The complexity of mosaics

This post follows, more or less, the content of a talk I gave at the BBVA Fundation in Madrid in April 2019. You can see the video (in Spanish, with English captions provided by YouTube’s Autotranslate) or you can check out the slides.

The view from Erice.

In 2013, I was attending a workshop on noise, information and complexity at the Ettore Majorana Center in beautiful Erice, Sicily, a medieval town sitting on top of a steep hill overlooking the western part of the island. The town, a network of tiny, winding streets lined mostly with medieval buildings, was foggy most days. The Center I was visiting, apart from its awe-inspiring location, is said to have played an important role in fostering relationships between scientists of the West and the East during the Cold War. As a proof of its openness to hosting even the most unexpected of visitors, the Center proudly displays a picture of Pope John Paul II seated behind a version of Dirac’s equation missing an all-important i, the unit of imaginary numbers.

Does this mistake have any theological implications? Did the Pope not like imaginary numbers? Luckily, the equation has been corrected by now.
(Credit: Ettore Majorana Center)

One afternoon, the hosts of the workshop drove us down to Palermo for sightseeing. We toured a number of churches, whose layered styles and decorations reflected the different cultures that flourished on the island over the centuries. The last stop on our tour was the Martorana Church, an Italo-Albanian church of the 12th century, where to this day Mass is held in ancient Greek (yes, it is a complicated history). And while everybody had their noses up in the air, admiring the golden mosaics on the ceilings and the late baroque decorations, I was mesmerized by what lied underneath my feet. I am not talking about some forgotten crypt or creepy burial vault: I was looking at triangles – colorful, 12th century triangles.

XII century triangles. And also my shoes.

What I was looking at, was a 12th century version of a fractal figure which is known today as the Sierpinski triangle, a geometric pattern named after Wacław Sierpiński, the Polish mathematician who studied it eight centuries later, in 1915.

The fractal structure is even more striking here, with three levels of Zelda’s Triforce nested within each other.

You might think this famous tiling pattern was a fluke back then, a random pattern appearing only on the floor of this particular church. It turns out that this type of decoration existed all over the floors of Italy and Europe and was due to a family of Roman artists known as the Cosmati. If you find this fascinating (and you definitely should), I recommend reading “Sierpinski triangles in stone, on medieval floors in Rome”, by Conversano and Tedeschini Lalli, J. Appl. Math 4 (2011). Or you can simply maze through the pictures of these pavements on Wikipedia.

These Sierpinski triangles look suspiciously hyperbolic. Was Poincaré a visitor? (Credits: Wikimedia)

Tiling periods (and the lack thereof)

Since I was a little kid, I was fascinated by tilings. I would spend hours looking at them (don’t all kids do?), trying to figure out which set of tiles was sufficient to reproduce the whole thing (which, to my great surprise, did not always coincide with the way the tiles were cut). I didn’t know at the time that what I was looking for was the period of the tiling, the minimum set of tiles needed to cover the whole space in a periodic fashion. To illustrate this concept, let’s have a look at these beautiful Ottoman tiles from the city of İznik, Turkey.

Tiles from İznik, Turkey, around 1580, from the collection of the Aga Khan Museum in Toronto

Here, we quickly realize that there are two different kinds of tiles: the top right and bottom left tiles are the same, whereas the ones on the diagonal are mirror reflections of the off-diagonal ones. The artist who made these had to actually paint two different kinds of tiles, preparing two separate stacks, one for each kind. If the tiles were made of thin, translucent glass, only one stack would have been necessary (why?)

While it is the drawings that make these tiles beautiful, if we wish to study how they can be composed, we might as well forget about the particular details of the drawings for a moment, and just focus on how each tile can be attached to its neighbor while preserving the continuity of the picture (this is something we do a lot in science, trying to focus on important features by filtering out unnecessary details). Since each square tile has four neighbors, we can think of these two different kinds of tiles in the following way:

From this new point of view, one kind of tile is just a square with four quadrants labeled 1,2,3,4 in a clockwise fashion, and the other kind of tile (the reflection of the first kind) has four quadrants labeled, -1, -2, -3, and -4, also in a clockwise fashion (as if looking at the first kind of tiles from the other side). The tiling rule is such that neighboring tiles sum to zero across their common edge. Now it is easy to see that, if we were given only one type of tiles, we could not do much with them, since the sum would always be positive (for the positive tiles), or negative (for the negative ones) across any edge, but never zero. But if we have access to both types, then we can cover an arbitrarily large surface.

The tiles attach as if they were magnetic, with opposite poles attracting each other.

But, how do we know that we can actually keep going and fill up any rectangular region, no matter how big it is? The trick is, there is a pattern which repeats: every second tile (both horizontally and vertically), the colors repeat, so we can keep making the same choice over and over again. There is a 2×2 square which is our period, and once we obtain it we can simply copy-and-paste this period as many times as we need. Notice that a period is the smallest tiling whose sum is zero along each of the two dimensions.

The Sierpinski tiling, on the other hand, does not have a period.

Try to focus on the pattern of the small drank green triangles. In the top row, they appear fairly often, but already in the second row they are spaced further apart, and then in the middle of the picture there is a big segment (the light green triangle) where they don’t appear. In other words, since we have larger and larger triangles appearing, there cannot be a period, since we would eventually find a triangle larger than the period itself! Tilings of this kind are called aperiodic.

The quest for a truly aperiodic tiling

While the Sierpinski triangle does not have a period that could cover the whole plane as the triangle gets bigger and bigger, if we use a Sierpinski triangle of a fixed size, we can actually generate a simple periodic tiling of the plane, as follows: Attach upside-down versions of the original triangle to its left and right, repeating the process in both directions ad infinitum. Then, take this infinite row of triangles, flip it upside-down and glue it to the original row below, stacking copies of these two rows on top of each other to fill an infinite plane. The aperiodicity of the Sierpinski triangle was a choice of how the smaller triangles tiled the inside of the Sieprinski triangle as it got larger and larger. The same set of triangles would tile the plane periodically if we used the procedure outlined above. In other words, aperiodicity was by choice, not of a necessity. But could there be a particular set of tiles for which no periodic tiling could ever exist?

In 1961, Hao Wang conjectured that, at least for the case of square tiles (which are now called Wang tiles), this is not the case: If a set of square tiles can cover an arbitrarily large rectangle, then there is a way to do so in a periodic fashion. Wang was not interested in floor tilings (at least, we don’t know of any floors decorated by him). Instead, he cared about the decidability of the tiling problem: given a set of tiles, is there an algorithm which can tell whether these tiles can be used to tile an infinitely large floor? If Wang’s conjecture about square tiles was true, we could set up a computer program that explored all the possible ways of covering a 1×1 square, then a 2×2 square, then a 3×3 square, and so on. The program would simply try every possible combination: while there are a lot of combinations, for any n-by-n square there is a finite number of tilings, so the computer could just check every single one of them. Specifically, at some point in the computation, one of two things would happen and the program would stop:

  1. The computer would find a square which could not be covered with the given tiles, or
  2. The computer would find a square which contained a period.

When either of the above happened, the program would stop. In the first case, finding a square which cannot be covered by our tiles implies that any larger square is also impossible to tile. In the second case, since we have found a period, just like in the case of the tiles from İznik, we can tile any rectangular region by repeating the period as needed. The computer might take a long time to decide whether 1. or 2. is the case for our set of tiles, but we know that we will always get an answer, with certainty, at some point. You may be thinking by now that there is a third possibility that I skipped over: The tiles could cover the whole space, but not in a periodic way. And you would be correct in thinking that.

