you are more of an expert in the field, hold on, because it gets a bit more technical towards the end of it. And hopefully, it is understandable to most people.

I will begin with quantum simulation. I get asked a lot whether it is useful. Whether I do this just because I enjoy using trapped ions and lasers and it is fun to play around with, or if it does have some real-world application. And the answer to that is yes, it does have a real application in the real world, hopefully, in the future. And the idea behind quantum simulation and quantum computation is really to be able to solve classic problems which classical computers cannot do. And these problems can be those kinds of problems which just take too long to Solve, even with the best supercomputers that we have. So,, in principle, the classical computer could Solve it, but if it takes 20,000 years to Solve that problem, then it is not really very useful for us.

And you can ask the question, "Well, do any of these problems really matter? Any of the problems that we're trying to solve or hope to be able to solve in the future with simulators or quantum computers, are they applicable to real life?" And yes, because even just basic chemistry can be really, difficult to simulate. And using a classical device, when you start to get quite a few molecules or some complicated interactions going on, then the problem just becomes far too difficult to solve, classically. And So,, if you can simulate even these basic chemical reactions, you can make a lot of processes look far more energy-efficient, for example, So, in the production of fertilizers which use ammonia and things like that. So,, it has the potential to really be useful, I would say, in a broad range of fields.

Then, you might ask, "Okay, do I want the quantum computer, or do I want a quantum simulator, and what are the differences between them?" And it is very similar to a classical world. So,, a quantum computer is kind of like the analog to your laptop. And you give it a problem, and it computes the answer to it and then gives you the answer out. With a simulator, it is a little bit different. It is more like a wind tunnel. If you are looking at aerodynamics and aerodynamic flow around an airplane, you do not solve that probably exactly. The wind tunnel, for example, does not solve that problem exactly, but it simulates the environment. And that is very similar to what we are looking at doing with a quantum simulator. It is the same kind of thing.

So,, we're using a small-scale quantum system, in order to imitate properties of a larger one that is of interest to us. And the advantage of the small-scale quantum system is that we can probe the dynamics and the states, really, particle by particle because we should, in the ideal case in designing these things, have really optimal control over all parts of our quantum simulator. And we can get a lot of insights into the workings of more complex systems then, which we can't do currently, using classical computing techniques. And the reason these computers and simulators have the advantage over the classical analogs to them is that they must be able to generate large amounts of entanglement in the systems. But we will come back to that a little bit later on.

So,, what does a quantum simulator look like? There's a lot of different platforms, such as superconducting qubits, which you might have heard of in relation to Google and IBM, with Google claiming quantum supremacy, I think a couple years ago now, and IBM still arguing against this. And there's trapped ions, and then, there's also, atoms and optical lattices. All three of these platforms are also, investigated at the University of Innsbruck. But I am going to talk about trapped ions because that is what I was involved in, and that is what Rainer Blatt's works on, mainly. I will go over some of the basics of quantum information, and then the experimental set-up that is used for the quantum simulator. I will then discuss a little bit more about entanglement.

I already mentioned that to get this advantage over classical analogs, you need to be able to create a large amount of entanglement in the quantum system. So,, to do that, we need to be able to characterize how much entanglement is there now. So, how do we characterize it, and then an overview of a new protocol which uses a relation between entropy and entanglement, to characterize it. And then, finally, I will show you some results of implementing this protocol on some 10 and 20-ion chains in our system.

So, just looking at the basics. In classical computing, we have the basic unit of information, which is 0 or 1. This can be represented by a voltage, for example. You can say 0 is zero volts, and 1 is 5 volts. That is quite a common way of doing it. And your computer or your laptop will work with long strings of these bits, such as this. And then, there is a quantum analog to this, which is the quantum bit or the qubit, and this also, has 0 and 1. But the cool thing is, a state can exist in any superposition, also,, of 0 and 1. So, you can have a large amount of 0 and a little bit of 1. Or here, I have shown a state where there's equal amounts of 0 and 1. It is not constrained just to be in 0 or in 1.

And you might ask, "Okay, So, I represented 0 and 1 with a classical bit as being a voltage. What does the 0 and 1 with the qubit maybe correspond to?" So,, you can use an atom, for example, and say that the ground state for the atom will be your 0 or your spin-down state. And an excited state, the atom will be your 1 or your spin-up state. And then, you can move between the two of these using the lasers. So,, you come in with your laser, and you just move the atom into the excited state. A useful way of viewing this is on the surface of what is known as the Bloch sphere. This is a nice geometrical way of representing this. So,, you have 0 out the South pole and 1 out the north pole.

