Bits are winning the war against black holes
The black hole information paradox told in ASCII characters
The black hole information paradox told in ASCII characters
My 14 year old is about as interested in physics as I am and wants to be an astrophysicist when he grows up. This means we get into a lot of conversations about the deep topics I write about in this publication. One day he asked about the black hole information paradox thanks to news about a recent paper purporting to solve it by proposing the existence of “quantum hair”. I went into a long and bumbling attempt to explain the paradox to someone who’s most recent math course was middleschool algebra and had no concept of quantum superposition, wavefunctions, unitary evolution, Schroedinger’s equation, or any of the usual vocabulary one uses in these discussions.
As frequently happens when my son asks questions about advanced physics topics, I realized I couldn’t explain it because I didn’t really understand it. Despite having read quite a bit about the topic and being pretty skilled at tensor calculus, I neither understood the problem nor the need for a solution at the level that I could explain it to a 14 year old.
As a teacher, I live by Richard Feynman’s principle as told by colleague David Goodstein in Feynman’s Lost Lecture: The Motion of Planets Around the Sun:
Feynman was a truly great teacher. He prided himself on being able to devise ways to explain even the most profound ideas to beginning students. Once, I said to him, “Dick, explain to me, so that I can understand it, why spin one-half particles obey Fermi-Dirac statistics.” Sizing up his audience perfectly, Feynman said, “I’ll prepare a freshman lecture on it.” But he came back a few days later to say, “I couldn’t do it. I couldn’t reduce it to the freshman level. That means we don’t really understand it.”
I quickly realized that I had failed Feynman’s principle on the black hole information paradox for I could no more explain the topic to a freshman than a 14 year old.
I’ll be the first to admit that the black hole information paradox is a theoretical problem and the solutions are likewise theoretical, but that doesn’t stop us from at least trying to understand the problem and potential solutions to it.
Most explanations of the black hole information paradox are pretty hand-wavy in trying to explain what it’s all about. If one does go into the details, it becomes unintelligible to the uninitiated pretty quickly.
The first thing you have to understand about the paradox is that it is about information. So what is information?
Succinctly, information is defined as any series of answers to yes or no questions. If you’ve ever played 20 questions, the guessing game where you are allowed to ask 20 yes or no questions to figure out what someone is thinking of, that is a process of gathering information. Anything that has information like a book, video, or image can be reduced to a series of yes or no questions.
In computers, yes’s and no’s are represented as 1’s and 0’s respectively. A single binary number is called a bit. Numbers made up of bits, of 0’s and 1’s, are called binary numbers, but sequences of 0’s and 1’s are far more than numbers. They are representations of information. The longer the sequence, the more questions can be answered yes or no, and the more information it encodes. Every piece of information on a computer is represented this way. For example, the lowercase letter ‘a’ has a standard encoding, 1100001, which translates to the number 97.
Since each letter’s encoding appears at a fixed location in a table of encodings called the ASCII (American Standard Code for Information Interchange) table, the binary representation answers yes or no questions about where in the table the symbol is. Thus, each 7 bit standard ASCII code is a game of 7 questions and that is the number of questions you need to precisely pinpoint the location of a single symbol in the table. For example, the first binary digit, 1, says that the letter is in the second half of the table. The second digit, 1, says the letter is in the second half of the second half, i.e., the fourth quarter, and so on. Go through this exercise with all the digits and you will find the letter. (You will also have performed what is called a binary search, one of the most powerful algorithms in all of computer science.)
This digression into computer science is necessary because information is about more than just how computers do things. It is fundamental to how the universe works in quantum theory.
The main difference between how computers and quantum mechanics work is how information is encoded. A bit on your computer can only be 0 or 1, never both, but in quantum mechanics a bit can be 0, 1, or both at once. This means that quantum mechanics doesn’t play 20 questions very fairly. For example, take my ASCII encoding for lowercase ‘b’, which is 1100010. Suppose that the last digit, 0, is both 0 and 1. That implies that my ASCII sequence is 1100010 and 1100011 at the same time. The latter is the encoding for the lowercase ‘c’. That means that my letter is ‘b’ and ‘c’ at the same time. Not one or the other.
