This post is a continuation of the previous one, which you should read first…
Now, what exactly are these wormholes that certain physicists claim to be trying to make or, at least, simulate? In this post I’ll explain what the scientists did to bring the problem within reach of our still-crude quantum computers. [I am indebted to Juan Maldacena, Daniel Jafferis and Brian Swingle for conversations that improved my understanding.]
An important point from last post: a field theory with quarks and gluons, such as we find in the real world or such as we might find in all sorts of imaginary worlds, is related by the Maldacena conjecture to strings (including quantum gravity) moving around in more dimensions than the three we’re used to. One of these dimensions, the “radial dimension”, is particularly important. As in the previous post, it will play a central role here.
Einstein-Rosen Bridge (ER) vs. Einstein-Podolsky-Rosen Entanglement (EPR)
It’s too bad that Einstein didn’t live long enough to learn that two of his famous but apparently unrelated papers actually describe the same thing, at least in the context of Maldacena’s conjecture. As Maldacena and Lenny Susskind explored in this paper, the Maldacena conjecture suggests that ER is the same as EPR, at least in some situations.
We begin with two identical black holes in the context of a string theory on the same curved space that appears in the Maldacena conjecture. These two black holes can be joined at the hip — well, at the horizon, really — in such a way as to form a bridge. It is not really a bridge in spacetime in the way you might imagine a wormhole to be, in the sense that you can’t cross the bridge; even if you move at the speed of light, the bridge will collapse before you get to the other side. Such is the simplest Einstein-Rosen bridge — a non-traversable wormhole.
What, according to the Maldacena conjecture, is this bridge from the point of view of an equivalent field theory setting? The answer is almost fixed by the symmetries of the problem. Take two identical field theories that would each, separately, be identical to one of the two black holes in the corresponding string theory. These two theories do not affect each other in any way; their particles move around in separate universes, never interacting. Despite this, we can link them together, forming a metaphorical bridge, in the most quantum sense you can imagine — we entangle them as much as we can. What does this mean?
In quantum physics, we are no longer limited to things being one way or the other. They can be in between. Let’s take a famous example. A computer bit in an ordinary computer can be off (“0”) or on (“1”). Two such bits can have four possibilities: (0,0), (1,0), (0,1), or (1,1). But in a quantum context there are an infinity of possibilities. First, even one quantum bit (qubit) can be somewhere between on and off: cos θ (0) + sin θ (1), where θ is any real number between 0 and 2π. Two qubits can be in any combination of the form
- a (0,0) + b (0,1) + c (1,0) + d(1,1)
as long as |a|2 + |b|2 + |c|2 + |d|2 = 1. In fact a, b, c and d can be complex numbers, too. (In the following I’ll often drop a, b, c, d to keep expressions shorter.)
This has huge implications. If the bits are in the state [ (0,0) + (1,1) ] , what does this mean? One thing it means is that although we don’t know what we’ll get if we measure the first bit, we do know that whatever the first bit is, the second bit will be the same. That is: the first bit might be 1 or it might be 0, but if we measure it to be 1, then we can be sure we’ll find the second bit is 1 when we measure it. (Naively, this is the same as saying that we don’t know what socks I’ll wear tomorrow, but we know that if I wear a red sock on the right foot, there will also be a red sock on my left foot. But that can’t be the whole story, because there’s a different state, [ (0,0) – (1,1) ], with the same naive feature but a minus sign, and that state is somehow be different. Maybe I’ll come back to the differences sometime; not today.)
If instead the bits are in a state [ (0,1) + (1,0) ], then whatever the first bit is doing, the second is doing the opposite. In more complicated states, well… it’s complicated.
These are the kinds of entangled states that are used directly in Einstein, Podolsky and Rosen’s demonstration of quantum physics’ “spooky action at a distance”. What’s spooky? Even if the bits are far apart, even as far as Pluto, in entangled states the measurement of one of the bits partially or completely determines what the other bit is doing.
The Thermofield Double State
Back to the ER bridge. To obtain the bridge in the field theory using Maldacena’s equivalence, we must entangle the two quark/gluon/etc field theories in the following precise way. Label every state in the quantum field theory by an integer n. (This is a small cheat, because the number of states is uncountably infinite, but we will sidestep this subtlety.) Now set up the “Thermofield Double State” (TFD) state of the two field theories, which is a sum over all of the states (n,n) weighted by a factor exp[- En / (2 kB T)]. Here En is the energy of the state n; the temperature of either one of the black holes (remember they are identical) is T; and k_B is a famous constant of Nature, named after Ludwig Boltzmann. (The exponential of energy divided by temperature is a famous expression, due to Boltzmann, that always arises in the physics of temperature.) The TFD state can be written more explicitly in the math of quantum physics:
where L and R stand for the two field theories and β = 1/ ( kBT). [“Z” is just there to get the probabilities to come out to one, and the asterisk in “n*” indicates that really we should use the conjugate of the state n in one of the field theories.]
