My name is Christine Mumry, I'm Professor of Developmental Biology at Leiden University Medical Centre. IPSCs are very useful for modelling disease and development because we can derive them from patients and we can turn the IPSCs into any cell type we like. So if the patient has a genetic disease, the mutation in the DNA, the differentiated cells from the IPS cells have the same mutation. What we hope is that they have the same phenotype, that means the same sort of symptoms as the patient has. And in that way we can screen for drugs that correct the symptoms in the cell and then hope that the same drug will work in the patient. So among the key findings are how, to what extent the IPS cells or adult stem cells can actually be made to be high throughput and a larger scale. And what I actually mean is it will be very expensive for an individual to have their own IPSCs or adult stem cells derived. And what we'd really like to see is that we can make thousands and thousands of IPS cells or adult stem cells from different individuals. So a kind of population study. That's also what pharmaceutical companies want to know. They're not really interested in testing, let's say, thousands of drugs on a single individual but they want to know how many, many individuals would react to the same drug at different doses. So that's a sort of population study and that's what they're hoping we will be able to develop. And we'll need robotics to do that. So we're going to need robots to make from thousands of patients, IPS lines or adult stem cells and then do the screening course. Academia is rather good at making different stem cell lines and looking for disease phenotypes and they also have access to patient material with the proper informed consent. But what academics are not very good at doing is making the assays we need to examine the cells very robust. And what I mean by that is that we can do an experiment three times as we are often required to by scientific publications. What happens if you do those assays for many years at an end using different reagents and different cell lines? And that's what industry does very well, interpreting the results properly, making the right software to analyze the data, putting the data together and actually ensuring that for many years on end they can repeat the assays with different drugs and different systems. So upscaling and validating is what we need industry to help us with. There are a number of challenges to the fields from you could say very trivial to actually quite major. So if I would go to the trivial ones in some sort of way, it's finding the right sort of personnel. So to do this type of work you need to be highly educated. You really need to understand what you're doing. You need the right training and it's quite difficult for academia at least to attract the right sort of young people into the field that they will stay. Many of them choose to go to different aspects of industry. They go to science writing, they go to work for ministries and what we need them to do is actually to stay for a bit longer at the bench and do the experiments in a very skilled way. In the next generation art we also need new leaders to bring the field forward and to develop new technologies and implement them. The more serious technical issues are related to how do we actually model aging. So from adult stem cells or IPS cells we can get let's say young adult differentiated cells. But we can't actually mimic aging. So if a disease is only evident when you're 60, 70 or 80 we can't get cells that are mimicking that sort of age. We also can't get cells that mimic diseases of aging. Let's say it's quite hard to do them for dementia or loss of vitality of cells. So aside from regular maturation we're going to need to have new methods to look at the let's say the challenges of life. What do we get from smoking, from sunlight, from drinking. How do we mimic those environmental factors that predispose to disease.


In some way, we can and we can't blame Franklin Delano Roosevelt. He did and he didn't make the decision. All he did was delegate this authority. Now at the time he did that, he and his advisors knew what the plans were. They didn't know the full scope and extent, but they knew the general tenor, that the idea was that what the military would then do is round up specifically people of Japanese descent. That's what was being talked about. That was the primary plan. There were some discussions, by the way. They initially said, well, shouldn't we round up people of German background, Italian background? Concluded quickly, no. They concluded no for many reasons, not the least of which it would be impossible. There were just too many. It was only a minority that you could round up, a discrete, insular, smaller minority. People of German descent then and now make up the single largest ethnic group in the United States. More people claim German ancestry than claim the ancestry of any other single nation. Not only that, these were important voting blocks. If you tried to round up every one of German ancestry and Italian ancestry, you can bet you wouldn't be re-elected. What Franklin Delano Roosevelt did was delegate this authority. If we continue reading, it's not just this authority. The next paragraph he says, I hear, by further authorize and direct the Secretary of War and the said military commanders to take such other steps as he or the appropriate military commander may deem advisable to enforce compliance. They don't just get to set up intern camps. They now have the authority to compel people, to force them into those camps. As you start to interpret documents like this, it becomes apparent how hidden away in what looks, if you just glance at this casually, you have no idea what it meant, how hidden away in documents that look like just so much bureaucratic legalese is power, is authority, is policy. It's not just this document. I would suggest that's generally true, that when people go looking for history, sometimes they think wrongly that history is going to be big, bold, dramatic, that you're going to see it and you're going to know that is history. Well, sometimes, often, history consists of things that, if you passed it by, if you just saw this document, you'd have no idea how momentous it is. It's not until it's put into action. Sometimes history is in these details.


Today we're in chapter 2. We're looking at verses 6 through 16. I'll begin reading at verse 6 and Actually, do I want to do that? Yeah, I do. You know what I just did? I was looking at 2nd Corinthians and I'm thinking why do I have that? I'm getting old. Yeah, verse 6 And I'll read verses Verses 6 through 10 and we'll get into our study 1st Corinthians chapter 2 beginning at verse 6 However, we speak wisdom among those who are mature yet not the wisdom of this age nor of the rulers of this age who are coming to nothing but We speak the wisdom of God in a mystery The hidden wisdom which God ordained before the ages for our glory Which none of the rulers of this age knew For had they known they would not have crucified the Lord of glory But as is written I has not seen nor ear heard Nor have entered into the heart of man the things which God has prepared for those who love him But God has revealed them to us through his spirit For the Spirit searches all things. Yes, the deep things of God as we begin our study here in chapter 2 We need to remember a few things or perhaps learn a few things about Corinth and and this letter to the Corinthians We need to remember that Corinth the city of Corinth, Greece was a was a city that was filled with pride Especially had pride in human wisdom It's been said that it was filled with the quicksand of sin Ready to draw them into shallow human philosophy The love of human wisdom and the sinfulness of the city of Corinth was legendary And God had spoken to the Apostle Paul God had directed him and said to him You're gonna go to the city of Corinth and you're gonna minister and So when we see him actually responding to that call As was his practice he would first go to the synagogue because Paul said to the Jew first and then to the Greek So he would go to the Jewish synagogue wherever it was that he went and he would first enter into the synagogue and reason with Those who were there and he'd take the gospel to those who are not Jewish And so he went into the synagogue and he would reason with the attendees But as he had gone into the synagogue there in the city of Corinth, the response was angry opposition In Acts chapter 18, it's recorded what happened. It says at verse 6 when they opposed him and blasphemed He shook his garments and said to them your blood be upon your own heads I am clean from now on I will go to the Gentiles And so what they did is he went into the synagogue as they began to to to blaspheme Christ. He's worthless He's not somebody that we're gonna pursue. Are you kidding me? They got very angry at the Apostle Paul now the opposition in Corinth at first was greatly greatly Antagonistic towards him and he actually began to harbor some concern even fear You see in Acts 18 9 and 10 it says the Lord spoke to Paul in the night by a vision Do not be afraid but speak and do not keep silent For I am with you and no one will attack you or hurt you for I have many people in this city Do not be afraid but speak So he was as brave as any man could be and yet you see in Corinth He had fear that's why in chapter 2 we saw this earlier at verse 3 That's why he told the Corinthians I was with you in weakness in fear and in much trembling And so there was a concern in the heart of the Apostle Paul because of the opposition that was filling that city and the rejection of God These again were people who were enamored with human philosophy with with man's wisdom It's reminiscent of what Paul said to the Colossians in chapter 2 verses 8 through 10 when he said to them See to it that no one takes you captive through hollow and deceptive philosophy Which depends on human tradition and the basic principles of this world rather than on Christ For in Christ all the fullness of the Godhead lives in bodily form And you have been given fullness in Christ who is the head over every power and authority So you're not to be yielding yourself to human philosophy, but you yield yourself to Jesus Christ Now for these Corinthian believers to survive their faith had to rest in something greater than human wisdom and Their own courage Their faith had to rest in the power of God and not the wisdom of men and They had to somehow confront the powers of darkness with the reality of their faith in Jesus Christ Because as they do that it's going to demonstrate the power of the message and it's going to demonstrate that the message of the gospel Has actually been communicated to us from heaven itself It's like what the psalmist said in Psalm 90 verses 10 through 12 The length of our days is 70 years or 80 if we have the strength Yet their span is but trouble and sorrow for they quickly pass and we fly away Who knows the power of your anger For your wrath is as great as the fear that is due you and this I Love this this phrase this this line teach us to number our days a right That we may gain a heart of wisdom Help us to realize that we have a limited time here on earth and to take each day into consideration Being aware that we have a temporary existence here but we have an eternity with you and Help us to order our steps properly in order that we might have a heart of wisdom And