# The mentors that shape us

I was in awe of Wheeler. Some students thought he sucked.

I immediately changed it to…

I was in awe of Wheeler. Some students thought less of him.

And next, when I saw John write about himself,

Though I’m 59, few students seemed awed. Some thought I sucked. Maybe I did sometimes.

I massaged it into…

Though I’m 59, few students seemed awed. Some thought I was not as good. Maybe I wasn’t sometimes.

When John published the post, I read it again for any typos I might have missed. There were no typos. I felt useful! But when I saw that all mentions of sucked had been restored to their rightful place, I felt like an idiot. John did not fire a strongly-worded email back my way asking for an explanation as to my taking liberties with his own writing. He simply trusted that I would get the message in the comfort of my own awkwardness. It worked beautifully. John had set the tone for Quantum Frontier’s authentic voice with his very first post. It was to be personal, even if the subject matter was as scientifically hardcore as it got.

So when the time came for me to write my first post, I made it personal. I wrote about my time in Los Alamos as a postdoc, working on a problem in mathematical physics that almost broke me. It was Matt Hastings, an intellectual tornado, that helped me through these hard times. As my mentor, he didn’t say things like Well done! Great progress! Good job, Spiro! He said, You can do this. And when I finally did it, when I finally solved that damn problem, Matt came back to me and said: Beyond some typos, I cannot find any mistakes. Good job, Spiro. And it meant the world to me. The sleepless nights, the lonely days up in the Pajarito mountains of New Mexico, the times I had resolved to go work for my younger brother as a waiter in his first restaurant… those were the times that I had come upon a fork on the road and my mentor had helped me choose the path less traveled.

When the time came for me to write my next post, I ended by offering two problems for the readers to solve, with the following text as motivation:

This post is supposed to be an introduction to the insanely beautiful world of problem solving. It is not a world ruled by Kings and Queens. It is a world where commoners like you and me can become masters of their domain and even build an empire.

Doi-Inthananon temple in Chiang Mai, Thailand. A breathtaking city, host of this year’s international math olympiad.

It has been way too long since my last “problem solving” post, so I leave you with a problem from this year’s International Math Olympiad, which took place in gorgeous Chiang Mai, Thailand. FiverThirtyEight‘s recent article about the dominance of the US math olympic team in this year’s competition, gives some context about the degree of difficulty of this problem:

Determine all triples (a, b, c) of positive integers such that each of the numbers: ab-c, bc-a, ca-b is a power of two.

Like Fermat’s Last Theorem, this problem is easy to describe and hard to solve. Only 5 percent of the competitors got full marks on this question, and nearly half (44 percent) got no points at all.

But, on the triumphant U.S. squad, four of the six team members nailed it.

In other words, only 1 in 20 kids in the competition solved this problem correctly and about half of the kids didn’t even know where to begin. For more perspective, each national team is comprised of the top 6 math prodigies in that country. In China, that means 6 out of something like 100 million kids. And only 3-4 of these kids solved the problem.

The coach of the US national team, Po-Shen Loh, a Caltech alum and an associate professor of mathematics at Carnegie Mellon University (give him tenure already) deserves some serious props. If you think this problem is too hard, I have this to say to you: Yes, it is. But, who cares? You can do this.

Note: I will work out the solution in detail in an upcoming post, unless one of you solves it in the comments section before then!

Update: Solution posted in comments below (in response to Anthony’s comment). Thank you all who posted some of the answers below. The solution is far from trivial, but I still wonder if an elegant solution exists that gives all four triples. Maybe the best solution is geometric? I hope one of you geniuses can figure that out!

# Ant-Man and the Quantum Realm

It was the first week of August last summer and I was at LAX for a trip to North Carolina as a guest speaker at Project Scientist’s STEM summer camp for young women. I had booked an early morning flight and had arrived at my gate with time to spare, so I decided to get some breakfast. I walked by a smart-looking salad bar and thought: Today is the day. Moving past the salad bar, I ordered a juicy cheeseburger with fries at the adjacent McDonald’s. Growing up in Greece, eating McDonald’s was a rare treat; as was playing video games with my brothers and reading comic books late at night. Yet, through a weird twist of fate, it was these last two guilty pleasures that were to become my Hadouken!, my Hulk, Smash!, my secret weapons of choice for breaking down the barriers between the world of quantum mechanics and the everyday reality of our super-normal, super-predictable lives.

I finished my burger, stuffing the last few delicious fries in my mouth, when my phone buzzed – I had mail from The Science and Entertainment Exchange, a non-profit organization funded by the National Academy of Sciences, whose purpose is to bring leading scientists in contact with Hollywood in order to elevate the level of science in the movies. I was to report to Atlanta, GA for a movie consult on a new superhero movie: Ant-Man. As I read halfway through the email, I grumbled to myself: Why can’t I be the guy who works on Thor? Who is this Ant-Man anyways? But, in typical Hollywood fashion, the email had a happy ending: “Paul Rudd is playing Ant-Man. He may be at the meeting, but we cannot promise anything.” Marvel would cover my expenses, so I sent in my reluctant reply. It went something like this:

Dear Marvel Secret-ary Agent,
Hell yeah.

The meeting was in three days time. I would finish my visit to Queens University in Charlotte, NC and take the next flight out to Atlanta. But first, some fun and games were in order. As part of my visit to Project Scientist’s camp, I was invited to teach quantum mechanics to a group of forty young women, ages 11-14, all of whom were interested in science, engineering and mathematics and many of whom did not have the financial means to pursue these interests outside of the classroom. So, I went to Queens University with several copies of MinecraftEDU, the educational component to one of the most popular video games of all time: Minecraft. As I described in “Can a game teach kids quantum mechanics”, I spent the summer of 2013 designing qCraft, a modification (mod) to Minecraft that allows players to craft blocks imbued with quantum superpowers such as quantum superposition and quantum entanglement. The mod, developed in collaboration with Google and TeacherGaming, became really popular, amassing millions of downloads around the world. But it is one thing to look at statistics as a measure of success and another to look into the eyes of young women who have lost themselves in a game (qCraft is free to download and comes with an accompanying curriculum) designed to teach them such heady concepts that inspired Richard Feynman to quip: If you think you understand quantum theory, you don’t.

