Do black holes store negative entropy? by Koji Azuma

Okay, so welcome to, uh, the afternoon session. And, uh, the speaker now will be Koji Azuma, and speaking about, "Do Black Holes Store Negative Entropy?" Okay, please.

Thank you for the introduction. And, uh, possibly, I'd like to thank the organizers for giving me the opportunity to give a talk here. I'm from the Research Laboratory in Japan. And today, the title of my talk today is, "Do Black Holes Store Negative Entropy?" This work was done with my cooperator, Dr. Satya Subramanian. And before starting my talk, I'd like to sincerely thank them. Let's start.

And today, I want to talk about the so-called area law for black holes. Precisely, and today, we would like to apply a quantum information theoretical viewpoint to black hole physics to resolve a paradoxical nature given by this area law for black holes. To achieve such a goal of my talk, in my talk, I started by briefly reviewing the key concepts of quantum information. And then I entered a brief review on black hole physics. And after that, I want to introduce some inconsistencies between the quantum mechanical viewpoint and the protocol mechanics. And finally, I will give some of our ideas to resolve such a political nature.

I don't think, you know, what that I, my talk started by briefly reviewing the key concept of quantum information. Okay. Actually, the quantum information theory is a pretty simple theoretical, uh, paradigm in some sense, like because of the quantum information theory is regarded just a mathematical framework based on the postulates of quantum mechanics. Quantum mechanics is based on just four postulates. The first postulate talks about the description of the physical system and the physical state. In quantum mechanics, how do you know that the physical system is associated with a Hilbert space in quantum mechanics? And its state description corresponds to a normalized vector in the Hilbert space. Or, uh, more precisely, the second postulate is just talks about the time evolution. The time evolution of the wave function or quantum state follows the Schrödinger equation. That such a Schrödinger equation can be written by this type of form, where the quantum state at some different time are connected by a unitary operator. The third postulate talks about observables. Means, like a momentum or position in physics. Such observables are associated with Hermitian operators in the Hilbert space. These are self-adjoint operators. They are always diagonalizable. These are laws of generalities. Such operators can be written by in this spectral diagonal spectral decomposed form, here that the $a_n$ represent the eigenvalue. Eigenvalues represent the possible outcomes which could be obtained if we perform the corresponding measurement for the system. The eigenstate represents a state such that if we apply the or such measurement for the system, this eigenstate correct applicants outcome always $a_n$. And here, and then in general, that if we performed such a corresponding measurement to the general state $\psi$ at the side, in this case, that the probability to getting the specific outcome $a_n$ is represented by this equation, which is called Born's rule. The fourth postulate talks about the composite system. If the physical system A is associated with Hilbert space $H_A$ and the physical system B is associated with Hilbert space $H_B$, then the composite system should be described by the tensor product of the Hilbert spaces. This is adjustable position. Why I included that the postulate is here like this is related to me. So I want to introduce a concept of quantum computation. Namely, the goal of the quantum information theory is to answer what kind of information processing is possible by combining this proportionally. This is the main concept of the quantum information theory. And as a result, we have already known how to describe a state in general. A state of a system, including classical and quantum, is described by a density operator in general. This operator is just a mixture of quantum states. Here, the $p_i$ represents the probability distribution. The $\psi_i$ represents a quantum state. Developed density operator is associated with an ensemble of states. And this situation is described by this type of cartoon. Like here, we assume that some source is sourcing with the quantum state $\psi_i$ with probability $p_i$. This situation can be described by using a density operator. This is a, this is a, like a description given by quantum mechanics. For instance, if the density operator is a pure state, then of course, that this formalism has no reducing to the normal quantum mechanics which appear in that normal textbook. On the other hand, if the density operator is a mixture of orthogonal states, in this case, that we also assume that we are just assumed that all the possible operations for such a system are also diagonalizing the orthogonal basis. In this case, this density operator, namely Mickey's job, that also another state is regarded as a classical state in our community. This is because all the phenomena in this situation can be explained by using conventional information theory. More precisely, in the conventional information theory, treated this type of paradigm memory in the classical information theory. Those emitter alphabet $i$ is probability $p_i$. We apply the classical operation which is governed by the transition matrix. Then the structure crash for operation output alphabet $j$. This is the common classical information processing. We have a promise that all possible operations are diagonal in the orthogonal basis. And, uh, one advanced operator is a Mickey Showdown also owner state. In this case, the quantum information theory is equivalent to the classical information processing. So the police notices there for the quantum information theory is a broader paradigm than the classical information theory.