If Wang’s conjecture were to be false, and there is a set of tiles which only generates aperiodic tilings, then our computer program would keep exploring larger and larger squares, without ever being able to give us a definitive answer whether we could tile the plane with this set of tiles. It would keep calculating, using more and more resources, until either it ran out of memory, or the heat generated by the computation boiled the oceans and the Earth and the tiles themselves.

So is Wang’s conjecture true? In 1964, a student of Wang, Robert Berger, showed in his PhD thesis that this conjecture is false: he constructed a set of 20,426 tiles which cover the plane, but can only do so aperiodically! Even worse than that, he actually managed to show that the tiling problem was undecidable: no computer ever built could predict with certainty whether a given set of tiles covered the plane or not!

Before I explain how Berger’s proof works, let me digress a bit and focus on his aperiodic tiling. Clearly, 20,426 are too many to be shown in a blog post, but since his result first appeared, other examples of smaller sets of aperiodic tiles have been found. Berger himself lowered the number to 104, Donald Knuth (of Computer Science fame) to 92, Hans Läuchli to 40, and finally, Raphael Robinson in 1971 produced a set of 6 tiles with the same property! Robinson tiles look like this (they are not depicted as exactly square tiles here, but they can be made into squares easily).

Robinson’s 6 tiles. (Credit: Wikimedia)

The pattern they create looks like this.

I believe that if the Cosmati had known about this tiling, they would have loved it.
(Credit: Wikimedia)

So, here we do not have triangles but squares, but apart from this it looks very similar to the Sierpinski triangle. Focus on the orange squares: there are some smaller ones, and they are sitting at the corners of slightly larger squares, which are in turn at the corners of even larger squares, and so on. While at a first look it might seem like a periodic pattern, it is not, since larger and larger squares keep appearing. We will come back to this orange squares in a while, keep them in mind.

In 1974, Roger Penrose found a set of just 2 aperiodic tiles, but which are not squares.

Penrose “kite and dart” tiles. (Credit: Wikimedia)

Penrose also had this cute idea that one could make a puzzle game out of these shapes, and he even got a patent for that! (“The tiles of the invention may be used to form an instructive game or as a visually attractive floor or wall-covering or the like”). At some point such a puzzle game was actually produced, but it is unfortunately out of production now. If you ever stop by the Newton Institute in Cambridge, UK, they own a copy (and they let you play with it!)

Gamebirds, a puzzle game based on Penrose’s aperiodic tiling. The aim of the game is to fill the circle using all the bird-shaped tiles. Once you have solved the puzzle once, you can notch up the difficulty of the problem by replacing one large bird with a special tile, shaped like a dog, and this will completely change the solution. You can keep replacing birds with dogs until you have 5 dogs, and each time you will have to find a very different solution.

One of the characteristics of Penrose’s tiling is that with it one can obtain patterns with a 5-fold rotational symmetry, which means that you can rotate the tiling by 72°, which is 1/5th of 360°. This is interesting because a beautiful, and elementary, argument from Linear Algebra shows that in periodic tilings you can only get 2-, 3-, 4- or 6-fold symmetries (which corresponds to all n-fold symmetries for which 1+2\cos(2\pi/n) is an integer), so having a 5-fold symmetry is a very unique thing! And just like the case of Sierpinski, there are traces of Penrose’s tiling in art, for example in the Darb-e Imam shrine in Isfahán, Iran.

You can see the 5-fold symmetry by looking at the white stars, or noting that the black lines form pentagons! (Credit: Wikimedia)

Aperiodicity and Undecidability

Going back to Wang’s problem of whether the tiling problem is decidable: how did Berger prove his undecidability result? There are a lot of technical details he had to take care of, but the essence of his proof was to map each step of adding tiles to an ever-growing tiling, to the steps taken by a computer when running an algorithm (also known as a computer program). Each step of running the algorithm would correspond to instructions on which tile to add next and where. Specifically, Berger was interested in simulating the behavior of a very simple, yet very general computer – a Turing machine.

A Turing machine is basically a model for a machine that can run a particular computer algorithm, reduced to the bare minimum. It comprises of four main ingredients:

  1. A tape of arbitrary length on which the machine can write (and overwrite) symbols,
  2. A “head” which
    • can read/write one symbol at a time (like a scanner/printer combo)
    • can move the tape left/right one position at a time
    • can store a finite amount of information (in internal memory)
  3. A program (table of instructions), which tells the “head” what to do next given the symbol it reads on the tape and the current internal memory state.
  4. An initial internal state (which tells the “head” how to start moving), as well as a final (halting) internal state (which tells the “head” when to stop).

While being a really simple object, Turing machines are capable of running any computer algorithm, no matter how complex, so they can come in handy when you need something simple and extremely versatile at the same time!

A model of a Turing Machine, by Mike Davey. A real Turing machine should have infinite tape. The central unit is the machine head, reading and writing symbols on the tape, as well as pulling the tape in either direction. (Credit: Wikimedia)

For example, we could have a Turing machine which can only read/write the symbols 0 and 1, has 6 internal states labeled with letters A, B, C, D, E, F, and has the following program:

 ABCDEF
01RB1RC1LD1RE1LAH
11LE1RF0RB0LC0RD1RC

Here is how to read this table: Assume the initial state of the machine is A and the tape is filled with the symbol 0. The head of the machine will check the entry in the table corresponding to (0,A) and find the instruction “1RB”, which instructs it to write the symbol 1 (flipping the 0 that was already there to a 1), move the tape to the right, and change the internal state of the head to B. The head will now look up the new instructions for (0,B) (since, after moving the tape to the right, the new symbol under the head will be a 0 again), find “1RC” on the table of instructions, change the 0 into a 1, move the tape to the right once again, and change the internal state to C. It will repeat this process, reading one symbol at a time, checking its table of instructions to decide what to do next, until it reads a 0 while being in state F. If that happens, the special instruction “H” tells the machine to stop its execution: it has reached the “halting” state.

You can try to simulate the execution of this machine on a piece of paper, at least for the first few steps (you might need quite a lot of paper if you want to keep going). Or you could use a computer to simulate it. But you may find that after ten, or a thousand, or a million steps the machine has not halted yet. What if we kept going for another million steps? What about a billion? Can we be sure that the machine will halt eventually?

In his landmark work of 1936, Alan Turing showed that analyzing the behavior of this type of machines is outside the reach of any algorithmic computation: there cannot exist any algorithm which, given the description of a Turing machine’s program, can decide whether the machine will eventually halt, or if the machine will keep running forever! This is known as the halting problem.

Berger’s idea was to simulate a Turing machine using a set of tiles. For each possible symbol, the machine could read or write on the tape, he associated a corresponding color for each edge on the tile borders, as well as one color for each of the possible internal states of the machine. As you can probably guess, for two tiles to be neighbors, their common borders had to have the same color. Then he defined a set of tiles which “implemented” the transitions of the machine’s program, in such a way that each horizontal line was one “time step” of the tape during the execution of the machine. The resulting tiles looked like this, and the rule for the arrow is: two tiles can be next to each other only if the head of each arrow matches with the tail of another arrow.

This is Robinson’s version of the tiles, from Undecidability and Nonperiodicity for Tilings of the Plane, Inventiones Math. 12, 177-209 (1971).

Imagine we start our tiling with a row describing the initial state of the machine, which means having a “blank tape” (for example, a tape filled with the symbol 0), and one tile where the head of the machine is. It would look like this.

