Lenny Susskind has a new book out: The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics. At first I was horrified by the title, but upon further reflection it’s grown on me quite a bit.
Some of you may know Susskind as a famous particle theorist, one of the early pioneers of string theory. Others may know his previous book: The Cosmic Landscape: String Theory and the Illusion of Intelligent Design. (Others may never have heard of him, although I’m sure Lenny doesn’t want to hear that.) I had mixed feelings about the first book; for one thing, I thought it was a mistake to put “Intelligent Design” there in the title, even if it were to be dubbed an “Illusion.” So when the Wall Street Journal asked me to review it, I was a little hesitant; I have enormous respect for Susskind as a physicist, but if I ended up not liking the book I would have to be honest about it. Still, I hadn’t ever written anything for the WSJ, and how often does one get the chance to stomp about in the corridors of capitalism like that?
The good news is that I liked the book a great deal, as the review shows. I won’t reprint the thing here, as you are all well-trained when it comes to clicking on links. But let me mention just a few words about information conservation and loss, which is the theme of the book. (See Backreaction for another account.)
It’s all really Isaac Newton’s fault, although people like Galileo and Laplace deserve some of the credit. The idea is straightforward: evolution through time, as described by the laws of physics, is simply a matter of re-arranging a fixed amount of information in different ways. The information itself is neither created nor destroyed. Put another way: to specify the state of the world requires a certain amount of data, for example the positions and velocities of each and every particle. According to classical mechanics, from that data (the “information”) and the laws of physics, we can reliably predict the precise state of the universe at every moment in the future — and retrodict the prior states of the universe at every moment in the past. Put yet another way, here is Thomasina Coverley in Tom Stoppard’s Arcadia:
If you could stop every atom in its position and direction, and if your mind could comprehend all the actions thus suspended, then if you were really, really good at algebra you could write the formula for all the future; and although nobody can be so clever as to do it, the formula must exist just as if one could.
This is the Clockwork Universe, and it is far from an obvious idea. Pre-Newton, in fact, it would have seemed crazy. In Aristotelian mechanics, if a moving object is not subject to a continuous impulse, it will eventually come to rest. So if we find an object at rest, we have no way of knowing whether until recently it was moving, or whether it’s been sitting there for a long time; that information is lost. Many different pasts could lead to precisely the same present; whereas, if information is conserved, each possible past leads to exactly one specific state of affairs at the present. The conservation of information — which also goes by the name of “determinism” — is a profound underpinning of the modern way we think about the universe.
Determinism came under a bit of stress in the early 20th century when quantum mechanics burst upon the scene. In QM, sadly, we can’t predict the future with precision, even if we know the current state to arbitrary accuracy. The process of making a measurement seems to be irreducibly unpredictable; we can predict the probability of getting a particular answer, but there will always be uncertainty if we try to make certain measurements. Nevertheless, when we are not making a measurement, information is perfectly conserved in quantum mechanics: Schrodinger’s Equation allows us to predict the future quantum state from the past with absolute fidelity. This makes many of us suspicious that this whole “collapse of the wave function” that leads to an apparent loss of determinism is really just an illusion, or an approximation to some more complete dynamics — that kind of thinking leads you directly to the Many Worlds Interpretation of quantum mechanics. (For more, tune into my Bloggingheads dialogue with David Albert this upcoming Saturday.)
In any event, aside from the measurement problem, quantum mechanics makes a firm prediction that information is conserved. Which is why it came as a shock when Stephen Hawking said that black holes could destroy information. Hawking, of course, had famously shown that black holes give off radiation, and if you wait long enough they will eventually evaporate away entirely. Few people (who are not trying to make money off of scaremongering about the LHC) doubt this story. But Hawking’s calculation, at first glance (and second), implies that the outgoing radiation into which the black hole evaporates is truly random, within the constraints of being a blackbody spectrum. Information is seemingly lost, in other words — there is no apparent way to determine what went into the black hole from what comes out.
