 Hello. Welcome. I'm glad so many people wouldn't be scared away from what I said five minutes ago. I'm going to talk about quantum information and there's nothing new in what I say today. I just want to give you a gentle introduction containing some technical details so you get a first idea how information in the quantum world is fundamentally different from information in our classical world that we're used to My day job is I work for theoretical mathematical physics master program at Ludwig Maximilian's University in Munich So if you want to hear a lot more about this Come to Munich and and roll in our master program Okay, so But before we get into quantum. Ah, I have an overview. So before we get into quantum physics I want to settle a bit of Almost trivial background on classical cryptography, and I should say also I mean the area of quantum information is very wide and I should also say it's not my own area of specialty it and Compasses lots of different things in particular quantum computing and quantum cryptography And I decided not to talk too much about quantum computing Because I mean basically there are a couple of algorithms that would be a bit hard to explain And I think it's much easier and much clearer to talk about quantum to cryptography And that's what I'm going to discuss today, and that's why I'm going to talk about classical crypto first The other thing is As I said, I'm a theoretical physicist, so I'm interested more in the mathematical structures that we encounter rather than the actual thing so But there's also a huge difference between quantum Cryptography, which is something that exists in the real world. You can buy off the shelf quantum crypto devices these days I know last year we had talks here about how they can be compromised But quantum computers do not exist. I mean there exists prototypes and as I'm informed Kind of the the largest job that has ever run on a quantum computer is the factorization of 15 into 5 by 3 As you can understand So this is also merely a demonstration rather than of practical use because on my MacBook Pro I can factorize much larger numbers So I don't gain a lot from the theoretical advantage of Quantumness in the case of the quantum computer, but for quantum crypto as I'm going to explain to you You can get a real advantage because you can prove That there's no eavesdropper I'm going to show you how that works Okay, so so I talk a bit about classical crypto then I have here my oops don't point with a laser pointer at people I Have here my little experiment Great hand of applause for my guinea pigs So so these two things will be my warm-up and then There'll be some heavy-duty mathematics machinery in the in the middle and I try to Make it as little demanding as possible But we need some formulas and otherwise I was just telling you pros and would just advertise the thing rather than explain it So I decided To torture you a bit with formulas For the price that you can gain actual understanding and hopefully of this and then from this I can then explain how quantum crypto works and then if there is time in the end I'm going to elaborate on the question whether the human brain is a quantum computer or not Okay, so classical crypto Basically the essentials of classical crypto work like this I mean the archetypical example of cryptography is a one-time pad So here are my friends Ellis and Bob and they want to transmit one bit of secret information. So what they do is One of them say Ellis Turns off the light in her bedroom and goes to a drawer and picks a pair of socks Okay, so the nice thing about the socks is that she ordered the socks when she after washing them So there are either two red socks red socks or Two green socks Right, but the thing is there's never a red sock and a green socks. She grabs At dark. She doesn't see the color. She grabs one pair and Gives one of the socks to her friend Bob. So here in this case you can see on my slide she has picked the red pair and The green pair is still in the draw Right, so that's their secret key. The secret key is the color of the socks And since she picked the red one, she knows the code is now if she picks a red sock She she transmits the opposite of the message. She says not message, right? And then Bob knows also the code. He sees her. He has also read sock Therefore, he knows Alice has a red sock pretty trivial. So he knows not not message is the message Okay, so much everybody knows, right That's not very hard. That's very effective. That's really secret Unless our eavesdropper Eve got hold of a copy of the sock Right, she also see has a red sock because she managed to put a third sock in the drawer and get hold of it And therefore she knows not not is the message that Alice wanted to transmit Okay, that's classical crypto one-time pat So Everything trivial so much, but let's abstract from this a little bit. So what what they use here is That the state the sock pair that they share is in a correlated random state Right, that's essential. So it has to be random. It wouldn't work if all the socks were red, right? Then there wouldn't be a code, right? It has to have this random element so Eve cannot guess which one they pick unless she gets a copy And it also has to be correlated that means in 50% of the probability They pick two green socks 50% of the probability they pick two red socks, but with 0% of the probability they pick one Red and one green Okay, so that is the essential thing. So it's random. So nothing is here 100% and it's correlated. So the two colors of the socks always the same and The important thing here is also that this state has to be I mean I'm using high-brow language for a trivial thing This state has to be prepared by a trusted entity. So either they do it themselves Oh, they have to do the sock distribution like saying it by mail. They have to trust the mail system Right if Eve is the mailman and looks into the envelope sees the color