We now move to two qubit systems. Consider two qubits now, so I write here two quantum systems and I label them, the first one is A and the second one is B. Each of them is described by a quantum states, which is a vector in Hilbert space. I write here H of A and HB, and the competent system state vector of AB, so I write here AB. This will be a vector of the two qubit space and the space is actually tensor product of two Hilbert spaces. Each of them is spanned by the basis 0 and 1, and here the span 0 and 1. The composite Hilbert space here is also spanned by the basis and the basis actually the product of them. This, we can see that the span by 00, 01,10, and 11. Here I wrote the notation, say 0 times 0, then I simply put 00 for convenience. The first system A and the second system in B, then we write here AB. Then I'm in the most general quantum state, a two qubit states in this space can be written as, I mean, the state must be spanned by the basis here, so B. I have Beta, Gamma, Delta. These are complex numbers and we have extra constraint that the vector here must be normalized. This is the most general form of two qubit pure states and important class or classification of two qubit state is entangled and product state. To clarify the meaning of internal state and let me introduce the LOCC scenario. Here LO means a local operations and CC means classical communication. Classical communication is a way of communicating with the parties by each exchanging bits. Classical in the sense that we use bits and quantum communication means we use qubits. Here classical communication is only with a bit. LOCC means A applies local operation here on A side only, so we call local operations. So A can prepare quantum states and many play quantum state as you like. B as well, so here B allow to perform a local operation. Between them, we allow classical communication to prepare stage or to ask or to tell that we solve between them. This arrow is a scenario. Alice can prepare quantum state, for instance, I'd write here Psi A and B may prepare quantum state Psi B. In general they don't have to be same, so it can be any state here. Overall the state Alice and Bob can prepare I write big Psi, which is this guy. This is exactly the LOCC scenario and so to state the form of the states that LOCC can prepare. We can imagine some other quantum states, so I write here I Psi prime which is perhaps this guy, A and B. This is also state that Alice and Bob can prepare by LOCC. Then these stage is called product stage. I made product in the sense that A system and B, they are completely vectorized. As we have seen in the case of single qubit transformations, you can always find a unitary transformation that link this guy to this guy. Why? Because the most general quantum operation for single qubit or general form of inter transformation or universal quantum gate forcing a qubit is simply rotation. This one can be characterized by the block vector and a block sphere, and the most general form transformation is to rotate the block vector within the block sphere, and that will be this guy. You can always find a unitary transformation that mount this guy to here. Which means you can find a unitary transformation, say local unitary transformation from this state to this guy. I write here UA and B as well. For B system, you can always find the single qubit rotation from psi-prime to a phi prime, so we write here UB. Overall, you can find the relation UA and UB tensor. The overall state here we have psi AB and they're the same. You can always find a local unitary transformation, are local in the sense that A perform unitary transformation and B perform unitary transformation. You can always find the local unitary theories that relate this KAIST and this KAIST. Basically product state. Then we call these two states and these two states are local unitary equivalent. Then we can write down here as LU. By writing LU, we can say that these two states, this guy and this guy, they are local unitary equivalent. This means that you can always find local unitary that relate these two states here. This is very important property of LOCC. This is the class of quantum state that LOCC can prepare. We can go in more general form. For instance, as good example would be this state, I'll write here, phi plus and this is a state 0, 0 plus 1, 1. A system and B system, they're both zero and they are both one and one. These two cases are superposed. Then we call this phi plus state. The question is, is this a product state? The answer is no. You can check the state is not a product state. You can start writing phi plus as a product state with some parameter Alpha, Beta, Gamma, Delta, and assuming that these stages products and you can find a contradiction. This is not a product state, and we call this entangled state. More precisely, I mean this cannot be written as psi and phi B for any choice of the local state A and B, so we write here psi and phi. This is clearly not product state and entangled state. Then given a quantum state like this state, in general, two qubit pure states can be written in general like this. Is there a way to find if a given state is a product state or entangled state? There is a simple criteria for this purpose and the answer is actually a Schmidt decomposition. I'm going to explain a Schmidt decomposition. By Schmidt decomposition, SD, there are four factors, some pages vectors that describe this KAIST, but after Schmidt decomposition, we don't need all of them, we only have say just two vectors. You can always find two vectors. You can always find a local basis, U and U_2, and V and V_2 that is equivalent to this guy, so I can just rewrite this one, but I choose different basis. Basis means U_1 and U_2. That these are normal basis they must be also going to and V_1 as well. V_1 and V_2, they should be normal as well. This is also known as basis for A system, and this is also known as basis for B system. You can always write down this state to this form, and then this will be equivalent to Lambda_1, 0, 0 plus Lambda_2, 1, 1, then local, unilaterally equivalent. This means that given this quantum state, you can always find the unitary transformation, U_0 and U_1 for A, and V part as well, V_0 and V_1. By finding the Schmidt decomposition of two qubit state, you can always came up with this guy where these basis are just after the local need to transformation, and then you have two coefficient. These two coefficients is called Schmidt coefficient. We can also say that once you have state Psi and this is described by the Schmidt coefficient, Lambda_1 and Lambda_2. This must be now entangled. If one of them is one, then you have either this guy or this guy, and this is clearly a product state. As long as you have the Schmidt coefficients are non-zero, you have two vectors here and superposition of two vectors and then your state must be entangled. Then, now, the next step is how to find the Schmidt decomposition of two qubit state. You can do something more general. Suppose you have a vector and it can be written as the coefficient C_ij. We can take the computational basis in i and j. Then this matrix, C_ij is actually maybe it's a big matrix and maybe the dimension of system A and B, maybe large, then you can take a C_ij. It's a big matrix. The matrix can be decomposed. Then we have U, D, V, and you can always find this decomposition. Then the elements i_j correspond to C_ij. Then we simply write down as k, and the element of this U unit of transformation is ik, and D is a diagonal 1, so we write down k. We suppose this D is a diagonal matrix with the entries Lambda_k. We [inaudible] and therefore, this means that free conjugate, and should be kj but it's transposed and therefore it's a jk. Now, you plug in these two here, and then we just rewrite this vector again. Then we have a k, and then we have a k here, then we write here k. There are two indices, i and j, and then you can expand i, and U_ik and i. This is for system A and system B. We can write down j and V_jk, complex conjugate and j. This defines a new basis and we can write down K because the summation of i, and here we can write down V_k as well. These are simply the basis and we have a single index. We started with two indices, i and j, but after all, we have a single index at k, so Lambda_k and these two vectors, and came back to this computational one after local unitary transformations. All the properties above the entanglement of this state, we found in this coefficient Lambda_k. Whenever one of them is one, you have a product state. Otherwise, the state must be entangled. This is canonical. Sometimes this is called canonical form of two qubit state or bipartite quantum system. Because once you have this guy, then you can decide fine, if the state is entangled. It's not program or product. All the information about the entanglement can be found in this Schmidt coefficient.