Okay, then, I also want to introduce a so-called purification. This is a very important notion in quantum mechanics. And quantum mechanics always allows us to regard a quantum system A in a general mixed state in general, but advanced operator as a subsystem of a larger system A in a pure state. Yeah, here that the pure state AE. If we take the partial trace over system E, in such a case, the density operator for system A is equivalent. We create equates such as the differential operator. What is this? With no concept like if there's some observer has a system A in a mixed state. In such a case, we can always imagine the existence of a purification partner somewhere in our universe. Tell us that even if we have some mixed state, and such a mixed state can be regarded as like a large subsystem of a larger system in a pure state. Other examples for this type of density operator, we can always consider this type of purification. At example, the water. Anyway, according to quantum mechanics, if your system is in a mixed state, there is a purification partner somewhere in our universe. This is associated with why the many-world interpretation is still possible in quantum mechanics. Developing principle, quantum information scientists maybe the universe should be in a pure state because of the thanks to the existence of this notion of purification.

Entropy plays a central role in the classical information theory. And entropy is defined for any probability distribution. The Shannon entropy is defined by in this form. By using such a Shannon entropy, we can define the mutual information and the conditional entropy. Are the problems like this one? Mutual information and conditional entropy are both positive quantities. You know what that we can write a Venn diagram. Some right to summarize this situation for a bipartite system. Actually, that in the classical information theory, that each term has each operational meaning in the context of the classical communication between Alice and Bob, etc. This is the important conclusion given by classical information theory. And such a theory is called nowadays channel theory in your community. And such a channel theory now generalizes the quantum version of the channel theory. In such a channel theory, we normally use the von Neumann entropy as a generalization of the classical Shannon entropy. By using von Neumann entropy, the mutual information and conditional entropy are defined by this form. Namely, that just the definition is pretty analogous to the classical ones, where we just defined this quantity by replacing the Shannon entropy in a classical world with a von Neumann entropy. Notified that this definition is adopted in the community. Like, suppose that the density operator $\rho_{AB}$ is in a classical state. I mean, if the density operator is a classical mixture of orthogonal states, then all the above quantities are reduced into the classical ones based on the Shannon entropy. Because this quantity alligators generalized version of the mutual information, conditional information entropy, von Neumann and functional entropy, etc. Again, I want to mention about the quantum information theory, the broader than the classical. However, in contrast to classical cases, in quantum cases, we cannot write a Venn diagram because the conditional entropy could be negative. Okay. And that, for example, this can be easily understood by considering this example. Suppose that the density operator is in a pure state. In this case, the von Neumann entropy for the system AB itself. Therefore, the conditional entropy becomes $-S_B$. This is general. In general, this could be negative. Developed in the quantum, more conditional entropy becomes positive and negative depending on the situation. They want to, by just reading this situation, one might notice that maybe conditional entropy might look to be related with entanglement. This is true. This is true. You need to minus the conditional entropy called the coherent information. It is associated with entanglement. Coherent information is defined by just minus the conditional entropy. Okay. And however, the coherent information is associated with entanglement, but this is not exactly an entanglement measure because the coherent information could increase under local operations and classical communication. However, that still coherent information is associated with entanglement. Actually, the positive coherent information has operational meaning. Clear operational meaning in the context of quantum state merging protocol. Quantum state merging protocol was introduced in 2005 by Horodecki et al. Here we consider accordingly situation, namely, additional how to copy of the density operator AB. But as I said, in the quantum information theorist, we can always imagine the purification partner somewhere in your universe. Therefore, that this site, if we can regard system E as just a purification partner of the density operator between Alice and Bob, then we assume that Alice and Bob share such a copy of the state and copies of such an initial state. Then some measurements after performing the unitary operation which is chosen by at random, after that you are performed the projecting measurement. The outcome is sent into public side. Will be performed on correction operation. In such a case, for example, arrival this type of situation. In this situation, namely, by using one-way classical communication between Alice and Bob, transform this state into this form. Here the output is divided as a foreign like an artistic system. Alice's state A is merged to the both sides. Here, namely, many number of copies of the Bell state. Here, the coherent information has a clear meaning in this quantum state merging protocol. More precisely, the ratio between M and N, Alice and Bob, multiplication, how many number of entangled pairs are obtained. So that this protocol and represents a number of copies of the initial state. The ratio between M over N approaches the coherent information in the limit of the N goes to infinity. Now that this coherent information is associated with how many number of pairs can be destroyed by performing a sample protocol between Alice and Bob.

