On November 30th, the US House of Representatives passed a bill that seeks to block a strike by rail workers in the country. These workers have been protesting for months against the cutting down of the workforce and seeking better scheduling and paid leave. Due to the prospect of a powerful strike, the Joe Biden administration was forced to intervene in September and a deal was struck between the employers and the unions. However, the membership of a number of unions rejected the deal, paving the way for a strike. If it was in this context that the House of Representatives intervened, the bill will now go to the Senate. Eugene Porir of Breakthrough News explains the nature of the railway industry in the US and the demands of the workers. Well, the rail strike, which is potentially looming here in the United States, the earliest it could happen would be December 9th after a cooling off period, is a huge, huge element of the labor struggle here in America. If people have a sense, almost everything moves by rail. So whether we're talking about food, fuel, medicine, everything, you know, good, bad or indifferent, including passenger rail, because most of the rail in America, the actual tracks are controlled by freight rail. So we're talking about the potentiality of a total shutdown of all of the freight railways and most of the passenger railways in the United States, which would bring the economy to a screeching halt. And the reason that is happening is because the Congress of the United States or at least the Democratic Party, which is running the lower house of Congress, we should say the House of Representatives, the White House has decided to side with these huge major giant companies, a handful of companies, four or five companies that control most of the freight rail in the United States. Those companies are completely denying the ability of workers to have access to adequate health care. And that's what this is really turning on. It's health and safety issues. So the big issue that is out there is the issue of paid sick days. The workers who are working on many of these railways oftentimes are away from their home for weeks at a time. Even when they are off, they are on call about 90 percent of the time. So it's the type of job that not only are there long extended hours, but it's also the type of job that you can be called in at any moment. And so it's a difficult job to handle things like health care, for instance, without paid time off, unless you want to lose money because scheduling appointments, doing follow-up appointments, if you have an emergency, if you just get sick. I mean, it's really one of the most difficult jobs scheduling wise to really be able to deal with those kind of up and downs. And the railway workers across the 12 unions that make up the broader sort of railway brotherhood, as they're traditionally called in the United States, have access to zero, zero paid sick days. They have no paid sick days. The deal that is on the table that will be imposed potentially by Congress, maybe, maybe not, we'll see, would give them access to three unpaid sick days. But again, that's just three days. They're unpaid and they'd have to be scheduled 30 days in advance. So what this is really coming down to is railroad workers want to have access to adequate health care. They want to be able to have a number, an adequate number of paid sick days. I mean, at one point they were attempting to get 15. They've said they're willing to settle for just four. They just don't want to have to schedule at 30 days in advance. So very reasonable demands by the workers. They also have been raising a number of issues around safety. The trains have become very long, sometimes miles long, but they've heavily reduced the workforce of the of the broader railway companies. So they're making about 85 percent more profits as opposed to the year 2000. But they're doing it with tens of thousands of thousands of fewer workers. So workers are working longer. They're working harder. The companies are making much more profits. But because they're working longer and harder, there are more opportunities for them to get hurt, to get sick or anything of that nature and they have access to no paid sick days. So this is what it's going to come down to. It would cost $688 million to provide 15 paid sick days for all the railway workers. The railroad industry made $20 billion in profits in 2020 and 2021 alone. So it's pure corporate greed. It's the pure desire of the railway companies to make the most amount of money possible, no matter what it costs the workers in terms of health and safety, family or whatever it may be. So that's the basic situation that we have here. The possibility of a total rail shutdown over the fact that workers are not granted one paid sick day off despite the fact that companies made $20 billion in profit in 2021. So the basic demands that the workers are presenting here are around their ability to essentially stay alive. Right now they have a situation where workers are and there's a lot of different types of workers. And that's an important thing to understand about these railway brotherhoods. The ones that are at stake, it's about four unions that represent the vast majority of railway workers, about 60,000 out of about 115,000. And one of the things that they're most concerned with is that since their schedule is so inflexible and so random that they're not able to actually access health care without having to take unpaid days off and thus lose income. So the main demands that they were putting forward were, you know, the normal demands that were out there in terms of people when people think of unions wage and hour demands. The contract did include a 24 percent increase over, I think it's a five year period, but the point in wages and a $5,000 bonus, but as the unions pointed out, that didn't even keep up to inflation. But the workers did not make that the sticking point issue, the issue of wages. They really made the issue of health care benefits and safety, the sticking point issues. So from a health care perspective, what workers want is access to paid sick days. They're willing to settle for about four paid sick days, although originally they were seeking 15 paid sick days. The company is only offering three unpaid. The companies are offering three unpaid sick days that have to be scheduled 30 days in advance. The Biden administration, Nancy Pelosi, and the Democrats in the House of Representatives are backing three unpaid days scheduled 30 days in advance. So back in the company's 100 percent workers are also have been raising in the alarm about the increase in safety hazards. The trains are getting significantly longer. They're working with fewer and fewer workers over time. The workforce has been dramatically cut on the rail. So even though profits are going up for the companies because they're able to ship more faster and pay people less and work the people they have longer and harder to do it. It's creating bigger concerns in terms of the checks on what's happening on the trains, the possibilities for derailments, the possibilities for workers to get hurt while they are doing the various work along the trains that was once done by multiple people, sometimes being done by one person in certain situations. So there's a lot of dangerous hazards that are there as well. And those are issues that they would want to be addressed. Those issues also totally unaddressed. And the bosses are claiming that there are no safety issues whatsoever with them making these trains longer and longer and having fewer and fewer people to look after them, despite wanting to also run them at faster speeds. So, you know, that's really what workers are laying forward. But they are willing to accept quite a bit. They're willing to accept wages that are below inflation in terms of their increases. They're willing to accept the fact that there's not going to be a 100 percent settlement in terms of the issues of safety that exist.  And we're willing to go back to work here. But ultimately, the greed of the corporations is meaning that there is the possibility for a strike on December 9th after the so-called cooling off period.


the Bible does not consider a baby in the womb less of a person. As a matter of fact, if you are a baby in the womb, you are considered, according to the Bible, to be a born person. In Matthew 1, when it says that Mary was found to be with child, the word that's used, there's a word in gospel, which means in the womb. That's just only pointing to the location of the person. When Joseph is told what was conceived in her, the word that's used for conceived is the word Gennethon, which is from Genomite, which by the way is the same word that speaks of what we are born of, when we're born again, or when we're born of blood or a flesh. And so to be alive, this same word is used of a baby in the womb. And the Bible describes that this baby that's born, that is alive, that is a human being, is just simply in the womb. But the Bible considers this baby to be the same sort of being that you and I are.


