Again, back to the arrow scenario. Initially we had non maximum interactive Psi. Psi is given cosine Theta and sine Theta. Then the state is transformed to Phi plus state, it's maximum impulse state. The first scenario we have considered is the case when Alice applies a single copy level operation. Alice and Bob, they perform a class casket communication on a single copy as well. Then the optimal conversion rate that transform Psi into Phi plus is given this number. But this is not the most general strategy. The most general strategy is to work with this end cubic coherently on a Psi. Then we apply the technique from the information theory, the law of large numbers. Then we found that the optimal rate to transform or convert this one to Phi plus is given by this number. Let's see what are the difference between these two numbers. Suppose this one, just for convenience, let's put p as sine squared Theta. Then we have the parameter p. First we have this number as H of minus p and p. Then the entropy is given like this. Then we have 2p. If p is one-half, then we have one, then this one is Hp. Then this is the line that is 2p. It's clear that this guy, Yn is larger than Y1, so applying this coherent operation, we always get advantages. We always get higher rate of converting Psi into Phi plus states. In other words, this is in fact the number of entangle state. Actually the highest number of entangle state that can be distilled or extracted out of given state Psi. Then we can say that, how much intangible is contained here. That was the main question about the quantification of entanglement. Given this state and given this guy, how can we compare entanglement between these two states? Assuming single copy level, we have this number, but in general allowing all the possibility and asymptotic limit, we have this number. In other words, the maximum number of Phi plus state, which can be distilled from here, the optimal rate is given by H cosine squared Theta. That is the optimal number. But in fact, let me clarify what this number means. This is a function of shimmy coefficient and then shimmy coefficient, so say cosine Theta and sine Theta. Any the basis they take, the local basis, then this shimmy coefficient is in fact the eigenvalue of reduced state of this guy. If I put ab, then I have reduced state a, this is exactly the shimmy coefficient of the state. Then H cosine Theta is precisely the Von Neumann entropy of 4a. How does it work? Von Neumann entropy is defined as minus trace Rho and Rho. Then these tells us the maximum or optimal rate of converting this state into Phi plus state. We can identify entanglement containing this state in terms of Phi plus to here. What does that mean? We want to know entanglement of Psi, and we want to know entanglement of Phi plus. Maybe it is very difficult to say what precisely this measure should be, it may be difficult, is similar a definition. What we can do is to compare entanglement between these two states. This is by LCC. Then we now define, just for convenience or as we understand, this is a unit of information. We can compare this scenario like some comparisons of values. I give you a simple question. For instance, I have a paper, I have eraser, a smaller one. Which one is more valuable? How can we compare the values of these two object? It is difficult. Sometimes this is more useful, so this is more valuable and sometimes it is more useful and so this is more valuable. This is subject to conditions or situations. There's a objective way or the way that all of us and people agree that which is more variable and this is by money. This maybe is three Euros, but this may be one Euro, then this is three times more valuable than this one. In this way we can compare the values, three pieces of papers is equal to one eraser,that's how we compare the values. Here are the same, then the unit is one Euro. So when one Euro is the unit of values, and here we do the same. Let's say this is the unit of entanglement, therefore we put this as well. This may be some number, then we need some number multiply this pie and to get this one and that is precisely s of y. Since this is one, this means s of trace b sine and sine. This is another entropy and this is the China's cosine squared theta and sine-squared theta. This is precisely the entanglement of sine and this is defined or can be seen in terms of cosine and sine theta. The meaning is that given this state, the optimal conversion rate or the maximum or the highest, number of five-plus states which can be extracted by LOCC from this state is given by this number. This is clear operational meaning and the name of this quantity is called Entropy of entanglement. I want to emphasize that entanglement measure is not from the beginning, we start from LOCC scenario and we impose some certain operation meaning. Then we drive entaglement measure, which has a clear interpretations, which is the number of maximal items which can be distilled. That's one remark we can learn about this one. This scenario precisely quantify entanglement of quantum state. The next one I want to tell is to generalize this single copy manipulation. We have seen that this conversion was done in terms of atomic coefficient. Let me compare two states,you can work with cross operators and single copy, conversion and etc. Then this is possible by LOCC with probability one, but the other way around is probabilistic so that we can conclude that this is more entangled than this state. This relation, LOCC order among entangled state can be quantified, the direction can be seen in terms of this shimmy coefficient. Now let me introduce some interesting terminology is called modularization. Suppose I write Mu, which is the vector in terms of Mu 1, and the other lambda which is i and i. We are saying that major rises Lambda if and only if we first have to write down Mu as a decreasing order or non-increasing order, and lambda as lambda, etc. Then if it happens that Mu 1 is larger than lambda 1 and mu1 plus Mu 2, let me keep the order. If this satisfies that we call major rises the lambda and If this happens, then there is a theorem that sine Mu and sine lambda. This direction is always possible by LOCC with probability one, this transmission always possible. Then we can put this 5 plus state in p dimension is always the states that can be transformed any other quantum states. So, this is a channel rejecting the single copy manipulation and entanglement of keeping contacts that can be quantified by a trophy entanglement. It has clear operational meaning. How many entangled state this five plus state the unit of entanglement can be extracted from a given state sine.