This possibility of a bit being 0 and 1 at the same time is called superposition. Some people also call it a Schroedinger’s cat state because of the famous thought experiment where the answer to the question: is the cat dead or alive? is both dead and alive at the same time. I won’t go into that experiment here since it has acquired a lot of baggage in the near century since it was proposed. What matters is understanding that the way we are used to thinking of questions as having definite answers is false.
If a letter can be a ‘b’ and a ‘c’ at the same time, then this has implications that can affect much larger things. Most of the time, if a word contains a ‘b’ and you change it to a ‘c’ or vice versa, it becomes misspelled nonsense, but in some cases it can change the meaning of the word: from “cat” to “bat” or “can” to “ban”. Sometimes these word changes are grammatically incorrect and render a sentence incorrect, again nonsense, but in some cases they can change the meaning of a sentence. “This company cans tuna” versus “This company bans tuna” for example have vastly different meanings. In quantum theory, these sentences can both be true at the same time just by the flipping of a single bit.
Again, a sentence change may mean nothing but sometimes it can have enormous implications at higher levels. Suppose this is a description of what a company is supposed to do at its founding? Am I proposing to investors to start a tuna canning company or a company that seeks to ban tuna fishing?
The point here is that usually quantum systems, when they encode information as 0 and 1 at the same time, don’t affect anything in the world that we ordinarily see. Small changes are canceled out or rendered nonsensical (in physics parlance a “nonsensical” configuration in language is equivalent to an unstable configuration of physical entities) and hence fail to propagate to larger sized entities. This is the default. They have no effect and so, if later on the 0 and 1 superposition falls out of superposition and becomes only a 0 or only a 1, i.e., the question gets a definite answer, the difference is unnoticeable.
Likewise, our observation of the world can force quantum information into answering questions in specific ways because those are the only ones that allow for the world we see. If those questions were answered differently, with 0’s instead of 1’s or vice versa, the world would be other than we see it. For example, if I see a company that owns a tuna canning factory, then I can reasonably assume that its charter says it “cans” tuna, not “bans” (not counting typos). That question must be answered for me and therefore the world retroactively.
But sometimes a critical bit of information gets in superposition that has a cascading effect to larger and larger entities and we do see these changes. That is what, incidentally, Schroedinger was trying to argue with his cat. Quantum effects usually stay small and unnoticeable, but they don’t have to.
Like the answerer in a game of 20 questions, the quantum world tends to be stingy about the questions it answers. In fact, whenever it can avoid answering a question yes or no, it does. It only answers when we question it directly and then typically only gives the information directly relevant to the question. For example, if I want to know where a quantum particle is at a particular time, I only get to know where that particle is at the moment I detect it. In other words, when asking, “where are you?”, the particle answers “here”, but, if we ask, “where have you been?” the particle answers, “everywhere”.
Now that we have an understanding of what quantum information is, we can talk about how it changes with time, which is critical to understanding the black hole information paradox.
Suppose I am playing a game with binary numbers. I have a long string of random bits like 0110011, the ASCII code for the digit ‘3’, called a scrambler. Now, I take any other binary number of the same length, like 1110001, the string for ‘q’, and compare the two. For each bit I go along and if the two numbers are the same, I put down a 0, if they are different, I put down a 1. This is called an “exclusive or” or “XOR” operation and is what we, in English, typically mean when we use the word “or”. That is, we mean one or the other but not both. For the two strings above, the answer is 1000010 or ASCII for the uppercase letter ‘B’. In other words, ‘q’ scrambled by ‘3’, gives ‘B’.