Welcome to video eight of this series on the theory of relativity. Here we want to investigate how we can form a yardstick for space-time measurements. At the end of the previous video, we had seen that the methods of differential geometry appeared ideal for expressing Einstein's idea that gravity represents curvature of space-time. But differential geometry is built upon the assumption that between any two points there exists an invariant distance and that in any coordinate system we can express that distance in terms of coordinate differences. However, special relativity has shown us that two reference frames in general won't agree on space or time measurements. The key insight is primarily due to Herman Minkowski. A translation of his 1908 presentation titled Space and Time appears in the Dover publication book of reprints that we've referred to several times in this series. In his introduction, he states, the views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. Well, let's see what he meant by that. Recall video two where we developed the equations of special relativity that relate time and space measurements and two inertial reference frames that are moving relative to each other with velocity v. Here's Einstein's version. We prefer to use units in which the speed of light is one so we can scratch the c-squared terms out of his equations. One system uses Latin letters t, x, y, and z for time and space, and the other uses Greek letters, tau, psi, eta, and zeta. Our problem is that the Greek time and space coordinates, tau and psi, are a mixture of the Latin time and space coordinates, t and x. To see what Minkowski did, consider an analogy. Suppose centered on some particular point you set up a system of coordinates. For a second point, you can then determine the coordinates x and y in your system that uniquely specify the location of that point. Suppose someone else chooses a system rotated with respect to yours. The position of the second point can also be expressed in their system, say by coordinates psi and eta, but clearly your x and y will in general not be the same as their psi and eta. The coordinates of a point are specific to a particular coordinate system. However, the distance between the points, let's call it s, has a physical significance independent of any coordinate system. Anybody can take a tape measure and determine that distance. Specifically, in the x, y system, you can draw a triangle with sides x and y, and the hypotenuse will be s. You can figure out s from x and y using the Pythagorean theorem, s squared equals x squared plus y squared. In the psi, eta system, you can also draw a triangle with s as the hypotenuse, but with sides psi and eta. The Pythagorean theorem will again give you s, but now in terms of psi and eta. s squared equals x squared plus y squared is the equation of a circle of radius s, and so is s squared equals psi squared plus eta squared. Any point on the circle in one coordinate system will also be on the circle in the other, although with different coordinates. It doesn't matter what angle theta a circle is rotated by, it always looks the same. We say circles are invariant under rotation. The equation of a circle of radius s doesn't change regardless of how we rotate the coordinates. With the little trigonometry, we can write each Greek coordinate as a combination of the two Latin coordinates with factors of the cosine and sine of the rotation angle theta. If you sum the squares of the right side expressions, you find the reason it reduces to x squared plus y squared is because for any angle theta, cosine squared theta plus sine squared theta equals one. Greek and Latin coordinates are related in Einstein's special relativity equations look a lot like the rotated coordinate expressions with beta, the dilation factor, and beta times v taking the place of cosine and sine of theta, and the plus sine replaced by a minus sine, similar but different. The sum of squares of the Greek coordinates does not equal the sum of squares of the Latin coordinates. In this case, the key fact is that beta squared minus beta v squared equals one. Because of this, it's the difference of squares that is the same for the two systems. We get s squared equals t squared minus x squared equals tau squared minus psi squared. The graphical representation we developed in video two shows the similarity and difference between rotation and the special relativity transformation. Here are the blue lines that note the Latin coordinates and the red lines, Greek coordinates. The green lines correspond to the speed of light. The relation between Latin and Greek coordinates looks like a rotation except the two axes are rotated in different directions. It's more of a squeezing and stretching operation. Still, we've found a coordinate relationship that looks the same in all reference frames. For rotation, s squared equals x squared plus y squared is a circle of radius s. What does s squared equals t squared minus x squared represent? If you choose a value of s, you get a plot like the black curve shown here. This is a hyperbola. This particular hyperbola has s squared equals five. Since three squared minus two squared, nine minus four equals five, the point t equals three, x equals two, or minus two, should be on the hyperbola. In the blue Latin system, if we go three units to the right and two units up or down, we indeed fall on the hyperbola. If we do the same thing in the red system, three units to the right and two units up or down, we again fall on the hyperbola. In special relativity, it's hyperbolas that are invariant. So, s squared equals t squared minus x squared. But what does s represent? Let's plot a patch of the tx coordinates in blue and the tau xi coordinates in red. The green line shows the speed of light and the black curve is a piece of our hyperbola. If the two events happen at the same place in the blue system, then x equals zero and we simply have s equals t. In the blue system, moving along the x equals zero line, it takes one, two, three, four units of time to reach the hyperbola. In the red system, this would look like five units of time. Likewise, if the events happen at the same place in the red system, then xi equals zero and s equals tau. Moving along the xi equals zero red line, we travel one, two, three, four units of time to reach the hyperbola. In the blue system, this would look like five units of time. Minkowski called s the proper time between two events. It's the elapsed time as seen in a frame where the two events occur at the same place. The invariant hyperbola has the significance that any clock leaving the origin at any velocity will reach the hyperbola in the same s units of time as measured by a clock in that system. Minkowski's proper time idea also gives us another way to understand the twin paradox as what we'll call the principle of extremal aging. Suppose in some reference frame we put a clock at the origin and let it sit there for two units of time. This defines these two blue events and the clock naturally moves between the two. Since x doesn't change, the proper time is just t equals two. Now suppose we choose another event at a different position in space at time t equals one. We move a clock from the first event to this intermediate event. The proper time it will record is s one equals the square root of one minus x squared. From there we change directions and move it back to the final event. The proper time of that interval will also be square root of one minus x squared because in each case the clock moved to distance x. Since one minus something positive is less than one, the two proper times s one and s two are each less than one. So the sum is less than two, which is the proper time on the blue path that corresponds to inertial or natural motion. We have the result that the natural inertial motion between two events is that which results in the longest elapsed time as seen by a clock that moves between the two events. This is just a twin paradox. There the earth twin was the one who moved naturally, that is without changing velocity, between the two events and ended up older than his space twin sister who experienced acceleration. Now for the flat spacetime of special relativity, Minkowski's invariant proper time is an interesting way to look at things, but we're looking for a yardstick for the curved spacetime of general relativity. Ah, but now back to the happiest thought of Einstein's life. If a person falls freely, he will not feel his own weight. So even in the gravitational field, proper time will provide a yardstick for a small freely falling reference frame. Any other reference frame is just a transformation of the freely falling frame so proper time must be the yardstick we're looking for. The only difference is the small caveat. As we've seen that means that we have to limit ourselves to small differential displacements.  But the final result follows the same principle. Consider earth's surface. Gravity pulls things down. Suppose a clock starts at t is equal to zero and we let it rest on the surface to say t equals 30 seconds. This clock will feel its own weight so 30 seconds can't be the proper time between these two events. After proper time, we need to throw a clock from the first event so it freely falls to the second event. At every instant of time, it's moving according to special relativity. By the principle of extremal aging, the proper time s that it shows will be longer than the time t on the clock that felt its own weight. This then is the yardstick of space time, flat or curved. No matter what the gravitational field, you could be near a black hole or in between two rapidly orbiting neutron stars. Remember, throw a clock from one event so that it falls freely to the second event. The elapsed time recorded by that clock is the proper time between the two events and all reference frames will be able to calculate this time using their metric tensors. We've seen that light plays a central role in relativity so we need to look at this special case. In special relativity, the speed of light, which is one in our units, is represented by the green line on this graph. Light moves such that in say three units of time, it covers three units of distance. In general then, dx will be plus or minus dt, plus or minus because light can move either up or down or left or right and so on. The resulting proper time is therefore zero. The proper time between two events connected by array of light is always zero. If you could move at the speed of light, apparently you could go anywhere in the universe in zero time, at least as measured by your own clock. We now finally have the complete general theory of relativity, at least for empty space outside of matter. And we can write it all down on one page. You blanket all of space-time with four coordinates, the first three for space and the fourth for time in any way you want. The catch is you cannot assume these coordinates have any specific physical meaning. We'll have to let the theory tell us what they mean. We might say that no prior geometry can be assumed. Then the generalized Pythagorean theorem allows us to convert small coordinate differences between events into the corresponding invariant proper time. The change in a coordinate x over the change in proper time s, we interpret as a space-time velocity u, the tangent vector of differential geometry. Particles move along geodesics, so the geodesic equations provide the equations of motion by telling us how to change velocity or space-time direction as we move along the path. The draw-square test for space-time curvature generates the Riemann tensor. Summing various terms of the Riemann tensor produces the Rieschi tensor that measures how the volume of a dust cloud changes as it falls in a gravitational field. Einstein's equation of gravity are that the dust cloud volume doesn't change, at least for a short time, so the Rieschi tensor vanishes. Finally, this gives us the equations we need to solve to determine the metric coefficients of the generalized Pythagorean theorem. And from there we can calculate how any object, and even light, will move in a gravitational field. We're now in a position in future videos to work out in detail the incredible predictions of Einstein's crowning masterpiece.