And then the vector, which is your state that you are interested in, is the green arrow. So, here, we can start at 0. And then, you come in with your laser and you rotate the state all the way around, up to 1. And if you are on the surface of the Bloch sphere, like we are here, then you are in a pure state. You can also, exist inside the Bloch sphere. Often, this will occur when there's unwanted interactions between your pure state that you started with and the environment, and then you will end up in what is known as a mixed state, and that is inside the Bloch sphere. So,, a very useful way then, of representing your state is through what is known as the density matrix of it.

So,, I have left up what I just showed on the last slide to the left here, for reference. And then, the density matrix is normally symbolized by rho, here, and it is given this strange mathematical notation. If you do not follow this bit, do not worry. But for those who do understand it, it might give a bit more context to what is going on. But the important thing is, if you know rho that contains all the information about your state psi. So,, rho is really what you want to know about the state of your quantum system, that is what you want to get out. And that is what we will be discussing later in the talk with this protocol, using entropy and entanglement to find out information about rho.

And then, finally, we have entanglement. This is what Einstein did not really like, and he called it "spooky action at a distance," quite famously. And if you have two particles that are entangled, then their measurement outcomes will always be correlated. If they're entangled such that the outcomes are always...the outcomes of their spins are always opposite, then if you make a measurement of one of them and you find it in the up state, then you know straight away that the other one is in a down state. And So,, these measurements are always correlated.

So, now, I will go on to the experiment that I was using. I already mentioned this before. So, now, we have an ion instead of an atom, because I work with trapped ions. And it is in the down state, and you come in with a laser and you can promote it to an excited state. And that is what we normally use as our 0 and 1. And in the Blatt group, in the specific experiment we use, calcium is quite a nice ion, it has a very simple-level structure. So, here, the down state or the 0 state is the ground state of the ion, and that is the S1/2 state. And then, you also, have the D5/2 state up here. And that is the excited state, So, that is the 1, essentially.

And you can couple between these two using a 729-nanometer laser, which we call the qubit laser. And then, you might ask, "Okay, So, how do I now detect which state I am in?" With a classical bit, you might assume you can take a voltmeter and take the voltage. "How do I do the same with my qubit?" We use a laser at 397 to do this on this transition. If you are in the S state when you are shining this laser, your ion with fluoresce like crazy on this transition, and you will see it here as being really, really bright on your camera. If you are in the D state, So, the excited state up here, you cannot fluoresce at all, So, you remain dark. And this is a picture we took at the bottom, here. It is just a long string of ions in our trap, where everything is in the S state because everything is fluorescing.

This here is quite cool. I just took this video on my phone. This is the ions in real time on the camera that are twinkling. I am driving them with the qubit laser and taking them from the S state to the D state and back again. And that is why you see them twinkling. And that is really, really cool to be able to see that in the lab on the camera, to be able to see ions. I can never quite get over that. Good. So, far, I have talked about the ion, but I have not really mentioned the trap. So, we use a linear Paul trap. This thing is a couple of inches in length, I think that is about six or seven centimetres, and it has four radio-frequency blade electrodes.

So one is here, and then one is behind it. And then, there is another one up here, and then again, there is another one behind it. And these oscillate between positive and negative voltage at an RF frequency, So, at about 29 MHz. And then, we also, have two DC electrodes. These are held at constant voltage there, and that provides confinement along the axial direction, So, along the tips of the trap. And what this does is it creates a time-varying saddle potential. So, here I have got another little video. This is rotated by 90 degrees. So, in this diagram, you would have the DC electrodes. One comes from the top, and the other one comes from the bottom down here. And this is the potential that is formed by those blade electrodes.

So, if it is not oscillating at all between positive and negative, your ion in the middle here will want to roll down the potential down there, that now if it is oscillating, it changes. So, once rolled down there, it then changes and so, on. So, your ion just makes very small little oscillations around the middle of the trap. And then, it can't escape in this direction or this direction because then, you have these DC electrodes providing the confinement. So, what you end up with is a nice, long, linear string in the middle of your ion trap. And then, we address the ions using a 729-nanometer laser, like I mentioned before, but we have quite a few of these. So, it is the same laser, but we split it into several different paths, depending on what we want to do.