so the wisdom that we have is not going to come from human philosophy that is Antagonistic to the gospel of Jesus Christ, but it's going to come From God himself as it's been communicated to us as a message from heaven And that's why Paul in verse 5 had said that your faith should not be in the wisdom of men But in the power of God The world must feel the impact of our faith the faith of Christ Because our faith rests on Jesus and what he has accomplished on our behalf again verse 2 chapter 2 He had said I determined not to know anything among you except Jesus Christ and him crucified And so Paul had come bringing this ministry of the word of God in order that he might be able to proclaim What God has done for us through Jesus Psalm 60 verse 12 with God we will gain the victory and he will trample down our enemies And so that's what happens through the preaching of the gospel of Jesus Christ This victory comes through Jesus triumphing over the enemy and he did so when he conquered the grave and We have victory in him because of that spin said Jesus left an empty tomb in order to fill our empty lives That's why in 1st Corinthians 15 Paul could say in verses 55 through 57 where oh death is your victory Where oh death is your sting The sting of death is sin The power of sin is the law But thanks be to God he gives us the victory through our Lord Jesus Christ and in 1st John 5 for John said it like this Everyone born of God overcomes the world. This is the victory that has overcome the world Even our faith and so we're overcomers Through the one who has loved us and has given us the power to overcome so the heart of our message will always be Jesus Christ and him crucified and the Proclamation of the message is with the intent for God to receive all the glory even as we saw in chapter 1 the last verse verse 31 He who glories let him glory in the Lord So God gets all the glory. We preach a message of the cross of Jesus Christ It's the message that saves man and we make sure that Jesus stays central again 2nd 2nd Corinthians 4 5 we preach not ourselves, but Christ Jesus the Lord and ourselves your servants for Jesus is safe So Paul is making it very clear our faith is not to be in the wisdom of men Now as we begin that was all in introduction as we begin He is not saying the gospel is without wisdom He is saying that the gospel is past human wisdom is Past human beings finding out unaided In the book of Job chapter 11 verses 7 and 8 one of the so-called friends of Job a man by the name of Zophar Said this he said can you by searching find out God? Can you find out the Almighty to perfection? It is as high as heaven. What can you do? Deeper than hell What can you know it is the height of human arrogance to think that we without help could find God It is God who finds us all the way back in the first book of the Bible the book of Genesis and God had given the command not to touch that forbidden fruit and Man had succumbed to the temptation to do so as we know the story and After doing so We see evidence of man hiding from God in the garden and God actually is the one who breaks into human history By calling out to Adam it is an atom who after failing called out to God It is God who calls out to Adam and asks the question. Where are you and from the Bible's first chapter to the last The spirit and the bride the last chapter say come there's an invitation It's God from the beginning saying Adam. Where are you and then concluding his book by saying come There's this call from God to lost man But we cannot search out God and discover him unaided. There's no way that I have the capacity to understand him You see if I could understand God unaided then I would have to be God myself You cannot find out God without God allowing you to discover him It's one of those incredible cosmic games of hide and seek if he decides to hide You're never going to find him. He's not like me when I used to play hide and seek with my kids when they were small Where I would hide in some place where they'd discover me eventually, you know I always hid in the same place all they had to do is look for me once and they'd find me every other time He was always in bed under the blankets. That's where I would hide And yet there we they would go all through the house looking for me and looking for me They couldn't find me and I always made sure to to play hide and go seek in the dark It would always scare them a little bit more. It was quite fun And they would come looking and always hit on the bunk bed on the top in the corner With some pillows around me and a blanket over me always did the same exact place And I can still remember hearing them come into that room It was the last place they'd look because they knew that they were gonna find me and they eventually and then I could hear I'm climbing up the ladder of that bunk bed And I would be looking and I could see just the top of David's head when he's little He just you just see his little head and his eyes would be looking right at where he knew where I was and I would sit There quietly until his little face popped over and then I'd jump out It was a lot of fun.  They didn't never found me If God decides to hide from you, do you think you can find him? No, of course not.  Oh God of Israel the Savior Isaiah 59 1 and 2 behold the Lord's hand is not shortened that it cannot save Neither his ear heavy that it cannot hear But your iniquities have separated between you and your God and your sins have hit his face from you that he will not Hear unless God answers the sin question sin that makes separation between man and himself Man is doomed to be forever lost doomed to be forever lost Man cannot find out God. God reveals himself to man Romans 11 33 oh the depth of the riches both of the wisdom and knowledge of God how Unsearchable are his judgments and his ways past finding out unless the Lord reveals himself Man will never know him Man didn't invent the gospel and because man did not invent this gospel message our faith doesn't rest on man's ingenuity Our faith rests in God's power to save a power to save that was displayed when Jesus died on the cross when Jesus was buried and Jesus was resurrected That's how his wisdom is demonstrated through salvation and so Paul has been speaking concerning that and that's why in verse 6 He goes on to say however we speak wisdom among those who are mature Yet not the wisdom of this age nor of the rulers of this age. Who are he says coming to nothing? We're not giving you the wisdom of this period of time the wisdom of his age We're not giving you natural wisdom We're not giving you a wisdom that that comes from a mind that hasn't been illuminated by the spirit of God Because the end results of human wisdom will always be hollow emptiness True wisdom comes from God and true wisdom is actually found in Jesus Christ Like it says in Colossians 2 3 and Jesus are hidden all the treasures of wisdom and knowledge and so Paul's exhortation to the church of Corinth as well as through the entire history of the the life of the church is simply this the church is to rest in the wisdom and power of God and We need to resist the temptation to find wisdom in any other source the wisdom that we have is actually found in Jesus Christ and Is revealed to us in the word of God in chapter 1 here in 1st Corinthians remember with me verse 24 How he says in chapter 1 verse 24 to those who are called both Jews and Greeks Christ is the power of God and the wisdom of God Well, the fact is we're supposed to rely on the Lord Jesus Christ and all that he is to us And so we've been called to to to rest in that wisdom, but he also noticed in verse 6 speaks of States that he is speaking wisdom among those who are and this is interesting who are mature He's speaking to those Who are saved He's now this is interesting because the Corinthian church when you read Corinthians you see there's they had so many problems It's interesting to note that Paul is actually saying but you haven't come behind in any gift You do have an actual relationship with Jesus Christ. It's just getting muddled right now And so as he's speaking and bringing correction the fact is he's speaking to people who have a spiritual discernment He's speaking to those who are saved You see the unsaved don't see the wisdom that's in the Word of God The unsaved do not see the wisdom that's found in the message of salvation to to the unsaved The Bible is a closed book that's filled with stories for children for me It has myths and it has just fantastic things about devils and and trees with certain kinds of fruit and Someone who says he was dead they say he was dead, but now he's alive and that's how the world looks at Christianity It doesn't make any sense But for those of us who believe we see it as the wisdom of God and the power of God disclose to us Here in the Word of God and that's what Paul is speaking about It's a wisdom that he's speaking amongst those who have spiritual discernment Again in verse 7 notice how he says we speak the wisdom of God the wisdom that originates with God is what he's saying In a mystery when you look at the word mystery the word mystery is interesting because for us Here in the 21st century and for longest time the word mystery if you use the word mystery like I'm reading a mystery book or I Saw a movie that was a mystery For us the word mystery speaks of something that is hidden from us. It's a mystery But the New Testament use of the word mystery is is a secret that at one time had been hidden But it's now disclosed so a mystery in the New Testament senses is as a secret that has now been revealed and he's saying We have something that was mysterious at one time that has now been opened up for you to understand And so we're speaking the wisdom. He said of God in a mystery Again, this is a mystery that needs to be revealed and has been to the message of the gospel of Jesus Christ and this mystery is disclosed to people and That's why a lot of people have a problem with this one because they say it's so simplistic The mystery is revealed through just a proclamation of a message called the gospel When you stand up and you say Guess what? There's a God and this God Loves you and this God has a son and this son's name is Jesus and Jesus God's son was born of a virgin and He dwelt amongst men for 33 years and Then ultimately was betrayed by a close associate placed on a cross as The Lamb of God who takes away the sin of the world Laid his life down voluntarily poured out every drop of his blood Died saying father forgive them for they know not what they do Dismissed his spirit was taken off of a cross Placed in a borrowed tomb remained there for three days and Then lo and behold during that day the first witnesses were women who were not valid witnesses in a Jewish court of law And yet God says I am going to validate women by having them as the very first witnesses of the resurrection So Mary and the other women who are there at the tomb on the very first ever say he's alive He's risen and so what he's doing right there is breaking the taboos amongst that culture saying you can have a Declaration that is official and recognized through these women And they see the fact that Jesus Christ is alive and he remains on earth ministering for many days And ultimately he ascends into heaven and his disciples are standing there on the Mount of Olives as he ascends and they watch him And they're so dumb founded as the clouds receive him that angels have to appear and say men of Israel What