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Hey, it's Bridget. Welcome to Above Life Channel. The purpose here is to inspire your spirit and to fill you with hope I'm working in some hot conditions. It's unseasonably hot right now in the fall here in Minnesota while I'm recording this in the greenhouse and it's the weekend so I need to kind of sneak it down to get some quiet time so I can actually channel for you. So I'm in the greenhouse room and it's hot. You might hear the fan in the background or whatever, but I'm trying to battle with like the sun. So it's like reflecting off the metallic coverings and the awnings. So it's a little weird. I get it, but that's just the way it's gonna have to be right now. So Yeah, let's channel. We're gonna channel today. So if we're gonna talk to you, I think I want to talk with Louise Hay. We've heard from her before. She is the former owner of the Hay House Publishing Company and she is someone who's got a lot of healing energy. She's been through a lot. She's a trailblazer and divine feminine for sure. And I want to talk with her and connect with her and ask her for some advice and guidance for us during these really tender times. There's really tender times. I personally feel a huge energy shift as an empath into the fall. For me, I really love the sun and the energy of the heat and the warmth. But where I live, there's seasonal changes that are pretty dramatic. Yeah, pretty radical. And it's getting harder and harder the last three years to really kind of be okay with that. So it's affecting my energy already and I'm trying to be positive and interact with the season change and the shift of energy proactively. So I want to talk with Louise Hay. No, not about seasonal affective disorder, but about being an empath and understanding the energy flows. Like when there's a change, when you're dealing with a change, you got a change of season, you got a change of a job or maybe not outcome-based. Let's do it this way. Let's ask Louise about change related to our personal power, the ways that we want to interact or engage with the circumstances or situations in our lives. Yes! All right, high five, you guys, high five. All right, so yeah, I know there's so many different patterns going on here, light going on here. Just pretend we're at the thrift store having a channeling session. Pokedots, you know, do something. All right, let's just talk. I don't know why all of a sudden my ears are like really itchy. I don't know if they got sunburned. I hope not. I was walking outside with all the sunscreen on my ears. Okay, okay, okay. Sunny place problems.  I have no idea why. That's very strange. Okay. All right. All right. Focus, focus, focus, focus. Let's talk about change, Louise.  She's such a lovely soft energy. That's why I need that today. Maybe do you need that too, a soft energy? Let's talk to a spirit guide that's soft. Soothing, nobody intense. Louise say soothing like a healer teacher. Okay, feel her energy. Do you feel that? Very kind of archangel handle like kind of got a splash of spice like archangel area and gentle, but strong, powerful, leaderful, kind and nurturing, loving that kind of thing. Give us some advice about change when we want to lean into what life is offering us and really find our power and a change in the change process. A lot of internal, she says, a lot of internal work is happening. She says, Bridget, this isn't just for you. There's a lot of internal work that is happening for many of you, she says. At a level that has been like something that I have not seen before, very unique to the time that you are in bodies on a planet. Remember, your bodies, the masses that you are, are not as well suited for the vibrational changes that occur in the atmosphere with the planets, astrologically and also quite metaphysically in other ways, like with the climate and the plan and the the weather and the elements and also the people. And so there is definitely this challenge that is happening because of a, it is not a break, she says. There's no break, there's no breakdown, there's not a problem. There's a series of problems that are being revealed and unearthed, but there already always have been there. And if you're an empath, which you are, you have felt the problems before and it has been something you've held inside and now it's outside external. So guess what? You can then just bring it to the dump.  We can let go of that, she says. We can bring it to the dump, which basically means bring it to a spot in the earth that is allowed or created as this. It literally looks like a bowl for repurposing and composting. So dump it, dump it. She says, like an altar of trash, energetic trash. Bring your energetic trash here to the earth and it's like an altar and it's like this bowl and we're going to have a big bonfire and we're going to burn all that and clear it away and it's not toxic.  There will be freedom not just for you, but for all. So this is maybe why I've been thinking about having a bonfire. I haven't had a fire in a couple of months you guys and I've been thinking about it and so maybe this is why. It's a good time to do that. There are great times by the way you guys during the month, during the year, during the seasons to purge, to clear and to go in deep within yourself. So on the new moons, go in deep, spend some time with yourself, connect to your desires, your wants, do a check-in. Where are you at right now? New things may have popped up for you. You may have been changing and not realizing how much you've changed. So now your tastes are different. Your food needs are different. Your, the things you like, the kind of music you listen to, the clothes you wear might be different and that's fine. That's what happens. Okay? So that's new moon. Full moon is clearing. Purging, detox, release, let go. At those times you can use that for clarity and clearing and release. Also the beginning of a month is a fresh start, a new start, optimism and a moment of energy. The ending of a month is a release and clearing, release and clearing. Ability to let go of any baggage, not just during the month, but any time in your life and something that is a reoccurring thing that's been happening maybe recently and a relationship or at work or in your own thoughts, in your own mind. Clear, clear, clear. You can at the end of the month. It doesn't mean it's going to be magically better and you're one and done for clearing.  The dirt comes back just not quite as bad. Whereas if you never vacuumed or swept it would be nasty. So use those natural times for that clearing. You can also use astrological things like retrogrades and different signs in different places based upon your astrological chart if you're into that. And then also seasonally. So right now I'm feeling that a lot, a great deal, a great, great, great deal. Okay. So Louise, continue please. Let me take a breath. Oh, that's kind of interesting. I literally heard the word money and then I heard the look at the sky. Interesting, like money is falling from the sky. So it's not those kind of just floating, almost like a kite. Like, look, this is very strange clairvoyant image, but I'm going to share with you exactly what I see because I think it's important to do that. So you understand how you make the links for you because it's happening for you guys now anyway too. So I literally see a kite that looks like money, like bills. Okay. And it's like floating in this beautiful picturesque blue sky. So there's a green kite, green money and then this beautiful sky. So I don't, is there, and it's weird because it's not even on a string. It's like a stick. It's like a stick, like a puppet show. Oh, being manipulated being, there's a performance that's happening, not manipulation. She corrects me. She says, no, no, no, no, no, that's not the word Bridget. That's not the word. There's a performance that's going on right now. So what does money mean to you? What does it represent? We're not talking about actual money. This is not a literal meaning. That's psychic stuff, signs, metaphors, images, not literal. Most of the time people, that's why intuition is not easy because it's hard to believe and make the connections to the metaphor, the signs and symbols, what it means to you uniquely. That's why I say, what does the money mean to you? What does that symbolize or represent for you? So for many of us, the response would be something maybe along the lines of general, this is just general, would be security, safety, the opportunity to be independent, sovereign and to make your own choices, freedom. Money represents freedom. So, or value and worth, self-worth. So Timmy, I just clicked in on freedom and self-worth, boom, boom, those two things. So for me, that's what comes forward. So forever comes forward for you. That's what you grab on to. You could write it down in your journal and then you could journal about it more later. Okay, so okay, so the money is represented of, I'm going to say freedom. Yeah, like sovereignty, like I'm getting that vibe. Self-worth, not quite as much, it's a part of it. It's behind the scenes for me, but the first thing is freedom, freedom. So let's use that. All right, so freedom. And the clear blue sky, as she says, you're always free. You always are. Freedom comes through the expression of your choice.  