So, we have a very, very broad beam, which we call the global beam. And this addresses all the ions at the same time and does the same operation on them. But we also, have a single line addressing beam, and it is very, very narrow, and it really addresses only one ion at a time. So, if you have a chain of 20 ions all in the ground state, and you want to put ion seven into the excited state, you can just address it with the addressing beam and do that, which is a really, powerful tool, and it is a nice system. So, now, I am going to move on to how we engineer the dynamics in our system, So, how we engineer the entangling dynamics. And So, far, I have been talking exclusively about the electronic state, the state of the ion.

So, when we are in the ground state and excited state, I have been representing that with a down arrow and an up arrow. We also, now have an extra degree of freedom, thanks to the trap, which is the motional state of the ion. So, if you are in the 0 motional state, the ion is just in the center of your trap, it is not moving at all. But you can put a quanta of energy in. It is a bit like if you have a pendulum and it is motionless, and then you go and perturb the pendulum, it will swing backwards and forwards. The ion can do a very, very similar kind of motion. So, you can write a photon into the system So, it does this motion. Or you can write more photons into the system, and in that case, it will swing with a bigger amplitude.

To do this. So, here, we have Ћω0 is just equivalent to our 729-nanometer laser. And we just sort of detune our laser by an amount plus Δω, which is resonant with this transition to excite the ion. So, we excite it to an excited state and, write a photon in. And likewise, if we are in a ground state...a down state, sorry, over here with the electronic degree of freedom and we have one photon in the system, we can take that photon out of the system by using a red sideband transition. Now, these transitions can then be used to implement entangling dynamics through using the Mølmer–Sørensen gate or the MS gate or interaction.

And this can be understood as follows. So, I have done it here just for two ions, but you can generalize it to an ion string. It just gets very, very messy, then with the diagram. So, here, we have two ions side by side, and they are both in the electronic ground state. And I have said, "Okay, they're in the state with infernons in the system." This can be 0, it can be 100, it does not really matter. And So, I say, okay, I now come in with two beams at the same time, one of which is resonant with the blue sideband transition, and one with the red sideband transition. And I say, Okay, the ion on the left absorbs the photon from the blue sideband transition, and then the ion on the right then absorbs a photon from the red sideband transition. And you end up going from down-down to up-up.

But you can say, "Well, the first ion might have instead absorbed a red sideband transition photon first, it doesn't matter. And likewise, because of the symmetry, the ion on the right might have done it first, So, you get this picture built up. So, from what I just said, when everything is resonant, if you stop these dynamics halfway through, what you will find is that you will have... Some of the state will be in down-down, some will be in up-down, down-up, and then some will be in up-up. Just quite messy and does not really help you with anything. But now, if you were to tune away from this transition, from the blue sideband transition and the red sideband transition by the same amounts...if you were to tune your lasers away, what you will find is that the overall resonance is still there.

So, you still go from down-down to up-up, but the probability to populate these up-down and down-up levels becomes very, very small. So, if you stop the dynamics halfway through, what you end up with is, your state of your two ions is partly in down-down and partly in up-up, but virtually no population is in up-down and down-up. And that is a maximally entangled bell state...one of the maximally entangled bell states. And So, now, you have created an entangling gate, an entangling interaction. This is the basis of what we use in our system. So, this is the Hamiltonian for it. So, this is the Mølmer–Sørensen interaction part, where you have a ΣX/ΣX, which is just as one spin goes from down to up, the other one must also, go from down to up. You can only have these pairwise spin flips. Or likewise, if you start an up-down, you will flip to down-up.

And what we do is, we just...very similar, but we have an extra term here, which practically we implement just by detuning our lasers again by an additional amount. So, it is very straightforward to implement. And if you make this very, very large...which we often do, it is quite a nice thing that happens. It becomes very unlikely that you will get down-down to up-up happening. You will just normally get up-down to down-up and vice versa happening. This has quite a few perks to it, but one of the major ones is that if you start with a 10-ion chain, for example, and you have 5 ions in the excited state to begin with and 5 ions in the down unexcited state, and then you apply your interaction, you know that the number of excitations in the system must be conserved. So, after your dynamics, you should still have five excitations left.

So, if you do not have five excitations left, you know that Something has gone very, very wrong, and you need to fix it and figure out what is going on. So, that is really, helpful for us. So, practically, we then do this as follows in the system. So, we have a beam that contains both the red and the blue sideband frequencies, detuned a bit, coming in at the same time. And we use our broad global beam to do this, So, it eliminates all the ions at the same time. And this has a coupling strength of JIJ, So, the more power you have in your beam, the stronger this coupling will be and the faster your dynamics will be.