of the sock then they're lost Right and of course there's a further assumption and that'll also exist in the quantum case because otherwise you're completely lost That public communication Is reliable. So I mean here on the back. She has to announce Not message and Bob has to be able to hear this message and to hear it correctly. Otherwise, it wouldn't work So, okay forward Okay, that is the classical situation now on to show you how to improve this in the quantum situation But to be able to understand this I have to explain a little bit of quantum mechanics first and here's our little experiment So I I give you the theoretical experiment first and then we do it in real life so my experiment consists of a box and the box has a Handle that has two positions a and b In this case the box is here and this is the handle So it has a position left and has a position right and left and right and left and right It has a Slot where something can be added. That's on top here. You can't see it And then it has a pointer. So and how does it work? So my experimenter here my little my guinea pigs. Thank you very much decide randomly To pick one of these positions a or b Okay, a or b or back a and then So these concepts these describe two possible measurements that my machine can do I can either measure I Can do a measurement of the a aspect of the thing I'm measuring or I can do a measurement of the b aspect of the thing I'm measuring and then I add the thing I Add the thing and then the pointer points either to the minus one or the plus one direction Okay, that's all I have to explain about I mean, this is how the experiment works And the thing is I do not do it once I do it three times and I do distribute the things to be measured from Central origin. So that is what we're going to do here and Maybe do this one so Philip has already pressed his button So let's reset the whole thing Okay, so because you probably can't read I mean here is an LED display saying either plus or minus one you can read this Thanks to Munich Chaos computer Club they provided me with these three lamps. So the lamps can be either red or green and it's It's green if it's plus one and it's red if it's minus one so you can better see what they're doing So if I may ask you now to to run your experiment measure your glass of water that I gave you So and you see it because the blinking stops Okay, so now you see So first of all you observe that all the pointers are pointing in the right direction So all pointers are measuring B and we measured minus one plus one minus one Okay, so that's pretty much random at this point but now we do this over and over again and We we change the switches and we memorize What the results are? Okay, and the thing I mean maybe I give away the the message is that we we are going to find a correlation between these measurements That I will argue is an impossible in a classical world But then later argue that it can be achieved I can build such a machine in a quantum world and then this proves if you like that The classical world is not quantum because I can build machines that are I can prove I'm possible in a classical world Okay, so let's do a couple more measurements gentlemen So here we have a Here we still have B and we still have B and you see Plus one minus one minus one. Let's do it again Maybe you leave the handles for a couple of times and just do more measurements. Just leave it with a Okay, now we have minus one minus one plus one over there and again Okay Okay, so and we keep a log book and we do this over and over again And once we do this we get a table like this. Okay, I've done this at home. I promise this is the real output of this Okay, so you see here I have I'm sorted the the results already here In this case everybody had their handle in the a position So we measure a three times a 1 a 2 a 3 and you see plus minus plus and then plus minus plus and then minus minus minus In this case the first two switches on the a position the last one is in the B position and so on up to here Where the last one where everybody had their switch in the B position so now the question is is this pattern completely random? Well, it's not I promise. It's not who sees an Aspect that is not random Okay, maybe you focus on these columns Okay, somebody spotted that there's no plus plus plus in this column very good. That's true very good Is there anything else that is missing in the first column? single pluses Okay, so there are no three pluses and there are no single pluses. So everything here is either a single minus Or three minuses Okay So And what about the green columns? Yeah, so you spotted the difference. Yeah No, no, well, this is plus minus plus. This is minus minus plus No, this is start off with plus plus plus. No, this one doesn't agree Yeah, so somebody said in the back. It's the other way around right so in the green ones There's either a single plus or three pluses Okay, does everybody see this so let's summarize this if I if I measure a three times I have I have an odd number of minus signs Okay, let's let's do this in real world. Let's measure a everybody everybody put their handle to the left To the left and then where's my reset button? Okay Okay, let's measure the thing Minus one plus one plus one a single minus sign. Let's do it again Mine minus one plus one minus one one more time. Hey, we get three minus signs Minus one minus one minus one. Okay, you see there is always whoops once more There is always in An odd number of rad lamps. There's always an odd number of minus signs Okay, and then these three green columns. These are the columns with one a and two B's Okay, so maybe two of you switch So now we have one a and two B's and we measure again Now we should have an even number of red lights So zero is an even number and one more Okay, and two is also an even number of red lights. So the real experiment does indeed what I promised to you Okay, so let's formalize