Then there is only one way we can extend this tiling further: for each of the tiles we have put down, there is only one tile that can go on top of that (try to check it yourself!). This is because the Turing Machine only has one possible transition, starting from the symbol 0 and state A. So after we add an extra layer, the pattern looks like this.

And then we repeat. Each time we put down a new tile, there is only one choice possible: we have to respect the transition rules of the Turing Machine, and our tiling will describe the state of the tape at the various steps of the execution.

The tiling generated by the Turing machine described above.

If the machine halts at some point because it has completed its task, then there will be no way to add new tiles. In order to be sure that we could tile an arbitrarily large area, we would need to know in advance that the Turing machine defined by these tiles (converted into a set of fixed Turing instructions via Berger’s, or Robinson’s mapping) never halted. But, as I mentioned earlier, Turing showed that no algorithm can ever tell us such a thing. Which means you might regret having chosen these tiles for your new bathroom floor (you definitely should have chosen the ones with the flowers instead).

So, why is the aperiodic tiling so important for Berger’s and Robinson’s proofs? We assumed that we started the tiling with a special line, representing the tape in the “blank” state, and this has forced every other choice in the tiling. But using only the alphabet tiles with a single symbol, we get a periodic tiling which can always fill any region! In order to really force our tiling to have a description of the execution of a Turing Machine, we need to guarantee that the tiling is started with that special initialization line. In Robinson’s construction, this is possible using the orange squares as guides (go back and look at the picture of Robinson’s aperiodic tiling if you don’t visualize them), forcing the initialization to happen along the lower edge of each orange square which appears in the pattern. But remember, the Turing machine needs to have access to arbitrarily long segments of tape (we cannot predict how much it will need in case it halts), so we need to have arbitrarily large squares in our tiling. And this means, we really need an aperiodic tiling in order to have all possible tape lengths at our disposition! Any periodic tiling would have restricted the maximum amount of tape the machine could have used before repeating itself.

Tiling a quantum system

You might be wondering: what does all of this have to do with physics (you are, after all, reading the Quantum Frontiers blog and not The IKEA Catalogue 2019). The answer is: tiling problems can be converted into Hamiltonian groundstate energy problems. Think of a square lattice, where to each edge we can assign one of the possible edge configurations of your set of tiles. We can force edges of a square to come from one of the valid tiles by defining a plaquette interaction which gives an energy penalty to non-valid configurations. In this way, we can tile a region of the plane with our set of tiles if and only if this Hamiltonian has a frustration-free groundstate: a groundstate which simultaneously satisfies all the local plaquette constraints, or in other words, one that has zero energy. Deciding whether of not this special kind of groundstate exists is undecidable!

You do not need quantum mechanics for this, as this is a completely classical problem, but you soon realize that the number of possible configurations of the edges in the lattice is arbitrarily large! If you want to write down the matrix which represents this Hamiltonian interaction, you have to resort to larger and larger matrices.

Here is where quantum mechanics comes to the rescue! In a celebrated result, Toby Cubitt, David Pérez-García and Michael Wolf proved that you can have a similar result, this time for the spectral gap of a local Hamiltonian (the problem of deciding whether the spectrum of the Hamiltonian has a constant gap above the groundstate energy), using only a fixed number of local degrees of freedom. Their result is definitely not easy to explain: the first version of the paper was 146 pages long – luckily they managed to simplify it down to 127 pages… But I can try to give a very minimal explanation of how they managed to do this. The key part of their construction is to encode the rules of the Turing machine not directly in the tiling, but in a complex phase (complex number of unit length) which multiplies a certain fixed set of local Hamiltonian terms. They then use the quantum phase estimation algorithm to read off this phase, feeding this input into a Universal Turing Machine (a programmable Turing machine which can simulate any algorithm). In this way, the number of degrees of freedom needed is fixed, and by varying the complex phase mentioned above, they are able to simulate all possible classical Turing machines!

Quantum tiles on a line

Now that we have entered the realms of local Hamiltonian problems, one might wonder if what is going on here is specific to 2 dimensional systems. Clearly, the same phenomena can happen in 3 or more dimensions, since we can simply take multiple slices of 2D systems and stack them on top of each other. But what about 1 dimensional systems? Can we make this construction work on a line?

Interestingly, Wang’s conjecture in 1D is true: every tiling of a line necessarily has a period. Since we are tiling a line, we can think of each tile as essentially a connection between its left-edge color and its right-edge color. Any set of tiles (and associated edge colors) then defines an oriented graph whose vertices are the colors and whose edges are given by the tiles. The rule is again that tiles can be neighbors if their corresponding edges are the same color. The longest (oriented) path we can find in the graph is then the length of the longest segment which can be tiled. It turns out that this length will be infinite if and only if there is a cycle in the graph. In other words, if there is a period.

So we can’t construct aperiodic tilings in 1D, and the tiling problem is decidable. One might be tempted to guess that the same should happen with the spectral gap of local Hamiltonians: We can look at the terms defining the Hamiltonian and decide if a uniform spectral gap exists, as the size of our quantum system increases. After all, in many cases, 1D systems behave “nicely”: we have the DMRG algorithm, polynomial time algorithms for computing groundstates of gapped Hamiltonians, area laws and matrix product state approximations, no thermal phase transitions or topological order, and so on.

But against all odds, in a paper with Johannes Bausch, Toby Cubitt, and David Perez-Garcia, we showed that the spectral gap problem is still undecidable in 1D. How did we get around the lack of aperiodic tilings in 1D?

The key idea was to construct a Hamiltonian whose groundstate would be periodic in the (state of the) spins of an arbitrarily long spin chain, but with a period depending on the halting time of an algorithm (modeled as a Turing machine) encoded (in binary) in the complex phase multiplying each Hamiltonian term. Roughly speaking, this is how we set this up: We partitioned the set of spins into segments. On each segment, we introduced a special Hamiltonian, known as the Feynman-Kitaev history state Hamiltonian, which made sure that the groundstate on that segment was a transcription of the tape during the execution of the classical Turing machine defined by the complex phase (as discussed above).

If at some point the machine has not halted and is running out of tape, so that the segment is not large enough to contain the complete transcription of its execution, then the machine can “push” the delimiter a bit further away, “stealing” some tape space from its neighbor (more technically: the resulting configuration with a larger tape segment is more energetically favorable than the previous one). But once the machine halts, the tape segment shrinks exactly to the minimal size required for the machine to reach its halting state. So, in case the machine halts, the line is divided up into periodic segments, whose length is exactly the optimal length for the machine to halt. If on the other hand the machine does not halt, then the best configuration is the one where there is a unique tape segment, and only one machine running on it.

To recap, the groundstate of this Hamiltonian looks very different depending on whether the Turing machine (encoded in the phase parameter) eventually halts or not. If it does, the groundstate will look periodic, with the period being determined by the halting time. It is therefore a product state, if we think of each segment as a single, huge, particle. If instead the machine never halts, then the groundstate will have a single, very long segment, with a big Kitaev-Feynman history state, which is a highly entangled state.

Even more interestingly, we can set up the different energy scales in the system to behave as follows: for system sizes where the machine has not halted (because it still does not have enough tape to do so, or because it will never do), the single tape segment groudstate has vanishing (but positive) energy, while after it halts, each segment has a small, negative energy. These negative energies in the halting case keep accumulating, so that the thermodynamic groundstate has strictly negative energy density. We can use this difference in energy density between the two cases to construct a “switch”: we introduce two other Hamiltonians to the system (introducing extra local degrees of freedom), one gapped and one gapless. We couple them to everything else we had already set up (the tape segment and the Kitaev-Feynman history state Hamiltonians), in such a way that only one of them controls the low-energy properties of our system. We can set up the switch based on the difference in the energy density in such a way that, before halting, the system is gapped, and it becomes gapless only after the Turing machine has halted (and we cannot predict if this will ever happen!) Hence, the spectral gap is undecidable!

