[MUSIC] So to approach quantum information processing, the first thing we have to do is to have access to quantum theory. And to do that we need to know the actions of quantum mechanics. So there are four elements already, states, dynamics, measurement and observables and the mathematical language to work with them is Hilbert space. So Hilbert space is nothing but a vector space with the inner product. Just inner product we know in a standard way. So in a standard vector space, we write down vector like this and it's a dual vector like that and inner product here. And in the Dirac notation, we use a slightly different way of writing the standard vector here. We write this vector in this way and this vector is called ket vector. And dual one, we convert this arrow to the other direction here and then we call bra vector. And so this notation is called bra and ket notation sometimes and inner product will be written like this. And this dirac notation is from Paul Dirac, the physicist in the UK who made a contribution to quantum mechanics and the metrics formulation. And this notation actually simplifies the meaning and sometimes deliver the messages of quantum mechanics. So nowadays actually this kind of standard and use a way of writing or describing quantum systems and here we take this dirac notation. So often I'll be writing here h, this one, as a Hilbert space and then this is a mathematical, say, platform where we work or we described quantum systems. So the first element in quantum theory is state. So we need this notion of quantum state to describe our given quantum system. So suppose we prepare our quantum system, for instance atoms or ions or photons in the laboratory. So we want to describe them. To describe this, our quantum system, we need quantum state. So this is purely mathematical language and physical interpretation remains open. So quantum states is simply a vector in hyperspace. We need one more constraint, it must be a unit vector. So we require the fact that is normalized or has a union norm. So this is simply quantum state. And I will get back to this point later when we want to make probability interpretations. The second element is quantum dynamics. So dynamics means the evolution in time and therefore is a state of transformation. So suppose we have a quantum state in the beginning at time T Zero. So we have a state like this and it evolved in time. And we get the researching state here, t. And this is from of course the initial state. And I'll write here initial state, we need some tool to describe this transformation. So therefore I write here U of t. And Ut shows the transformation from the initial state into the state at time t. And this is quantum dynamics and the quantum action tells you that these dynamics must be a unitary transformation. So literally means that this transformation satisfy the following condition. So that transformation satisfying this condition, it's called unitary transformation. And unitary transformation fulfill this condition. And the next one is quantum measurement. So this has been unique in quantum theory. And to describe quantum measurement, we need POVM. It's called positive operator valued measures. So this means a set of operators, and we write here Mi. And they satisfy the following two conditions. The first one is they are non negative. The second one, they satisfy. So some of all of them must be identity. So we need these two conditions. And from there we have a looseful computing measurement or computing probabilities. The rule is that, Pi means that the probability that we see the detection events on the detector described by this POVM element is given by this formula. Sometimes this is called a measurement action or, Born's rule. So these actually probabilities, I mean proper probabilities in the sense that they are all the negative. And if I compute this number Pi, Then I will plug in this one here, then certainly we have, This equation and where here is identity and therefore we have this guy. This one is a norm of the vector we already have defined this condition and therefore we have this one. So sum of all the probabilities is one and therefore these numbers corresponds to proper probability. And I want to also tell you that this PMIs. I mean physically or in a physical scenario, they corresponds to detector, I mean more precisely they corresponds to the description or characterization of detector. So, I have a first detector, then I will describe my detector as M0, the second one, M1. And the probability that I will have a detection event in the first detector, I will write a P0 and here P1. So far I've explained the state's dynamics and measurement. The last one is observables. So observables are defined or just from the axioms. Observables corresponds to say self or the joint operators or permission operators. So I will write down A is observable and it corresponds to a self or the joint operator. So self for joint operators satisfying this relation. So if an operator satisfied this equality, then this can be factorized into Ai and Mi. And they don't have to be Eigen decomposition and Ais are real numbers, you can be positive or negative. And Mis, they are POVMs. And this is important. So it means that the self or a joint operators, I mean, these are physical observables. They can be estimated from this detection event, repeating many times. Once you get probabilities, you can compute the expectation value of this observable. How does it work? So,suppose that we set up the experiment like this in terms of this POVM element, then you look at statistics. So this is what given initial state, you perform measurement and that we're got a detection event here as well. And now, From the beginning I knew what I want to estimate so I knew what this A is. And from there, I also know this real numbers and I want to know my expectation of this number A with respect to these probabilities obtained from experiment. So this is the quantity I can compute from the data from the experiment, then word. This is what? Ai and Pi. And Pis are given here. So I plug in these numbers. In the end, what I have is this guy and this I mean is the quantity that we have known from the beginning, so this is A. Overall, I can rewrite this guy as sign and A and we call this is the expectation of the observable A with respect to given stateside. In information theory or probability and statistics, we call this quantity as the average of A with respect to this probability or expectation value of A with respect to P. The same happening in quantum scenario, in quantum. Instead of these numbers, this is a quantity we can see in quantum theory, the self or joint operator. And this is the preparation instead of probability, we have a state here. So what are we can sometimes write down here sign, expectation value of A with respect to quantum state. So this is identified like this in quantums here. So observable is precisely I mean the quantity, we can, I mean the general way of describing the quantity that we can see about. I mean we can see about quantum systems. So whatever mentioned what do you do, in the end the quantity you're expecting to have is this guy, expectation value. And before measurement, so before we have our probabilities and these observables, there are dynamics like this and states like this. These are not something we can see directly. All these can be say, inferred or guessed from our measurement statistics or this estimated values.