this now comes the math so we have an When I measure three a's I have an odd number of minus signs So if I say if I call the variables a 1 a 2 a 3 this just means that the product of 8 1 a 2 a 3 is minus 1 Right and similarly for an even number of minus signs a product of 1 a and 2 B's is plus 1 That's what I just said or in the other two combinations Now we can if a 1 times b 2 times b 3 is plus 1 and everyone is either plus or minus 1 I can say that's the same as a 1 being the product of b 2 times b 3 Either a is plus then the product has to be plus so a is minus and then the product has to be minus That's pretty trivial right, but now from these three formulas here. I can multiply these three equations So I get a 1 times a 2 times a 3 then you see that every b b appears twice So this is b 1 times b 2 times b 3 squared and any square isn't is a positive number So this the product is minus 1 a plus 1 But we said it should be minus 1 Okay, so this is my contradiction that I was talking about You can compute from these experiments You can compute the values of the product of 3 a's and what you compute is plus 1, but when you actually measure it It's minus 1 Right, therefore something is wrong right this cannot be true That there are these variables that have these properties Okay, so in particular this cannot happen for local realistic Measurement so what realistic means that's a technical term. I'm going to explain later But local I have to explain so local means that what Phillip is measuring here on his first station only depends on the position of his handle and the contents of his water glass Right so everything that and it could contain a random element could here Maybe there is a coin flip inside the box that determines the outcome But it can take only into account the position of the handle and his water glass So that is the assumption that goes into my little calculation that he really what he measures when he puts the handle in the a position Is the a measurement on his on his water glass Okay Now I explained to you that this experiment is impossible, but I showed it to you Right so something must be wrong Well, of course what is wrong here is that might I mean this is not a quantum experiment, right? There's an Arduino inside and blah But it's not local they're connected by cables Right, so in reality my my true experiment This box knows about the handle position of the other two boxes and therefore I can show you this on stage Even if it's not a quantum computer because it's not local this box Measures if you like also the aspect of the handles on the other two devices But if I assume that I only do a local measurement then I just Derived for you that this is impossible. Okay, is everybody confused now more than you would like to be confused No, great. Okay, so let's go on So now let's let's get into Quantum mechanics as I said I have to explain a very little bit of quantum mechanics To explain to you how this works. So again I'm going to use the abstract language that I started out using with the socks so first of all there are states in my system and in quantum mechanics a State is represented by a vector. So what is a vector vector is is two numbers written Well, in the simplest case is two numbers written on top of each other and put parentheses around that's a vector That's enough for you to know and it's So and and it's usually use Greek names. So let's call this one psi And for the case that you just write two numbers on top of each other. This is called a qubit I mean for quantum bit if you like and there's one further condition And that is that the absolute value of the first entry squared plus the absolute value of the second entry squared has to be one So let's look for examples The examples I'm going to use in the following are the ones with one zeros and this one squared plus zero squared is one so this fulfills this condition and Usually I want to call. I want to call this state. I want to call up. So I draw an up arrow Okay, and I could also have a zero and one instead of a one into zero And I want to call this right and these are kind of the two states the two simplest states that my system can be in I haven't talked about how I realized this the state of what kind of thing this is if this is an electron spin or Something else There's one thing you might know that has exactly these states and that is polarized light So if you have light, which is an electromagnetic wave then it can either Oscillate the electric field can oscillate. I mean the light that from from this light comes to me This can either oscillate in the vertical direction or it can oscillate in the horizontal direction, right? And if I can I can use a polarization filter to filter out I mean this true light is a mixture of both but I can use a polarization filter to block out these two components So either have the filter in this direction then only the horizontal polarization gets through or I put it that way And then only this component gets through this is used in old-style 3d cinemas for example I hope everybody has seen a polarization filter in their lives at one point or the other if you have an LCD display Your own one Okay, so so that is this how I describe the state of whatever system I want to discuss And then of course the system is not alone there in the world. I want to I want to observe it I want to do measurements on this and I also have to formalize how these measurements work And I of course I can do several different measurements like with a socks. Remember I had socks. I can measure the color or I Can measure the size Right, and so also in the corner world. I have different measurements and all my measurements Like the states were vectors and they're represented by matrices here. I fit two by two matrices again with the condition The technical terms that it has to be her mission, but let's not worry about this condition Because we only need three examples