Question: Okay, I just like to understand how do you can you explain this protocol without introducing an environment? And how and how does one do that?

Answer: Actually, that this is the definition. Like why we introduce a purification partner. The goal of this line is that quantum state merging. The goal of the quantum state merging is just to merge the advantageous system into both sides. So we will keep coherence with the environment. This is the goal of the protocol itself. From the initial point, we assumed implicitly assume the existence of an environmental system. The goal of this merging protocol is to merge such a system into Alice and Bob's side, keeping quantum coherence with such an environment. That's why this is a, like an assumption. And so if we, if we forget about the environment, actually that such a generalized quantum state merging is sometimes considered in your community, but this is a pretty difficult problem.

Question: Just just a very quick follow-up, then, um, but nevertheless, the quantity is independent of the choice of purification?

Answer: Yeah, yeah. Actually, the purification has always has a freedom. Like, if once we are given some density operator, we can imagine the purification partner. The, if we, even if we apply the unitary operation for the purification partner, the same purification. Without even if we either perform some unitary operation, of course, the state description is a bit different, but it's terrible entanglement. It doesn't change.

Question: So you see that this conditional entropy has the operational meaning and area that it counts certain number of pairs? But does that mean that the conditional entropy is always negative?

Answer: No, no, no. That's a nice question. Like, as I said, if we consider like this type of classical state case, or the quantity is reduced to the classical conditional entropy or mutual information. Therefore, that the conditional entropy in such a case becomes positive. But that this protocol has a works only when the tab such a conditional entropy, namely initial state. If this protocol works when the, the display. Okay, so this certain type of state, I mean, that, okay, this is the number of pairs.

Question: Okay, yes, thank you.

Answer: Yeah. Okay. And I go proceed further. Okay, okay. And, uh, this is very important to result in your community. Okay. And I also want to mention about the different terminology between the quantum information theory and black hole physics. For example, in the inner from our viewpoint, like if the density operator is a maximum entropy ensemble, such as a canonical ensemble state, such as a microcanonical ground summary state, it's from our entropies. They're used to the coarse-grained entropy, which satisfies some dynamic relation, etc. Namely, from our community, von Neumann entropy is just a general measure for that defined, which is defined for arbitrary state. But in the black hole community, maybe sometimes that this entropy is called fine-grained entropy or etc. But from our viewpoint, just like we normally use a von Neumann entropy as a measure of that any kind of state. And if the density operator has a restriction, such that such a density should be a maximum entropy ensemble, in such a case, its von Neumann entropy is reduced into the coarse-grained entropy. This is our viewpoint. Okay. Here I want to stress that only my entropy is like a perfectly generalized bargain for any kind of entropy. In principle, statistical mechanical entropy or classical entropy also right. This is I want to mention why I give you this type of basic stuff in the in this complex.