This led to one of those intellectual scuffles between “the general relativists” (who tended to be sympathetic to the idea that information is indeed lost) and “the particle physicists” (who were reluctant to give up on the standard rules of quantum mechanics, and figured that Hawking’s calculation must somehow be incomplete). At the heart of the matter was locality — information can’t be in two places at once, and it has to travel from place to place no faster than the speed of light. A set of reasonable-looking arguments had established that, in order for information to escape in Hawking radiation, it would have to be encoded in the radiation while it was still inside the black hole, which seemed to be cheating. But if you press hard on this idea, you have to admit that the very idea of “locality” presumes that there is something called “location,” or more specifically that there is a classical spacetime on which fields are propagating. Which is a pretty good approximation, but deep down we’re eventually going to have to appeal to some sort of quantum gravity, and it’s likely that locality is just an approximation. The thing is, most everyone figured that this approximation would be extremely good when we were talking about huge astrophysical black holes, enormously larger than the Planck length where quantum gravity was supposed to kick in.
But apparently, no. Quantum gravity is more subtle than you might think, at least where black holes are concerned, and locality breaks down in tricky ways. Susskind himself played a central role in formulating two ideas that were crucial to the story — Black Hole Complementarity and the Holographic Principle. Which maybe I’ll write about some day, but at the moment it’s getting late. For a full account, buy the book.
Right now, the balance has tilted quite strongly in favor of the preservation of information; score one for the particle physicists. The best evidence on their side (keeping in mind that all of the “evidence” is in the form of theoretical arguments, not experimental data) comes from Maldacena’s discovery of duality between (certain kinds of) gravitational and non-gravitational theories, the AdS/CFT correspondence. According to Maldacena, we can have a perfect equivalence between two very different-looking theories, one with gravity and one without. In the theory without gravity, there is no question that information is conserved, and therefore (the argument goes) it must also be conserved when there is gravity. Just take whatever kind of system you care about, whether it’s an evaporating black hole or something else, translate it into the non-gravitational theory, find out what it evolves into, and then translate back, with no loss of information at any step. Long story short, we still don’t really know how the information gets out, but there is a good argument that it definitely does for certain kinds of black holes, so it seems a little perverse to doubt that we’ll eventually figure out how it works for all kinds of black holes. Not an airtight argument, but at least Hawking buys it; his concession speech was reported on an old blog of mine, lo these several years ago.
The important question is how can the universe be said to be a well-defined number of qubits? To make such a count, wouldn’t you have to dig deeper than sensible physics would allow?
One of the strange things about q-bits is they can in certain entanglements be associated with negative entropy. This is a bit subtle to go into here. Yet it does suggest that lots of positive entropy in our local region of the universe is due to how EPR pairs become inaccessible across cosmological horizons, or e-foldings. We might then suspect that the total number of actual qubits in the universe is very small — maybe zero. This might go somewhere in telling us why the universe started out with a particularly low initial (initial wrt. inflation) entropy.
Lawrence B. Crowell
Interesting way of looking at it. But the horizon isn’t necessarily a boundary in actual space. It could also be a boundary between what can and cannot be known. If the universe contains more information than can be stored in its qubits, it seems to me it might be stored in your dreaded “inner classical”. Just how I see it anyway.
Event horizons occur when there is some region or congruence of geodesics which have zero length. This of course occur where a light rays exists. An event horizon is different from a light cone, which is also a type of null surface, where a light cone is a projective blow up of a point. An event horizon is not a system of projective rays in quite that sense, but is a congruence of null geodesics.
Further, an event horizon has different structure for different spacetimes. A black hole event horizon is one where energy is conserved. The timelike Killing vector K_t defines K_t*K_t = g_{tt} = (1 – 2M/r) and maintains a system of isometries which conserve energy in the Schwarschild metric. For comsologies there is no such system of Killing vectors or isometries. The cosmological event horizon at
$latex
L~=~sqrt{3/Lambda},~Lambda~=~cosmological~const
$
defines regions of spacetime beyond which we can’t communicate. In a spacetime diagram this involves a system of null rays which connect to the big bang or the initial quantum event (so we can see all the say back!) but where these null rays spread out and then converge to our past light cone. This is because the lightcones for different frames point in different directions. In other words a spatial surface in the universe may be flat, but it is not embedded in a flat spacetime. As a result from a q-bit perspective there may be EPR pairs in one observer’s frame which is entangled with another region where any teleportation fidelity is limited by the cosmological horizon.
Lawrence B. Crowell
If there are no Killing Vectors, how can there be a preferred definition of L?
And what about the “homogeneous distribution” ansatz of FRW? Isn’t that an isometry?
The cosmological event horizon is frame dependent. A galaxy “way over there” sees a different bubble than we do. The cosmological horizon exists in a way similar to how the Rindler horizon exists for an accelerated frame.