Okay, my first example is I The so-called identity matrix which reads one zero zero one this fulfills this condition this matrix when I apply to vector You all promise to me you applied matrices to vectors before right then, you know This matrix leaves the vector as it is. It does nothing. That's the identity matrix Then I have this matrix that I want to call Z Z looks like this one except it has a minus one in this position Well, this does when it when you apply to a vector is it leaves the first component and talked and the second component gets a minus sign Okay, that's the operation that I do with the Z operator and Then I have an X operator that looks like this has the ones in the zero zero swapped And what it does with the vector it flips the components It puts the upper component lower position at the low opposite component in the upper position, okay? These are my three Measurements or observables that I need in this talk Okay, so now I have a state and I have I have measurement and of course you want to know I mean we want to predict stuff right so if I have a system in a certain state Represent by a vector and I do a measurement represented by a certain matrix you want to know what is the outcome and the problem is That's difficult But but you can be lucky And you're lucky particularly If the state psi Happens to be what is it called an eigenvector of M. So what is an eigenvector? An eigenvector is such a vector that when I apply the matrix to the vector I get the vector back Multiplied by number. I mean lambda is just a number So so the only thing the vector does if you think of the vector as as an arrow It can only change its length, but it doesn't change its direction Right, so I apply the matrix to the vector and what I get out is lambda. There is this number lambda Such that the result is lambda times you want when it was v1 first and it's lambda times v2 when it was v2 first So so the first line I mean I mean this number and this numbers are always multiples of each other But the important thing is that the second the lower numbers are Multiples with the same multiple lambda Okay, if this happens to be the case if if psi happens to be such a vector for this matrix M Then the outcome of the experiment is very simple because the merit measurements yields lambda whenever I measure This matrix M in this eigenvector psi. I always find the result lambda Okay, let's see examples of this So let's take our matrix z which was one minus one and I take my vector up And I see that clearly this is an eigenvector because remember that multiplies for it leaves the first component intact and multiplies The low on by minus one, but the low one was zero. So nothing happens here So indeed this up is an eigenvector with lambda equals one Okay, so I will only always measure one Similarly my vector that I wanted to call right the one was zero one Measured in Z gets here a minus sign. So lambda is minus one So in this measurement I was my result will always be minus one but if I if I have a general vector then I get v1 minus v2 and Though unless one of them is zero. They don't have a common multiplier, right? Everybody got that? Okay, in short that applied to up is up and that applied to right is minus, right? Okay, next next example is my axe remember axe switches the component So here you see we want v2 gets mapped to v2 v1. Okay, how? Can it be a multiple of itself when I swap the components well trivially if both components are the same, right? Then if I switch the same numbers, I don't do anything at all. So Okay, here a square root of two whoever is afraid of a square root you can safely ignore all the square roots in this talk They're just there to make the thing correct on in the formulas that are right. You can completely Blind them out. They're not important for what I'm going to say Right, so you can read this as one one if you're afraid of square roots It won't change the meaning Okay, and I want to call this because this thing is kind of half up half right. I want to call it upright Okay, this has the same components here and therefore if I've switched them if I flip them I get the same thing so alumnus one in this case and for up left I Have opposite equal components here and if I flip those I get minus one. So so these To diagonal arrows These are the eigenvectors for my matrix X for my flipping matrix X Okay, so that was the case when I have eigenvectors, but not every vector is an eigenvector, right so but in the more difficult situation there's Hope in terms of the spectral theorem and that tells you whenever I have some some M and And then I can always find two such eigenvectors Right there are always psi one and psi two and corresponding numbers lambda one and lambda two such that these are eigenvectors and then Any I mean the most general state can always be expressed uniquely as a weighted sum of these two eigenvectors So these are the two eigenvectors I want so I to and I multiply them by some weight by some number and then add them and I every psi I can write in such a way for for the correct values of C1 and C2 Okay, so what I have to do for my general state I have to write it in this way and then if I do the measurement now comes the fundamental difference in quantum mechanics Then the result is probabilistic. I measure sometimes Lambda one and I measure sometimes lambda two with a certain probability and the probability is given by these Numbers C1 and C2 particular so it has to be C1 squared and C2 squared these are the two probabilities for measuring these two outcomes and That's true. What true randomness? There's I mean you cannot look inside it and discovered the mechanism how the randomness works. This is true quantum randomness Okay, by the way mathematics makes sure that the sum of C1 squared and C2 squared is always one so With probably one you get one of these two results Okay, and then when I measure let's let's assume I measure the