In short, we perfectly correlate the two field theories — if one is measured to be in a state n, then other will also be in that same state [actually its conjugate n*] — and we weight the correlated states by a factor which is 1 for low energy states and exponentially small for high-energy states, so that it’s more probable to find the two field theories in states with energy below kBT than above.
Importantly, these two quark/gluon/etc theories otherwise do not interact at all! They may as well live in different, disconnected universes. None of their particles ever meet. Only the state in which they are placed relates them to each other. This is the spookiest of actions — arranged for non-interacting field theories that have nothing to do with each other. Not only are they at a distance, that distance is effectively infinite — or better, not even meaningful.
Thus, the Maldacena equivalence implies that an Einstein-Podolsky-Rosen entangled state of two quark/gluon/etc field theories without gravity, suitably chosen, is physically equivalent to an Einstein-Rosen bridge joining two suitable black holes in a string theory on an appropriately curved space; remember this space has more space dimensions than the field theories have. [See the figure at the end of this post for an illustration.] As I emphasized before, nothing can travel from one end of the bridge to the other. But you can still do many interesting things with this bridge. For example, two objects that enter the bridge from opposite sides can meet in the middle, even if they can’t cross the bridge or return to their origins. In the equivalent field theories, this is described as producing two small disturbances, one in each field theory, which can engage with each other even though the two field theories do not interact. The entanglement between them produces effects that no pre-quantum physicist could ever have imagined.
I will soon have to write about traversable wormholes — ones where something actually can cross from one side to the other. That’s part of the current hullabaloo. Because that story is a bit intricate, I will come back to it in a later post. For the moment, suffice it to say that in order to have something cross the bridge, we must allow communication between the outsides of the two black holes — not through the ER bridge but around it. From the point of view of the equivalent field theories, we must arrange for them to interact, in a simple, prescribed way. No longer can they really be an infinite distance apart; otherwise the needed communication would take forever.
With this in mind, let’s ask a simple question. Can’t we just check and study this conjectured relation between ER and EPR — between a physical spacetime bridge and a metaphorical, almost-metaphysical quantum bridge — just by putting the field theories on a computer, setting up the thermofield double state we want, and seeing how they behave?
Yes, in principle. No, in practice; it’s too hard. Modern computers can’t do it.
So we should ask — is there a simpler version of this problem where, perhaps, it might not be quite so difficult to simulate how this all works?
The answer is yes, to a degree. Although black holes and wormholes in three or more spatial dimensions are too difficult, there is a sort of analogy — a cartoon of a wormhole — in just one spatial dimension (along with the usual time dimension). ONE SPATIAL DIMENSION.
That seems awfully limiting. In fact, gravity in one spatial dimension has barely any content at all; there’s no gravitational force, no gravitational waves, not much remnant of anything we’d call gravity at all. One dimension is a line that stretches from one point at left-infinity to another at right-infinity, so in a sense there’s always a bridge between one end and the other. [See the figure at the end of this post.] What does it mean to make a spacetime bridge in this context? Well, the trick is not to use the standard version of Einstein’s gravity. If we add a spinless field to gravity, we get something called “JT gravity” (named for its inventors, Jackiw and Teitelboim) and this theory has something that resembles, in cartoon form, a wormhole.
In what sense is this a good cartoon? I’m not the expert here, so I’m giving you my impression; maybe I can give a better answer when I learn more. There are imaginable wormholes in three or more spatial dimensions that would connect black holes that are particularly stretched out along the radial direction — that’s the one that points toward the black hole’s horizon. In such cases it can sometimes be shown that the most important physics involves only that radial direction, and that certain physical questions can be answered by focusing only on that one dimension. In that context, even though the full theory in Maldacena’s conjecture has nine spatial dimensions (of which typically three or four are large, depending on context), only one of them actually matters for certain physical questions. The description of the physics involving that dimension, in an appropriate limit, reduces to JT gravity.
Great! It’s a cartoon, but sometimes one can learn general lessons from cartoons. All we have to do, then, is perform the equivalent of this operation on the quark/gluon field theories that lie on the other side of Maldacena’s equivalence, reducing them down in dimensions — to zero space dimensions, so that the particles do not move in space, and experience only time — and we’ll find the exact description of this wormhole.
Unfortunately, that’s not practical.