And the dynamics itself sort of looks like this. So, at the top here, we have a 14-ion chain, and So, everything that is blue is in the down state. So, these are all...seven ions that are in the down state. And then, one in the middle is put in the up state, So, the expansion state. And then again, all these ions here are in the down state. And then, the Ising Hamiltonian is turned on. In this excitation, it then flip-flops out through the system, kind of like a wave in a way. And you can make the more interesting dynamics happen if you have a more interesting initial state. So, here, we've got the Néel state where all the odd ions are in the excited state, So one, three, five. And all the even ions start in the down, the unexcited state.

And then, when we turn on the Hamiltonian, these excitations start spreading out and it becomes quite closer here. They interfere, you can see, and you get a very large excitation on this ion. Here, you get pretty much no excitation. And what it transpires is that after a few milliseconds' evolution of this...So, for example, about this point here, you get interesting, entangled states start forming, which is good if you are trying to build a quantum simulator, like we are. So, this is what the lab itself looks like. And So, this table is full of lasers for cooling and everything. This is the table with the qubit laser on it. The trap itself is housed in here.

So, this is the vacuum chamber you can just see, and the trap is in the middle of it, So, it is kept under about 10-10 vacuum. And then, this metal shielding around it, which helps keep out any fluctuations from magnetic fields and things like that. So, now, I am going to change gears a bit and go on to characterizing entanglement through using what is known as the second order Rényi entropy. So, I mentioned before that we will not have a large amount of entanglement in our system, in order to build a realistic quantum simulator. And now, how do we know what is going on in our system? Right now, with less than 20 ions, you can classically simulate this quite straightforwardly. But that Sort of defeats the whole point of building a quantum simulator.

You do not want to be in a regime where you can simulate classically, you want to go to more interesting regimes. So, how do you then know that your simulator is performing as you would expect it to do and it is doing the right calculation, essentially, for you? And So, you want to characterize certain quantities that can help you understand what is going on in your simulator. And one of these is obviously characterizing how much entanglement is there. There are several ways in which you can already do this, and quantum state tomography is a very powerful method, and that reconstructs the entire density matrix, rho. So, if you have rho, you do not just have what is going on with entanglement in your system, you know everything about the state there.

So, that is very good except that it scales very, very badly with several qubits. So, above about eight qubits, it is pretty much unfeasible to realistically implement, and eight qubits is very easy to simulate with a classical computer. I can simulate that in a couple of seconds, pretty much, with my laptop. So, that does not really help us very much. And you can have, instead, efficient tomographic methods, which is an extension to get set tomography. This is great because it can be used on large numbers of qubits, So, 14 to 20 qubits, but it only works well on weakly-entangled states, which is a limiting factor for two reasons.

It means, one, well, you are throwing a huge amount of the potential states that might be interesting. And two, it means you must know that you are only going to be working with weakly-entangled states in the first place. And ideally, you do not want to know anything about the state that you will be preparing. That is what you want the simulator to tell you about. So, with this, you really need to know that you will only be making weakly-entangled states for it to work well. We would like something that we can use on many qubits, but we do not have to know anything about the state that we are preparing. And this is where measuring the second order Renyi entropy because there was a previous protocol that used this back in 2016, and it used two identical copies of the system.

They used optical lattices to do this, So, atoms and optical lattices. And they had two of them, side by side, and they made collective measurements on these two identical copies of rho. And it was great because it made no assumptions, obviously, to the system, and it scaled quite well. But it required that you have the two optical lattices side by side, which, for trapped ions, is not So, feasible. You do not really want to have multiple ion traps side by side next to each other. So, this motivated looking at using this entropy, but in a way that did not require you to have two copies of the system.

So, first, what is this entropy? You have got the second-order entropy on the left, and then you have -log2Tr(ρ)2 on the right. And it is quite powerful because of this Tr(ρ)2 term. Because of the non-linear dependence on rho, you are not reconstructing the full state of the system, you are not reconstructing rho. What you are doing instead is you are getting out important information, useful information from the system that you can use, which means you need less measurements to get that. But it is still useful information that you can use to characterize the system.

And just a little bit of terminology. So, the Tr(ρ)2 is also, called the purity, and I will be using that later. So, the entropy is just related to the purity through -log2 of the purity. So, you might think, "Okay, great. We can get this entropy, but why do we care about doing that? How does that help us with entanglement?" And there is a very, very nice link between the two. If you consider this a chain of ions and look at the first five of them...and let us call that subsystem A, then what you can show is that this bipartite entanglement existing between this subsystem A and the rest of the system, if the entropy of the subsystem is greater than that of the entropy of the whole system.