outcome is lambda one Then that's also new in quantum mechanics. The measurement changes the state I cannot do a passive observation of the system when I do this measurement M and I get lambda one afterwards the state is not the same anymore, but the state has changed to the state Psy one Psy one that corresponds to lambda one and if my outcome Was lambda two then after the measurement the state has been changed to Psy two Okay, so that is how measurements work in quantum mechanics Okay, let's see this in the real world in an example we saw that the state upright For X this was an eigenvector for X that was the one with with the two equal components So it always yields one, but now I want to measure Z on this. So this one Upright was this vector and remember these two guys where the eigenvectors for the Z measurement And this is the way to write them as a sum of These two vectors and you see I mean these one over square two and one over square two These are this the numbers that are called see in the slide before So the result is either plus one if I measure this one or minus one if I measure this one each with Probability 50 percent because if I square one over square root of two I get one half and that's 50 percent Okay, so if I in this state the eigenstate of X and measure Z The result is like a coin toss. It's half of the time I get plus one half of the time I get minus one and After this measurement, I'm in either in the state up on the state right Okay, because again, I Write them in this way in terms of the eigenvectors and if I after doing this Z measurement I then measure X again I'm either in Well, I'm either in the state this plus this or in the state this plus this and then also if I measure the Z Then also the X measurement will be random again Right because I changed the state and after measuring Z the state is in this half plus one half minus one state With regards to the measurement of X and that shows us that X and Z if you like I measure the X value and the Z value of the thing They cannot have an exact value at the same time right if It's in this state then I have an exact X value But but a but a random Z value, but after the measurement I have an exact Z value by the random X value Okay, so that's the crucial property of quantum mechanics that that is possible that things That certain measurements cannot be exact at the same time Okay, so let's generalize this quickly to two bigger states that have more than one component So I could have end of these systems that I was talking about before and then the state looks like this So I each each individual one is in the state psi So I want no, so I choose up to psi n then I write this fancy tensor product sign in the middle, which doesn't have to concern you And then I can have several combinations and again I can add them with weights And then I can if I have several things like in this case I had three for example I can do a joint measurement so I measure something on each component So on the first component and measure M1 on the second component and measure M2 blah blah blah up to Mn right and the result the Result is then just the product of of the individual measurements Okay, but in particular I can choose this trivial matrix that does nothing if I if I have a three particle system But I just measure something here on the first one I can say I measure something on each station, but these two always measure one They always do the trivial measurement that always yields one. That's the same as just doing a measurement here Okay Okay, that is the formalism for three copies or in general n copies okay, now The new thing is I can prepare something that's called the singlet state. So this is now a two particle system And I can prepare a state like this. So that is the first one is in the upstate and the second Particles in the upstate, but it's also a sum of the first particle being in the right state and the second particle in the right state So it's again the sum of these two states Notice so so now we couldn't measure Z the Z component on the The we do the measurement of Z on the first On the first system so just on this one or on this one and then you see that here it would yield plus one here It would yield minus one And in the total thing it's each with probability one half Right if I just do one that measurement the result is random. It's either plus or minus one but if I measure Z on both components then this would yield plus one and this would yield a plus one and the product is what plus one And here it yields a minus one and a minus one and the products again plus one so The product is always plus one so this state is in fact an eigenvector that always use the result plus one For two measurements of Z that means this state is a random state if I look at each individual component But they're always the same right If I measure Z on both they can either be both plus or both minus Right, that's like with the socks The socks could either be both red or both green and it's like this But the important thing is it's also an eigenstate of xx remember what x does x flips the two components If I flip the one down and this one down Then I flip it with this thing where I get back when I flip the Lower one to the upper position and again, so again, this is the plus one eigenstate of xx So also if I do two x measurements I Measure plus one plus one with 50% probability or I measure minus one minus one with 50% probability But they're always the same Okay, but remember That each individual one I cannot have sharp values on both x and z But on the combined measurement of two x's or two z's I can have sharp measurements on both of them That's the new thing here Everybody looks a bit confused Okay, so I can both yield random results, but they always see the same thing For both types of measurements okay, so this is now my quantum version of socks these are my quantum socks because This observation that I have