Ned Wright’s website has some diagrams of what I am about to describe, in particular the page 3. Draw a vertcal line connecting a point and draw little light cones along it. Now draw two slanted lines connected to the vertical lines at the starting point. Draw little light cones which are skewed away from the vertical line. Keep doing this with more and more lines which diverge away closer to the horizontal and with light cones that are skewed further away.
Now choose a point on the vertical line — that is your “now” or time = 0. Draw a null line from your now which is tangent to all the skewed light cones on the other lines. This will define a football shaped curve which connects up with the origin = big bang. This defines the radial distance to which you can observe the past universe. Anything outside this teardropped or football shaped null ray is outside of your capability to observe. What is interesting is that you can observe virtually all the way to the beginning, where this includes the earliest regions of the universe that are fantastically distant. The CMB region is about 80 billion light years out on our current Hubble frame! This is because the spatial points have been comoved outwards — which is why we skew those light cones on the off vertical lines! If we could observe neutrino or gravity waves from even earlier regions of the universe we will be seeing regions of the universe that on the Hubble frame are now vastly distant from us.
Now with the diagram draw future directed null rays, which will define an open cone. The region in the cone defines the future region of the universe we can send a signal to. It is clear there is a whole lot of universe we can never send a signal to, which includes regions that we can see in the past. So a distant galaxy out there we see could be outside or domain of future causal contact.
The cosmological horizon is a distance which defines this past and future domain in any local observer’s domain of observation or signal sending. This distance
$latex
L~=~sqrt{3/Lambda}
$
defines loosely the region where the comoving of points on a spatial manifold begins to shift these points away fast enough to prevent future contact. It is similar in a way to the Rindler horizon where an inertial observer is no longer able to send a message to an accelerated observer. Of course there are differences here, for here the local horizon is induced by the comoving of points on a Hubble frame.
Lawrence B. Crowell
I was going to ask you what would happen if an entangled pair of qubits was sent from an emitter to a pair of receivers that had moved out of the range of ever being able to compare their results, but it occurred to me that by then the Hubble expansion might have stretched the qubits so widely that they wouldn’t be detectable anymore. Is that actually a mitigating issue, or is there something about decoherence that I’m missing?
Entangled pairs can exist across event horizons. However, there is a loss of fidelity in teleportation. Alice and bob may share an entangled Bell state |B), and with it Alice may teleportate another state |Y) to Bob. So the two states are coupled to each other eg ~ |Y)|B) (I am using “)” because the carrot symbols have problems. Alice may then pass the |Y) state through a CNOT gate and pass the output state on |Y) through the four possible projections. Bob’s state will also be similarly rotated, but needs the classical signal from Alice to assertain the value of this output. In an ideal case this can be accomplished and the teleporated state to Bob is then correctly deduced. Instead of there being just 4 possible projections this can in principle be 2^N projections and the teleporation process can parallel process many states.
BTW, in a quantum computer a CNOT can be arranged at simply with ions in a linear laser trap. If an ion is vibrating due to trapping phonon interactions with neighbors, the no pi pulse rotation is performed, but if the ion is not vibrating then a pi rotation (change of q-bit state) is performed. This was outlined by Cirac and Zoller over 10 years ago.
Now if we introduce horizons into this picture things become more difficult. A classical case with the Rindler wedge is where an inertial observer is unable to transmit data to an accelerated observer in the region II outside the region I bounded by the particle horizon. In this case if Alice is the inertial observer she can’t send the classical result to Bob on the accelerated frame. So even though Bob has the teleporated state he is unable to read it without completely adulterating it without Alice’s classical key.
You might win the lottery, but if you can’t find the ticket you can’t claim the winnings.
In the case of cosmology we can observe regions as far back to the initial event (quantum tunnelling, bounce, D3-brane collision etc). The further back you look the further out that stuff is on the “current” Hubble frame. In fact it is so far out that we can’t observe it in its later state. We can only observe galaxies up to about .7billion years after the big bang. So at a later stage of evolution this stuff is not causally connected to us. Similarly we might see some galaxy cluster way way out there with a z ~ 7, but if it is beyond the cosmological horizon distance these bodies are being comoved away such that we could never send a signal back to them. With inflation and now the accelerated evolution of the universe local regions are being more rapidly isolated from each other in this way. So suppose that in the above galaxy we (Alice) happen to share an entangled pair with an observer there (Bob). So we make our projections on the state, and we try to send our signal to Bob. Bob never receives the information and so can never cipher the teleported state correctly according to this prescription.