As is the case for 2D system, we need a very large local Hilbert space dimension to make this construction work (so large we did not even care to compute an exact number – but we know it is finite!) On the other extreme end, we know if the local dimension is 2 (we have qubits on a line), and the Hamiltonian has a special property called frustration freeness, then the spectral gap problem is easy to solve. Contrast this with the aperiodic tiling constructions: first Berger found a highly complicated case (with 20,426 tiles), then his construction was refined and simplified over and over, until Robinson got it down to 6 and Penrose showed a similar one with only 2 tiles.

Can we do the same for the undecidability of the spectral gap? At which point does the line become complex enough that the spectral gap problem is undecidable? Can we find some sort of “threshold” which separates the easy and the impossible cases? We need new ideas and new constructions in order to answer all these questions, so let’s get to work!

Long live Yale’s cemetery

Call me morbid, but, the moment I arrived at Yale, I couldn’t wait to visit the graveyard.

I visited campus last February, to present the Yale Quantum Institute (YQI) Colloquium. The YQI occupies a building whose stone exterior honors Yale’s Gothic architecture and whose sleekness defies it. The YQI has theory and experiments, seminars and colloquia, error-correcting codes and small-scale quantum computers, mugs and laptop bumper stickers. Those assets would have drawn me like honey. But my host, Steve Girvin, piled molasses, fudge, and cookie dough on top: “you should definitely reserve some time to go visit Josiah Willard Gibbs, Jr., Lars Onsager, and John Kirkwood in the Grove Street Cemetery.”

Laptop

Gibbs, Onsager, and Kirkwood pioneered statistical mechanics. Statistical mechanics is the physics of many-particle systems, energy, efficiency, and entropy, a measure of order. Statistical mechanics helps us understand why time flows in only one direction. As a colleague reminded me at a conference about entropy, “You are young. But you will grow old and die.” That conference featured a field trip to a cemetery at the University of Cambridge. My next entropy-centric conference took place next to a cemetery in Banff, Canada. A quantum-thermodynamics conference included a tour of an Oxford graveyard.1 (That conference reincarnated in Santa Barbara last June, but I found no cemeteries nearby. No wonder I haven’t blogged about it.) Why shouldn’t a quantum-thermodynamics colloquium lead to the Grove Street Cemetery?

Building

Home of the Yale Quantum Institute

The Grove Street Cemetery lies a few blocks from the YQI. I walked from the latter to the former on a morning whose sunshine spoke more of springtime than of February. At one entrance stood a gatehouse that looked older than many of the cemetery’s residents.

“Can you tell me where to find Josiah Willard Gibbs?” I asked the gatekeepers. They handed me a map, traced routes on it, and dispatched me from their lodge. Snow had fallen the previous evening but was losing its battle against the sunshine. I sloshed to a pathway labeled “Locust,” waded along Locust until passing Myrtle, and splashed back and forth until a name caught my eye: “Gibbs.” 

Entrance

One entrance of the Grove Street Cemetery

Josiah Willard Gibbs stamped his name across statistical mechanics during the 1800s. Imagine a gas in a box, a system that illustrates much of statistical mechanics. Suppose that the gas exchanges heat with a temperature-T bath through the box’s walls. After exchanging heat for a long time, the gas reaches thermal equilibrium: Large-scale properties, such as the gas’s energy, quit changing much. Imagine measuring the gas’s energy. What probability does the measurement have of outputting E? The Gibbs distribution provides the answer, e^{ - E / (k_{\rm B} T) } / Z. The k_{\rm B} denotes Boltzmann’s constant, a fundamental constant of nature. The Z denotes a partition function, which ensures that the probabilities sum to one.

Gibbs lent his name to more than probabilities. A function of probabilities, the Gibbs entropy, prefigured information theory. Entropy features in the Gibbs free energy, which dictates how much work certain thermodynamic systems can perform. A thermodynamic system has many properties, such as temperature and pressure. How many can you control? The answer follows from the Gibbs-Duheim relation. You’ll be able to follow the Gibbs walk, a Yale alumnus tells me, once construction on Yale’s physical-sciences complex ends.

Gibbs 1

Back I sloshed along Locust Lane. Turning left onto Myrtle, then right onto Cedar, led to a tree that sheltered two tombstones. They looked like buddies about to throw their arms around each other and smile for a photo. The lefthand tombstone reported four degrees, eight service positions, and three scientific honors of John Gamble Kirkwood. The righthand tombstone belonged to Lars Onsager:

NOBEL LAUREATE*

[ . . . ]

*ETC.

Onsager extended thermodynamics beyond equilibrium. Imagine gently poking one property of a thermodynamic system. For example, recall the gas in a box. Imagine connecting one end of the box to a temperature-T bath and the other end to a bath at a slightly higher temperature, T' \gtrsim T. You’ll have poked the system’s temperature out of equilibrium. Heat will flow from the hotter bath to the colder bath. Particles carry the heat, energy of motion. Suppose that the particles have electric charges. An electric current will flow because of the temperature difference. Similarly, heat can flow because of an electric potential difference, or a pressure difference, and so on. You can cause a thermodynamic system’s elbow to itch, Onsager showed, by tickling the system’s ankle.

To Onsager’s left lay John Kirkwood. Kirkwood had defined a quasiprobability distribution in 1933. Quasiprobabilities resemble probabilities but can assume negative and nonreal values. These behaviors can signal nonclassical physics, such as the ability to outperform classical computers. I generalized Kirkwood’s quasiprobability with collaborators. Our generalized quasiprobability describes quantum chaos, thermalization, and the spread of information through entanglement. Applying the quasiprobability across theory and experiments has occupied me for two-and-a-half years. Rarely has a tombstone pleased anyone as much as Kirkwood’s tickled me.

Kirkwood and Onsager

The Grove Street Cemetery opened my morning with a whiff of rosemary. The evening closed with a shot of adrenaline. I met with four undergrad women who were taking Steve Girvin’s course, an advanced introduction to physics. I should have left the conversation bled of energy: Since visiting the cemetery, I’d held six discussions with nine people. But energy can flow backward. The students asked how I’d come to postdoc at Harvard; I asked what they might major in. They described the research they hoped to explore; I explained how I’d constructed my research program. They asked if I’d had to work as hard as they to understand physics; I confessed that I might have had to work harder.

I left the YQI content, that night. Such a future deserves its past; and such a past, its future.

WIP

With thanks to Steve Girvin, Florian Carle, and the Yale Quantum Institute for their hospitality.

1Thermodynamics is a physical theory that emerges from statistical mechanics.

“A theorist I can actually talk with”

Haunted mansions have ghosts, football teams have mascots, and labs have in-house theorists. I found myself posing as a lab’s theorist at Caltech. The gig began when Oskar Painter, a Caltech experimentalist, emailed that he’d read my first paper about quantum chaos. Would I discuss the paper with the group?

Oskar’s lab was building superconducting qubits, tiny circuits in which charge can flow forever. The lab aimed to control scores of qubits, to develop a quantum many-body system. Entanglement—strong correlations that quantum systems can sustain and everyday systems can’t—would spread throughout the qubits. The system could realize phases of matter—like many-particle quantum chaos—off-limits to most materials.