And that is really quite powerful, because it means if you can make a measurement of the entropy, then you can get information about the entanglement in the system out. So, we now have this condition for getting entanglement in the system. We need to measure that the entropy of the subsystem is greater than that of the whole system. And that is where this new protocol comes in. So, this protocol is based on local random unitaries. So, bear with me on this for a moment. Unitaries drawn from this local unitary ensemble. And that was meaningless to me when I first heard it, and So, I will try and explain it how I understand it now. What this means is, these unitaries are just randomly drawn from an ensemble, such that if you start... If a pure state... For example, let us start in the 0 So, at the South Pole of the pure Bloch sphere, which is here, given in blue.

Then, you will be randomly rotated... If you randomly draw on the unitary, then you will be randomly rotated anywhere on the surface of the Bloch sphere with equal probability. And likewise, instead, if you started in a bit of a mixed state, So, the red sphere here, you will be randomly rotated to any other place with equal probability on the surface of this little shrunken Bloch sphere there. And what the protocol does is...it is quite simple. It just applies these local random unitaries to each ion. So, you randomly draw u1 to the unitary 1, and it'll randomly rotate ion 1. Likewise, u2 will rotate ion 2, all the way to uN, which will rotate your ion N. But it is important that really does rotate it to any other place on the surface of the sphere with equal probability.

Why does this work? There is a bit of a physical understanding you can get if you just consider one ion or one qubit to start with. So, imagine, you prepare again in S, the S state, the ground state. And you randomly rotate somewhere, and then you measure σz. And that means you really make a measurement of, is your ion...is it bright or is it dark? You shine in that 397 laser, and you see whether it fluoresces or not. And then, you prepare again in the ground state 0 and then you make a different random rotation, and you measure. And then, you do this many, many times, again and again. And what you can do, So, you can express your state, rho, in terms of the length of your Bloch factor.

If you trace Tr(ρ)2 squared of this, and then providing it really does rotate to any other place on the surface of the sphere with equal probability, you can get a relation between the Tr(ρ)2, which was the purity, and the σz measurement outcome. So, if you have a relation between σz and the purity, then you naturally have a relation between σz and the Rényi entropy. And for those of you who know tomography, you might say, "Okay, but that is kind of crazy. If I have just got one qubit, I can make three measurements on it, and then I can basically reconstruct rho." And you would be right. It would be madness to do this protocol just with one qubit. That is not what it was designed for, it was designed for tens of ions. And that is where it becomes powerful, and that is where the scaling really improves.

So, the protocol itself, it was designed by Andreas Elben and Benoit Vermersch of Peter Zoller's group in Innsbruck. You just apply these random rotations. So, you prepare your state of interest, then you apply these random rotations to each ion in turn. And then you get this hideous expression. So, on the left, you have, the entropy is -log2 of... Here, it is called X-bar, but that was previously the purity or the Tr(ρ)2. And X itself, X-bar is just X averaged over the unitaries, and X is given by this awful expression. And you can kind of ignore this bit and this bit because it is pretty much the scaling factors, actually. And the important part is this bit over here. And this is where the information is held in statistical cross-correlations between the outcomes of the measurements.

And that means it is asking the question, "If ion 2 is down, is ion 5 always up?" Or, "If ion 3 is in 0.4, is ion 9 always in 0.7?" And it looks at all these correlations between the outcomes and averages over them, and then it gives access to this entropy down here. And if we have the entropy then, as seen before, we have the entanglement. So, to practically implement this in the experiment, we prepared the Néel state, which is the up-down, up-down, up-down. And then, we evolved it under our Hamiltonian, the different time steps of τ. And then we applied these random local unitaries to each ion, and then we made a σz measurement, So, we asked if it is bright or dark. And then, we repeated this 500 times. Each time, we changed the local unitaries we were applying, So, really randomized them completely.

And now, So,me results. So, on the left, we have the purity, and on the right we have the Rényi entropy. And this was for a 10-ion chain that we first looked at. So, you can see, if we start on the left... And zero milliseconds' time evolution is in purple, So, this means we just prepared the Néel state. And then, we did the protocol, So, we did the random rotation straight away on that. And now, with the Néel state, it is a product state, So, there should be no entanglement in the system. And if we then look at the Rényi entropy for the zero milliseconds, we see that the subsystems have roughly the same entropy then as the whole 10-ion system here, which is what we would expect for a product state.