do you do in standing around didn't you hear what he said go out and tell people about the gospel and To us that make well to me at least and I'm sure most everyone of us that makes perfect sense That's the gospel That's what he did. I didn't invent it. You didn't invent it Human beings didn't originate this this came from God himself This is the mystery of God revealed to man. It's called the gospel and guess what? believing that message Believing that message that God loves and God forgives Can absolutely with the power of the spirit in your life Absolutely transform your life to be Unrecognizable to those who knew you best Speaking to one of our brothers in our fellowship today after church It's been with us a long time He said I he said you can share this if you want. I think he just wants his name mentioned He's a good brother. He serves here, and I love him Fernando He said pastor. He said I don't I don't know if you know this He says I came to church here years ago now years ago He said and I went to what is called the lion tamers He said I went to lion tamers when it used to be not on this property, but in Ontario He said and when I went to lion tamers He said I heard what they had to say He said but I I just I just didn't grab hold of it. He said and I was into drugs He said I got arrested and I spent five years Five years in prison in Corcoran And I said Corcoran, huh? He goes. Yeah, I said I got relatives in Corcoran not the prison Well, maybe in the prison. I don't know probably think about it I said I've got relatives. I used to go to Corcoran as a little boy and we began to speak He said he said well, I wanted to tell you that so I could tell you this he said for five years I did my time five years in Corcoran He goes I received weekly tapes from the ministry here He said I wanted you to know that you have prison ministry in Corcoran You just didn't know it he says because all the tapes that I received I donated to the prison library and He said and it's just an amazing thing. He says that your your your ministry is there in the Corcoran prison He goes, but he said can you imagine? He said I did my time. I grew in the Lord by listening to to tapes He says can you imagine? He says and now I am the leader over the lion tamer ministry here he's the one who leads our lion tamers and so That's what God does God grabs hold of a guy Who came to the lion tamers because he was a druggy and The guy didn't really care very much what was being said continued in his drugs Gets arrested Spends five years in prison Comes out But he comes out five years of Bible studies.   He believes all of that and he's a testimony as all of us in this room Who loved Jesus Christ talk? We all have testimonies every one. I Don't encourage you to go out and get arrested spent five times unless you want to start a prison ministry in Alcatraz That's fine with me I'll send you tapes What a God we serve What a God we serve this message this message of love this message of transformation Wisdom that has been hidden so that the natural Philosophic mind is not going to be able to receive it It makes no sense to so he says we're speaking wisdom amongst those who are mature and this wisdom while we speak it this wisdom of God in a mystery and The result of believing this message he said it results in our glory and Romans chapter 8 verses 17 and 18 Now if we are children Then we are heirs heirs of God and co heirs with Christ if indeed we share in his suffering In order that we may also share in his glory. I Consider that our present sufferings are not worth comparing with the glory That will be revealed in us Romans 8 29 continuing says for those God for new he also predestined to be conformed to the likeness of his son That he might be the first born among many brothers this message results in our glory First John 3 2 and 3 dear friends now We are children of God and what we will be has not yet been made known But we know that when he appears we shall be like him for we shall see him as he is Everyone who has this hope in him purifies himself just as he is pure So God Produces lives that bring glory to him notice verse 8 which none of the rulers of this age knew None of the rulers of this age knew that God would give his son for the ransom of human souls because in their eyes Jesus was only useful for death But he goes on in verse 9 But as is written I has not seen nor ear heard Nor have entered into the heart of man the things which God has prepared for those who love him God has revealed them to us through his spirit for the spirit searches all things. Yes, the deep things of God it's interesting as he gives us verse 9 Paul is actually giving us three scriptures Isaiah 64 verse 4 Isaiah 65 verse 17 as well as Isaiah 52 verse 15 he mixes these three scriptures and And he's given to us some insight now when I first read this I Thought that what he was referring to was heaven and I guess the average person upon reading that might think the same thing I Has not seen nor ear heard nor have entered into the heart of man the things which God has prepared for those who love him So at first reading it I thought I thought within myself. Well, it's well. Yeah We don't have a an idea of how glorious heaven is going to be But in reality, that's not exactly what he's saying. This doesn't seem to be a picture of heaven In context it has to be that it's referring to the natural man who doesn't understand God's ways In the life of a believer Because see God has already put eternity into our hearts. We already have this sense that Jesus is prepared to place for us And we're in that direction. We're moving in that direction already So we already know that Lord is going to do something to us And what it's going to be within us is is going to give us the joy of knowing where we're where our final destination is So what he's trying to say here to us or what he is saying and we're trying to understand is God's wisdom is only Comprehended by those who love him The eye and the ear are a natural means of obtaining information the heart Speaks of the mind to a Hebrew and so what he's saying is unregenerated rational thought will never never fathom God's ways That's why We're to trust in the Lord with all our heart and lean not into our own understanding That's why in all our ways we acknowledge him and he directs our path We I'm glad to be speaking to a group of people who appreciate what I'm saying about this when I say You can just never argue somebody into the kingdom of God You just can't I've tried I have tried I Tried a long time and a lot of times argued with a lot of people try to argue them into the kingdom of God He discovered something guys that you have to if you can argue them in someone can argue them out If I can give a logical case as to why the things I believe are right and true and you should embrace There's somebody who can come and contradict that argument But one thing I have discovered about God is when he convinces you no man can convince you otherwise No man can convince you otherwise You can come up to me and you can argue with me all day long and you and I'm not I'm not the smartest You know I'm not the sharpest knife in in you know in the drawer That's for sure, but I do know one thing for sure God doesn't lie. I Have I have a real strong belief that he didn't lie I may not be able to fathom him and I can't and unless he reveals more to me I'll never will and this side of heaven. I won't understand completely on the other side. Yes, then I shall know even as I'm known but There's one thing I do know for sure and that is that God's word is true God's Holy Spirit brings conviction God transforms human lives and God has done that through his word in my life Which has given to me a great strength of belief that really can't be shaken in other words There's just no way that somebody would ever come up and argue me out of out of the kingdom of God Why it isn't because somebody argued me into it It's because God's word and God's Holy Spirit convinced me of its truth And God keeps those that belong to him and no man can take them out of his hand And so the Lord has grabbed hold of my heart and that's why when you have that kind of sense when you say nope There's no doubt about it. I don't understand everything about God So the best thing I do is hold on to the things that I do understand a whole fast of the things I do understand and the things I don't understand it's put in a little box in my mind that is you know to be looked at later And if I have to address those things later, I will and I've had all kinds of questions over 41 years Lord May I speak to you about your judgments? I want to ask you why you do this and The Lord has said you know to be continued This is a conversation You don't really need to engage me in and I'm not going to give you an answer that satisfies you at the moment because you're not ready So why don't you just put this aside for a little while and let's deal with what you do know What are the things that you do know? Well, I do know that the Bible says that God loves me. Okay, then why don't you hold fast my love and trust me in the rest? All right, that makes sense to me, and that's what I've done for 41 years Trusted that he loves me and everything else has worked its way out when you know that God loves you Then you're secure when you know he cares about you. You're secure. You may not understand everything that's going around you But guess what? Some things are to be answered later. Some things may never be answered this side of heaven So I hold fast to what I understand and I leave the mysteries in his hand if he wants to reveal to me the answer I bless him for it. So I trust him with my own with all my heart, and I don't lean on to my own Understanding I do acknowledge him in all my ways, and he directs me So these things that God has prepared are prepared for us because we're his children That's what he says in verse 10 notice God has revealed them to us through his spirit For the spirit searches all things yes the deep things of God for what man knows the things of a man except the spirit of the man Which is in him even so no one knows the things of God except the spirit of God So these things are revealed not to the unbeliever But to the believer to whatever degree that we can receive them and as our maturity continues to increase But he reveals these things to us by his spirit because it's a holy spirit who communicates his truth to us Jesus in John 14 26 said it like this he said but the comforter which is the holy ghost whom the father will send in my name He shall teach you all things and bring all things to your remembrance whatsoever I have said unto you so it's a holy spirit who brings the word to us It's a holy spirit who gives to us insight, and he reveals to us by his spirit these things now verse 12 We have received not the spirit of the world But the spirit who is from God That we might know the things that have been freely given to us by God These things we also speak Not in words which man's wisdom teaches But which the holy spirit teaches comparing spiritual things with spiritual You can go to school and You can take classes The class that you would see in your syllabus as you were signing up for Bible classes Would be homiletics all of us have heard that term if you're not the word homiletics just teaches you how to preach So if you want to learn to be a preacher you take a class called homiletics, right? And as you take the class called homiletics, they will teach you how to do your outlines They will teach you how to prepare a Bible study They will give you an opportunity in this homiletical class to stand up in front of the class and give a 10 minute or 15 minute devotion The students will all be seated in there with their notebooks open and their pens out correcting you Once you've given your message, they'll all kind of respond and say well, you know I thought your illustration was good But you were real weak on your exegesis and that's what they do in your homiletics class and afterwards you can become a polished speaker you can learn how to communicate and You can go out and now you've got your little degree and it says master of homiletics.  