How could Oskar’s lab characterize the entanglement, the entanglement’s spread, and the phases? Expert readers will suggest measuring an entropy, a gauge of how much information this part of the system holds about that part. But experimentalists have had trouble measuring entropies. Besides, one measurement can’t capture many-body entanglement; such entanglement involves too many intricacies. Oskar was searching for arrows to add to his lab’s measurement quiver.

Mascot

In-house theorist?

I’d proposed a protocol for measuring a characterization of many-body entanglement, quantum chaos, and thermalization—a property called “the out-of-time-ordered correlator.” The protocol appealed to Oskar. But practicalities limit quantum many-body experiments: The more qubits your system contains, the more the system can contact its environment, like stray particles. The stronger the interactions, the more the environment entangles with the qubits, and the less the qubits entangle with each other. Quantum information leaks from the qubits into their surroundings; what happens in Vegas doesn’t stay in Vegas. Would imperfections mar my protocol?

I didn’t know. But I knew someone who could help us find out.

Justin Dressel works at Chapman University as a physics professor. He’s received the highest praise that I’ve heard any experimentalist give a theorist: “He’s a theorist I can actually talk to.” With other collaborators, Justin and I simplified my scheme for measuring out-of-time-ordered correlators. Justin knew what superconducting-qubit experimentalists could achieve, and he’d been helping them reach for more.

How about, I asked Justin, we simulate our protocol on a computer? We’d code up virtual superconducting qubits, program in interactions with the environment, run our measurement scheme, and assess the results’ noisiness. Justin had the tools to simulate the qubits, but he lacked the time. 

Know any postdocs or students who’d take an interest? I asked.

Chapman.001

Chapman University’s former science center. Don’t you wish you spent winters in California?

José Raúl González Alonso has a smile like a welcome sign and a coffee cup glued to one hand. He was moving to Chapman University to work as a Grand Challenges Postdoctoral Fellow. José had built simulations, and he jumped at the chance to study quantum chaos.

José confirmed Oskar’s fear and other simulators’ findings: The environment threatens measurements of the out-of-time-ordered correlator. Suppose that you measure this correlator at each of many instants, you plot the correlator against time, and you see the correlator drop. If you’ve isolated your qubits from their environment, we can expect them to carry many-body entanglement. Golden. But the correlator can drop if, instead, the environment is harassing your qubits. You can misdiagnose leaking as many-body entanglement.

OTOC plots

Our triumvirate identified a solution. Justin and I had discovered another characterization of quantum chaos and many-body entanglement: a quasiprobability, a quantum generalization of a probability.  

The quasiprobability contains more information about the entanglement than the out-of-time-ordered-correlator does. José simulated measurements of the quasiprobability. The quasiprobability, he found, behaves one way when the qubits entangle independently of their environment and behaves another way when the qubits leak. You can measure the quasiprobability to decide whether to trust your out-of-time-ordered-correlator measurement or to isolate your qubits better. The quasiprobability enables us to avoid false positives.

Physical Review Letters published our paper last month. Working with Justin and José deepened my appetite for translating between the abstract and the concrete, for proving abstractions as a theorist’s theorist and realizing them experimentally as a lab’s theorist. Maybe, someday, I’ll earn the tag “a theorist I can actually talk with” from an experimentalist. For now, at least I serve better than a football-team mascot.

Symmetries and quantum error correction

It’s always exciting when you can bridge two different physical concepts that seem to have nothing in common—and it’s even more thrilling when the results have as broad a range of possible fields of application as from fault-tolerant quantum computation to quantum gravity.

Physicists love to draw connections between distinct ideas, interconnecting concepts and theories to uncover new structure in the landscape of scientific knowledge. Put together information theory with quantum mechanics and you’ve opened a whole new field of quantum information theory. More recently, machine learning tools have been combined with many-body physics to find new ways to identify phases of matter, and ideas from quantum computing were applied to Pozner molecules to obtain new plausible models of how the brain might work.

In a recent contribution, my collaborators and I took a shot at combining the two physical concepts of quantum error correction and physical symmetries. What can we say about a quantum error-correcting code that conforms to a physical symmetry? Surprisingly, a continuous symmetry prevents the code from doing its job: A code can conform well to the symmetry, or it can correct against errors accurately, but it cannot do both simultaneously.

By a continuous symmetry, we mean a transformation that is characterized by a set of continuous parameters, such as angles. For instance, if I am holding an atom in my hand (more realistically, it’ll be confined in some fancy trap with lots of lasers), then I can rotate it around and about in space:

A rotation like this is fully specified by an axis and an angle, which are continuous parameters. Other transformations that we could think of are, for instance, time evolution, or a continuous family of unitary gates that we might want to apply to the system.

On the other hand, a code is a way of embedding some logical information into physical systems:

By cleverly distributing the information that we care about over several physical systems, an error-correcting code is able to successfully recover the original logical information even if the physical systems are exposed to some noise. Quantum error-correcting codes are particularly promising for quantum computing, since qubits tend to lose their information really fast (current typical ones can hold their information for a few seconds). In this way, instead of storing the actual information we care about on a single qubit, we use extra qubits which we prepare in a complicated state that is designed to protect this information from the noise.

Covariant codes for quantum computation

A code that is compatible with respect to a physical symmetry is called covariant. This property ensures that if I apply a symmetry transformation on the logical information, this is equivalent to applying corresponding symmetry transformations on each of the physical systems.

Suppose I would like to flip my qubit from “0” to “1” and from “1” to “0”. If my information is stored in an encoded form, then in principle I first need to decode the information to uncover the original logical information, apply the flip operation, and then re-encode the new logical information back onto the physical qubits. A covariant code allows to perform the transformation directly on the physical qubits, without having to decode the information first:

The advantage of this scheme is that the logical information is never exposed and remains protected all along the computation.

But here’s the catch: Eastin and Knill famously proved that error-correcting codes can be at most covariant with respect to a finite set of transformations, ruling out universal computation with transversal gates. In other words, the computations we can perform using this scheme are very limited because we can’t perform any continuous symmetry transformation.

Interestingly, however, there’s a loophole: If we consider macroscopic systems, such as a particle with a very large value of spin, then it becomes possible again to construct codes that are covariant with respect to continuous transformations.

How is that possible, you ask? How do we transition from the microscopic regime, where covariant codes are ruled out for continuous symmetries, to the macroscopic regime, where they are allowed? We provide an answer by resorting to approximate quantum error correction. Namely, we consider the situation where the code does not have to correct each error exactly, but only has to reconstruct a good approximation of the logical information. As it turns out, there is a quantitative limit to how accurately a code can correct against errors if it is covariant with respect to a continuous symmetry, represented by the following equation:

where epsilon specifies how inaccurately the code error-corrects (epsiloneqzero means the code can correct against errors perfectly), n is the number of physical subsystems, and the DeltaT_logical and DeltaT_physical are measures of “how strongly” the symmetry transformation can act on the logical and physical subsystems.

Let’s try to understand the right-hand side of this equation. In physics, continuous symmetries are generated by what we call physical charges. These are physical quantities that are associated with the symmetry, and that characterize how the symmetry acts on each state of the system. For instance, the charge that corresponds to time evolution is simply energy: States that label high energies have a rapidly varying phase whereas the phase of low-energy states changes slowly in time. Above, we indicate by DeltaT_logical the range of possible charge values on the logical system and by DeltaT_physical the corresponding range of charge values on each physical subsystem. In typical settings, this range of charge values is related to the dimension of the system—the more states the system has, intuitively, the greater range of charges it can accommodate.