These are the things that you have freedom and are expressing your freedom by choosing to spend time in these ways. And she says, I understand, I know, I do understand, I can hear the structure and the formalization of the mind saying that, this is Louise Hay, she's saying that I can understand the structure of the work and I have to work and there are things I have to do on the tasks and the to-do list and the structure in order to pay the bills, etc., etc. I know that there are expectations of you, she says, that you have chosen based upon that job, that family situation where you live, the career, etc., you have made choices that have led to this place, that have created the job, that also then affords you the opportunity for freedoms. So maybe going to the job at a certain time or having to do the certain tasks are not freedoms, but the opportunity to be at the job is a freedom and to choose a different job is a freedom. It might not be an ideal job, it might not be an ideal place to live, it might not be an ideal X, Y, or Z, but you have the freedom to choose those things. And she says, it's not a freedom, instant manifestation, that's not what freedom is about, that is not what this is about, that is not the meaning of this or the intention, the intention is to understand that change is a natural, ever flowing, always occurring process, but you don't recognize it is happening so often until it shows up in the manifested world. But then to look back and recognize the series of choices that created this moment where you're at, shouldn't make you feel bad or judgmental about yourself, but it should give you the insight to understand that you are then free to start another chain of change, another chain of intentional manifested choices linked together to create a path or walkway to the next thing. So we take Bridget for example, okay, she likes a warm weather, okay, likes warm weather and she knows this so there's an awareness and yet she chooses she lives in Minnesota, she was born here and throughout her experience and many times in her life, she could take a job someplace else, she could have went to college someplace else, she could have married someone from someplace else and she has not and the someplace else could be the warmer place. However, this was a recent realization in the last three years, she's changed, which is normal, she's acknowledging her body likes the warmth. Therefore, now that she's an awareness of it, it's happened for a while, it's not a passing thing, it's a present thing. She has the opportunity to start to actually think about what it might look like to live in a warmer climate. She has thought about that people, I'm like Florida, there's some things I'm not really fond of about Florida just at the moment, so I will be okay here. However, there are some things I could do. There are definitely some changes I could make that would get me closer to the energetic realization or manifestation of an actual move to Florida and there's different ways of living in Florida. It doesn't have to be a move outcome, this is the end result. It could be a timeshare scenario where I'm there three months out of the year. There's lots of choices that result in being in the warmer climate, but the goal and the vibe is the warmer climate. That is really good. So there's little things long way and then she literally showing me like I'm getting your TSA pre-check so that when you do travel, you can whip in and out of there. I'm like okay, because I don't have it, I should have it, I travel enough to have it, I have in the passing way and then she's also like, you know, the passport is something you've been thinking about. That's an indicator of change. If you, I have to renew my passport, that would help because then if I wasn't Florida I could go on cruises or if I needed to go to other warmer places like the Bahamas or something, the Caribbean or something, I could use that. So that's interesting. So she's just showing me little steps that I could take, little changes I could start to make and I mean there's so many other things I could think of. I don't need to have a project plan or anything but that but she's making me aware of this. So it starts as a thought or an idea and it's consistent. It kind of keeps coming up. So pay attention to that when it keeps coming up and then recognize, ask yourself legitimately, is this something I'd actually want? Is this something I could see for myself and I can totally, when I close my eyes, I don't have to close my eyes you guys. When I use my third eye I can see myself in a condo, in a condo on the beach or near the beach where I just walk a little bit and I'm on the beach where I can see the beach ocean will be really great so might as well think of it the best way I possibly can and on the beach. That's what I see but and I say condo on purpose because I'm like I don't want to take your post up. So see I'm already like woohoo. I happen to have a super Florida shirt on right now. Oh this is awesome. That is a great example. Thank you. Okay so the changes that you make your life looks the way it does now because of changes that have been occurring below the surface all the time and if you start to feel restless energetically, she's saying the emotions are indeed a signal for you. They're not to be ignored. They're not there to cause problems or make waves like be a troublesome energy or distraction. The emotions are not meant to be a distraction. Oh my gosh, talk about this. Talk about this. Talk about related to change especially. Emotions. Water. This is Disney World. That's in Orlando, Florida my friends. Oh yes we are manifesting as we were thinking right here. I don't actually have a trip planned for 2021 anymore. I've been there twice this year already and so 2022 I got to plan some trips so maybe I'll actually like see my ears are itchy now all of a sudden. Okay I need my hair cut too. Oh my god I gotta wait like two weeks. Okay okay okay. Talk about emotions and our feelings and how they're not a distraction. This is interesting. I'm gonna pull up a chair and listen to this one. She's like oh that's sweet. She just literally said you're so cute Bridget. I'm like um okay I'm a professional psychic. Thank you very much and a life coach. A great intuitive life coach. You're telling me I'm cute. Okay. Hey it's Louise. Hey okay so okay. Emotional freedom is something that many of us strive for although we don't realize it and we do use emotions connected to our trauma and because of that there's a fear of emotion and emotion is connected to your intuition. Emotion is connected to an alliance between your heart and your spirit as you've talked about in other videos and it's important for people to recognize that emotions are not meant as a distraction. They are not meant to create disruption in your life so that you feel punished or defeated or like you're doing something wrong. She said emotions are not meant to be judgments. Oh that somebody needs to type that in the comments because I need to remember that. Emotions are not meant to be judgments. Emotions are not meant to be judgments. They are connected to your freedom. Your freedom. The freedom to change. The freedom to change. The freedom to change. Emotions are not meant to be judgments. They are not. We use them as distractions. We've misunderstood the emotional energy and freedom of emotion and expression externally as something that is not allowed or that we have to contain or tame and that is not the case. This is why you're seeing explosive emotions for people.  People have held energy in for so long anger and hurt and pain and it comes out in this ferocity that lights fires and destroys things and trashes people. And she's saying it's over it's like this overindulged overindulgence in hiding your emotions overindulgence of hiding emotions. That's interesting overindulgence is like this almost like I'm behaving badly. And it's like the excessive let's say it that way the excessive use of hiding emotion for years and years and years and years trying to fit in trying to belong trying to find value self-worth from other people seeing that you have your act all together and the truth of the matter is nobody has their act together. People pretend some days are good days and you do other days are not so good and you don't but guess what it's all part of the whole and she says emotions are not meant to derail you or diminish you or judge you. Emotions are meant to free you from toxic energy that is holding you in the past toxic energy that is holding you in the past. Emotions are meant to free you from toxic energy that is holding you in the past. Your most powerful place is in the present and being loving and kind to yourself requires you to be at a new level of psychic awareness and personal development she says. The reason why so many people are getting very in tune and intuitive and are asking difficult questions and working on themselves right now is because that's what needs to happen in order to understand that emotions are not here to harm you they're here to free you and they support your change process they support the freedom to change and it's happening it's inevitable all the time you are evolving and changing you just don't always recognize that and you don't consciously participate in the process and she's saying consciously participate in the process of change constantly show up for your own change if you notice she says if your emotions are getting hot uncomfortable hot like literally you guys I am like so sweating right now it's I'm dying it's already it's hot in here but I'm like I think it's the energy like change I need change no no but then at the same time it's like slow your roll slow your roll so be aware be present consciously I literally feel dizzy all of a sudden oh my goodness constantly participate in your own change process show up for it and she says don't wait until it's things are flooding over your everything's overflowing and just making a freaking mess don't wait don't wait and then try to sort things out and she's like lean in she's like just step right in she says and let the emotions be there and don't tie them to one thing don't expect the emotion to fix something or give you some magical answer it's just there to flow give you information and let you practice being free and allowing yourself not to attach to that one emotion or that one part because when you attach then you go into the judgment your mind steps and it starts to judge you to try to put things together to make sense to make a plan or to say how stupid these things are so you need to stop feeling because