They're very good But there's no anointing there There's a lot of information and I appreciate it, but there's no passion there There's a lot of stuff that's being said, but there's no love there and people will come and listen and say Well, that's interesting, but if I want information, I'll read the same books There's got to be something deeper than that and that comes by the spirit of God In the early days of Calvary Chapel Ministries people used to look at pastor Chuck and used to say always a simple-minded man leading simple-minded men and His messages are so simple and and they thought they were insulting pastor and he said that's exactly what I want I want the message to be understood by every person inside that room We were taught this put the cookies on the shelf where the children can reach them It's not like it's not like Like we're so brilliant, but God knows we're not and I hope it doesn't come off that way because look at Raul and you get my point You know people tell him what I say, you know that don't you I used to teach a home Bible study That's where I started to learn to teach I Was 23 years old my dad Went to eighth grade and quit school in the eighth grade, which was common by the way when he was growing up and because he came from a family of 12 children He started working at the age of 13 At the age of 17 he went into the US Navy and served honorably on the USS Pennsylvania during World War two he got out of the Navy and he met this young woman at a dance, I believe in Manabello East L.A. He spoke Spanish not knowing that she was a Mexican girl and He said whoo, she's a cute little thing and This cute little thing answered in Spanish back.  I've never forgotten that again. I need to hasten to say this my father was a very intelligent man My dad was not unintelligent my dad was just not educated beyond the eighth grade And so I Learned a long time ago as all of us have in this room That we ought to keep the word of God as simple and plain as is possible So that people are not impressed with us, but impressed with him So that they actually walk out of a Bible study saying I've learned something about God that I needed to know That's what Bible studies are intended to do. It's not difficult guys at all To to say impressive things. It's not hard Anybody who's been teaching for a while can structure Throw in some Greek throw in some history throw in all kinds of things that is very impressive My heart has always been but when you walk out of the room, do you love Jesus more? Do you know him better? That's the key and so the Lord Actually gives to us his word his spirit in order that we might be able to communicate these things to man and Understand that it's the Holy Spirit who brings these things to us that we might know those things that have been freely given to us We can't find God's ways out on our own God reveals his ways It's like it says in Isaiah 55 8 and 9 my thoughts are not your thoughts neither are your ways my ways Sayeth the Lord for as the heavens are higher than the earth so are my ways higher than your ways and my thoughts than your thoughts And so what he wants us to know is what it's been freely given to us again Romans 8 32 He that spared not his own son but delivered him up for us How shall he not with him also freely give us all things if God gave me his son? What's he gonna hold back from me? What's he gonna hold back from me if he gave his son? his son What is he gonna hold back from me? Nothing Because what is more valuable in heaven's treasury than the son of God? And so I should know that my God loves me He gave his son for me and will not hold anything back Well in verse 13 these things we also speak not in words which man's wisdom teaches us But which the Holy Spirit teaches compare and spiritual things with spiritual, but the natural man Does not receive the things of the Spirit of God for they are foolishness to him Nor can he know them Because they are spiritually discerned But he who is spiritual judges all things yet he himself is rightly judged by no one for Who has known the mind of the Lord that he may instruct him? We have the mind of Christ the natural man That's the title of this message The term natural man Speaks of the unspiritual or the unregenerate man The man or woman is just a generic term meaning human being The individual who has never been born again is called in scripture the natural man But when you take that scripture apart It says the natural man receive it the not I memorized it in the King James. It's easier to come off that way The natural man receive it not the things of the Spirit of God that word receives or receive it Means the natural man the unspiritual man the unregenerated man The man who has not been born again does not welcome that word receive is a Greek word that speaks of welcoming Does not receive the things of the Spirit of God. It's a picture of the Spirit will say knocking on a door and a natural man Looking through a window at will say the Spirit Jesus will say standing at that door knocking And he's in the house not wanting to open the door to allow Christ Access the natural man receives not the things of the Spirit of God The natural man does not welcome in the things of the Holy Spirit white They are foolishness to him that word foolish is is a root word where we use the English word moronic We also use the word imbecilic The natural man the unspiritual man the person who's never been born again Does not welcome in the things of the Spirit of God because they're stupid to him. It makes no sense And so that's the man who will look at you when you get saved and say why'd you do that? Well, didn't you grow up going to the church? Why'd you do that? Listen, I went to church as a little boy. Yes I went through the things of my church that that were supposed to make me as full citizen in the church I received the baptism. I received the communion. I received the confirmation. Yes, I did that I Did all of those things But I was still a natural man because the water baptism didn't save me because the Bible teaches It's not the washing of the flesh, but it's the renewing of the mind The Bible never teaches that water itself is what God uses to cleanse the human soul the Bible teaches it's the blood of Christ that cleanses the soul and So water doesn't save you and so though I was four months old Getting in water baptized in a small church in Los Angeles. I was still an Unregenerate baby growing up into an unregenerate little boy who went to my catechism class Who got my first communion and then later on I was an unregenerate, you know middle schooler junior hire receiving my confirmation Taking my confirmation name getting slapped on the face by the bishop and the whole nine yards learning to turn the other cheek and all that we were taught and I am unregenerate and At 15 when I have the opportunity to start drinking I do I start taking my drugs a little later And then for the next few years, that's my life yet you argue with me and say to me I'm not a Christian. I would say I was baptized. I was I received communion and I've been confirmed I'm recognized as an adult in the church I just haven't been to the sacrament of confession for a while haven't done my penance But the bottom line is is I am a member of the true and original the first church And I knew my catechism well enough even as a drunk and a druggie to argue like many of you I did not know Jesus Christ And so when my friends would tell me that Jesus could save me and Jesus loved me With the flick of the wrist I would say are you kidding me? I already know I was raised in the church You weren't and you're coming to tell me about God.  I just that had not really put it into practice Can you imagine How when the Holy Spirit that day December 27th 1970 and the Holy Spirit grabbed my attention and said You are not saved how surprised I was that day I was literally surprised by God Because I thought I was a Christian. I just wasn't practicing my faith But the Holy Spirit's conviction in my life was a question and he said you're uncomfortable and I said yes I am and he said why and I said because I'm not like these people And he said and what is making the difference between you and them and for the first time that I can remember ever admitting it To the Lord who is prompting my heart. I said because I'm not a Christian I was the first time that I ever admitted I wasn't a Christian because I would have argued hammer-and-tongue that I was All the way up to that Maranatha concert. I would have said I am a Christian I'm just not doing what I'm supposed to do right now, but when I got there That's the Holy Spirit. You see the natural man receive it not the things of the Spirit of God They are foolishness to him neither can he know them for they are Spiritually dessert. It's the Holy Spirit who does the work within you The natural man these things are foolish. Are you kidding me? What do you guys do it on Sunday night at church? The natural man, are you kidding me you serve over there in that cult?  That's the natural man They don't understand. They don't understand. Why don't you do what you used to do? You were a lot of fun You were a lot of fun my cousin said that to me my cousin Ray David what happened to you and you were a lot of fun He said now you've changed what happened to you.    I Was your clown? I said, but guess what now? I'm not your clown Now I'm saved. My life has changed. I have purpose. God has forgiven me I'm new and that is what God does through salvation And it didn't come through my baptism as a baby It didn't come to memorize in the Ten Commandments and getting the little golden silver stars on my catechism It didn't come when I got my confirmation name and I changed I gave my name Richard because I liked Richard I wanted to be I wanted to be Ricky Rosales That's true. I Even put that on my Social Security card when I went into the military. I gave myself a middle name David Richard Rosales That's on my military record. I don't have a middle name My mom was all mad at me. She says you don't have a middle name. I say side to that Richard. No, you're not the natural man But he was spiritual judges all things yet. He himself is rightly judged by no one you can You live a life that the world if they don't know Christ just cannot relate to But God reveals to us and to the body of Christ things that we all unite on He was known the mind of the Lord who has known the mind of the Lord that he may instruct him we have the mind of Christ Well, what has happened is God has given to us a new mind Somebody once said you're brainwashed brainwashed Someone answered that yeah, my brains were dirty and they needed a good washing and They've been washed by the blood of Jesus Christ So absolutely they're brainwashed. I am brainwashed by the blood.