Because the first string of bits I am using is random, the resulting sequence will be scrambled. But, if I know what that random sequence is, then I can reconstruct the original. This is because exclusive or is an information preserving operation. Thus, if I know my scrambler is ‘3’ and my output ‘B’, I can determine with certainty that the input was ‘q’.
Suppose, however, instead of using an exclusive or I used an ordinary logical or operation, this is the operation where if any of the digits are 1, then the result is 1, but, if both are 0, then the result is 0. The logical or is not an information preserving operation because, even if I know the output string of bits and the scrambler string of bits, I cannot reconstruct the original string. If the output has a 1 and the scrambler has a 1 in a particular position, I cannot say if the original had a 0 or a 1 there. In our case, ‘3’ OR ‘q’ is 1110011 or ASCII lowercase ‘s’. But if I know my scrambler is ‘3’ and my output is ‘s’, I cannot say with certainty that my input was ‘q’. For example, the sequence 1000001, ASCII uppercase ‘A’, is another option for the input that gives the same result.
Another information preserving operation that will become important as we discuss quantum theory is a slight modification to the exclusive or called the controlled not or CNOT. The CNOT has two outputs instead of one. The first output is the same as the first input while the second output is the XOR of the two inputs. For example, if I take the CNOT of 11 the output is 10 since the XOR of 11 is 0 and the first input is 1. Thus, the CNOT is also information preserving. It may seem redundant to parrot the first input, but it becomes important when we apply it to quantum information.
Now that we know what information preserving means, we can ask: is quantum theory information preserving as well? The answer is a resounding yes!
This means that a sequence of quantum bits, called qubits, that includes not only 0’s and 1’s but the case of 0 and 1 at the same time, a superposition, when evolved over time, will always obey rules such that you can work out what those qubits were at any time in the past no matter how scrambled they become. In other words, time evolution in quantum theory works like the exclusive or, not the logical or.
This all makes sense when we talk about ordinary 0 or 1 qubits, but, when we talk about the superposition, 0 and 1, it becomes a problem because we have to be able to work out from an observation of a 0, that at an earlier time, that was a 0 and 1 state, not just a 0 by itself. In other words, if I see a cat that is walking down the street, how do I work out that it was part of a physics experiment where for some time it was both alive and dead and not just alive the whole time?
Quantum mechanics implies that we can indeed do this and, to throw some jargon at you, this is called the “no-hiding theorem”. It is not a “theory” in a scientific sense but a corollary of quantum theory. In other words, as long as quantum theory is correct, the no-hiding theorem must be.
This idea suggests that, even if you have a 0 and 1 superposition state change to a 0 by itself state, you can still recover the fact that it was a superposition state before by looking at the environment it interacted with.
Think of it like this, imagine that I have my scrambler sequence above, but now I make it twice as long: 0110011 0000000, ‘3’ and the null character. This is my “environment”. Now suppose I apply my exclusive or operation against the first seven bits to a sequence of qubits. Let’s say that I observe that the output is 1000010 or ‘B’ again, but now my environment changes to record the input that appears to have been lost. Let’s say those lost bits are recorded in the first seven bits of my environment. If my environment that I observe is 0110011 1110001 ‘3’ and ‘q’, then I can work out that I started with seven superposition, 0 and 1 qubits. (I can also work out what environment I started with.)
The no-hiding theorem tells us a lot more than this however. It not only says that the environment preserves those superpositions but the output doesn’t have to be related to the input for you to recover the information. In other words, I could get a completely random sequence of bits out and still be able to recover the input sequence just by looking at the environment alone.
Hence, quantum theory always preserves information including superpositions at all time and in all cases even if that information appears lost.
Quantum theory requires the no-hiding theorem to be correct because of two principles that quantum theory obeys: linearity and unitarity. The exclusive-or is a linear operation while logical-or is not and this is why the logical-or causes information loss while the exclusive-or, if we don’t include superposition, preserves information.
