We have simplified expression of qubit states. Qubit state is described by two parameters like this, Theta and Phi. One phase vector here and the angles Cosine Theta over two and sine Theta over two here. This define a qubit state and just knowing two parameters, Theta and Phi, we can characterize the qubit state. Once we define this qubit state, the next thing we want to know about dynamics, how these qubit state evolve in time. To describe qubit dynamics, we need a unitary transformations for qubit state, the important class of unit target transformation is Pauli matrices. These are the four metrics, two-by-two matrices. Identity, which means this guy, an X operation. This is 0,1, 1,0, and Y, 1,0, 0, -1. These four matrices are called Pauli matrices. We always take this base as 0,1 and these bases, 0, and 1 is called standard basis in quantum information, or sometimes we call it computational basis. This is always our reference in quantum information processing. Whenever we write down state and dynamics, this is our reference. We always rely on this computation basis. The next important class of integer transmission is Hadamard transformation. We write H and this is 1,1, 1, -1. This Hadamard transformation is unitary. You can easily check that H and H. In fact this all Hadamard transformation and Pauli matrices are unitary and Hermitian. They are both unitary and Hermitian. For instance, I can write here x, this is in fact x Theta, so this self-adjoint. Or equivalently, you can call Hermitian. This is also unitary because if you check this one, this is the identity. These are the observables and unitary transformations. If you apply Pauli matrices, if you have a zero then x, then you will have one. If you have a one and x, then you will have a zero. This x operations flip the value of the computational basis. Sometimes this is called NOT operation for quantum called qubit state. If you apply hadamard transformation, then you have states 0 plus 1 is a linear combination of them. We write down as plus a state. You apply Hadamard transformation to the state one, then you have minus here. Interestingly, you can write down Z operation as the computational base zero and minus one. X basis is, this is plus, plus, minus and minus. Therefore, you can see the relation between these two guys a in Hadamard transformation. Therefore, this would be Hadamard and Z and Hadamard. This is very useful relation and how we can convert this operator Z into X. If you apply X operation and Hadamard transformation to X operation, then what happen here is Hadamard and Hadamard and Z and Hadamard and Hadamard. Then we apply Hadamard twice, then you will get identity and therefore this [inaudible]. Hadamard transformation provide a useful at transmission between 01 computational basis and plus minus basis. Sometime this basis is called Hadamard basis. The next useful class of single-qubit operation or unitary transformation for single-qubit is phase operation. I will write here pi Alpha, this is one, zero,zero to e i alpha. By applying this guy to this quantum state, you will see the change in the phase effector. We have a quantum qubit state here, and we can see how to transform the qubit state to some other states, so this introduce transformation of oil dynamics of qubit state, and I want to tell you one more thing. The physical theory tells you that prediction, so in the mechanics, if you know the initial condition, so you know where your particle is, and you know the environmental conditions, and then you can make the perfect prediction about the future, and this happens the same in quantum mechanics. If we know the initial state, then we can get the final state at time t, and the dynamics precisely characterized by this unitary transformations, I will write here UT. This is from the Schrodinger equation, so I can write down the Schrodinger equation, as this guy, and this H is a Hamiltonian, so this is observable, Hamiltonian is also observable Hermitian operator, but it has interpretation to energy, so expectation value of this Hamiltonian gives you energy of your given quantum system. This is the Schrodinger equation, then this just shows you the equation of motion above that quantum stage, then if you solve them, you will see the transformation of the states and on the level of quantum state, that transformation or dynamics is precisely described by a distributor transformation, and how they are related to each other, so how this transformation and this Hamiltonian are related to each other. This U is given as this equation, so given Hamiltonian. Then you solve this first-order differential equation, then you will get the solution, and the solution is actually this guy. If Hamiltonian is time-independent and this will be simple. That's the relation, this Hamiltonian or energy, or the observable that gives you expectation value of the energy, it generates the translation or generates the motion, so motion on the level of quantum states. Therefore, this gives you the description how quantum state or system is evolved in time from initial state into the final state on the level of a quantum state. On the level of a quantum state, this is actually the right description and it has the origin from a Hamiltonian, so the point I want to make here is that the dynamics of a quantum state is deterministic. We know that we can profitably describe how quantum state evolves in time, we know the initial state and final state. What is uncertain, or what is the probability is actually measurement part. After then, suppose you have a quantum state, psi, and this is Alpha and Beta, and then you perform measurements. We will get probabilities and quantum theory provides you the right prediction about the probabilities, so I prepare my first measurement. Well, my first detector is this guy and this guy. This one is, Alpha square and the probability that you will get detection event in a second detector is Beta square. We fail to make this perfect prediction about outcomes here, so we never know with certainty which detector shows you that detection events, but we do know about the probabilities, so we do make perfect prediction, right prediction, about the probabilities in terms of losing quantum mechanics.