It is time for an update a bit more light there you go much better actually a bit more light take this and turn this off more light solar charger anyway I'll talk about this later just as a quick update because of the fires that have been going on this year last year PG&E the electricity company that is around this area has decided to turn off the power for two and a half days what is today it's almost midnight on Tuesday it's gonna turn off the power in 24 minutes for all of Wednesday and maybe until Thursday afternoon or up to five days so I might have no power until Sunday that still applies power just may or may not come nothing happened here so far this was a while back we had some power that went off and then back on little power surge type of thing so that's so far so good knock on fake wood or whatever that being said it has now escalated but now fires are going on again there's nothing around here but up north we have evacuations we have mandatory vacations we also have voluntary evacuations that being said my in-laws and the family that are on my wife's side they are preparing just in case it may or may not happen just like last time they came here we all camped around here until the fires were close to here and I was ready to go I had all my stuff packed in the car but that was back then so that being said long story short if you see a bit of a dip in the output in terms of the channel that's why I'm not stopping the channel there's nothing going on it's just that we have bigger problems to deal with cross fingers that the fires are stopping the winds are pretty bad though it's not looking good so far nothing bad here hopefully but maybe just like last time we have to prepare put things in the car get ready to pack and go somewhere where it's safe so that's it just a quick little channel not content update but just an update in terms of what's going on around here and how it may or may not affect the channel my outputs everything else but at this point if something happens I don't really care about the channel obviously it's going to be just about people being safe and us getting somewhere where it is safe all right that's it and I'll leave it at that


Okay, I think we can start. We will have one lecture from Bayanima, from the Institute, about, well, I live to say, yeah, please. All right, well, it's absolutely wonderful to be at the school again. It's great to see so many of you here. Originally, I was going to give two lectures on various basic aspects of quantum entanglement. Unfortunately, I only have time for one lecture, but I decided to spend the one lecture on a topic that's really possible to cover in one lecture. The topic of quantum entanglement has been a very, very important and interesting one for many, many decades, and, of course, it's something that people thinking about quantum gravity have also thought about for many decades, but as you all know, people have been thinking about it a lot more. Again, recently, you're going to get, I'm sure, a beautiful set of lectures by Hiroshi later in the school, but there's a few nontrivial facts about the very basic objects. The very basic objects are von Neumann entropies for density matrices, or if you trace out parts of a Hilbert space, von Neumann entropies for density matrices in various subsystems, and there is a few quite deep results about these entropies that were really established in the 1970s, mostly by the work of Eliot Lieb together with a number of collaborators, and for example, there's the famous statement about the strong sub-editivity of entropy that plays such a big role. Essentially every nontrivial fact that you're familiar with, which involves these entropies boils down to the strong sub-editivity statement, and so let me actually quickly remind you what the strong sub-editivity statement is. If you have some density matrix rho, we define S of rho to be negative trace rho log rho, as usual, but the statement of strong sub-editivity is that if you have a density matrix for these systems, A, B, C minus a density matrix, and I'll tell you how to remember this, it's always hard to remember the statement, this is less than or equal to S of rho A, B minus S of rho B, or I hope the notation is obvious, we're starting with a giant density matrix A, B, C, and then rho B, C is the density matrix you get by tracing over a sub-subsystem A and so on and so forth. This is the famous statement of the strong sub-editivity of entropy, and in fact this statement and all of these related statements follow from one master fact, and it's this one master fact that I want to spend this one lecture explaining, which has to do with another object which I think is a little more fundamental even than ordinary von Neumann entropy, this notion of relative entropy, and the fact that relative entropy is monotonic under course graining, it always goes down when you trace out parts of the system, so let me just tell you what the definition is, and we'll motivate it, we'll see the four parts of the lecture as I'll motivate it classically, and then we'll talk about some very simple statements about convexity that lifts from just ordinary functions to operator value functions, and then I'll just in order to make the proof a little more understandable and motivated I'll introduce some nice quiver notation as a model builder, we don't call them quiver diagrams, we call them moose diagrams, but anyway there's some quiver or moose notation for talking about quantum states, and then I'll go through the proof, so that's the outline. I should say that the reason I'm giving this lecture is that when I talk to people, grad students, even some postdocs working in this field, I find that most of them don't know the proof, and the reason is that the proof is buried in the back of textbooks, and the original proofs of Lieb and Russovsky are somewhat complicated, they involve some relatively heavy machinery about operator convexity. Now these proofs have been simplified, so if you Google, which is what I did when I was trying to learn the subject, simple proof, strong sub-editivity, you get a list of papers, most of which are superficial rewritings of the deep results of Lieb and Russovsky to kind of make them superficially look simpler, but I ran across this very beautiful paper by Nielsen and Pat in quantum pH rule from 2004, which is, I think, this was, you know, when you're looking for an understanding of something, you want an understanding so that when people come to you in the middle of the night, they put a gun to your head, they wake you up and they say, why is strong sub-editivity true, that you can remember why it's true, there's a moral reason why it's true, there's a reason it makes sense, and I didn't see that in any of the other works, and this is something where it's possible. After this lecture, if someone puts a gun to your head in the middle of the night and asks you to prove the monotony of relatively entropy, you'll know why it's true. And it involves some very simple ideas, but somewhat deep ideas as well, very simple, but deep ideas about ways of thinking about operators and states in somewhat interesting ways. Anyway, but before getting into it, let me just tell you what the master statement is, and at least how the strong sub-editivity follows from it. The master statements involve this notion of relative entropy, which is something that involves a density matrix rho and another density matrix sigma, and the relative entropy between rho and sigma, so I'll use this notation for, is, and we'll motivate this in a bit, but let me just write it down for a second, it's this interesting object, okay, it doesn't just depend on one thing, it's a measure of how different two density matrices are, and it's asymmetrical between rho and sigma, so I'm going to hand you sigma, it's a measure of how different rho and sigma are, oops, no, that's fine. Now, the claim is that if you have two density matrices, rho and sigma, and a big Hilbert space, and you trace out any part of the Hilbert space to go to smaller Hilbert spaces with some new densities, rho prime and sigma prime, that this object always goes down, okay, it's always monotonically goes down under any tracing out, okay, so that's cool, that's sort of built into the structure of Hilbert space in quantum mechanics, you know, this is nothing, you know, quantum mechanics is nothing other than the linear algebra of operators with positive eigenvalues, okay, so that's the, built into that structure is a certain, is something that's monotonic, it always goes down under course grading, and of course, as many of you know, in the past number of years there have been beautiful proofs of things like the C theorem that follow from, instead of, you know, complicated planes around with correlation functions, they follow more simply from looking at the things like strong sub-editivity, and that ultimately boils down to something which is monotonic in any quantum mechanical system, okay, so that's the, that's the really deep fact, so this is always goes down under course grading, so tracing out any part of the Hilbert space, and now, yes, yeah, let's just assume finite dimension, it doesn't really make a difference, the only thing you have to worry about is some of the eigenvalues of rho or zero, but that's not, yeah, it doesn't really, it's not going to make a difference, okay, but let me just relate this statement to strong sub-editivity quickly, so if we talk about this statement now, but for a case where I'm looking at the relative entropy between rho ABC, that's one guy, and another guy is just the identity matrix in Hilbert space A, tensor rho BC, okay, this guy on the one hand, this is equal to trace rho log rho, so this is negative s of usual von Neumann entropy of rho ABC, and then the minus log sigma, because there's the identity there, I just end up, I just end up tracing over a, so I just get a trace of rho BC log rho BC, so that's the entropy of system BC, okay, so this is plus s of rho BC, and monotony tell me that this is always greater than or equal to, now let me trace out the system C, if I trace out the system C here, this is now just rho AB together with the identity here, and just rho B, which by exactly the same logic is s of rho AB plus s of rho B, okay, so you see that monotonicity tells you that this is greater than or equal to that, and that's exactly the statement of strong sub-editivity, okay, all right, so what we're going to do in the rest of the lecture is motivate this notion of relative entropy and prove it, classically where it's trivial and quantum mechanically where it's more interesting, any questions right now, very good, so first, and there's lots of many ways of motivating relative entropy, Shannon had some very, very nice reasons for motivating it, which I won't have time to talk about, so we'll just get into the heart of it quickly, imagine you're given two probability distributions, so these P's add up to one, and the Q's also add up to one, okay, so imagine we're giving two probability distributions, and we'd like a measure of how different they are, okay, how different is this distribution from that one, then just classically we can define s of P and Q to be the sum over i of P i log P i minus P i log Q i, okay, and this turns out to be a great measure, and we'll see why it's a great measure in a second, again it's a measure of how different the two distributions are from each other, we'll see right away that it's always greater than or equal to zero with equality only when P equals Q, so it's not symmetric, okay, but it's greater than or equal to zero with equality only when the distributions are identical, A, and B we'll also see really, really in a trivial, deeply trivial sense that this object is monotonic under course graining, okay, so that's, so it's a real measure of the sort of the information content of the difference between P and Q, when you course grained you have fuzzier goggles and it's harder to tell them apart, so the difference between them is smaller, okay, so this is a quantity that's sort of built to do these two things, so let's see why that's true, okay, so in fact the way that we'll show that it's, the way that we'll show that it's positive is by actually showing that it's monotonic under course graining, okay, so we'll do that statement first, so what do I mean by course graining, so imagine that these are some bins, right, I have some, I have some bins with probabilities in each bin, so imagine I have some bin I, there's some bin I, and there's a bin J, so in one picture this would have probability P I and probability P J, in the other distribution it's Q I and Q J, and what course graining means is that I'm going to combine them into one bin, right, so I'm going to take these two bins and I'm going to combine them into one bin I plus J and whose probability is P I plus P J for one distribution in Q I plus Q J for the other