that's what the mind's doing to try to protect you from what future pain it's future pain okay so let's be present with Louie say that that's like profound I'm like whoa her energy is usually like really kind of chilling mellow and then maybe that's why I feel like literally this dizzy almost like whoa lots of crown chakra do you guys feel that lots of crown chakra work purple crown chakra work let's just take a few minutes to let that energy seep into your brain clear out some of your cobwebs in your mind recognizing some of your thought processes that do not allow the emotional freedom that you need emotional freedom by the way doesn't mean venting to other people letting them have it or telling them what you think because nobody needs to know what you think necessarily but how you feel honoring your feelings in a relationship or in a situation or circumstance could go miles in freeing you so that you can see the the the the pebbles on the path the little stepping stones for their change processes right for making your relationships better for giving you more better opportunities at work to delegate to share to have life be less stressful and not as intense and really supported like there's opportunity here this is not all emotions aren't going to come through and be all awfully intense all the time at first they probably will be if you've been holding them back for a long time understand that when emotions come out they're not usually face value either so anger is really oftentimes connected to grief a fear of loss fear in general or sadness I get mad instead of getting sad because it's easier it feels more powerful and so the truth is under my anger oftentimes it's disappointment sadness sometimes regret sometimes this kind of being hard on myself energy and so giving yourself the opportunity to be present with the emotions I think is important and it's not to feel all the bad stuff it's to also be present with the good stuff like that oh my gosh I feel creative and great even though I'm maybe I have a friend that's dealing with some really heavy stuff which I do right now with a parent like end-of-life kind of stuff which is really sad and I'm in there without feeling her doesn't mean that I can't have moments of happiness and creativity and excitement and prosperity and abundance and fun and humor and neither and she can too like we can talk and still laugh about something crack a joke you know make a funny share a memory and think something hopeful about the future and talk about hey remember remember when we went on this last trip wouldn't it be great if when we go to Paisley park again and have fun and the after you know after party the dance party stuff and you know that kind of thing like all of it belongs here because you belong here so that's the vibes I'm getting from miss louise say from the afterlife always a pleasure always informative I can't wait to listen if there's another thing that comes in and that's courage that's courage and my ears are itching what is up with this itchy okay I don't know I'll figure that out maybe I need to listen more if I need to pay attention maybe my ears sunburn it might very well be I have sensitive skin even when it's the fall I still need to use sunscreen I'm gonna need to practice that sunscreen sunscreen especially if I'm living in a warmer climate all year long at some point on my journey right change undertones of change you know it's not about the end result the outcome the no no no it's about acknowledging you're in the process and feeling the hopeful positive energy of the process as much as you're feeling the clearing in the depth and the shadow side of it it's a wholeness you are whole just because you feel bad doesn't mean you're a bad person okay it means you're in a place honor yourself love yourself lean in cuddle up and recognize it as it's a changing ever changing process so you're not gonna feel bad forever it might sometimes feel like that but it gets better and there's different times in the day that you do feel better even if you're dealing with heavy stuff okay so okay you guys thanks for watching above life channel I hope we've inspired your spirit today and filled you with hope before you leave make sure you subscribe never miss a weekly channeling video they come out on mondays by the way if you don't get a notice come here on mondays you'll see it there's also a facebook page for above life channel that I post every week when this video comes out sunday morning coffee is a podcast I offer it's an audio podcast it's also here on sunday so sunday and monday are the active days here at above life channel don't forget to be here all right so if we've inspired your spirit and filled you with hope good check check now what's next well what's next is for you to live your life this is your life after all so live it just live it thanks for being here


OK, we were mainly, if not that everything was focusing on bipartite entanglement, OK? So today, we will continue with the bipartite entanglement, and we will see that many of the things there can be generalized to multi-partite systems. Can you hear me? Better now? OK. OK, so what we were discussing yesterday was, first of all, a separability problem. So we were looking at mixed states, and then we moved on to pure states. Sorry. And we were convincing ourselves that entanglement theory is a resource theory where the free operations are LOCC. So everything that we can do locally and the free states are the separable states. So just to remind you, LOCC means that we can apply local operations on LA systems. You can apply some operations. You can communicate some result to Bob. He can then, depending on this outcome, make some measurement, communicate the result back, and so on. OK, so this is what we call LOCC. And entanglement theory considers this for free. So this doesn't cost anything. If you want to do something more fancy than that, then you need entanglement. We were saying yesterday also that you can measure entanglement. And if I'm not wrong, this was point B. And we saw, for instance, that one measure of entanglement for pure state, for pure bipartite state, would be the entropy of the reduced density matrix. So now, the next question that we should address is, how can we sort states? So this was actually C. So now we are at D. How can we sort states or states according to their entanglement that they contain? Well, we know already this answer because we have discussed yesterday that if I can go from some state via LOCC to some other state, then the entanglement in that state has to be larger equal to the entanglement of phi for any entanglement measure E. So this means that LOCC induces some kind of order in the set of states. So if I can go from phi to phi via LOCC, it means that psi is a better resource. So I get some kind of ordering. The ordering, however, is only partial. And this is important. It's only partial because there are states where, so there are pairs of states, psi and phi, such that you can neither go from psi to phi via LOCC, nor can you go the other way around. So nor can you transform phi to psi via LOCC. So this means that those two states here are not comparable. You cannot go from one to the other. So this means that you cannot say that this is a better resource than that or the other way around. So you cannot order them. And this is why we have a partial order in contrast to a total order where all two elements are related. Partial order means that it's not for the whole set of states. But for some of them, you have that. And so what we have learned yesterday is that LOCC induces such a partial order. And regarding entanglement theory as a resource theory, this is the most natural choice. It's the only thing that we can say, because LOCC is for free. And so this is why this state is for sure more a better resource than this state here. Now, what we have also seen yesterday is that this implies also that if two states are LU equivalent, which I write like this. And this means that, as we said also yesterday, that I can write, can find in a bipartite case. So I have two local unitaries. The U1 and U2 are unitaries. And they act on the first and the second system, respectively. And so I can transform, say, psi into phi with local unitaries. If this is the case, then we have seen that e psi has to be equal to e phi for all of them. Because simply because I can go from psi to phi with local operations. And I can go from phi to psi with local operations. I can undo the local unitary, because it's unitary. And so this means that we have this inequality in both directions, so we have an equality for whatever entanglement measure we have. OK, now you might wonder whether the opposite is also true. If this is true for all e, is it true that then the two states must be a LU equivalent? That's a very interesting question, not so easy to answer. Especially, we need some more information. We need some more definitions and knowledge about entanglement, monotones, and so on to be able to answer that. And if we have time, then we will come back to this question. So is the opposite also true? OK, so far we know. And this is important that LOCC induces this partial order. And this is also the reason why one studies LOCC, because this is a way of telling whether a state is a better resource than another state. So we get an order in the whole Hilbert space, which is, in general, very large, as you know. So this also means that, of course, we have to study this LOCC. So let's discuss what LOCC is. OK, so very weak. And roughly, we were talking about this already yesterday. So let's do it more precise. Because I have a system. I have a part A that holds some system. Part B that holds some other system. Now, what is the most general transformation that they can do if they are only allowed to act? So if they're not allowed to use entanglement? Well, Alice, without loss of generality, would start. And she applies some P of M. So she applies some