The exclusive-or with two inputs and one output, however, is not unitary which is why we lose information when we do include superposition. A unitary operation preserves the size, dimension, angles, etc. of the input and is reversible. A good example is rotation. When you rotate something you aren’t changing its size, angles, dimensions, or any of that critical information. So it is like changing your point of view. You still have all the same information from a different vantage point.
Time evolution in quantum mechanics is a lot like rotation. You aren’t really changing the universe. You are just changing the way you look at it, but, because there are so many dimensions, possibly infinite, rotations can make everything look completely different. If you see a sculpture that changes from a bunch of hanging trash to an image, you get the idea.
Of course, with exclusive or, XOR of 11 is 0, so the dimension of 2 has been lost. The XOR isn’t like a rotation. It is flattening the input, removing a dimension completely. This is like the illusion where we see a picture when we stare at a sculpture from the right angle. We have lost the 3D information and replaced it, in our view, with a 2D image. That is why we think of it as being different from that point of view when in fact it hasn’t changed at all.
If I replace the XOR with the CNOT, however, I have two digits as an output instead of one. The CNOT takes the XOR, which is linear, and makes it unitary, so it is both linear and unitary. With CNOT I have the 3D perspective where, despite all the rotations that time evolution carries out, we know that nothing has really changed.
If you take a moment, admire this prediction of quantum mechanics. Linearity and unitarity mean that the universe is no different now than it was at the Big Bang. All that has changed is our perspective. The universe continually rotates, along with ourselves, in a space with a huge number of dimensions (which we model as being infinite) presenting different points of view without losing anything of the whole.
Enter Stephen Hawking.
Hawking convinced himself in the mid-‘70’s that this beautiful picture could not hold up against the unrelenting gravitational pull of a black hole. His reasoning was two fold: first nothing can escape a black hole and second black holes all eventually evaporate into nothing, a result he calculated in the early ‘70’s. He calculated that when this happens black holes takes all the quantum information that fell into them while they existed and leave nothing behind. They are the ultimate cosmic exception to unitarity.
Having gone through all the arguments above, it seems a shame that black holes might do this. Rather than the universe preserving its integrity in completeness for its entire existence through linearity and unitarity, Hawking was arguing that the universe arbitrarily erased bits of its past and future within black holes. From a purely aesthetic point of view, this strikes me as an abomination.
A black hole acts, according to Hawking, like an exclusive or operation rather than a controlled not. When information evolves in time so that it falls into a black hole, the information disappears from view, which is fine according to quantum mechanics. What is not fine, however, is that the environment, Hawking argued, does not contain any clues to what the information was. In other words, black holes violate the no-hiding theorem. Since they violate the no-hiding theorem, by extension, they violate quantum theory itself. What Hawking had shown, and likely he didn’t at the time realize the implication, was a proxy war between the two greatest discoveries of 20th century physics — a contradiction between Einstein’s general relativity theory that predicts black holes and quantum theory which predicts the no-hiding theorem.
First of all, we want to ask why black holes should violate unitarity, that is, why are they not information preserving? If black holes did not evaporate, then this wouldn’t necessarily be a problem. If something fell into the black hole, it would still be there forevermore, hidden from view but still part of the universe. The problem really comes because of Hawking’s prediction that these black holes evaporate into nothing without leaving a trace.
Black holes evaporate because of the way that quantum fields interact with curved spacetime. All of space and time are filled with quantum fields which, when you add energy to them produce particles. This happens in particle accelerators. We “poke” the vacuum and out come particles. Hawking predicted that black holes, because of their intense gravitational field, are constantly poking the vacuum around them, causing particles to appear. These particles carry off energy from the black hole. This energy has to come from somewhere and that place is the black hole’s own mass, which Einstein showed us was equivalent to energy times the speed of light squared.