The above equation states that the inaccuracy of the code must be larger than some value given on the right-hand side of the equation, which depends on the number of subsystems n and the ranges of charge values on the logical system and physical subsystems. The right-hand side becomes small in two regimes: if each subsystem can accommodate a large range of charge values, or if there is a large number of physical systems. In these regimes, our limitation vanishes, and we can circumvent the Eastin-Knill theorem and construct good covariant error-correcting codes. This allows us to connect the two regimes that seemed incompatible earlier, the microscopic regime where there cannot be any covariant codes, and the macroscopic regime where they are allowed.

From quantum computation to many-body physics and quantum gravity

Quantum error-correcting codes not only serve to protect information in a quantum computation against noise, but they also provide a conceptual toolbox to understand complex physical systems where a quantum state is delocalized over many physical subsystems. The tight connections between quantum error correction and many-body physics have been put to light following a long history of pioneering research at Caltech in these fields. And as if that weren’t enough, quantum error correcting codes were also shown to play a crucial role in understanding quantum gravity.

There is an abundance of natural physical symmetries to consider both in many-body physics and in quantum gravity, and that gives us a good reason to be excited about characterizing covariant codes. For instance, there are natural approximate quantum error correcting codes that appear in some statistical mechanical models by cleverly picking global energy eigenstates. These codes are covariant with respect to time evolution by construction, since the codewords are energy eigenstates. Now, we understand more precisely under which conditions such codes can be constructed.

Perhaps an even more illustrative example is that of time evolution in holographic quantum gravity, that is, in the AdS/CFT correspondence. This model of quantum gravity has the property that it is equivalent to a usual quantum field theory that lives on the boundary of the universe. What’s more, the correspondence which tells us how the bulk quantum gravity theory is mapped to the boundary is, in fact, a quantum error-correcting code. If we add a time axis, then the picture becomes a cylinder where the interior is the theory of quantum gravity, and where the cylinder itself represents a traditional quantum field theory:

AdSCFT-01

Since the bulk theory and the boundary theory are equivalent, the action of time evolution must be faithfully represented in both pictures. But this is in apparent contradiction with the Eastin-Knill theorem, from which it follows that a quantum error-correcting code cannot be covariant with respect to a continuous symmetry. We now understand how this is, in fact, not a contradiction: As we’ve seen, codes may be covariant with respect to continuous symmetries in the presence of systems with a large number of degrees of freedom, such as a quantum field theory.

What’s next?

There are some further results in our paper that I have not touched upon in this post, including a precise approximate statement of the Eastin-Knill theorem in terms of system dimensions, and a fun machinery to construct covariant codes for more general systems such as oscillators and rotors.

We have only scratched the surface of the different applications I’ve mentioned, by studying the properties of covariant codes in general. I’m now excited to dive into more detail with our wonderful team to study deeper applications to correlations in many-body systems, global symmetries in quantum gravity, accuracy limits of quantum clocks and precision limits to quantum metrology in the presence of noise.

This has been an incredibly fun project to work on. Such a collaboration illustrates again the benefit of interacting with great scientists with a wide range of areas of expertise including representation theory, continuous variable systems, and quantum gravity. Thanks Sepehr, Victor, Grant, Fernando, Patrick, and John, for this fantastic experience.

Why care about physics that doesn’t care about us?

A polar vortex had descended on Chicago.

I was preparing to fly in, scheduled to present a seminar at the University of Chicago. My boyfriend warned, from Massachusetts, that the wind chill effectively lowered the temperature to -50 degrees F. I’d last encountered -50 degrees F in the short story “To Build a Fire,” by Jack London. Spoiler alert: The protagonist fails to build a fire and freezes to death.

The story exemplifies naturalism, according to my 11th-grade English class. The naturalist movement infiltrated American literature and art during the late 19th century. Naturalists portrayed nature as as harsh and indifferent: The winter doesn’t care if Jack London’s protagonist dies.

The protagonist lingered in my mind as my plane took off. I was flying into a polar vortex for physics, the study of nature. Physics doesn’t care about me. How can I care so much about physics? How can humans generally?

Peeling apart that question, I found more layers than I’d packed for protection against the polar vortex.

Tweet

Intellectualism formed the parka of the answer: You can’t hug space, time, information, energy, and the nature of reality. You can’t smile at them and watch them smile back. But their abstractness doesn’t block me from engaging with them; it attracts me. Ideas attract me; their purity does. Physics consists partially of a framework of ideas—of mathematical models, of theorems and examples, of the hypotheses and plots and revisions that underlie a theory.

The framework of physics needs construction. Some people compose songs; some build businesses; others bake soufflés; I used to do arts and crafts. Many humans create—envision, shape, mold, and coordinate—with many different materials. Theoretical physics overflows with materials and with opportunities to create. As humans love to create, we can love physics. Theoretical-physics materials consist of ideas, which might sound less suited to construction than paint does. But painters glob mixtures of water, resin, acrylic, and pigment onto woven fabric. Why shouldn’t ideas appeal as much as resin does? I build worlds in my head for a living. Doesn’t that sound romantic?

Painters derive joy from painting; dancers derive joy from moving; physics offers outlets for many skills. Doing physics, I use math. I learn history: What paradoxes about quantum theory did Albert Einstein pose to Niels Bohr? I write papers and blog posts, and I present seminars and colloquia. I’ve moonlighted as a chemist, studied biology, dipped into computer science, and sought to improve engineering. Beyond these disciplines, physics requires uniquely physical skills: the identification of questions about the natural world, the translation of those questions into math, and the translation of mathematical results into statements about the natural world. In college, I hated having to choose a major because I wanted to study everything. Physics lets me.

Easel

My attraction to physics worried me in college. Jim Yong Kim became Dartmouth’s president in my junior year. Jim, who left to helm the World Bank, specializes in global health. He insisted that “the world’s troubles are your troubles,” quoting former Dartmouth president John Sloan Dickey. I was developing a specialization in quantum information theory. I wasn’t trying to contain ebola, mitigate droughts, or eradicate Alzheimer’s disease. Should I not have been trying to save the world?

I could help save the world, a mentor said, through theoretical physics.1 Society needs a few people to develop art, a few to write music, a few to curate history, and a few to study the nature of the universe. Such outliers help keep us human, and the reinforcement of humanity helps save the world. You may indulge in physics, my mentor said, because physics affords the opportunity to do good. If I appreciate that opportunity, how can I not appreciate physics?

The opportunity to do good has endeared physics to me more as I’ve advanced. The more I advance, the fewer women I see. According to the American Physical Society (APS), in 2017, women received about 21% of the physics Bachelor’s degrees awarded in the U.S. Women received about 18% of the doctorates. In 2010, women numbered 8% of the full professors in U.S. departments that offered Bachelor’s or higher degrees in physics. The APS is conducting studies, coordinating workshops, and offering grants to improve the gender ratio. Departments, teachers, and mentors are helping. They have my gratitude. Yet they can accomplish only so much, especially since many are men. They can encourage women to change the gender ratio; they can’t change the ratio directly. Only women can, and few women are undertaking the task. Physics affords an opportunity to do good—to improve a field’s climate, to mentor, and to combat stereotypes—that few people can tackle. For that opportunity, I’m grateful to physics.

World

Physics lifts us beyond the Platonic realm of ideas in two other ways. At Caltech, I once ate lunch with Charlie Marcus. Marcus is a Microsoft researcher and a professor of physics at the University of Copenhagen’s Niels Bohr Institute. His lab is developing topological quantum computers, in which calculations manifest as braids. Why, I asked, does quantum computing deserve a large chunk of Marcus’s life?