generalized measurement with some number of outcomes. She obtains some outcome. Let's say that she obtains outcome I. So she has implemented AI. And she transmits this information to Bob. Now Bob uses this information. And depending on I, he applies some measurement, which I denote now by Bi, and obtains some outcome J. So he applies this generalized measurement, and he obtains some outcome J. Now he transmits this information again back to Alice. Now Alice, depending on I and J, chooses a new P of M and gets some outcome K, and so on, and so on. So this continues. Now, the problem is that these have to be, first of all, local measurements. So we have to make sure that they obey the completeness relation. And second of all, one can show that this doesn't end. So there are protocols. There are problems where you actually need infinitely many rounds of classical communication. So this makes this very difficult. This is why starting LOCC is so difficult to do. Let's write down what I mean by why is it difficult. Let's write down what is the whole operation. So what is lambda? What would be the corresponding CP map? So this lambda LOCC corresponding to our protocol would be what? If I act on some state row, I would have what? Well, I start with row. I first apply AI. Then Bob applies B, J, I, depending on the outcome of Alice. Then Alice applies A, I, J, K. Bob applies B, I, J, K, L, and so on, and so on. There's a tensor product. We go on, and we have here a sum over all I, J, and so on. And here's our state row. And the whole thing, dagger. So this is our transformation. This would be our map. Now, there are two things that we see. One is that given such a map to show that this is LOCC, it's very difficult. Because you have to show that each of these operators here factorizes in operators where each of them is a complete measurement. So AI has to fulfill the condition that AI, dagger, AI is equal to the identity if I sum over I. The same holds true for AI, J, K. If I sum over K, this has to be the identity, and so on. So I have all these conditions because these must be valid measurements. This is what they do here. So even though you see that these operators here factorize, so I can write this as a sum of some operators AI tilde, tensor, BI tilde. Because I can call this whole operator here AI tilde, and this operator BI tilde. So it will for sure be of this form, which looks pretty innocent. The problem is that for each of these operators, I have to make sure that it factorizes in those. So this looks simple. And these maps is what is called separable operations. It's called separable because you write a map as a tensor product of local operators. And recall the definition of separability. So this is the same thing, but for operators, for maps. So separable operations are those which can be written in the form that we have up there. So it's just a sum of AI, tensor, BI, so some local operators, row AI dagger, tensor, BI dagger, where this has to be a trace preserving map. So this means that the whole operators have to obey this condition of completeness. So this has to be true. So separable maps are defined like that. Any separable map can be written like that and the other way around. All our LOCC operations, as we see, are contained in a set of separable operations. This is an important message. So LOCC is contained at the moment we have it like this. Now, if everything that I can write like that can be implemented locally, so if every separable operation would be LOCC, then our life would be much easier. Because then we just need to deal with those maps and not with these nasty conditions here. These nasty conditions come from the LOCC condition. But the problem is that this is not true. So LOCC is strictly contained in set. It has been proven that there are separable operations which cannot be realized with LOCC. So this means that if we want to study LOCC, then there is no way that we just study, or it's not directly, that we just study the much easier set of operations, the separable operations, but we really have to deal with this complicated structure here. And again, this is not only complicated because I have here some products. This is complicated because this doesn't end. So this goes up, goes on until infinite. So there are protocols where you need infinitely many rounds. And so now I have a product of infinitely many operators, and I need to check that I can write it as a product of operators that obey all these conditions. That's, of course, a mess. It's very difficult. Yes, it has to be realizable with LOCC. So Alice does something here, so she gets all the possible outcomes. So because here I'm talking about really deterministic transformations, I will come to the stochastic transformations later on. So this is really the set of LOCC, is this set of maps with the additional conditions that this is really realizable, like I explained before. Yes, please? That's a very good question. But the answer is because separable operations cannot be So there are separable operations which cannot be realized with LOCC. So if entanglement theory from its very origin is a resource theory. So we want to call a state entangled if it cannot be prepared locally. But this implies that the free operations are LOCC and not set. But good question. Yes, so if this was the case, then things would be much easier. But it's not the case. So now the question is, of course, well, if we want to order our states, so we want to get some partial order, we want to say what is a better resource, then what should we do? We have to study LOCC, but LOCC is very difficult to do. Fortunately, for the bipartite case, one can show that LOCC is very simple. So in the bipartite case, if I study LOCC transformations among pure states, among pure bipartite states, which is what we consider here. We consider transformation from side to side because we want to sort our states in the Hilbert space, our pure states. Because if I consider these, then one can show a very nice theorem, which tells us that all this mess here can be replaced in that case by something very simple, which is just Alice does some operation, as we had it before. She sends the outcome to Bob. And Bob, depending on the outcome, applies just a unitary, and that's it. So we don't have to consider classical communication back to Alice. We don't have to consider that Bob also does some measurement. Just needs to apply a unitary that depends on the outcome of Alice. So this is a theorem by Juan Conglo and Sandoval Bescu. I don't remember now which year, but I don't remember. And this means that everything, or roughly speaking, it means that everything that Bob can do can be done up to local unitaries. OK, so this means that this nasty kind of structure mathematically boils down to something that is very simple, because here our maps simply look like this. So this is just some of AI. This is the measurement that Alice does, and the unitary by Bob. That's it. So I only have to check, then, that these AIs, this set, for this completeness relation. So this is very easy to do. So this means that in the bipartite case, we are very lucky, because these kind of transformations, the ones that we have to study in order to study entanglement become very simple. So this means now what? Well, I want to use that now, because I want to go back to the original problem. I want to study which state is more entangled than the other. So how can we use that? Let's talk about the state transformations. So does anyone know what the outcome will be? So when can I go from a state psi to a state phi? Can you speak up? I cannot. Can you speak up? I cannot hear you. Maturization? Yes, exactly. So this is Nielsen's criterion, which tells us when it is possible. So this problem has been solved. So we know in the bipartite case, when is it possible to go from psi to phi? Now, what I would like to do is to tell you how you can derive that using this result here, because this is something that I will also use later on for the bipartite case. And it's going to be very simple to derive it. So let's see when I can go from psi to phi. So I have a state psi. Let me just draw this picture also. Now, what do I do? Well, L is applied some measurement, right? So I get here some a1 tensor identity applied to psi. This would be the first outcome. The second outcome would be a2 tensor identity applied to psi, and so on, and so on. Now, if I want to reach the state phi deterministically, it means that at some point of this protocol, I have to end up in the state phi for all the outcomes, for all the branches of this protocol. They must be all transformed to psi. But now, I know that the LOCC protocol in the bipartite case looks that simple, and I'm using that now. So this means that the states that I have here must already be a u equivalent to phi, because there's still a local unitary that Bob has to apply in order to get the state phi. OK, so let's write this condition. So I have that ai tensor identity applied to my initial state psi has to be up to some proportionality factor and a unitary that is applied by b to my final state phi. For each of the outcomes, I have to obtain the final state, because I'm considering here deterministic transformations, not with a certain probability, but all of them have to end up in the same state phi. OK, let's write this condition. So the first thing, we have seen yesterday that a bipartite state has the Schmidt, so there exists the Schmidt decomposition. And this means that I can always write a bipartite state up to some local unitaries as some diagonal matrix applied to the phi plus state. So the phi plus state is, as we had yesterday, just a sum over ii. This diagonal one is simply the one that contains the square root of the Schmidt coefficients. If you apply that to this state, you see that you end up with any state that you want. OK, so I write my psi like this. And now I use here instead of a d, because I also want to include the local unitaries, I