distribution, okay, so is it clear what I'm doing, you know I have like a hundred possibilities for one, a hundred possibilities for the other, they each have some probabilities, but I'm just going to take these two possibilities I'm going to say I'm merging you together in one bin, so their probabilities at, I get P I plus P J and Q I plus Q J, and now the reason this thing is monotonic is that just under this elementary operation of binning it's trivially monotonic, the statement of monotonicity in this case becomes simply P I log P I over Q I plus P J log P J over Q J is greater than or equal to P I plus P J log P I plus P J over Q I plus Q J, okay, so that is a statement, now why is that statement true? This is a direct, it's almost a definition of a direct consequence of the fact that the function of the function f of x equals x log x being convex, so if we have any convex function, let me put it, if we have any convex function at all then it's true that if I have Q I f of x I plus Q J f of x J is greater than or equal to Q I plus Q J f of Q I x I plus Q J x J, and if these aren't normalized, if the sum isn't one I just have to normalize it here, okay, so this is just the definition of a function f of x being convex, since we'll need it again in a bit let me just remind you of trivial facts about convexity, so when you have a function whose double derivative is positive we say that it's convex up, if it's double derivative is negative in some region we say that it's convex down, so suppose you have some f of x with f double prime of x positive, then clearly from pictures this is what it looks like and if you have some a and some b then it's clear that the average of f of a and f of b, any weighted average of it anywhere in the middle is going to be above this curve, so if I have a point here just somewhere in the middle t a plus 1 minus t b then I have that f of 1 minus t a plus t b is less than or equal to 1 minus t f of a plus t f of b, okay, minus sign there so that f double prime is positive, okay and and and this this follows, I mean it's obvious from the picture but another way of saying is if I look at the quantity g of t which is the difference between the left and the right hand side clearly g double prime of t is positive by assumption, okay so because g double prime is just f double prime, so g double prime is is is everywhere positive, g is equal to zero at t equals zero and it's equal to one at t equals one and therefore it can't be positive anywhere in here, if it was positive anywhere in here it would have to have a maximum somewhere in there and if it has the maximum g double prime would have to be negative, all right, so I'm so it's obvious from the picture but we're going to come back to that quantum mechanically in a tiny bit, okay so the function negative x log x is convex you can trivially take this double derivative and that fact is everywhere in stat mech, right and so we see that the fact that this relative entropy decreases under rebinning is literally the statement of the convexity of the function negative x log x just to finish it all I'm doing here is writing this expression by putting a qi here and a qi and then over qj and a qj and then that's literally the statement of convexity where xi and xj are just pi over qi and pj over qj okay so this notion of relative entropy then has the property if you rebin if you bin them it always goes down right and so in particular if I bin everyone in one giant bin what is it it's zero right the relative entropy is zero in that case and that's why it's always greater than or equal to zero all right is this clear so this little this this tells you why this object deserves to be thought of as a measure of the how much information you have in the difference between p and q right again as you make your goggles fuzzier foggier you resolve less and less the difference you bin them the the amount of difference you can see between them always goes down all right any questions about this no if I had more time I actually it's a very nice exercise I'll leave for you since I probably won't have time to do it here we talked about it in the language of rebinning okay you can always you can also say it in the language where you where the number of bins stays the same but what you do is you take you take a given bin and another one and you chop this one into a big piece in the small piece which is of order epsilon and then you take that order epsilon piece and you add that to the next bin over okay so you first increase number of bins by chopping it into two pieces but in this way obviously you're not changing the relative entropy and then you add this little bin over to some other some other bin J if you do that that gives you a simple derivation of what are essentially Boltzmann equations rate equations for how probability distributions can change smoothly now keeping the bins the same and this proves that that always goes down so you know the the the convexity statements in the second law of thermodynamics are the same fact so that okay so so let's now move on to the quantum statements so now what we want to do is do exactly the same thing except except in quantum mechanics and of course it's non-trivial so we're going to define I'll switch notation a little here back to the one I used before is trace rolog row minus trace rolog sigma and now it's non-trivial of course in general row and sigma don't commute and so we would like to show that this is monotonic before showing that it's monotonic let me at least quickly show you that it's positive okay and there is an interpretation for what this there's a very simple interpretation about what this guy is that we can give right away and we'll see a slightly more sophisticated version of this interpretation than the actual proof but if we just first write this down I mean let's just let's just compute it let's imagine that we're diagonalizing row okay so so s of row and sigma this is equal to the sum of if these eigenvalues are p and the other ones are q it's pa log pa minus let me call it u alpha a mod squared pa log q alpha okay so I'm assuming that row has eigenvalues pa sigma has eigenvalues q alpha and u is the matrix that is unitary matrix that takes us back and forth between the two bases so u alpha a is just q alpha the overlap between q alpha and pa now notice that because this is a because it was unitary we know the sum over alpha of u alpha a mod squared is equal to one also the sum over a of u alpha a mod squared is equal to one and so there's a slightly cool thing that we can do we can imagine classical probability so I'm going to imagine a classical probability big p alpha a in other words not just big p that depends on n variables but something that depends on n squared variables okay alpha and a which is pa u alpha a mod squared and q alpha a which is q alpha times exactly the same thing u alpha a mod squared all right so these are classical probabilities but if we had n probably n eigenvalues for the peas these are n squared classical probabilities we can call them probabilities because they add up to one right if I sum over because of this if I sum over alpha first let's say for the top guy I get one then I summer the pa's and I get one all right so so these individually add up to zero and then it's extremely easy to see that as classical for this big p and this big q is equal to this relative entropy for the quantum states okay so and therefore it's equal greater than or equal to zero and equality I didn't say this before it's obvious but equality if and only if row is equal to signal so we can think of the relative entropy itself as a classical relative entropy but in this doubled space right not not in the at the Hilbert space is n-dimensional and an n-squared dimensional space we can think of it as a classical relative entropy we're gonna see that n- squared dimensional space coming up in the proof and as I said a little bit more interesting sophisticated way but this is already a hint that's that we should be thinking not in the Hilbert space itself but not in an n-dimensional space but in an n-squared dimensional space okay any questions about this all right so now let's move on to part two which is some quantum mechanical statements about convexity obviously convexity is going to be important it was important it was important classically already but quantum mechanically so we saw that classical convexity or ordinary convexity is a statement that if you have f of t a plus one minus tb it's less than or equal to tf of a plus one minus tf of b okay you could say that a function f is now quantum mechanically convex there's many notions of convexity but this is this is a simple one so so quantum mechanically we say that f is convex or operator convex in this sense the same thing is true as a quantum mechanical statement where a and b are operators but of course then we have to take expectation values of this in any state so if I choose any state psi then psi f of t capital A plus one minus t capital B is less than or equal to the expectation value of tf of a plus one minus tf of b okay now just because a function is classically convex most certainly does not guarantee that it's quantum mechanically convex okay but there is a special function where it's easy to check that it's true and that's the function negative log x so f of x equals negative log x is operator convex and this follows if we can show that this this this follows if we can show the t squared of this sucker psi f of tA plus one minus tB psi is always greater than or equal is always greater than or equal to zero okay so we just have to be able to show that and that is indeed a property of of the logarithm function so let me just the demonstrate such a simple proof let's just demonstrate it quickly so let's say we have the function negative log a and since I want to manipulate this I'll give an integral representation for it just the definition of the log as one over a plus u minus one over one plus u so this is a convenient representation so if I look at negative log a plus epsilon delta then if I expand this out this is negative log a and then I have integral zero to infinity the first order term du which is negative epsilon one over a plus u delta one over a plus u and then the quadratic term plus epsilon squared integral zero infinity du then one over a plus u delta one over a plus u delta one over a plus u plus dot dot dot but we care about this piece because we care about the double derivative right and so we see that over here so we see that the epsilon squared of log a plus epsilon delta is just this integral just the last piece which I can write the integral du of du dagger du where this matrix du is just is one over a plus u delta one over a plus u one over square root a plus u okay okay and so so a is assumed to have positive eigenvalues of course u is positive everything is positive the square root makes sense and so f of x is operator convex sorry negative log x is operator convex okay now now we're going to see another elementary consequence of convexity which is again very similar to the rebeening idea that we talked about here but said in a slightly different way since we're going to be interested in having big Hilbert spaces and small Hilbert spaces and tracing things out let's explore a little bit consequences of operator convexity when you have big spaces and small spaces so so simple consequence so if f is operator convex then suppose we have an operator of the form suppose we have an operator oh but now we have a big Hilbert space in a small one so I knew we have a tensor product of two factors and so let's say it looks like this some a and a dagger so let's say I have some operator that looks like that then clearly f of x again as an operator statement I'm going to not put this not gonna sandwich it anymore is less than or equal to half f of so sorry and y is 0 0 so this is heading towards tracing something out right so so something where there's no communication between the two sectors is less than or equal to a half of f of x a a dagger y plus a half f of x negative a negative a dagger y so that's just that's just the trivial consequence and so now let's take this statement and I'm gonna sandwich it between some vectors so let's say I have a vector of the form v dagger 0 ok so I have v dagger 0 f of x y zeros here so this says that this is less than or equal to v dagger 0 f of x y a and a dagger p 0 actually I should have it's a half this plus a half that but it's clear because I can I can get to this matrix from this one conjugating with a you that's only minus one in this part of the Hilbert space that term does not these two terms sandwiched in this state are going to give me exactly the same answer is that clear okay and so this has a very nice interpretation so let's let's interpret this it tells us that that let's say that we have some unitary operator you that maps from so sorry some operator you