It gets more interesting as you look now to five milliseconds' time. As you can see then, the subsystems have a much higher entropy than the entropy of the whole system over here at 10. And if we remember from a few slides ago, if the entropy of the subsystems is greater than that of the whole system, you know that you have bipartite entanglement in the system. And we looked at this for all of the subsystems at 5 milliseconds in that 10-ion chain. So, here, all the one-ion subsystems are just ion 1, and then ion 2, and then ion 3, and So, on. So, all permutations of the subsystems and the lines here are at 3Σ below the mean, and the dotted line is 3Σ above the 10-ion entropy.

You can see none of this overlap, which means that within 3Σ, we have entanglement between all the bipartitions in the system. And then, finally, we decided we would look at a 20-ion chain. So, unfortunately, back then, our single ion addressing was not as good. We couldn't really apply it to all 20 ions. So, what we did, we prepared 20 ions in the Néel state, evolved those 20 ions under the Hamiltonian, and then applied the protocol to the 10 middle ions. So, from ion 6 through to 15. But we did it for a bit longer this time, we evolved it out to 10 milliseconds. And you can see that there's low entropy...initial Néel state is evolving into very high-entropy sections in this. This is consistent with the formation of highly entangled states in the system.

I am coming now, pretty much to the end. So, this is the Blatt group, with the people who contributed to this significantly highlighted, Benoit and Andreas. So, thank you very much for your attention. And thank you to Ines. And I will hand you back to Ines and Colin.

Ines: Thank you very much, Tiffany, for this great and insightful presentation.

Colin: Thank you, Ines, and thank you So, much, Tiffany, for that excellent presentation. I am going to give a brief, high-level overview of some detector solutions that Andor have within the umbrella of quantum optics. Now, quantum optics is quite a broad umbrella, So, the way I have chosen to do this is just to select a few different fairly popular types of optical systems, optical experimental setups within quantum optics, outline the challenges from the detector point of view, and then state what we recommend, our quantum optics solutions.

So, the first one I have chosen is where you want to image entangled photon pairs. Now, Tiffany's already done a really good job of defining what an entangled photon pair is, So, there is no need to go over that again. But this type of system is one where you would have a bi-photon entangled pair where you would detect each photon of the correlated pair on the same imaging detector. So, there's a couple of key requirements for this, and it largely relates to, ultimately, the throughput or the measurement throughput of the experiments, which have the potential to take up to years in some cases, to do. But we do not have years to wait, normally, So, we want to find ways of multiplexing it and rattling these things along as fast as possible.

So, there's a number of things that feed into this. On will be that when you see an event, you know it is from a photon and not from a dark background event from the detector itself. So, it is the ability to capture and discriminate spatially-correlated photon pairs with good statistical confidence, and that is key. And yes, of course, fast counting is needed as well, to get through these neat experiments and build up useful imagery at the end of it. You did have to do quite a number of iterations at times, So, fast counting will greatly accelerate the measurement process, as well. And why do you want to this, apart from just...well, aside from just pure fundamental quantum physics research? There are potential real-world allocations to it as well. For one thing, you can use quantum imaging approaches with entangled photons to actually get better signal-to-noise ratio. Here's a good recent example here from the Miles Padgett group.

You can also use quantum entangled photon approaches to enhance Resolution. So, more beside that is a couple of key potential actual real-world benefits of following this type of quantum path. So, the recommended tried-and-tested solution for many years in the market is to use an electro-multiplying CCD outdoors – Andor’s iXon Ultra EMCCD is our key recommendation here. An EMCCD is a signal-amplifying technology which can register a single photon event. To do single photon counting experiments, you need a very, very sensitive technology. This is great in that it combines single-photon sensitivity with what is called back-illumination, which means that it can collect more than 90% of the incident photons as well. So, you are wasting very few of the photons that are incident on the detector you are registering as events.

Another benefit of using, basically, an imaging array approach as a detector rather than a single-point detector is just this massive multiplexing advantage you get when counting - simply by virtue of the fact that you should be using a megapixel array rather than single-point. So, that hack can take your measurements from years down to hours by itself. Even though the detectors are single-photon sensitive, there are other potential sources of noise in the detectors which are known as dark current, and clock induced charge. And they are amplified by the very same mechanism that we use to amplify the photons.

We must do something quite careful in that EMCCD camera to make sure that those sources of noise are reduced to as low as we possibly can, to a small fraction of a percentage chance that if you see an event that is from a dark event, we want to know with a really good sense of assurance that these events are from photons - that is important to build that statistical confidence. I mentioned the count rate. So, technically, EMCCDs operate full resolution in the tens or maybe 50 frames per second. But once you start to separate them down into smaller regions of interest, of course, you can access then hundreds of frames per second. So, that basically much defines the rate that you can count photon pairs at.

