[inaudible] majority of cases we need more than one bit to perform our computation. For quantum computing it is also true. So, in this lecture, we are going to learn the mathematical description of systems with multiple qubits. So, here you see a table for now, imagine that we have two particles each carrying one qubit. In this table we have the possible measurement outcomes for these two qubits in our ordinary buses zero, one. We have four possibilities of such measurement here. Now, it is convenient for us to describe the system, to describe this measurement outcomes. Again, in some Hilbert space and some linear space over the complex numbers. But since we have four different measurement outcomes, here this space has to be four dimensional. Each measurement outcome will be again, one of the basis vectors of this space. So, let's name these vectors. This measurement outcome we'll call 00, this 01, this 10, and this 11. So, we just have named for basis vectors of this space in which we'll describe the system with two qubits. Good. Now, imagine that one of the qubits have this superposition before the measurement. Can we describe this state before the measurement in the space we and just constructed? Yes, we can. We can describe it like this. So, we used two of the vectors we're just introduced on the previous slide to describe the state of this system. This system is still in the superposition state. So, we can measure it and the probabilities of the measurement outcomes are still the same. So, it's alpha squared for the measurement outcome 00 and beta squared is the measurement outcome for 01 since measurement of the first qubit always gives us zero and on the measurement of the second qubit is probabilistic for us. Again, the sum of the coefficients before the basis vectors and the state gives us one so it is a unitary vector. Now, let's consider a more complicated situation when both qubits are in some superposition. We still can describe this state in our newly constructed space like this. It's easy to show that the sum of squares of all the coefficients here gives us one. So, it is again a unitary vector. For any state, for example, this one. The probability of obtaining this state after the measurement is alpha, delta squared. So, we have everything we got used to in the case of one qubit, and this is true because we believe that measurement of two different particles are independent events, so I can multiply the probabilities. I have alpha squared for obtaining this zero, and we have delta squared for obtaining this one. So, the probability of 01 is alpha squared multiplied by delta squared. Okay. Now, the interesting part is, we just constructed the description of the state of two particles like we just did. But the amazing part is that we cannot construct the whole space like this because there are vectors in this four-dimensional space. For example this one which cannot be constructed through this process just as I just described. But physically, this state which is described like this can exist and on two separate particles. We will return to this fact very soon. Now, it's very common misunderstanding about all this. The real quantum systems, the qubits, they don't situate in some Hilbert spaces, and if you take two of them, they don't initiate some tens of product of these spaces. It is our way of describing them. It is our choice. We decide to describe it as some vector in Hilbert spaces, and it is convenient for us to describe two qubits in the space of four dimensions. We already named our buses. Now, we have to show how we enumerate it. We have to decide which column vectors will represent these buses. You remember that we have this 1 0 column for the vector 0, and this 0 1 column for the vector 1. Now, we have to choose for each buses vector of our newly constructed space, which column vectors represent it. There are many rules we can use for this, and the most basic rule for us will be the following. Let's just take the two vectors. This one is 0, and this one is 0 so they form this vector 0, 0. We take a tensor product of this vector so the Kronecker product. So, we have here 1 multiplied by 1,0 and 0 multiplied by 1,0. This is our tensor which is 1,0,0,0 tensor. We can substitute with vector in four dimensional space like this, and just remove these brackets inside of our tensor. So, this is the way how we are going to construct the cones for our newly defined buses. For example, if you have vector 0,1, we will have to do the same thing. So, 0 is 1,0 vector multiplied by 0,1 which is equals to vector 1, and this again we take this 1 and multiply it by the whole vector here so that's 0,1. Now, if we take this 0 and multiply it again by this vector, so this gives us this tensor which we can naturally map to this four dimensional vector. Of course we can apply this rule to all the buses vectors and we obtain this table. We can choose other rules to enumerate our vectors. But this rule is very convenient because if you have already noticed that if we use the binary notation here, we look at these as some, numbers the zero zero is zero in binary notation. This is one in binary notation. This is two and this is three. So it's numbers of our vectors. Here when we have zero we have one on the first place and we have one on the second place. Here we have one on the third place and on the fourth place. So, the position of this single one in the column it is defined by the number of the vector which we can read from its name in binary notation. So, this is why this rule is very convenient. Now, if you consider for example the state of three qubits like these and this state will be represented by a vector eight dimensional space. So, it will be a cone with eight coordinates in it. We can see that this number is five in binary notation. So, we can easily understand that this single one will be on position six in the vector and all others will be zeros. Okay. So, of course, for the systems with multiple qubits with more qubits than two if you have for example, n qubits, it will have vectors in the spaces with number of dimensions two in the power n. So, for n equal to one we have this one qubit which is the vector and then space with two dimensions. If n equal to two, we have four dimensions and n equal to three, we have eight and with 10 we have 1,024. If we have like 1,000 particles, we have these enormous number of dimensions. So, it's like 1,024 and the power 100 which is approximately this number. So, the state of this system with only 1,000 particles is described by this number of complex numbers. This is why Richard Feynman was so optimistic about quantum computing. You may argue that if you take only 100 of flipping coins then the number of outcomes of reading these coins will be the same and it will also be this number of outcomes. But for some reason nobody builds a computer on these flipping coins. In the next lecture, we are going to understand why.