use here some operator g. Now, of course, the same is true for my final state phi. And I write it as an h tensor identity applied to phi plus. OK, now I look at this condition here. So let's write it down. We have ai applied to g tensor identity, psi has to be square root of pi h tensor the unitary phi plus. So I have now on both sides, I have the same state phi plus, OK, which I was just using the Schmidt decomposition. OK, so now let me consider the case where h is invertible, which just means that I have here, so the Schmidt coefficients are not 0. If you want to consider also the other case, you have to consider something like a pseudo inverse. Let me skip that for a moment. So h is invertible. This means that I can multiply with h minus 1. So I simply do that. OK, it's just rewriting the equation. And now I'm using that the phi plus state has a symmetry, because you might know that this is true. So if I apply an a tensor identity to the phi plus state, then this is the same as identity tensor a transpose applied to the phi plus. OK, so you're familiar with that? If not, then just write it down. It's a simple exercise to prove that this is true, and it's something very important. In fact, this simple relation, which can be proven very easily, implies this very interesting result. OK, so using this relation, you can easily prove that this is true. And we are going to use that now in order to show which transformations we can do to obtain deterministically in other states. So what I do now is, from here, I get that if I consider any unitary, then u tensor, u conjugate applied to phi plus, will be phi plus. But for any unitary, this is true. This simply follows from here. Just multiply with another u or a, and you get this relation. And so this means now that if you look at this expression, this means just multiply with u i dagger here. I'll get u i dagger, and get identity. This means that this operator here has to be just a conjugate of this operator here. So I have that my, and I'm not sure you don't see this. So I have that h minus 1 ai g has to be equal to the square root of pi times u i dagger conjugate, so u i transpose. And this is actually an if and only if, because you can prove that this is an if and only if. This is the only symmetry that you have from a state phi plus. And this is why you need this to be true. OK, now let me recall what these operators are. So this is the one that comes from psi. This is the one that comes from phi. These are the Schmitt coefficients of psi and the phi. And this is our measurement that we do. So for the measurement, we still have to impose the completeness relation. So I have to write now ai as a function of the other operators, which is easy, because this is just square root of pi h u i transpose g minus 1. And I have to impose that the sum of ai dagger ai is that entity. But now what is that? If I insert this expression, then I get the sum of pi.  And this has to be equal to that entity. Now you see that I'm summing here over i, and this g does not depend on i. So I can multiply with g dagger. So I get g dagger g has to be equal to the sum of pi u i conjugate h dagger h u i transpose. This is a necessary and sufficient condition now. So I can go from psi to phi, if and only if, there exists a probability distribution, pi, and unit there is u i, such that this relation holds. But now what is this relation? Does this sound familiar to any one of you? We have heard the answer already before, but this is exactly the condition that the maturation is fulfilled. So this is exactly equivalent to the lambda, so the vector that contains the eigenvalues of g dagger g is maturized by the vector that contains the eigenvalues of h dagger h. So let me summarize this. What we have proven here is that I can go from psi, which I write as g applied to the phi plus to a state h applied to the phi plus, if and only if these operators g and h fulfill these conditions. So if and only if there exists a pi probability distribution and u i unitary, such that this is true. Now this relation is something very well known in matrix analysis, but also in probability theory. And it's equivalent to this maturization condition on the eigenvalues of our Hermitian operators, the ones that we have here. So this problem is solved. So we know that there exists a probability distribution and unitary is u i, such that this is true if and only if the eigenvector containing so the sorted, but this is implicit. So the sorted eigenvalues of g dagger g maturized by the sorted eigenvalues of h dagger h. So this tells us, just a second, this tells us now that we have solved the question, when can I go from psi to phi? Because now I just have to translate what is this g as a function of psi, right? But this g is just the square root of the reduced density matrix. So this thing here actually is just the reduced density matrix of our state psi. And because of that, we have that this is true if and only if lambda psi, so the Schmidt vector of psi is maturized by lambda phi. Yes, please. Where do you have unitary maps? No, no, no, no. Yes, yes, yes. OK, so we have obtained now Nielsen's criterion, which says that I can go from psi to phi if and only if the Schmidt vector of psi is maturized by the Schmidt vector of phi. Let me write this as a conclusion from what we have seen. This is Nielsen's criterion. And this is a necessary and sufficient condition, so it's indeed a criterion. So psi can be transformed to phi if and only if lambda psi is maturized by lambda phi. Yes, yes, yes. So I'm always talking here about deterministic. So this means deterministic transformation. Of course, in a deterministic transformation, I also have branches. But deterministic means that at the end, all these branches end up in the same state. So I don't have here a state phi prime with a certain probability. I get this probability 1 this state at the end. But of course, I have branches. I might have branches. OK, now what is this relation actually? So this means that the following. This means that if I take the sorted eigenvalues, the sort of Schmidt coefficients of psi, and I sum up the smallest ones. So I take L going from some k to the dimension of the Hilbert space. And this has to be larger or equal to the same expression for phi. So you might also recognize these expressions here, because these are some very well-known entanglement monotones that we will discuss a little bit later again. OK, so but this is a relation that is very easy. It's very easy to check also. So you just take the Schmidt coefficients of your state, psi and phi, and you compare the relation. This is a set of inequalities. You have d minus 1 inequalities. You compare if this is true. If it is true, then you know that the state psi can be transformed to phi. And if not, you know that it's not possible. No, this depends on how you, so which ones you sum. So if I start here now from 1 to k, then I would need to write it the other way around. But I start from k to d. So I'm summing the smallest ones. OK. So what would be an example? So for instance, let's consider the state phi plus. And let's consider two qubits. OK, so what is then the Schmidt vector of our phi plus? Everybody knows it, so I'm considering two qubits. What are the Schmidt coefficients of the phi plus state? Up and down? Yeah, Schmidt vector of phi plus is? For qubits, yes, for qubits. Half, half, exactly. OK, so I'm wondering now whether I can go from phi plus to some state a 0, 0 plus b 1, 1. OK, so this is some other state in the Schmidt decomposition. And I assume now that the a, with others of generality, I assume that a is larger than b. So the state, the Schmidt vector of this state would be a b. So now, when can I go from this state to that state? Well, if and only if this is true. So what is this condition now? So I start with k equal to 1. If I start from 1 and I sum up all of them, then I get identity for both of them. I get 1 for both of them, because it's a normalized state. So I start from 2. So I take the smallest Schmidt coefficient. So I take the second 1. And I have to check whether 1 half is larger or equal to b. OK, so comparing the two smallest Schmidt coefficients. But of course, I sorted it. This is sorted. So a is larger than b. And they have to sum up to 1. So clearly, b is smaller than 1 half. So this is obviously always fulfilled. So this means, as always, and therefore, we have that the phi plus state can be transformed to any other state of that form via ILOCC, because it's true for all of them. No, it's a square of them. So the Schmidt coefficients are the square of the square roots. Why? So I have the smallest Schmidt coefficient of my phi plus is 1 half. And the smaller Schmidt coefficient of my phi state is b. And b has to be smaller than 1 half, because this has to be sorted. And they have to sum up to 1. So it is like that. OK, and now, so what we see is that it doesn't matter, right? Because it doesn't matter which state this is, actually, because this is the Schmidt decomposition of an arbitrary state, right? So from the phi plus state, I can go to any other state. So what we see from here is that some very important message, namely that from studying ILOCC transformations among pure states, we learned that the phi plus state can be transformed to any phi. But this means that the phi plus state is the optimal resource. Because given the phi plus state, I can go to any other state. So the entanglement of the phi plus state is larger equal to the entanglement of any other state. So if I want to do an experiment, then clearly I just prepare the state phi plus, because if I want to use another state, I just do this LOCC, which is for free. So from here, we see that the phi plus state is maximally entangled. It's the optimal resource. So it's nothing better than that. Now you might wonder, why do I say the phi plus state and not the psi minus state? Why do I say the phi plus state? Is the psi minus state also an optimal resource?  