Okay, you know what, I want to check and change my topic a little bit because I, I have already introduced our concept in our field. So what I jumped into the brief review of the black hole physics. Maybe this is well-known in this audience in this workshop, but I want to brief, I want to make a brief review on the black hole physics from my understanding. That black hole mechanics start by finding even by Bekenstein and Hawking in the night 1773. Bekenstein, uh, considered just a general level under the classical general relativity. They derived for a stationary black hole. This is the essential disaster energy conservation law for black hole. This time, the change of the mass of the black hole should be associated with the change of area, change of the angular momentum, and change of the charge, etc. So this is just a first energy conventional law for black hole space-time. And as well as well as Bekenstein Hawking also provides the second law for any classical process except for whole ingredient. Namely, as long as we consider classical processes such as black hole merging or etc., the area of the black hole monotonically increases. At the same time, they did wonder whether that maybe this first and second law looks very analogous to the first and second law for thermodynamic physics. In the paper, they wonder whether that maybe also time on the right hand side in the first row may be associated with temperature times enthalpy change of entropy. But they are, they were at that point very conservative. Namely, they just considered under classical general relativity. There were at that point temperature should be dealt because the black hole seemed to absorb only in the classical process. And at the same time, consideration just getting experience was given by Bekenstein. The Bekenstein constant following simple geranking experiment. Vacation just threw an entropic particle to the black hole. Of course, the black hole absorbs such a particle and the black, such a particle disappear outside the black hole. Then they can start asked us if black hole has no entropy, namely, if the pure state entropy disappear from our universe, namely, for example, if we, if we find some entropic particle outside the hole, we just throw everything to the black hole, then we can delete entropy outside the port. The word which is incompatible for from our universe product, which is incompatible with the second law of thermodynamics. Also, maybe conjectured that maybe black hole should have entropy, but the entropy should be associated with parameters appearing in the first row. Because of the no-hair conjecture for black holes, maybe if the black hole has enthalpy, the entropy should be associated with proportional to be the area. But she cannot determine the proportional constant. Temperature was determined by actually Hawking radiation. Hawking paper, and the Hawking finds approachment that the Hawking radiation of black hole radiates something thermal radiation from the hole. Okay, then Hawking considered focus in the first paper, first focus on the subject black hole. In this case, the first law of thermodynamics is simply written by this type of form. And change of the mass is associated to the change of behavior. In this case, because the temperature or has no angular momentum and charge. Then the Hawking expands the semi-classical theory. Then by considering that or space-time described like this one. In this space-time diagram, the pink line represents infinity. Orange line corresponds to future infinity. Green line corresponds to light cone, which becomes an event horizon after the black hole has made. And they're focusing focus on the possible trajectory which passes near the event horizon and finally comes back to the future infinity. This is just a muscle relaxer. They consider there for such a possible trajectory passing near the event horizon. And how can you find that this trajectory creates an entangled pair, which is now regarded as the two modes. Previously in your community, even starting from the vacuum effect here, namely, even the possible trajectory starts from the vacuum state by passing through the event horizon near the event horizon, this entangled pair about the future infinity receives some thermal radiation from the black hole. And the clarification partner maybe falls down into the black hole. Since from the about the future infinity or special infinity of Java, the black inside the black hole has only negative energy spectrum, such as negative and $H$ minus pathway $H$ minus should have negative energy spectral force. Okay, then, what is a surprising fact? Like these two modes scale states case, the registration operator. This $H$ plus particle is a proportional given state. Give state only has one parameter associated with temperature. And by considering this type of quantum computing over black hole space-time, he concludes that this temperature of the thermal gives state of the $H$ plus is in the Kabbabi over to pi. Okay, then determines that the temperature of the radiation from another core. And besides Hawking, the guard this in the as a this analogous relationship as an unknown parameter. However, the such a parameter now is determined by by regarding some radiation temperature is equivalent to the temperature of the black hole. As a result, stay alive at so-called Bekenstein-Hawking equation. Okay, then what the fact that someone receive a positive energy part from the black hole. Therefore, the mass of the black hole should decrease according to the first law of the black hole mechanics. Of course, the area of the black hole should decrease. This is a, like a story understood by even by quantum information societies like me. Yeah.

And the next, I want to introduce inconsistency proposed by some quantum information scientists in our community. Okay. And then the, okay, this is a conventional story. But, uh, from quantum information science, three points this looks strange in the sense that actually that as Hawking suggests, Hawking state is transformed into two modes. Previous state. However, the $H$ plus and the $H$ minus infinity of pure state. In this case, the entropy of the $H$ plus is equivalent to the enthalpy of the negative energy particle falling into the black hole. In order, the entropy should be positive. Barrier, even for such a negative energy particle falling down into the black hole is correct. Of course, that I brought some entropy or dimensionality of the black hole increases. This through the generation of the Hawking pair. There was the entropy increases. Black hole entropy should increase, which means that the area of the black hole increases according to the Bekenstein-Hawking equation. There was that these are incompatible, which was mentioned by Brownstein et al. in our community. Okay. What is a key observation here from the just a quantum information theory viewpoint? Okay, let's go back to the consideration given by Bekenstein again. Bekenstein's consideration is very simple, like just a Bekenstein for the entire particle to the hole. Of course, as I said, in the quantum mechanical quantum mechanics, we can always imagine the purification partner somewhere in your universe. There was a purification partner exists somewhere outside the hole. They hold the backstage ranking experiment is regarded as just the entanglement generation between the black hole and the outside. On the other hand, what is the whole congratulation? Hawking radiation argues someone at the future infinity receives a summer luggage, which is a mixed state. Because we can always, we should we should imagine the existence of purification partner even for this state. Such a purification partner actually appears inside the hole, which is proposed by Hawking, right? The whole purification partners appear inside up here inside the black hole. Then what, what is the question here? Anyway, that microscopic from the microscopic viewpoint are not so different between the Bekenstein experiment and the Hawking radiation. Because anyway, entanglement generation between the black hole and the outside. However, the action for the area are entirely different. Namely, in the first experiment, Bekenstein argued the area should increase. Radiation area should decrease. From the microscopic viewpoints, anyway, both processes talk about entanglement generation between the hole and the outside. However, the change of the area are completely opposed.