All the details about how this works are better saved for another article. The point is that black holes evaporate away their mass without revealing the information they contain. Hawking was able to calculate the information that evaporating mass contains and showed it could not depend on the details of what fell into the black hole. Instead, it could only depend on basic parameters about the black hole like mass, angular momentum, and electric charge. It was, essentially, pure white noise emanating from the black hole’s exterior.
His calculations, however, were simplified. He assumed that he could treat the gravitational field of the black hole as a non-quantum field because the black hole is so big. Hence, he ignored any theory of quantum gravity that might exist. This is unfortunate because, by the no-hiding theorem, the environment, including the gravitational field, would be a good place to store all that information.
By assuming it was not a quantum entity, Hawking reduced the interaction between the black hole and any quantum information falling into the event horizon to an interaction where quantum information that ends up in the quantum aspects of the environment is not accounted for. In other words, his paradox is a consequence of his assumption that he could ignore quantum gravity. At least, this is what proponents of the information preservation properties of quantum theory believe.
By analogy, this is like using a logical or operation, which is a non-quantum logical operation that violates both linearity and unitarity, instead of the controlled not’s quantum operation that preserves both, for how quantum information interacts with black holes. The logical or is a good model of the time evolution of information containing matter as it passes the event horizon, never to be seen again. A logical or gloms all information together, as a black hole gloms matter into its singularity, losing the information in the process. A logical or operation is irreversible both for non-quantum and quantum information.
The reason Hawking made this mistake, a brilliant mistake, is that he assumed that quantum effects of the black hole itself didn’t matter. But we have seen that, when it comes to information, we can’t discount quantum effects in large objects, especially when we are talking about information. That information may not affect things very much, but its being there makes all the difference.
Over the years there have been numerous attempts to resolve the information paradox, but the details of those attempts do not really matter to this discussion. What matters is that, in order to avoid Hawking’s prediction, the black hole or its environs must take on the information that any quantum object falling in contains. A quantum theory of gravity may predict that the black hole’s gravitational field contains that information. Hence, as the quantum matter falling into the black hole interacts with the black hole’s gravitational field, its information is scattered into that field, preserved for all to see. This means that the black hole has “quantum hair”, i.e., it is not completely bald and featureless. This would be a good demonstration of the no-hiding theorem because, while the quantum information that fell in completely disappears, the environment enables us to, hypothetically, reconstruct all that information.
The recent quantum hair paper that started all of this manages to predict that information is preserved in the gravitational field, despite not having a complete theory of quantum gravity, which is quite a feat. There have been quite a few papers on that topic in the past however so whether this is the solution remains to be seen.
A small number of theorists, Sir Roger Penrose being one of the most prominent, believe that Hawking was right. Of course, we can’t really know, but to assume Hawking was right is to assume that quantum theory is wrong or at least incomplete. Most theorists think he was wrong and Hawking himself conceded it in 1997 because of an argument based in string theory and the holographic principle. While that argument may not itself be correct, it seems as if any quantum theory of gravity will likely obey the unitarity and linearity requirements of quantum theory and that a quantum black hole will somehow preserve the information.
It may be almost impossible to determine experimentally if this is actually true. We would need incredibly sensitive means of detecting minute quantum details in a gravitational field. This is unlikely for a large, stellar mass black hole. More likely if a quantum theory of gravity receives experimental confirmation, we will just be able to determine from it whether the paradox is resolved satisfactorily.
The important take away here is that in quantum information theory, like the black hole information paradox and no-hiding theory, many of the physical details that are so important in other areas of physics do not really matter. We can reduce complex time evolution to simple binary operations to understand what is going on and recognize that some binary operations on qubits may be inadmissible in our universe. A black hole destroying information is little different than any other interaction with or scrambling of particles destroying it. There is nothing really special about black holes in this regard nor should there be. Where and how that information is preserved is an open question of course, but we can be sure it is somewhere in the environment of the black hole.
Calmet, Xavier, et al. “Quantum hair from gravity.” Physical Review Letters 128.11 (2022): 111301.
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