simultaneous Correlated states both in x and in z Is the key to quantum cryptography because now My friends Alice and Bob share these two qubits in the state that I just talked about and then later on They decide Randomly if they want to do the x measurement on each I mean everybody does the x measurement on their copy of the qubit Or they both do the z measurement on the qubit and I just argued that they will get the same result The result will be random, but they get the same thing And then they can use this as the color of the socks to run their Classical crypto protocol on this this this matter the result of this measurement is their shared key Okay, and each one is random But they cannot be both determined so because they have the choice of either measuring x or z Until they do the measurement it's random and that is what helps against eavesdropping Okay, each one is random with both cannot be determined at the same time and that's different Like the two measurements I can do on my socks I can I can determine that this is a red big sock and This is a red little sock I Cannot say whether what the ax and the z component are at the same time Okay, so why is this better than the classical socks? Why what about an eavesdropper? So I could have the idea. Let's do the same thing as in the classical case Let's make a third copy Okay, let's prepare this three qubit state with the same pattern either they're all up or they're all down Right, and I can do your math and see okay if you do the Zett measurement indeed these two The first two always yield the same Z value and also the second two or the first the first and this Third yield the same Z value So in the end if they all measure Z, they all get the same result They all get plus plus plus or they get minus minus minus that's not better than the socks where I can make a copy, but if I do If I decide not to do the Zett measurement on this but the X measurement Remember X flip the two components if I flip this component down and this component down I do not get this vector here because the last one I would get Right right up. Whereas this is right right right. So if I just flip the the first two components That means I do the X measurement here and the X measurement here and nothing here I do not switch the components and in fact, this is not an eigenvector and I get random results for two X measurements That means in this state if Alice and Bob do the X measurement They will not necessarily get the same result anymore for the extra measurement Sometimes they do sometimes they don't and this is the result of making this third copy So Alice and Bob can determine that there is this third copy by both doing the actual X measurement and Announcing the result and they do it. I mean they don't have just a single Set of qubits, but they have many of this so they do this many times and if they have the true Secret two particle state, then they will always measure the same thing But if they have this one with a third copy Then they there will be results where they measure two different things and they can this way detect That there's a third copy and then can reject this state and say we can't do cryptography with this There's an eavesdropper Okay, everybody got that So let's let's put this in form of a protocol. Okay, so Alice and Bob I mean, this is even better. They can they can even ask Eve to supply their state Right because she can either be honest and just give them the state they want and then have no information about the state They have or they can prepare the state with the third copy So she has her copy of the state, but then Alice and Bob can detect it Right and that they so you can even assume the bad person To to run the protocol if you like So they ask Eve to distribute n of these copies in this they want wanted in this singlet state the two particle states I too But they don't know if Eve complies to their wish or not okay, now they pick a subset of these n copies let's say M and You pick the subset randomly and do either and and do Z and X measurements on these and they always do the same either They do both X and they boot both to Z and they announced the results publicly And if Eve has really prepared several psi two states, then the results will always agree So they do this M times and they announce the results and they can check if they agree no matter whether they measure X or Z If they all agree they can be sure or at least sure with a certain probability that Eve has honestly prepared the state and not prepared something else and then they So they announced real publicly if they agree they know they really have psi twos and not the ones with the third copy and then they can use the remaining qubits as and do that Now do the measurement both of X or both of that without announcing the result as their shared key and run their classical one-time pat crypto protocol on this Okay, so that is important so that this protocol is two stages and in the first stage They can check whether whether Their state is correlated with the state that Eve has or not because the correlation with Eve state in the corner where it leaves traces behind in The two particle state you all look confused Okay, but this was the main message, right? I have a protocol Okay, so let me repeat it. I have a protocol and the thing is I can do a test the the quantum states when I when if copies the information She has to hurt the state and this can be detected by Ellison Bob and that can detect that there has been a copy and then They know they cannot use this key for secret communication Okay, this is complicated, but this is the essence of quantum quantum cryptography and You might have heard other explanation of this before and if it didn't have this step then it wasn't a Explanation of quantum photography because then was just an explication of the thing with the socks the classical protocol So thing is really You can did the new thing is you can