Two reasons, he replied. First, quantum computing straddles the border between foundational physics and applications. Quantum science satisfies the intellect but doesn’t tether us to esoterica. Our science could impact technology, industry and society. Second, the people. Quantum computing has a community steeped in congeniality.

Marcus’s response delighted me: His reasons for caring about quantum computing coincided with two of mine. Reason two has expanded, in my mind, to opportunities for engagement with people. Abstractions attract me partially because intellectualism runs in my family. I grew up surrounded by readers, encouraged to ask questions. Physics enables me to participate in a family tradition and to extend that tradition to the cosmos. My parents now ask me the questions—about black holes and about whether I’m staying warm in Chicago.

Beyond family, physics enables me to engage with you. This blog has connected me to undergraduates, artists, authors, computer programmers, science teachers, and museum directors across the world. Scientific outreach inspires reading, research, art, and the joy of learning. I love those outcomes, participating in them, and engaging with you.

Chain

Why fly into a polar vortex for the study of nature—why care about physics that can’t care about us? In my case, primarily because of the ideas, the abstraction, and the chances to create and learn. Partially for the chance to help save the world through humanness, outreach, and a gender balance. Partially for the chance to impact technology, and partially to connect with people: Physics can strengthen ties to family and can introduce you to individuals across the globe. And physics can— heck, tomorrow is February 14th—lead you to someone who cares enough to track Chicago’s weather from Cambridge.

 

1I’m grateful that Jim Kim, too, encouraged me to pursue theoretical physics.

Humans can intuit quantum physics.

One evening this January, audience members packed into a lecture hall in MIT’s physics building. Undergraduates, members of the public, faculty members, and other scholars came to watch a film premiere and a panel discussion. NOVA had produced the film, “Einstein’s Quantum Riddle,” which stars entanglement. Entanglement is a relationship between quantum systems such as electrons. Measuring two entangled electrons yields two outcomes, analogous to the numbers that face upward after you roll two dice. The quantum measurements’ outcomes can exhibit correlations stronger than any measurements of any classical, or nonquantum, systems can. Which die faces point upward can share only so much correlation, even if the dice hit each other.

einstein's q. riddle

Dice feature in the film’s explanations of entanglement. So does a variation on the shell game, in which one hides a ball under one of three cups, shuffles the cups, and challenges viewers to guess which cup is hiding the ball. The film derives its drama from the Cosmic Bell test. Bell tests are experiments crafted to show that classical physics can’t describe entanglement. Scientists recently enhanced Bell tests using light from quasars—ancient, bright, faraway galaxies. Mix astrophysics with quantum physics, and an edgy, pulsing soundtrack follows.

The Cosmic Bell test grew from a proposal by physicists at MIT and the University of Chicago. The coauthors include David Kaiser, a historian of science and a physicist on MIT’s faculty. Dave co-organized the premiere and the panel discussion that followed. The panel featured Dave; Paola Cappellaro, an MIT quantum experimentalist; Alan Guth, an MIT cosmologist who contributed to the Bell test; Calvin Leung, an MIT PhD student who contributed; Chris Schmidt, the film’s producer; and me. Brindha Muniappan, the Director of Education and Public Programs at the MIT Museum, moderated the discussion.

panel

think that the other panelists were laughing with me.

Brindha asked what challenges I face when explaining quantum physics, such as on this blog. Quantum theory wears the labels “weird,” “counterintuitive,” and “bizarre” in journalism, interviews, blogs, and films. But the thorn in my communicational side reflects quantum “weirdness” less than it reflects humanity’s self-limitation: Many people believe that we can’t grasp quantum physics. They shut down before asking me to explain.

Examples include a friend and Quantum Frontiers follower who asks, year after year, for books about quantum physics. I suggest literature—much by Dave Kaiser—he reads some, and we discuss his impressions. He’s learning, he harbors enough curiosity to have maintained this routine for years, and he has technical experience as a programmer. But he’s demurred, several times, along the lines of “But…I don’t know. I don’t think I’ll ever understand it. Humans can’t understand quantum physics, can we? It’s too weird.” 

Quantum physics defies many expectations sourced from classical physics. Classical physics governs how basketballs arch, how paint dries, how sunlight slants through your window, and other everyday experiences. Yet we can gain intuition about quantum physics. If we couldn’t, how could we solve problems and accomplish research? Physicists often begin solving problems by trying to guess the answer from intuition. We reason our way toward a guess by stripping away complications, constructing toy models, and telling stories. We tell stories about particles hopping from site to site on lattices, particles trapped in wells, and arrows flipping upward and downward. These stories don’t capture all of quantum physics, but they capture the essentials. After grasping the essentials, we translate them into math, check how far our guesses lie from truth, and correct our understanding. Intuition about quantum physics forms the compass that guides problem solving.

Growing able to construct, use, and mathematize such stories requires work. You won’t come to understand quantum theory by watching NOVA films, though films can prime you for study. You can gain a facility with quantum theory through classes, problem sets, testing, research, seminars, and further processing. You might not have the time or inclination to. Even if you have, you might not come to understand why quantum theory describes our universe: Science can’t necessarily answer all “why” questions. But you can grasp what quantum theory implies about our universe.

People grasp physics arguably more exotic than quantum theory, without exciting the disbelief excited by a grasp of quantum theory. Consider the Voyager spacecraft launched in 1977. Voyager has survived solar winds and -452º F weather, imaged planets, and entered interstellar space. Classical physics—the physics of how basketballs arch—describes much of Voyager’s experience. But even if you’ve shot baskets, how much intuition do you have about interstellar space? I know physicists who claim to have more intuition about quantum physics than about much classical. When astrophysicists discuss Voyager and interstellar space, moreover, listeners don’t fret that comprehension lies beyond them. No one need fret when quantum physicists discuss the electrons in us.

Fretting might not occur to future generations: Outreach teams are introducing kids to quantum physics through games and videos. Caltech’s Institute for Quantum Information and Matter has partnered with Google to produce QCraft, a quantum variation on Minecraft, and with the University of Southern California on quantum chess. In 2017, the American Physical Society’s largest annual conference featured a session called “Gamification and other Novel Approaches in Quantum Physics Outreach.” Such outreach exposes kids to quantum terminology and concepts early. Quantum theory becomes a playground to explore, rather than a source of intimidation. Players will grow up primed to think about quantum-mechanics courses not “Will my grade-point average survive this semester?” but “Ah, so this is the math under the hood of entanglement.”

qcraft 2

Sociology restricts people to thinking quantum physics weird. But quantum theory defies classical expectations less than it could. Measurement outcomes could share correlations stronger than the correlations sourced by entanglement. How strong could the correlations grow? How else could physics depart farther from classical physics than quantum physics does? Imagine the worlds governed by all possible types of physics, called “generalized probabilistic theories” (GPTs). GPTs form a landscape in which quantum theory constitutes an island, on which classical physics constitutes a hill. Compared with the landscape’s outskirts, our quantum world looks tame.

GPTs fall under the research category of quantum foundations. Quantum foundations concerns why the math that describes quantum systems describes quantum systems, reformulations of quantum theory, how quantum theory differs from classical mechanics, how quantum theory could deviate but doesn’t, and what happens during measurements of quantum systems. Though questions about quantum foundations remain, they don’t block us from intuiting about quantum theory. A stable owner can sense when a horse has colic despite lacking a veterinary degree.