Bob is like Napolean, “Josephine (Alice) why don’t you ever write me?”
Things also become complicated due to Hawking-Gibbon-Unruh radiation from horizons. This can further introduce noise into the communication channel, which can with appropriate quantum error correction codes can probably be managed if the Hamming distance in any interval of time (sampling time) is not terribly large. This gets into a whoe different kettle of fish for later.
Lawrence B. Crowell
Suppose an emitter sends out a signal
cos a |B=0)|Y=0)+sin a |B=1)|Y=1)
in two opposite directions. Each branch of the signal arrives at a detector. I would think the state would become
cos a ( |B=0)|Y=0)|exists)+|B=1)|Y=1)|does not exist) )
+e^iz sin a ( |B=0)|Y=0)|does not exist)+|B=1)|Y=1)|exists) )
for some phase angle z.
Then an observer at each detector would record a result, and they would meet in person to compare them. I would think they would find B=Y=0 with probability (cos a)^2 or B=Y=1 with probability (sin a)^2. The important point here is that the observers are the classical keys.
I then consider a similar experiment in which you wait so long before sending the first signal that the detectors have receded from causal contact with each other.
If the detectors actually do generate results, then in the first experiment, the results would have to both a classical 0 or both a classical 1. And if those results were never brought together (neither by the observers meeting nor by any other means), there would be no evidence for the “anomalous” event of classical results appearing without being observed.
In the second experiment, however, each observer knows that, if the other observer is at his detector, he must see the same result. However, it is possible that one or both of the observers might not be watching the detector. The only necessary difference between the two expermients is that in the second the detectors themselves are out of causal connection.
If free will exists, then a strictly Bohmian interpretation is untenable. All other interpretations, as far as I know, agree that if there are any hidden variables, at least until detection they are mixed in the same proportion as the physical states they accompany, so that nothing more definite than the probabilities exists before detection. So the detection of an eigenstate is the beginning of that eigenstate’s existence.
Therefore, in the second experiment, the detection of an eigenstate would violate either causality (both the same without knowing what each other are) or the Born Rule (results B=!=Y possible, even though both have zero amplitude). The only conclusion is that in this case the detector does not display an eigenstate. Instead, it displays a non-ensemble mixture of the states. For example, a needle would break in half, or a screen would display two messages overlapping.
It’s interesting to consider what the analogous result in terms of Hawking radiation would mean. Such a mixture could be described algebraically as a violation of the Completeness Principle, or semantically as a violation of the Axiom of Restricted Comprehension. If this exists, it would be an obvious candidate for Dark Matter.
If I understand you (Collin237) right you appear to be confusing the classical information with some sort of “inner causal” connection.
Suppose that you and I have a quantum state which are in an entangeled state. Say these states are due to the decay of a spin-0 boson, and which decay into two fermions. You and I are separated by some considerable distance. So we both then have this Bell state. Now suppose that I have another state |Y> = a|+) + b|-) that I want to teleport to you. So I then entangle this state with the Bell state so I have |Y)|B). By such an entanglement since you hold part of the EPR pair your state is similarly entangled. I then make a measurement of the state I want to teleport and find one of the four outcomes
|+)|-) – |-)|+)
|+)|-) + |-)|+)
|+)|+) – |-)|-)
|+)|+) + |-)|-)
What I do is then communicate each of these possible outcomes with the numbers 1, 2, 3, 4 as classical information. If you then take your Bell state and perform the following operations:
1: 0 rotation
2: pi rotation about the z axis
3: pi rotation about x axis
4: pi rotation about y axis
you will reconstruct the state I am intending to transmit because of its entanglement with your part of the EPR pair.
The point of going through all of this is that without the classical information you will not beable to reconstruct the state properly. Further, the total information communicated involves 2 bits, but from that the state you find exists in the whole H^4 state space or points on the Bloch sphere.
Lawrence B. Crowell
What is the definition of “classical information” in a quantum universe?
We do have a bit of the classical/quantum split. Classical information are bits which are not defined according to quantum states, nor do they have any fundamental unit of action (hbar). One of the big questions underlying this is “why the classical world?” We all have a sense that the classical trajectory or orbit of even the biggest object is built up from quantum paths.