that maps from the small Hilbert space to the big one and in the obvious way so it takes a vector v here and it takes you v is equal to something that just as v in a zero underneath okay and then I can also clearly have something that goes backwards a dagger of it a you dagger which goes from big to small such that it takes vw and it just throws out the bottom component leaves me with v right so clearly you dagger you equals one you was not a square matrix right it's mapping from one Hilbert space to another Hilbert space but you dagger you is equal to one and so we learn already just about simple functions here is something that looks a lot like a coarse-graining statement that psi dagger for any psi f of you dagger OU psi is less than or equal to U psi dagger f of O U psi so this is a this is a statement that we have just learned okay is that clear so O is this big O is the O is the operator that acts on the big space so if I map a state in the small space into the big space and I take the expectation value of f it's less than or equal to had I done it the other way around okay that's all we're going to need about operator convexity just these very very elementary facts all right so now let's move on to the third part before launching into the proof let me introduce some nice quantum notation this is not strictly necessary but but I think you'll find that this kind of notation is useful of course it's very standard people use it all over the place cancer networks I mean in quantum information they use it all over the place of course we use it all over the place everywhere I find that it's a more grown-up way a better way of dealing with quantum states than just writing bras and cats all the time makes it easier to see the relationship between objects especially when you have Hilbert spaces that are product of many factors so let's leave that so three let's have the quantum notation okay and so your anyone who's played with quiver gauge theories or anything is very familiar with this notation instead of having a state psi well think about this as like a vector with downstairs index on which UN acts right so if I'm in an n-dimensional Hilbert space there's there's no reason there's no reason to privilege I mean these are just these are just representations of the UN so we should talk about them the same way we talk about representations for anything else we should use a nice tensor notation for them so this is really that's equivalently what they are but we can also I'm also gonna represent that I'm gonna remind myself that it's a that it's the representation of the downstairs index by drawing it with an arrow coming in okay so from that point of view a cat hi this would be some hi with an upstairs index and I'll represent that with a guy with an arrow going out so what are what are operators from this point of view an operator oh is something with an index coming out and going in and so an operator oh can take a state and return another state right so if I take I'd take this thing and I multiply it by psi then I get a new object with an arrow coming in which is the state u psi so this is all this is all trivial the density matrix is just again it's just something like this expectation value an expectation value of oh would be row oh row and so they just we just contract the indices everywhere we can when you have a when you have a Hilbert space which is a tensor product of a bunch of little Hilbert spaces then you can just have arrows for the different pieces so if there's two pieces then this could be a state in the tensor product of the two Hilbert spaces I can think of this I can think of this as a giant composite arrow right one big arrow which is really made out of two little arrows alright so this is really this is completely trivial the first place but but where this notation starts getting interesting is when you start swinging arrows around which we can of course trivially do with the pictures and and interpreting the same object in slightly different ways it so it's still pretty trivial what is going on but already here you see I can either think of an operator I can think of it as I can think of as an operator as an operator like this or I can swing this arrow around if I swing it around I can think of the operator actually as a state but as a state in a Hilbert space that's n-squared dimensional okay so this is exactly like double line notation for gluons right and then in and out arrow but again I think of these as one giant composite arrow so I can think about so so I can think about operators as operators on an n- dimensional Hilbert space or I can think of them as states in an n-squared dimensional Hilbert space okay and the second point of view where you're thinking about the mistakes is going to be the one that's really important for our proof and as I said we saw that already very early on in the fact that even the positivity of the monotonicity of the monotonicity was arguing that we should think about things in an n-squared dimensional space and not the n-dimensional one all right but let's just play with this notation a little bit more then I will tell you the precise moment where something interesting happens for now we're just screwing around and everything is trivial and just transcription nothing so for example what if I want to think about operators as states what is unitary evolution unitary evolution is some operate so some some operator big you but I want to think about this unitary evolution as an operator on this n-squared dimensional space but what is this operator this operator is just you and a you dagger in terms of the ordinary evolution operators guys UT UT dagger so if I take this and I and I acted on oh I will get you oh and so so this so this thing is this big operator UT what about other things let's say I have a row AB okay so it's a density matrix and a two-dimensional in a Hilbert space with two factors then how do I trace out B well I just do this then that's equal to row a okay and so on now actually before getting to the proof it'll be useful for us to play a little bit more and this is the one place where something a little interesting happens where the interesting things happen is is when with these pictures you move the arrows around so that you end up thinking about as an object sometimes as an operator and sometimes as a state if you always think of it as an operator always think of it as a state then it's totally stupid you can go back and forth between the two pictures that will what's interesting is that sometimes it's useful to think of it as an operator sometimes as a state and that's where these these pictures make that something very easy to do so let me give you an example of this so let's say that we have an operator B that lives in a big Hilbert space with two factors so the dotted line this so this is an operator and now let me write down something trivial of course this is equal to B let me now have an operator in the small Hilbert space S and I multiply by S inverse and then S okay so this is total triviality right but now we're gonna do something interesting now we're gonna take this arrow and we're gonna swing it around so that it points in the opposite direction okay so I'm doing absolutely nothing but let me now rewrite this identity right as saying that B now now I'm gonna think about B as a state so that's why I'm gonna put the arrows like this okay but B is equal to again and the line coming the other way so is it clear what I've done I've just swung that line right I also swung these things to make it look like a state and I swung this line from this side to make it look like it's going in the opposite direction okay but now this is kind of cool because this is telling us that we found a way to make a state in the big space in a natural way given a state in the small one okay so this is a state in the small Hilbert space and this sucker is an operator U that maps us from the small state into the big one so this is this we can define to be again let me just so yeah so this thing I will call you from S into B so I'm gonna need these I'm gonna come back to them so let me let me draw this one here so so there's this sucker you from S into B so it takes something from S it converts to something in B and this is just what I drew there it's also we can also take naturally a dagger of this U right how do we take a dagger of the U well just the definition of dagger right so what what we want we want to define so if I have in general if I have Y and U of X this is equal to this is defining what U dagger is is U dagger of Y with X so so if I put okay so if I just go back I just do this I imagine I put a big Y there I replace S by X then I can interpret exactly the same picture as X U dagger of Y right so the dagger is exactly the same picture but viewed in this direction so I have to flip the arrows all right is that clear okay so what is so U dagger then U dagger which is now doing the opposite so it takes something in the big space and converts it to something in the small S inverse the I'm just I'm just reading that picture backwards and flipping the orientation of the arrows and finally of course it's natural to look at U dagger U what is U dagger U let's compute U dagger U U dagger U which is okay this is just gluing those pictures together this is just S inverse now this thing is not in general equal to the identity but it is equal to the identity if we choose S appropriately okay so notice that this object this B this is just the trace over the this is just the trace over the small Hilbert space of B squared so if I choose S is equal to the trace over small of B squared to the minus to the plus a half then then U dagger U equals one so so in this way the picture is giving us a nice canonical way to map now between between operators in two different Hilbert spaces ultimately we interpreted the states we can go back and think about it as the operators as well but as you see in the intermediate parts of this process sometimes it was useful to think about those O's as operators and sometimes as states and that's why there was content to these to these drawings and to this idea of thinking about states as operators all right any questions about this okay so now now that we've played this game I have five minutes left I might take a little more not much more let's go through the proof the key idea of the proof the reason I went through all this and this is really the deep aspect of the proof is that the statement of monotonicity looks like a statement about operators in a Hilbert space the key idea is to interpret it as a statement about expectation values in the n squared dimensional Hilbert space okay so to not think about them as operators but thinking about states and exactly the way that we talked about here so so let's finally go to the proof and the key idea is the following so we're going to think about these density matrices as states so we're going to think about things like this but we're going to define an operator delta to just be it looks a lot like the unitary evolution operator that we talked about before which had a you on one side and you inverse on the other side except now we'll do these with these matrices row instead it'll be something that's just row inverse up here and sigma down here so the idea is that we're fixing row and sigma they're given and we're interested in their relative entropy okay but what I'm going to do is construct the following operator I'm going to construct the following operator on the on this n squared dimensional space so this is delta a beautiful thing about this is that the log of delta is also just negative log row plus log sigma right these two parts just don't talk to each other so the operator the operator that corresponds to log delta is just nothing here and log sigma here minus log row here and the identity on this side okay and so here is the gorgeous point is that the relative entropy between row and sigma can be actually thought of as just the expectation value in the state square root of row of the operator minus log delta that's exactly just because it's that that's how it breaks up that's what the relative entropy is so the relative entropy is an expectation value psi dagger convex function f psi right and we've already seen from from our discussion of convexity that in states there's this natural notion of coarse-graining that that statement that I erased over there but let me just say now what the logic is and then there's just one little tiny computation that we'll do with the diagrams and then we'll done I should say and this is where all of the you know all of the non-triviality and complexity in the original proofs come from they have to do with the fact that you're not talking about convex functions of one operator we have to talk about joint convexity of two operators and it's and it's more a lot more complicated the whole idea is that by going to the double Gilbert