Moreover, quantum-foundations research has advanced over the past few decades. Collaborations and tools have helped: Theorists have been partnering with experimentalists, such as on the Cosmic Bell test and on studies of measurement. Information theory has engendered mathematical tools for quantifying entanglement and other quantum phenomena. Information theory has also firmed up an approach called “operationalism.” Operationalists emphasize preparation procedures, evolutions, and measurements. Focusing on actions and data concretizes arguments and facilitates comparisons with experiments. As quantum-foundations research has advanced, so have quantum information theory, quantum experiments, quantum technologies, and interdisciplinary cross-pollination. Twentieth-century quantum physicists didn’t imagine the community, perspectives, and knowledge that we’ve accrued. So don’t adopt 20th-century pessimism about understanding quantum theory. Einstein grasped much, but today’s scientific community grasps more. Richard Feynman said, “I think I can safely say that nobody understands quantum mechanics.” Feynman helped spur the quantum-information revolution; he died before its adolescence. Besides, Feynman understood plenty about quantum theory. Intuition jumps off the pages of his lecture notes and speeches.

landscape

Landscape beyond quantum theory

I’ve swum in oceans and lakes, studied how the moon generates tides, and canoed. But piloting a steamboat along the Mississippi would baffle me. I could learn, given time, instruction, and practice; so can you learn quantum theory. Don’t let “weirdness,” “bizarreness,” or “counterintuitiveness” intimidate you. Humans can intuit quantum physics.

Chasing Ed Jaynes’s ghost

You can’t escape him, working where information theory meets statistical mechanics.

Information theory concerns how efficiently we can encode information, compute, evade eavesdroppers, and communicate. Statistical mechanics is the physics of  many particles. We can’t track every particle in a material, such as a sheet of glass. Instead, we reason about how the conglomerate likely behaves. Since we can’t know how all the particles behave, uncertainty blunts our predictions. Uncertainty underlies also information theory: You can think that your brother wished you a happy birthday on the phone. But noise corroded the signal; he might have wished you a madcap Earth Day. 

Edwin Thompson Jaynes united the fields, in two 1957 papers entitled “Information theory and statistical mechanics.” I’ve cited the papers in at least two of mine. Those 1957 papers, and Jaynes’s philosophy, permeate pockets of quantum information theory, statistical mechanics, and biophysics. Say you know a little about some system, Jaynes wrote, like a gas’s average energy. Say you want to describe the gas’s state mathematically. Which state can you most reasonably ascribe to the gas? The state that, upon satisfying the average-energy constraint, reflects our ignorance of the rest of the gas’s properties. Information theorists quantify ignorance with a function called entropy, so we ascribe to the gas a large-entropy state. Jaynes’s Principle of Maximum Entropy has spread from statistical mechanics to image processing and computer science and beyond. You can’t evade Ed Jaynes.

I decided to turn the tables on him this December. I was visiting to Washington University in St. Louis, where Jaynes worked until six years before his 1998 death. Haunted by Jaynes, I’d hunt down his ghost.

WashU

I began with my host, Kater Murch. Kater’s lab performs experiments with superconducting qubits. These quantum circuits sustain currents that can flow forever, without dissipating. I questioned Kater over hummus, the evening after I presented a seminar about quantum uncertainty and equilibration. Kater had arrived at WashU a decade-and-a-half after Jaynes’s passing but had kept his ears open.

Ed Jaynes, Kater said, consulted for a startup, decades ago. The company lacked the funds to pay him, so it offered him stock. That company was Varian, and Jaynes wound up with a pretty penny. He bought a mansion, across the street from campus, where he hosted the physics faculty and grad students every Friday. He’d play a grand piano, and guests would accompany him on instruments they’d bring. The department doubled as his family. 

The library kept a binder of Jaynes’s papers, which Kater had skimmed the previous year. What clarity shined through those papers! With a touch of pride, Kater added that he inhabited Jaynes’s former office. Or the office next door. He wasn’t certain.

I passed the hummus to a grad student of Kater’s. Do you hear stories about Jaynes around the department? I asked. I’d heard plenty about Feynman, as a PhD student at Caltech.

Not many, he answered. Just in conversations like this.

Later that evening, I exchanged emails with Kater. A contemporary of Jaynes’s had attended my seminar, he mentioned. Pity that I’d missed meeting the contemporary.

The following afternoon, I climbed to the physics library on the third floor of Crow Hall. Portraits of suited men greeted me. At the circulation desk, I asked for the binders of Jaynes’s papers.

Who? asked the student behind the granola bars advertised as “Free study snacks—help yourself!” 

E.T. Jaynes, I repeated. He worked here as a faculty member.

She turned to her computer. Can you spell that?

I obeyed while typing the name into the computer for patrons. The catalogue proffered several entries, one of which resembled my target. I wrote down the call number, then glanced at the notes over which the student was bending: “The harmonic oscillator.” An undergrad studying physics, I surmised. Maybe she’ll encounter Jaynes in a couple of years. 

I hiked upstairs, located the statistical-mechanics section, and ran a finger along the shelf. Hurt and Hermann, Itzykson and Drouffe, …Kadanoff and Baym. No Jaynes? I double-checked. No Jaynes. 

Library books

Upon descending the stairs, I queried the student at the circulation desk. She checked the catalogue entry, then ahhhed. You’d have go to the main campus library for this, she said. Do you want directions? I declined, thanked her, and prepared to return to Kater’s lab. Calculations awaited me there; I’d have no time for the main library. 

As I reached the physics library’s door, a placard caught my eye. It appeared to list the men whose portraits lined the walls. Arthur Compton…I only glanced at the placard, but I didn’t notice any “Jaynes.”

Arthur Compton greeted me also from an engraving en route to Kater’s lab. Down the hall lay a narrow staircase on whose installation, according to Kater, Jaynes had insisted. Physicists would have, in the stairs’ absence, had to trek down the hall to access the third floor. Of course I wouldn’t photograph the staircase for a blog post. I might belong to the millenial generation, but I aim and click only with purpose. What, though, could I report in a blog post? 

That night, I googled “e.t. jaynes.” His Wikipedia page contained only introductory and “Notes” sections. A WashU website offered a biography and unpublished works. But another tidbit I’d heard in the department yielded no Google hits, at first glance. I forbore a second glance, navigated to my inbox, and emailed Kater about plans for the next day.

I’d almost given up on Jaynes when Kater responded. After agreeing to my suggestion, he reported feedback about my seminar: A fellow faculty member “thought that Ed Jaynes (his contemporary) would have been very pleased.” 

The email landed in my “Nice messages” folder within two shakes. 

Leaning back, I reevaluated my data about Jaynes. I’d unearthed little, and little surprise: According to the WashU website, Jaynes “would undoubtedly be uncomfortable with all of the attention being lavished on him now that he is dead.” I appreciate privacy and modesty. Nor does Jaynes need portraits or engravings. His legacy lives in ideas, in people. Faculty from across his department attended a seminar about equilibration and about how much we can know about quantum systems. Kater might or might not inhabit Jaynes’s office. But Kater wears a strip cut from Jaynes’s mantle: Kater’s lab probes the intersection of information theory and statistical mechanics. They’ve built a Maxwell demon, a device that uses information as a sort of fuel to perform thermodynamic work. 

I’ve blogged about legacies that last. Assyrian reliefs carved in alabaster survive for millennia, as do ideas. Jaynes’s ideas thrive; they live even in me.

Did I find Ed Jaynes’s ghost at WashU? I think I honored it, by pursuing calculations instead of pursuing his ghost further. I can’t say whether I found his ghost. But I gained enough information.

 

With thanks to Kater and to the Washington University Department of Physics for their hospitality.