Question: Just to make sure, are you talking about the physics before the Page time or after the Page time?

Answer: Please, please tell me. Okay. Yeah, so, so before the Page time. Yeah, yeah. Two notions are different, right? Yeah. In the case of Page time. Okay. After Page time, he got it as a purification partner also start to appear outside through the island lights, etc. Yeah. Okay. And then, but as I said, just that and before the Page time, after it's this type of situation. Okay. Okay. In that case, it's a black hole entropy and this, I mean, the entropy of this actual QFT state would be a different notion. Um, so there's for me, it seems like there is no inconsistency. Just like that this is not a, like a just an intuitive discussion. Also, okay. The phases have been completely different performance. One. So let's talk about playing. Okay.

Answer: Then therefore, in this case, I thought, okay, but, uh, you are, you may misunderstand black hole physics. Of course, okay. Bekenstein's someone argued maybe entropy appearing the Bekenstein Hawking equation should represent the coarse-grained entropy. Okay. Such a partner say such a thing. Naming she's a black hole should be a microcanonical ensemble. It should follow the some dynamic relation described like this. Therefore, once you admit the decrease of the mass of the hole by Hawking radiation, the coarse-grained entropy should also decrease. Hence, there is no problem. But what are you considering? There is no paradox here. However, please notice even in this case, I, I think there is a part of if we believe the Hawking calculation picture is correct. Namely, according to Hawking's formulation, this $H$ minus is the negative energy bosons, right? There was the $H$ minus is the bosonic system with the unbounded negative energy spectrum. On the other hand, black hole should have also positive energy of object because otherwise the black hole, black holes must should be positive. There was a black hole in some sense should have a positive energy part also.

Answer: So, a stable system or micro. In this case, microcanonical ensemble is not well-defined because of that. In this case, that we have unbounded energy spectrum bosons inside the hole. Our strategy is just we just want to do not want to change the Hawking pair creation picture. Okay. Our key ideas are. Instead of throwing out Hawking's pair creation picture, we explored the possibility to keep the picture holding the picture as it is by utilizing the fact that Hawking radiation is special in the sense of it uniquely generates unusual negative particle inside the hole. Okay. Therefore, we want to modify the Bekenstein-Hawking equation rather than believing that rather than believing that. Okay, like this, the present coarse-grained entropy, etc.

Answer: Oh, actually, if we, I'm usually when we talk about simultaneous or evaporating rapport, we we have this area law, area time as you write it $A/B$, but also we need to add matter contribution. Ah, yeah. Some of them. Then it always increasing. Okay. Even though, but still, so I just wonder why you are only looking at area. Just like we focus on that this quantity on this. But of course, I know that in the past, it should be actually correspond to the area process. I think also at that time. Yeah, yeah. So, yeah, I think no, no products. Yeah, yeah. This is just a I focus only on this. This $S_V$ represents just some entropy. Information theoretical entropy. Of course, there is many interpretation. What is meant by $S_V$ here, right? But in this talk, I just focus on the $S_V$ just a represent only my entropy. And maybe von Neumann entropy. Maybe they're listening to the coarse-grained entropy if we consider the state is equal to area. Yeah, yeah. This is the like guys. Starting point. If we change that extra hand side, maybe such a part of the main no tapia. Okay.