And then, the other aspect for this particular type of system, very often when you are looking at these correlated-value photons, they can appear spatially, very, very close to each other on the array, maybe one or two pixels apart. So, we must be quite careful to some subtleties of images that there is a chance that when you have an event, a photon on a single pixel, that charge has a mechanism to spread across to your neighboring pixel. So, that is obviously a bad thing if you are trying to detect actual events which can be in neighboring pixels. So, we must pay extra attention to make sure that is going to...there's a very, very small probability of charge spreading across neighboring pixels during the readout of the detector.

So, that is a real fast summary of why we recommend that detector. There is obviously a lot more detail to that, but hopefully, that gives you a reasonable idea. The other type of entangled photon system I thought I would introduce is the quantum ghost imaging. Tiffany already raised this term of "spooky action at a distance," this Einstein term. That is pretty much what we're seeing in these types of systems where you would have an entangled photon pair but send it in completely different directions. And only one of the photons is going to the imaging array, the other will be in directing and attributing very different wavelengths. The other will be in directing photon at a time with an object, and then being detected by what is called a heralding detector, which is usually a single-point detector.

But that single-point detector then tells the imaging detector, "Yep, get ready to receive an event yourself from my identical twin," if you like. And that is what happens. So, that basically means that there is a mechanism to build up an image on the imaging detector of an object, which those photons that went that direction never did encounter themselves. So, that is the mechanism. And because of this temporal synchronization capability, the detector must have an additional type of technology which enables that temporal synchronization with very short shuttering times.

And before I move on, I should say, the potential real-world application of this could be, for example, when you want to interrogate an object, say, with an infrared wavelength and detect indivisible. That could be one actual practical means of deploying this type of ghost imaging system. So, the type of detector which we recommend for this is an intensified camera. So, this could be applied to either CCD or what is called scientific CMOS technology. But basically, it is different because it has this additional front end which is a fast-shuttering system. So, we do not have time to go through the details of what this involves, but it is basically capable of shuttering down to about two nanoseconds. So, that defines the optical shuttering time, and then you can synchronize that to the heralding detector.

So, that means you get this precise measurement of correlation in the temporal domain as well as the spatial domain. And then, of course, again, you want your statistical confidence of photon detection to be as high as possible. But there are potential sources of single-pixel events in this technology as well, and it is not called clock induced charge this time. It is called EBI, Equivalent Background Illuminants. But there are mechanisms through cooling to reduce these sources of false positives, as well. So, moving on to the next challenge quantum gases and trapped ions as a general field here – this is a really broad area.

But I have noticed over the years that some of the recurring things that are asked of our detectors... Well, first, it is good often that they are capable of very, very broad spectral response across a wide wavelength range. And that is more to introduce the flexibility for usage in the detector, that we can use them for ytterbium ions, right up to rubidium absorption experiments at 70 nanometers. So, UV right through to the infrared. And then, it depends on the system, but people either use, say for both sides condensate, they very often use absorption imaging. The benefits of that are, that it affords them the whole density-distribution information from within the atom clouds. Sometimes, as in the case we just heard from Tiffany's presentation, there can be very few atoms or ions involved. Fluorescence can be a much more effective module or means of interrogating these trapped species in such small quantities.

But then, of course, with fluorescence, you have less scope for getting information on local density distribution. So, very often, these approaches are considered complementary, absorption and imaging can be quite often combined. And very often, fast dynamics are needed. It depends, again, on the type of experimental systems. But very often, we are looking for change, time resolution in milliseconds or even microseconds time regimes. So, let us break it down with absorption fluorescence. Absorption, first of all, the most common type of CCD used for absorption, if you are only using absorption, is a CCD, a straightforward, slow-scan, deep cooled CCD.

And again, benefit there is at least a day you can get CCDs that extend very well into the new infrared and right down into the hard UV. They have very, very broad mechanisms for affording very, very broad spectral response. So, that can't be great for doing things like rubidium absorption imaging, which uses a 780 9-millimeter laser. You would use what is called a deep depletion type of CCD then. There is actually a means, even though these are relatively slow read out technologies, there is a means to do on-hand burst dynamics, which means you fill up the sensor itself with dynamic information, and then slowly read it out. And you can get dynamics quite readily into the microsecond and millisecond regime by doing this.