detect whether they had been a third copy Okay, let's let's quickly go through this experiment that I did here So I claim there is no classical machine and I want to argue that but there is a quantum machine So you remember we had these three gentlemen here operating the apparatus and I claim the following quantum state does what I want The state that is either up up right up right up or right up up minus Right right right and when they put the the handle in the left position they measure They measure Z and in the right position they measure X and if you go through this if if all three of them Remember the product of three a measurements was always minus one. So if they measure Z Z Z They should always get minus one should be Z Z Z eigenstate. So So this one so this one gets plus times plus times minus one This time one gets plus times minus times plus this one gets minus times plus times plus So all all of them have two pluses one minus product being minus and here you have three minus ones multiply together give another minus one So indeed if they all measure Z in this state They all get minus one. So and the other thing was if two measured If two measured B, which now is X to measure X and one measures Z They should all the product should always be plus one. So what happens if I do X X Z? So X and X. So I get this this one gets flipped down. This one gets flipped down And this this one gets a minus one So minus I have minus down minus right right right and this is minus Minus right right right. So these this one gets flipped with this one and you can also check that these two get flipped under this operation So indeed as you can see here plus one minus one minus one and I have one a and B and B Indeed they always the product is always plus one, okay? and so this experiment this quantum experiment realizes what I claimed before for my experiment what the outcome was and Classically argued it's not possible, but here it's a local measurement each one just measures on their own qubit and They get this result three Z measurements always plus one one one Z measurement to X measurements always plus one Okay, so this experiment Really shows that you can do things in a quantum world that are impossible in a classical world And in particular if you're philosophically minded this proves that quantum mechanics is not secretly a classical theory You could imagine that deep inside the microscopic's are little we I mean wheels and things that turn and Operate electrons and atoms and everything in a way that they look Like they have this weird quantum randomness, but this shows Whatever the gears inside are you can never reproduce Such an experiment. I mean with the secret classical mechanics Okay Okay, I said I'm not going to talk about Quantum computing. Let me just mention three things. Well entry points to Wikipedia if you like three famous algorithms Grover's algorithm is a database search which gives a quadratic speed up compared to the best known classical algorithm Charles algorithm is about factorization. That's the one with 15 beings three times five That gives an exponential speed up And similarly Simon's algorithm. So I copied this from from the web. So P I mean now we're talking complexity classes, right? Bqp is bounded quantum Polynomial these are the things you can do with the quantum computer and this set is bigger than the P problems the one with polynomial runtime on a Turing machine But and this This contains some NP problems even contains some problems that are just in P space But not an MP, but it does not contain NP complete So even with existing quantum computers, you're not gonna solve. I mean that's conjectured, right? I mean nobody knows whether these are really different Okay, but but current state of the art current believe is that the quantum computer does not help you with solving NP complete problems Okay, I said I'm not going to talk about this rather for the last hopefully a bit amusing 10 minutes I want to discuss this question is the human brain a quantum computer Because I have been discussing this question with friends over at least the last 15 years and Maybe you've seen this book by Roger Penrose or one of the other popular books by Roger Ponderos We advocate I mean Roger Penrose an important physicist in the theory of general relativity and and quantum Gravity, but he has written lots of popular books really thick ones And he advocates the idea that the human brain is such a great machine can solve so different difficult problems like proof theorems and whatnot that it No way can ever be a Turing machine It has to use the quantumness because deep inside the brain quantum processes blah, okay, so let's investigate this So let's summarize this 600 page book and the brain can do more complex things you can do on any computer Right, so so let's investigate this idea a bit So is this the human brain a quantum computer? Well, the answer is obviously not right This is a brain. This is a quantum computer Okay, so but maybe that wasn't the answer he was looking for maybe Maybe we should be in the next obvious answers obviously yes, right? It consists of atoms and molecules that are governed by the laws of quantum physics Basically, that's the argument that Penrose advocates, right? But so there's a silicon-based computer. I mean this is a memory chip and this is the band structure that you use for the semiconductor properties of a year of the silicon or Whatever technology use use semiconductors and this thing is the band structure and that's the result of a quantum physics calculation Right, so so you might my Mac book that runs this presentation is a quantum computer because use quantum properties of silicon Well, that's also not the answer then every computer is a quantum computer. Okay, so next answer is It's obviously not so so I mean I mean the lesson we learned from from from the previous slide was This question is not really about how the