Answer: And I want to we want to change the application. Maybe we normally more precisely, we would like to really modify the Bekenstein-Hawking equation by to keep the Hawking pair application picture as it is. Okay. To achieve such a goal, we made a two assumptions. Record being actually composed of not only positive energy particle $B$ plus, but also negative energy path $B$ minus. $B$ minus is like generated by just passing through the Hawking radiation because the Hawking appears Hawking radiation generator negative energy particle inside the hole. But a particle contribute to increase the system size of $B$ minus here. Okay. $B$ plus represents some some normal matter inside the hole, such as mass. That's why black hole mass becomes positive.

Answer: There exists in the outside outside the victim $B$ bar. In my talk, and namely $B$ plus, $B$ minus, $B$ bar is in a pure state. This is just a notion of purification. If you want, one can regard $B$ plus represented this region, which includes the singularity in the extended credit card test time. And the $B$ minus may represent that this secondary flat region. If you want. Okay. So what is my, well, our main idea? Just we make us we assume to think. Then the argument is the following. The area of the black hole is proportional to the coherent information, rather than simple entropy. This is our argument here. The coherent information is a $B$ has a direction. Actually, but we take that coherent information from the outside to the positive energy particle or inside the black hole. That we modify this. We can start Hawking equation. As I said, this coefficient information related with the entanglement. This is because of the this coherent information becomes positive only when then conditional entropy becomes negative, which happens only in the case of quantum logic between outside and the inside. The size of the black hole may be associated with the size of entanglement between inside and outside. Actually, that's why we chose this type of formula, which comes from the following fact. As I said, the black hole purification should exist outside because the $B$ plus, $B$ minus, $B$ bar is in a pure state. In this case, we have this type of mathematical relationship. This formula in this form. This simple form. In this case, the area over 4 is equivalent to the positive energy part of the black hole entropy minus entropy of the negative energy part of the black hole. Just the contribution is very small. Just we added this term here. In the first term is just rephrasing the this the original Bekenstein-Hawking equation to to keep the Hawking exploitation picture as it is. We added system. This is similar to the Maxwell lead or what Maxwell did for to establish Maxwell equation. We just add this term. Please notice that this definition looks pretty good because this this Con is sometimes related with the entirety. It looks like a area may be defined by correlation between the inside and outside. However, this definition defined by local. Namely, once we, we understand the description of the black hole as a density operator, we can calculate its area independent of the state of the outside. There was this type of ice properties satisfied. Thumbs up. The only thing that I said the second time, if the, as long as we consider the process where that there is no exchange, there is no Hawking radiation, in such a case, the second time always tell the whole of this formula is reduced into the original Bekenstein-Hawking formula. And, uh, in addition, that this is consistent with the original Hawking pair creation picture. I finally mentioned about this one. This is very pretty simple. Just like we considered some Bekenstein experiment. We are that black hole, the observer system, positive energy particle from the outside, while there are calling it's a Hawking pair $H$ plus, $H$ minus. Then the change of the black hole energy and the coherent information are described like this form. Because of the in this process, black hole receives a positive energy particle from the outside. On the other hand, black hole emits a whole Hawking radiation to the outside. So change of business is described by this form. And at the same time, this black hole receives a positive energy part from the outside. On the other hand, also black hole receives a negative energy particle through the Hawking pair creation. The water, this pair changing, this is a change of the coherent function. Let's imagine the situation where the system A is in a perfectly copy of the Hawking radiation $H$ plus. In this case, for example, if the system A is in the thermal state at the Hawking temperature $T_B$, in this case, $E_A$, energy of this particle A and the particular $H$ plus is the same. Of course, it does. Therefore, they entropy the same. There was a change of the mass and the change of the coherent information itself. In order to either from the our new modified adlo or either from faster or black hole mechanics, we can conclude the area change doesn't look up be surrounded by some other system with the Hawking temperature $T_B$ is exactly an equilibrium state, which is consistent with Hawking's original argument. However, that for such a system, we just consider small partitions. Such as for the equilibrium state. Then in this case, that let us consider the change of the coherent information. In this case of coated, this is and the $S_H$ becomes some becomes a change of the enthalpy of the system. In this case, but the air system A on the $H$ plus is a given state. There are this $G_{SA}$ and the $S_{VSH}$ follows the thermodynamic relationship. So that we can replace change of the coherent information by using the change of the month according to this expression. Then the letters assume that our equation of force. In this case, a total equation is actually it produces the first law of the black hole mechanics. The water, if we assume that coherent information is $A B$ over 4. In this case, that the total equation reproduces the first law of the black hole mechanics. It's okay. There was a result changing the Hawking pair creation picture, we can reproduce. We have we can avoid to pick a bypass the power I have mentioned in this talk. I also, as I said, the coherent information has a clear operational meaning like, uh, here and the information replaces it represents how many ebits are distributed between the outside wall. In this situation, if we apply this quantum state merging initiation, this suggests that above 4 represents the in such a case, if this formula is correct, then $A B$ over 4 represents the size of maximally entanglement terrible between the usual positive energy particles inside the hole and it's outside by a process at the outside which does not change the area. If this formula was correct, area over 4 clearly represents how many distributed entanglement exist between the inside hole and outside.