For the fluorescence applications, it is back to the detector we talked about a little earlier, the iXon Ultra EMCCD. This is a brilliant detector for low light fluorescence imaging. And we have examples here, of an MOT with very discrete amounts of atoms in it as they enter and leave the trap. And of course, we have perfect examples up there of lines of trapped ions as well. And you can imagine, these are very, very low-light circumstances where we have photons and any given exposure time coming from each of these imaged ions.

So, finally, within the umbrella of quantum optics, I wanted to touch on experiments that are to characterize what you call quantum materials or quantum imagers. And examples I have given here could be nitrogen vacancy centers, quantum dots, or even development of any type of quantum light source. Very often, spectroscopy is used as a modality for this type of characterization. And for the luminescence, spectroscopy with that is pretty much the key work that you will associate with this. So, this is a typical experimental layout here. And you will notice a few things, that there is a spectrometer in here for the spectroscopic dispersion, and there is a detector and a spectroscopic detector.

There is a cryostat in the system here, because quite often, a lot of these experiments are done at low temperatures. So, the types of solutions (we do not really have time to go through the full portfolio of Andor spectroscopic solutions) but there is quite a sizable one, as you can see from this layout here. That is worth going over. We have a section of our website that is devoted entirely to spectroscopic solutions (https://andor.oxinst.com/products/spectrographs-solutions). It is worth going on there for a browse because it is quite in-depth and quite broad. And finally, as you know, Andor are part of Oxford Instruments and produce an optical cryostat range. And you can see an example here called the Optistat, and there is the optical section at the bottom.

Ines: Yeah. So, thank you So, much, Tiffany and Colin, again, for your great presentations. And I think I would also, just like to mention again, because you showed on your previous slide, Colin, iXon Ultra EMCCD, So, with...give an example, this is also, the camera that was used in the experiment from the group with Tiffany. So, to just kind of close the circle. And yeah, now I would like to just go through some questions. There were some good ones that came through. So, we will start with a question for Tiffany.

How does your protocol scale with the size of the subsystem you want to determine the entropy for?"

Tiffany: Okay, I think this is a good question. So, I am just going to steal the screen back from Colin, sorry, because I have... It is easier to show on a slide because it is ugly. So, this is how it scales with the size of the subsystem you are interested in. So, here is the subsystem and it scales as 0.2 to the 0.8 NA. And then, the 7.7 is effectively a constant scaling factor, at the front of it. It is interesting because for entangled pure states, the scaling can actually be better, and it can go down to about 0.6 NA. So, for more separable states, it scales a little bit worse.

Ines: Okay, thank you, Tiffany. I will then ask you another question. Are you aware of any experimental protocol that does not rely on quantum state tomography and is able to measure the Von Neumann entropy and not the Renyi entropy?

Tiffany: So, I think there is a paper that came out last year by Institute of Physics. They suggested a protocol that can measure the Von Neumann entropy, So, not through randomized rotations but through other means. But what is interesting, though, is that Renyi entropy I have mentioned is actually one of many. So, there's a huge spectrum of these Renyi entropies. We looked at the second order one. And the limit where it goes to the first-order one it actually reduces to the Von Neumann entropy. So, the Von Neumann entropy is actually a limit of one of these Renyi entropies, which is quite interesting.

Ines: Colin, can detector be used to measure optical plasma imaging in vacuum chambers full of excited gas mixtures?

Colin: Yeah, the most common type of detector for plasma imaging is the intensified cameras that we showed in the context of ghost imaging. So, for sure, these types of intensified detectors for fast plasma imaging.

Ines: Okay, thank you, Colin. And I have one last question for Tiffany. "Might the entropy ennuis in the system contain useful information that has been thrown away?"

Tiffany: I would say the entropy is sort of the useful information. So, by making the measurements... We make the measurements and extract the information about all the subsystems and the portal system. And then, by looking at whether the entropy of some of the subsystems is higher, then you can infer the presence and entanglement from that. It is not necessarily that we are throwing things away or using the higher entropy to infer information about the system.

Ines: "Is it possible with your protocol to measure the entropy of ions that aren't spatially connected?" And it says, "For example, the Renyi entropy of the reduced density matrix of rho?"

Tiffany: So, the answer to this is yes. So, these were all really, really god questions. Yes, So, here on this plot, that is what you see. So, for example, you have given the example of a two-cubit system. Here, we have all the two-cubit systems, So, one and two, one and eight, two and eight, and everything like that. So, this here is those reduced density measurements plotted. So, all permutations of all subsystems.

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