space you reduce everything to just simple statements about about convexity of single operators and expectation values okay so so what what what we want to do is use this fact so if we have a map u if we have a map u from from a small Hilbert space into a big one let me just draw it like this then the statement that we had before is that psi dagger this is just a general fact right this because of the because the function log is operator convex psi dagger log x u conjugated with u psi is less than or equal to psi u dagger negative log x psi u right so that in the notation that I introduced in this notation that's what the old statement is that we proved right so what is it that we want to do to prove monotonicity then what I want is I want the big Hilbert space obviously to be let's say the big Hilbert space I know let's say it's a b I'm gonna trace over b so the small Hilbert space is a so so small as a so what I want is I want x I want x because I'm trying to make a statement about monotonicity of relative entropy I want x to obviously be the operator delta in the big Hilbert space a b right that's what I want on this side I want psi I want the state psi to correspond to row a to the half so if I have if for x I choose delta a b and psi I choose row a to the half then if we can find then if we can find such that psi u such that the map from the small Hilbert space to the big one such that psi u is just row a b the big one to the half okay oh and and sorry and x u is equal to delta a then we're done so there's nothing non-trivial going on here I'm just in this having having transcribed relative entropy into this expectation value statement I'm just saying what what we need what we need to show okay now in fact everything everything is fixed for us because we know what this u has to be we just figured out in general how we're map canonically map states from small Hilbert spaces into big ones so we know what this u is so we know exactly what the u is which which maps which for which psi u is equal to row a b to the half and so on all that remains is to show that in fact so we've checked u dagger u is one all that remains to show is that in fact x u is equal to delta a so we have to look at we have to look at u dagger delta a b u so we now want to show that this is equal to delta a and then we are done okay so so let's just do it so let me remind you what u is again so I'll write it down again u the u maps big to small right map small small to big and this is equal to oh a b to the half oh and sorry there's a little comment I need to make here and I'll just explain when slightly drawing this notation very slightly differently so remember before we had an s there but we said that we have to choose the s to be we have to choose the s to be the inverse of the trace of that thing squared over the b Hilbert space right now for this thing we have row a b so when we square it we get row a well when we square it we get row a b when we take the trace over b we get row a so we have to take so we have to take the square of its inverse okay so that's how we get a row a to the minus the half there u dagger I'm choosing to order which ones are dotted and dashed just to make my resulting diagram in a moment very slightly cleaner of course it doesn't have I can move them around that will and finally what is delta but is delta well we just so delta now is on the big space and so I oh sorry so I'll write delta like it has two indices this one and the other ones so this was just row a b inverse if you remember and then the arrows going the other way for sigma a b so again no again it's not symmetric between row and sigma because of the direction of the arrows and because there's an inverse so it's a row inverse with the arrows this way and sigma with the arrows the other way all right so now that's it u dagger times delta times you so let's draw what they look like the final big picture you dagger let me I guess I could have done this more intelligently but to save me drawing a lot of drawing but so here is you dagger one more time okay this whole line to the bottom now I have to put a delta in the middle okay so and the delta in the middle the arrows going out of the you dagger have got to hook up with the row inverse here so this part let me put this coming out this part is gonna go into row inverse a b this part is gonna go into the sigma a b those going out that and this part of just going into sigma so maybe just draw it like this okay so that part is going into sigma all right and now what's going on on this side is for you I have row a b to the half that part just goes in then I have row a to the minus a half the identity okay so this is the computation of you dagger delta you and now of course everything collapses okay because this row a b to the half row inverse row a b to the half is just the identity right so that whole chain this whole chain I can this whole chain here so let me let me write let me draw it so row a b to the half in verse a b row a b to the half and then I had these other things now let me draw it let me draw that arrow down on this end here actually let me just draw it where it was no reason to switch it sorry this is nothing other than that's just the identity and so what do I get from so on the top line what I get is row a to the minus a half and another row a to the minus a half so on the top line I get row a inverse okay so sorry from the solid line which goes through so this is equal to on that side but what about on the other side on the other side I have these lines going into sigma I have the lines going to sigma a b right this was a sigma a b coming out but these lines these lines going in and out are now getting contracted that identity so they just go around in a loop that of course is nothing other than the definition of the reduced density matrix for sigma in the space b in the space a delta a exactly as needed so that's it that is the that is the that's the proof I think something that would be interesting to think about perhaps in holographic context and other context is in in general and as you'll hear about from a Hiroshi I'm sure there are many many more inequalities that are satisfied by these density matrices than simply monotonicity than simply strong sub-editivity okay so so we saw that monotonicity is a sort of deep fact that lies underneath all these statements about that are used all the time but even classically people discovered in the late 1990s I'm sure Hiroshi will talk about this we even possibly people discovered in the late 1990s that if someone just hands you that they hand you some probability distribution and they give you all of the entropies for all possible subsystems how can you check whether or not it could in principle come from some underlying probability distribution of course they have to satisfy things like strong sub-editivity and so on but is that it how can you intrinsically categorize the space of all possible entropies for subspaces and people discovered in the late 1990s that there's much much more than strong at sub-editivity and that in fact there's an infinite number as you go to larger and larger spaces there's an infinite number of inequalities that are satisfied in a sense the space of part of allowed possible entropies of subsystems is probably not polyhedral if you want to describe it as being so it cannot be described by giving a finite number of inequalities of linear inequalities it's an interesting curved curvy-shaped space and there's a lot of progress made on the subject that's classically in the late 1990s surprisingly as late as that the quantum version of those statements is complete virgin territory very very little is known and people are starting to learn more and more about it so I think something that would be extremely interesting given that we know that when geometries emerge and Hiroshi will talk about it but when geometries emerge much more is true than just sub-editivity of entropy you have many many more relations are satisfied then perhaps better and more intrinsic understandings of where these facts come from might tell us more things so here I haven't done anything particularly I mean this is just pedagogically interesting and I hope you got something out of it but as far as research goes it doesn't yet lead as far as I can tell to new things that are true and you should always be suspicious in physics of just proving old things okay you can do a lot of work pushing things around it's very elegant and fun but there's nothing better than a new formula that has not been written down before that's vastly vastly more important than pushing things around and understanding them better so what will be fantastic is if this kind of thinking could lead to new formulas and new understandings for what the actual space of allowed possible entropies looks like nonetheless I think it's such a it's such a beautiful subject and there's a lot of my second lecture if I was going to give one was going to be all about the sort of convex geometry and polyhedral geometry associated with Hilbert spaces in this way but it's a very beautiful geometric subject and I encourage you to read some of this literature and enjoy playing with it more all right thanks a lot okay thanks team for your talk so now there are two issues on the one end would be nice to let people ask questions on the other end we cannot make nima missing his flight so the compromise is one question okay yeah okay so two questions three questions questions or comments you used the minus log delta as one particular case of a convex function but can you have just chosen some other functions and prove some other results that could be right and how can you extract information because there I mean it's it's the number of convex functions modular Hamiltonians and stuff stuff stuff like that I mean that's a little bit of a question do you think there'd be holographic analogs for each of the diagrams or each of these steps that you drew yeah I don't know of any I mean part of the problem is that there aren't still as far as I know there aren't very good holographic understandings of what relative entropy is relative entropy is used I mean that this trivial fact about it just as positivity of relative entropy is proved by for example Cassini in the proof of the of the of the of the first law right but but yeah as far as I know there's there isn't a good there isn't a simple holographic understanding of what the relative entropy is it's too bad I mean that having played with it or thought about it not terribly deeply but I love relative entropy it seems like a much more I mean it seems like a more fundamental notion to me than entropy itself and so yeah it'll be it'll be nice if it had a holographic interpretation but a little bit more generally I think you know we know that there's a famous picture with geodesics giving in a nice geometric understanding of strong subeditivity right I think that's that's part of the part of the dream that I'm sure Hiroshi will talk about but but part of the dream is that is that the emergence of geometry would somehow be correlated with something geometric which is sitting there already in the structure of the of the patterns of entanglement okay let's relative entropy is related to expectation values of modular Hamiltonian can you does your proof teach something about modular Hamiltonian that's that's that's that's what I was saying I mean I think I um I think the monotonicity the positivity of relative entropy certainly makes an appearance in the proof of the of the in the casini's proof of the of the second law so I don't remember if monotonicity does it must have it must have made an appearance somewhere um I can't remember off the top of my head if it adds but but yeah but it's just a statement of the it's just indeed that we can think of relative entropy as as giving us the entropy relative to the vacuum state and then we can uh the the the yeah and so it's it's very much related to expectation values of the of the modular Hamiltonian ah good question um I think there are there there I think they're relatively um uh yeah so there I mean that um I think there are to the extent that um uh that there are other functions that are like that are powers of operators that are also operator convex and so that could probably be used uh that that might be that might be useful for uh many type statements but but I haven't seen anything I haven't seen anything but I think it's true um I think it's true that uh I'd have to double check but I think it's it's a result not a not a complete trivial result um uh sorry let me let me not say say say something wrong I think that I think that there is there is a statement involving powers for sure there there's an operator convexity statement involving powers I don't remember what it is exactly but that that that should be useful for the retinas uh I think we have to stop here now so let's thank him again and uh