thing is made, right? Question is what kind of problems can I solve with it? Are the problems that I can solve with a brain Different from the problems that I can solve with a classical computer and you know, I mean for a classical computer I mean, I don't want to discuss Mac book pro specs, right? I want to discuss whether I can solve the problem with the Turing machine So the question is whether I can simulate one thing with the other, right? So the question is is the human brain a quantum computer is really the question Can I simulate the human brain? I mean in theory, right on on a classical computer on a Turing machine kind of Or can I and and thereby solve all the problems that human brain can solve or Kind of do more right and then the end the answer to this question Can I say I mean if I interpret this question is the human brain a quantum computer as kind of simulate human brain on on a big enough Turing machine if I have enough time Then the answer is again, obviously. Yes, right because as I said, oops the Sorry, sorry, let me put it again if if this question really reads can assimilate If I can simulate it with the Turing machine Then it's not better than a classical computer and then in this in the sense of this question It's not a quantum computer, right then it's as good as a classical computer Or at most as good as a classic computer. I would have to show that I can emulate a Turing machine on my brain I'll just show the the opposite direction, right, but the answer is obviously not This is the this training as equation. I mean in short I mean this is the equation governs the physics of my brain, right? This is how the quantum state psi evolves in time This is now a bit more than a two qubit stated. You have to simulate 10 to the 23 electrons in my brain, but I'm Terribly convinced that I mean this is a part of differential equation. It's huge Right, but I can always in America solve it to any accuracy that I like given my computers big enough enough enough time so Whatever quantum processes happen inside the atoms of my brain I can always simulate all in principle all the electrons in my brain and that on on the classical architecture therefore, I Won't be able to do anything fancy more fancy than I can do with the Turing machine But again Then I could apply this to any quantum computer, right? I Can also I showed you the picture of this optical setup, which is supposed to be quantum computer And I could also simulate this on my classical computer and okay So the question is not really so any The question really whether I can at all solve the problem or not Like I mean even if I have a traveling salesman problem with two million cities It has a solution and it takes a finite amount of time on a Turing machine. It's long right? It's a huge I mean it takes very very long, but it's Pacific problems always all one in time Right, it's certain times bounded in time So it's just when I vary the number of cities the questions how it scales. So the question is kind of effectively simulate kind of effectively simulate a brain on a classical computer or not and they're the the typical question is whether it scales polynomial and to that answer I have no idea, but At least get the question right I think the quest the correct question if you want to ask is the human brain a quantum computer the correct question you should think about is Can I simulate n brains effectively with the number of Turing machines that is polynomial and n right? So I need send n to infinity And study this problem. This is really the the problem That you want to investigate if you want to decide if you have upon if brains a quantum computer or not any all the other Understands of the question as I try to explain a completely meaningless and therefore. I think also paneros misses his point Because he's not investigating this question Okay, that's all I had to say Okay, thanks for Okay, I should I should say Many thanks for for Schneider from Munich CCC for setting up these red and green lamps yesterday at night at 2 o'clock in the morning If you I mean we can have Probably one or two minutes of questions now, but if you have more urgent questions, so I mean email Twitter DM me or go to my this is my blog That's terribly out of date, but it discuss all kinds of physics question You can find also the discussion of the is the brain quantum computer question in this blog. Okay, any urgent questions now Urgent questions now Okay One one person is not asleep enough to ask a question Yeah, so I read you shout and I repeat your question Okay, just shout I repeat what you say to exchange keys. You usually want to you said Thanks You usually want to exchange keys because the public communication chain is not real reliable and you won't know if drop us so isn't it's there a possibility to improve that key exchange theme to even on an under reliable public channel Where the man in the middle can also Modify the bits you want to compare in order to check whether you receive the same Measurements, okay, did everybody get the question? What about when I cannot trust the public channel? So remember I had to announce my results publicly like I put them in the newspaper Ellis says Berlin CCC Ellis says I measured x and got all I got it was minus one right and have to trust that this is really What Ellis measures when Bob reads this and the question is can I do without this? And my answer is not that I know of and I think you're completely screwed if you I mean that effectively means you cannot exchange any information right and then if even if you can't exchange the crypto text then Right, that's what you want right you have to exchange the crypto text if you can't do this Yeah, then I would say you're lost. I mean you can you can always authenticate your public channel and stuff But you can sign it and but I mean if it's unreliable in the end Okay, thank you. Sorry. We don't have time for other questions