But nevertheless it turns out there is a theory of quark-like objects in zero spatial dimensions and one time which seems to capture much of the physics of this cartoon wormhole. (A field theory in zero spatial dimensions is called “quantum mechanics”, studied by every college physics major.) This is called the Sachdev-Ye/Kitaev (or SYK) model. Caution: unlike the Maldacena conjecture, which proposes an exact relationship between field theories of quarks/gluons/etc and certain string theories, we have now moved to a relationship which is no longer exact. [In the figure at the end of this post, this is indicated by replacing an “=” sign with a “~” sign.] Instead, one obtains some kind of relationship with JT gravity only in a special regime of the SYK physics. It is hoped that that this regime captures something universal — i.e., independent of details — about the wormhole. That is, we may hope/pray that what we learn from the SYK model teaches us about the JT gravity wormhole, and that this in turn might teach us some lessons about more realistic wormholes.
This is somewhat analogous to the way, as I described in the previous post, that real-world quarks and gluons seem to capture some of the physics of string theories to which they are not precisely equivalent. It seems that there is something universal about hadron formation, because similar physics (Regge trajectories and something resembling KK towers) appears both (a) in string theories that don’t match the real world but in which calculations are easy, and (b) in the string theory that does match the real world but which can only be studied in experiments on hadrons — in natural simulations in the lab.
So here, too, we can hope and pray for the best. All we have to do is artificially simulate two copies of the SYK models, put them in the appropriate entangled state — the TFD state — and, if desired, add the required interactions to change the cartoon non-traversable wormhole into a cartoon traversable one.
How might we do that? Well, classical computers are still more powerful than quantum computers, so clearly that’s the best way to proceed. Use a standard computer to calculate the physics of the SYK models; if you put that computer into your lab, then, well, I suppose you’ve simulated/made/studied a cartoon wormhole in a lab. However, don’t get confused; it’s still a computer simulation.
But it sounds more exciting to do the computation using a quantum computer (oooh, cool!) because then you really do need a lab just to make the computer in the first place. So now, if you succeed in doing a simulation, you can say more seriously that you did it in a lab. Note, however, that the lab was needed for the quantum computer, not the wormhole. And it’s still a computer simulation, just a less powerful one than you could do with an ordinary computer.
There’s just one more problem. You can’t do the desired simulation with existing quantum computers. Quantum computers aren’t that good yet. This problem is just too hard for them.
So what do you do? You simplify the problem again, and you use classical computers (which, being more powerful, can handle this problem) to help you figure out how best to do it. This leads you to a simpler cartoon of the SYK model, called a “Sparsified SYK model.” Again, you can hope (and there are reasons to expect it) that a Sparsified SYK model, if sufficiently rich, can capture the most important physics of the SYK model in the required regime.
Let’s summarize where we are at. [See the figure at the end of this post.]
- The basic idea is to do build a quantum computer so it can do a simulation of two cartoons of SYK models, entangled and perhaps interacting.
- That in turn hopefully tells us about the behavior of two real SYK models, entangled and perhaps interacting.
- This in turn hopefully tells us about the behavior of cartoon non-traversable and traversable wormholes in JT gravity.
- (Notice we now have a cartoon-squared, and no precise equivalence as we had in the original Maldacena conjecture.)
- This in turn hopefully captures the physics of particular effects in very special classes of wormholes in certain string theories.
- This in turn hopefully captures the physics of wormholes in more general contexts in string theory.
- This in turn hopefully captures the physics of wormholes in the real world (assuming wormholes can actually exist.)
Got that? In the first two stages, one could have used a classical computer, and perhaps skipped the first step. But both because quantum computers are cool and because someday they will be more powerful than classical computers, it’s a nice exercise to see that it’s possible to use a quantum machine to carry out this set of calculations.
Extremely Important Caveat [similar to one as in the last post]: Notice that the gravity of the simulated cartoon wormhole has absolutely nothing to do with the gravity that holds you and me to the floor. First of all, it’s gravity in one spatial dimension, not three! Second, just as in yesterday’s post, the string theory (from which we ostensibly obtained the JT gravity) is equivalent to a theory of quarks/gluons/etc (from which we might imagine obtaining the SYK model) with no gravity at all. There is no connection between the string theory’s gravity (i.e. between that which makes the wormhole, real or cartoonish) and our own real-world gravity. Worst of all, this is an artificial simulation, not the natural simulation of the previous post; our ordinary gravity does interact with quarks and gluons, but it does not interact with the artificially simulated SYK particles. So the wormholes in question, no matter whether you simulate them with classical or quantum computers, are not ones that actually pull on you or me; these are not wormholes into which a pencil or a cat or a researcher can actually fall. In other words, no safety review is needed for this research program; nothing is being made that could suck up the planet, or Los Angeles, or even a graduate student.
Finally, the path is set. The artificial simulation is carried out using a quantum computer; it passes a couple of important consistency checks; a paper is sent for publication in a famous journal; and when it’s published, someone calls the New York Times.
In my next post I’ll tell you more about what was actually done in this quantum computer experiment, and what was achieved scientifically, by this group and by others who’ve tried similar things.