So of course, as I said yesterday, we care about a U equivalent classes. Because of course, I could write here any state that is a U equivalent to phi plus. But I don't need to, because I know that all these states have the same entanglement. So I just pick one representative. I could have picked any other, but I pick one representative and say that this is the optimal resource. Of course, all the U equivalent ones are also an optimal resource. But I don't need to consider them, because this is the same thing, where this phi plus or psi minus has met, that this is locally unitary equivalent. But this state allows me to go to any other with LOCC. And this is why it is the optimal resource. And the optimal resource for entanglement theory means maximal entanglement. But now we have proven that phi plus is not only maximally entangled, because there is some measure which is optimized by phi plus. But we have proven that for this state, it holds the entanglement of whatever measure you choose. This is larger equal to e of phi for all phi and for all entanglement measures. This is why I can say this is the optimal resource. Oh yes, yes, you're right. Yeah, you're right. OK, so this is an important message. And we will come to this when we talk about multipartite systems, because there we will see that it's not going to be that easy. Right, OK, so now from the theorem that I had here, which was telling us that these complicated LOCC transformations boils down to something very simple in the bipartite case. We have some more consequences. And let me stress here that this is only true if I talk about LOCC transformations among pure bipartite states. Whenever you relax some of these conditions, either not pure or not bipartite, it doesn't hold. It's just for pure bipartite states. You have this simple characterization of LOCC transformations. And from there, it follows. And write it now for short just that this LOCC is equivalent to Alice doing some operation. And Bob doing some unitary, depending on the measurement outcome. From here, we get the following. We get that for bipartite pure states, we have that LOCC is equal to LOCC finite. So I use only finitely many steps. That's obvious. It's obviously a consequence of this thing here, because this is finite. There are no more rounds. There's one round. Alice does something. Sends the classical communication to Bob. Bob applies the unitary. That's it. One round. So clearly, everything that I can do, even if I consider infinitely many rounds, I can do with finitely many rounds. I can actually do it with one round. So this is clear that this is true. And now I wrote it on the wrong side. Sorry. Let me put it here. And what has been proven for this case for the bipartite transformation is that this is also equal to the separable operations. Remember that I was telling you that the separable operations are mathematically much more tractable than the LOCC. In this case, it turns out to be the same. Can be proven to be the same. And on this side here, we have something else, which we called all deterministic protocol. For short, all that. And this is the following. This is a protocol where I start with my state psi. I do some transformation like we did there. So I get some a1 psi. Here, I get some a2 psi, and so on. And that means that in each step, I do a deterministic step. So this means that all these outcomes are LU equivalent. I might then go on, do something further. But in each step, they are LU equivalent for all the possible outcomes that I have. Now, we have seen in the bipartite case, this is always like that, because we end here. So Alice does some measurement. And then Bob only has to apply a local unitary to obtain the same state phi. So these states are all LU equivalent. So different local unitaries, but all local unitaries. So in the bipartite case, we have this. So we see that in the bipartite case, this all deterministic protocol is equal to LOCC finite, finitely many rounds, which is equal to the general LOCC, also allowing for infinitely many rounds. And this is equal to the separable.  So whatever can go wrong here, because this makes it very simple. So this is the reason why everything is really very simple in the bipartite case. In the multipartite case, everything that can go wrong goes wrong. So first of all, in the multipartite case, this will not be true. So you have protocols where you cannot do it with all that. You need a probabilistic step. This we don't know, and this is also not true. So this is what makes it much, much harder in the multipartite case. And this also shows very clearly the big difference between bipartite and multipartite. We will come to this later on. I just wanted to tell you here that the reason for why starting the bipartite case at the end is really very simple, because you have here everywhere an equivalence. So you can consider the most simplest protocol which there is in order to deal with the most difficult ones which there is. Now the reason why I also write this equation is because, of course, when you want to study this now in the multipartite setting or for mixed states, if you want to study LOCC, then because of this complicated mathematical structure that we saw, there is no way that you study it directly. So I don't know how you could possibly do that, because you have to take infinitely many rounds into account. So what you want to do is to consider a set that contains LOCC, which is the separable set, and something that is contained in LOCC. So then you have a bound on both sides, and you might be able to prove that the bounds are the same, which means that you also know it for LOCC. So this is the general idea. OK, we'll come back to this when we talk about multipartite states. Are there questions so far? OK, so then let's go on to the next chapter, which is entanglement measures and entanglement monotones. So let's talk how we can actually measure entanglement. So we were now using it always, just as a word, basically, and having in mind some examples. But let's look at the definitions. So entanglement measures, which I will for short call enmesse and entanglement monotones. OK, so first, what is the definition also in the case of mixed states? Enmesse and monotones for mixed states. So the definition, what is the definition of an entanglement measure? We have seen that already. You recall? When I can call a functional entanglement measure, it matches a resource, no? Exactly. So the condition is that E is an entanglement measure if E of rho is larger or equal to E of lambda rho for all. Let me write lambda LOCC. Lambda LOCC, this is a map that I can realize with LOCC for all states rho. And all lambda LOCC being contained in LOCC. As you were saying correctly, this is the free operation. So this I can do. This doesn't cost me anything. So this can only decrease entanglement. And now if I have an entanglement measure, then it has to obey this condition, nothing else, just this condition. OK, this makes perfect sense. It's very physical. This is exactly what we want to have. Now there is another very useful concept, which is entanglement monotones. Does any one of you know the definition of entanglement monotones? Somebody was mentioning it yesterday already. So entanglement monotone is something that does not increase on average. So this means that E of rho is larger equal to the sum of Pi E of sigma i for all rho and ensemble Pi sigma i, such that rho can be transformed via LOCC to this ensemble. OK, so here indeed I consider different outcomes. I get this probability Pi to state sigma i. And what I'm asking for now is not only that E, I should not say not only, but it's not that E of rho is larger than E of the sum, which would be the lambda of light to rho. But I'm asking that it's not increasing on average. OK, so it's something different. This is non-increasing on average. In the literature, this condition here is called monotonicity condition. So it's monotonic under LOCC. This condition here is called strong monotonicity condition, which might be a bit confusing because this is not necessarily stronger. We will see that this is stronger in case we consider pure states, or in case we consider convex functions. But otherwise it's not necessarily stronger. Nevertheless, this is called the strong monotonicity and the monotonicity condition. This has historical reasons for why it's like that. OK, so now we have these two definitions. We will see that both of them are very useful. This is really the one that comes from the physical picture. So this is what we have. This is our resource theory, and we cannot increase with LOCC. This is exactly what that means. This definition is slightly different because this means that we cannot increase on average. But we will see that these entanglement monotons are very useful. They can lead to entanglement measures, and they can be used in many different contexts. OK, so unfortunately my time is already over. Are there some questions? Yes, please. Yes, so LOCC is different than ZEP. So there are examples, even for bipartite mixed states, going from a mixed to a pure state. This is a result by Eric Chittambar and I think Juan Conlo, that you can achieve. So that proves that you can do it with ZEP, but you cannot do it with LOCC, even bipartite. This is a state transformation going from a mixed state to a pure state. Well, this is contained in that. So you're asking for an explicit example for mixed to mixed in the bipartite. Sorry? OK, so it's unanswered. So for mixed to mixed, it's unanswered. It's very difficult. What is answered? And we will, I might talk about this briefly, is from pure to mixed in the bipartite case. Are there more questions? Do I still have some time more? Do I still have some time? No.


Play fake, Tannehill rolling right, throwing back to the left. There's Smith opening the end zone. John Oosmith touched down tight. That's a bootleg throwback wheel. I love the concept. Great, great, great catch by Smith.