And I will briefly mention about the coherent information of those paradox. But this is not so pretty different station. So we, I want to just mention. But our formula is free from the information loss paradox. Normally, the information paradox starts from saying like our universe starts by in a pure state. There are the total system is a pure state. And the entropy with such a system. Intel, namely our entropy or university starts from there. Then we have one black hole, one big black hole or ingredient. Stock at some point. QR code disappear. As a result, our universe is fulfilled by thermal radiation only. As a result, if the black hole has no entropy, in such a case, area of the black hole result besides that. However, that our universe is now crucial to a summarization, which is which is a violate, which is contradict with the unitarity of the quantum mechanics. This is an information normally explained in your community. And that however, as I said, this story based on the regarding the black holes is composed of the monotonic particle. But, uh, you know, in your case, as I said, black hole is composed by not only positive energy particle, but also negative energy particles. Perfectly different. The water, our protocol is our our proposal is free from the information loss paradox. And we also free from the firewall paradox because the firewall paradox appear when we consider the Page time or Page is the formulation of popular core physics. However, in our case, that as I said, Pages once pages also regardless our universe is made by radiation and hold and radiation composed of single pieces. But in your case, black hole is composed to a system plus radiation. In our formulation, we regard black hole and radiation is composed a tripod type system. Type system. And maybe which is related with technician's question. If we assume that this even if we assume that even if we change this formula into this formula, maybe from the quantum information theoretical viewpoint, there is no problem. Even if this type of generalized second row force. But this is just a naive formulation. We cannot derive directly. But from naive notion, even if we replace the foreign could hold even from the quantum mechanical mechanical viewpoint, there is no by no paradox even if it is taken generalized second row. Of course.

Okay, maybe I'm consumed. And zero is finally, I just briefly mentioned like, and so far we have discussed only on the Chevrolet black hole. However, this formula looks okay even for the for any stationary black hole, which are the first row of the black hole mechanics. This is because that maybe we, even if we want to generalize this according to general general record, in such a case, we just need to replace the free mode frequency $\omega$ into this type of form. According to Hawking's paper, just by following such a replacement, we can we can and we we can show that maybe our formula looks old even for that any stationary black hole, including car or charged black holes. Okay. Then what that this is my talk. And that was the final message of my talk is, uh, area law is allowed to be modified a little bit. We may be able to we may not need to change Hawking pair creation picture. In this case, that we just added second families in the original formula. In this formula, the negative energy particle produced by Hawking radiation behaves as if they have a negative entropy. Thank you for your attention. Thank you very much.