Lawrence B. Crowell
1. Why would you find mixtures like that?
2. I assume “rotation” refers to an SU2xSU2 group. Is that correct?
3. Are the x, y, and z axes only an isospin basis, or do they have actual directions in a physical frame?
4. What is a zero rotation?
It occurred to me that Sean’s “horror” at the title of Susskind’s book may not be that far off the mark. Not only does a black hole have a horizon that hides things from observers; it also has a force that destroys any observers that get too close to it. So the universe is made “safe” for the laws of physics, because the violations a black hole commits cannot be known about.
Any attempt to deduce the structure of a black hole from the laws of physics is a total waste of time.
To transmit information about the state you are teleporting you must communicate on the classical channel information about the the entanglement of the shared Bell state and the state to be transmitted. You then communicate the outcomes. The rotations performed by the recipient will rotate their state into a form which recovers the teleported state. This can be shown with rotations by Pauli matrices. It is not excessively difficult to do, but does require a page or two of calculation.
Lawrence B. Crowell
PS, by rotations with Pauli matrices I mean generators of the form
$latex
U(theta) = exp(isigma_atheta_a)
$
for the index around the axis to be rotated.
Susskind’s black hole complementarity and holographic principle indicates that states which are identifiable on a membrane which wraps the black hole horizon are dual to states inside the black hole. This does suggest some deep connections with quantum gravity, for it the black hole becomes very small the horizon might become “blurred” by quantum fluctuations and the fields associated with the horizon and those in the interior will exhibit new physics.
Lawrence B. Crowell
Is the Holographic Principle a higher-dimensional version of Cauchy’s Integral from complex calculus?
I think the Holographic Principle is just a way for people to pretend they’re developing a profoundly deep understanding, when really they’re just superficially scratching the surface. 😉
The holographic principle has connections with the S^5 ~ AdS duality in superstring theory. It on lower dimensions tells us that field amplitudes in three dimensional space at a given “time” are projections from fields pinned to event horizons.
Lawrence B. Crowell
Isn’t that just replacing the “inner causal” with an “outer causal”?
And what validity does it have for a non-stringist like me?
I am not sure what is meant by inner or outer causality.
If the holographic theory is correct then metric fluctuations of spacetime should manifest themselves on a scale larger than the Planck length. Y. Jack Ng has shown that the fluctuations in a three dimensional volume are equated to those on a two dimensional bounding region by
$latex
(Big(delta L}{L}Big)^3~ge~Big(frac{L_p}{L}Big)^2, L_p~=~sqrt{Ghbar/c^3}
$
and so by solving for the delta L fluctuation these occur on a scale considerably larger than the tiny Planck length L_p.
The AdS/CFT is largely a stringy result. However, aspects of string theory shows up in a number of forms — such as sphere packing and quantum codes. It also sneeks its way into Jordan exceptional algebras in loop quantum variables. So physics probably has both stringy and loopy aspects to it.
Lawrence B. Crowell
It looks like I messed up the tex on that equation. Ng’s fluctuation equation for a volume of length scale L bounded by an area is
$latex
Big(frac{delta L}{L}Big)^3~ge~Big(frac{L_p}{L}Big)^2
$
L. C.
I mean how can an interior field be controlled from a surface it can’t classically communicate with?
If the surface is the cosmological horizon, this is clearly not a moot point. Without a non-relativistic causality, it would mean everything we observe has already been decided billions of years ago.
The black hole duality indicates that an exterior observer, at “infinity,” will see fields which have entered a black hole according to harmonic oscillator modes (or string vibrations) on a membrane a Planck unit length above the event horizon. The Russian for a black hole is a frozen hole, where because of time dilation nothing is ever seen to cross the r = 2M. Then as the black hole decays away these modes end up being radiated away in three space (or 4-dim spacetime). Conversely for an observer which enters the black hole their quantum information is not seen pinned to the event horizon, but taken in by the singularity, or some type of quantum-mawl in the interior.
This is the matter of black hole complementarity, which is that the observers outside and inside a black hole observe the same field amplitudes, but in complementary forms. Loosely put, the exterior viewpoint has fields on this membrane above the horizon, similar to a type of D2-brane, equivalent to fields in space removed from the black hole. So there is a 3 + 2 dimensional perspective on fields as measured outside the black hole. The interior is given by a D5-brane, and the duality says that the amplitudes on the two are equivalent.
Lawrence B. Crowell
By “3+2 dimensional” do you mean that the two interpretations (internal and external) are interpolated by a parameter that behaves like an extra time-like dimension? (As if the internal and external observers are looking at the surfaces of bread surrounding a 5-dimensional sandwich?)