So, so especially you should download the lab diagram of which emit the which emission the particle book in registration paper as well. Yeah, so today depicted. Yeah, this is a just a okay, okay, go ahead. This one. Okay. Um, no, no, no, no, no, no, right. Maybe okay, so you have a downloaded diagram in this ride. And, uh, $H$ minus is a partner particle of the Hawking particle $H$ prod. And then so you said that $H$ minus is a negative energy particle. While this is a negative particle. This is a comment from that general relativity thing. Like we, we here at the top, we regard energy represents energy up to infinity, future infinity. But eight months ago, I want to say that all the picture is based on assuming that this picture is given by the observer standing at the future infinity. Okay. So $H$ minus the Cobalt by the probability. So outside the observer cannot see $H$ minus. Indeed. But if you use transitional quantum fields in a public space-time, then $H$ minus does not have a negative energy. You may be confused. So we have another mode. So so in this details, $H$ plus, if the $H$ plus mode is very close to the horizon, then you can compute the expectation value of energy density by using the traditional scheme of the quantum field theory. Then you notice that the energy density does not happening positive or negative. So when so $H$ minus take the positive body density energy is just arrived a few. So a few times I've been a block radius. And then so the space-cell curvature give the protest the energy. Then $H$ plus becomes the near part. So near horizon $H$ plus is just a batch of particle, just a quantum for application with zero energy. Similarly, $H$ minus other or also zero zero point energy horizon. But when $H$ plus exchange of the particle mode, which is a and go into the horizon board. This is not $H$ minus, but the extend energy. Then its plus began the apartic with the positive energy. And then the extensive modes are the negative energy, which propagate to now with the negative energy outside of line. This is not $H$ minus. Okay. Plus $H$ minus is the problem. Yes, I agree. But $H$ minus does not carry and negative energy itself. So I'm wondering so this is your start point. So $H$ minus negative energy, right? But this assumption is not correct. Rising. Yeah, actually, we borrowed time lots of specialists of black hole physics. Actually, I confess a time lapse. I'm not a specialist for quantum physics over carbon space-time. And, uh, but I just, uh, our combination is based on the the accommodation given by David given the review paper given by Tovey and of course, maybe people. So maybe Long and, but, uh, we just assume that the whole story formulation. Okay. I I need to skip this one. I I just show the difference. What kind of Hawking's picture is adopted in my talk meter. This is written. This if you order the review paper 96, I will just follow that formulation in the given in this paper. And so of course, maybe other combination could be possible. But, uh, no, no, no. So this panel diagram you have $H$ minus in the top of the diagram. But it is edge minus propagate tuna to the lower side of the left side of this panel drive. So the trajectory is just in a light strategory, which is starting a left side, the right side, and which is a combat by The Horizon. Then if you can compute the energy density, then $H$ minus, it doesn't have any advantage. So you you may think so $H$ minus belongs to the left side region. And then so you may think $H$ minus has a negative energy boost becomes negative minus $H$ bar omega. But the energy the same energy should be defined by the some dynamics. Okay. Okay. So you want to consider black hole lineups. Then inside the B, such a notion of the negative energy particle is needed only when that divided the system. Only as you can notice that, actually, I we do not use that this particle should be negative energy particle. Like this particle should be different from the outside. This is the only assumption because of the this energy conservation law is based only on the what kind of particle was the black hole and what kind of positive energy particle goes out. The water, we do not expression that $H$ minus should be negative particle energy. I'm worried about so you studied the motivation. So negative energy is ring. So because due to energy on Mars, anyway, and, uh, mechanics, we do not need to use a negative energy particle, right? Because the fact that someone receive a positive energy at the future infinity means that the decrease of the mass of the pole. Also, you mean, and so you do not mean just use the 14 particle does not need child in a negative balance. The point, but we do not need. Yes, it's actually a posture rate. Yeah, the partner should not have negative. Just using the purification partner according to the black hole. But you assume that Delta $A B$ should be decreases. This one can be derived by using the first row of the black hole mechanics, right? What kind of property is held by $H$ minus? Like, just we assume that somebody receive our energy after a huge infinity. If we believe that the energy conservation law or the energy of infinity, in such a case, most of the black hole should decrease according to pass through or drug core mechanics. Then area decrease without assuming that $H$ minus the energy spectrum. Okay. Yeah, so you do not need assume the $H$ minus is in a negative when it stop. Yeah, just like that. Okay. In the name of the particles, you can eliminate the condition. Yeah, the minus can be positive energy. Okay. Just like we want to differentiate such a purification pattern of the Hawking here as a particle which order outside like a big A particle, considering that against ranking experiment. That's it. What does you do not need any existence of a negative energy frocks outside of horizon, right? So we assume that we compute the certain energy blocks. How to surprise them? And then we can we may have such a negative energy products outside of horizon, which is also without right? Okay. But such a negative flux does not fry any law in your argument, right? We do not need certain negative energy products outside of horizon in your argument, right? Okay. Okay. If there are other questions or comments from home, anyone? Okay. Okay. So if not, let's thank the speaker again.