My visit to Charlotte was so wonderful that I nearly decided to cancel my trip to Atlanta in order to stay with the girls and their mentors until the end of the week. But Mr. Rudd deserved the very best science could offer in making-quantum-stuff-up, so I boarded my flight and resolved to bring Project Scientist to Caltech the next summer. On my way to Atlanta, I used the in-flight WiFi to do some research on Ant-Man. He was part of the original Avengers, a founding member, in fact (nice!) His name was Dr. Hank Pym. He had developed a particle, aptly named after himself, which allowed him to shrink the space between atoms (well now…) He embedded vials of that particle in a suit that allowed him to shrink to the size of an ant (of course, that makes sense.) In short, he was a mad, mad scientist. And I was called in to help his successor, Scott Lang (Paul Rudd’s character), navigate his way through quantum land. Holy guacamole Ant-man! How does one shrink the space between atoms? As James Kakalios, author of The Physics of Superheroes, puts it in a recent article on Nate Silver’s FiveThirtyEight:

We’re made of atoms, and the neighboring atoms are all touching each other. One method of changing your size that’s out: Just squeeze the atoms closer together.

So the other option: What determines the size of atoms anyway?

We can calculate this with quantum mechanics, and it turns out to be the ratio of fundamental constants: Planck’s constant and the mass of an electron and the charge of the electron and this and that. The thing that all these constants have in common is that they’re constant. They don’t change.

Wonderful. Just peachy. How am I supposed to come up with a way that will allow Ant-Man to shrink to the size of an ant, if one of the top experts in movie science magic thinks that our best bet is to somehow change fundamental constants of nature?

The shrinking

Naturally, I could not, umm, find the time last summer to read last week’s article during my flight (time travel issues), so like any young Greek of my generation who still hopes that our national debt will just go poof, I attacked the problem of shrinking someone’s volume without shrinking their mass con pasión. The answer was far from obvious… but, it was possible. If one could convert electrons into muons, the atomic radius would shrink 200 times, shrinking a human to the size of an ant without changing any of the chemical properties of the atoms (muons have the same charge as the electrons, but are 200 times heavier). The problem then was the lifetime of the muonic atoms. Muons decay into electrons in about 2 millionths of a second, on average. That is indeed a problem. Could we somehow extend the half-life of a muon about a million times? Yes, if the muon has relativistic mass close to 20 TeV (near the current energy of the Large Hadron Collider in Geneva), the effect of Einstein’s relativistic time-dilation (which is how actual high-energy muons from cosmic radiation have time to reach our detectors before decaying) would allow our hero to shrink for a few seconds at a time with high probability. To shrink beyond that, or for longer periods of time, would require knowledge of physics beyond the standard model. Which is precisely what the Mu2e experiment at Fermilab is looking at right now. It’s like Christmas for Ant-Man fans!

So the famed Pym particle is a neutral particle that adds mass to an electron, converting it to a high-energy muon… When did I become a particle physicist? Oh well, fake it until you make it. Oh, hey, they are bringing pretzels! I love these little pretzel bites!

Enter Pinewood Studios

The flight was longer than I expected, which gave me time to think. A limo was waiting for me at the airport; I was to be taken directly to Pinewood Studios, luggage in hand and all. Once I was at my destination, I was escorted to the 3rd floor of a nondescript building, accessible only through special authorization (nice touch, Marvel). I was shown to what seemed like the main conference room, an open area with a large conference table. I expected that I would have to wait an hour before the assistant (to the) general manager showed up, so I started fiddling with my phone, making myself look busy and important. The next time I looked up, Paul Rudd was tapping my shoulder, dressed in sweats after what seemed like a training session for The 300. I am not sure what happened next, but Paul and I were in deep conversation about… qCraft? Someone must have told him that I was involved with designing the quantum mod for Minecraft and suddenly our roles were reversed. His son was a huge Minecraft fan and I was the celebrity in this boy’s eyes, and by parental transitivity, an associative, but highly non-commutative group action, in his dad’s eyes. I promised Paul that I would teach him how to install mods in Minecraft so his kids could enjoy qCraft and teach him about quantum entanglement when I wasn’t around. To my delight, I found myself listening to Mr. Rudd talk about his son’s sense of humor and his daughter’s intelligence with such pride, that I forgot for a moment the reason I was there; instead, it felt like I was catching up with an old friend and all we could talk about was our kids (I don’t have any, so I mostly listened).

The Meeting

Within five minutes, the director (Peyton Reed), the writers, producers, VFX specialists, computer playback experts (I became friends with their supervisor, Mr. Matthew Morrissey, who went to great lengths to represent the atoms on-screen as clouds of probability, in the s, p, d, f orbital configurations you see flashing in quantum superposition behind Hank Pym at his lab) and everyone else with an interest in quantum mechanics was in the room. I sat at the head of the long table with Paul next to me. He asked most of the questions along with the director, but at the time I didn’t know Paul was involved with writing the script. We discussed a lot of things, but what got everyone excited was the idea that the laws of physics as we know them may break down as we delve deeper and deeper into the quantum realm. You see, all of the other superheroes, no matter how strong and super, had powers that conformed to the laws of physics (stretching them from time to time, but never breaking them). But if someone could go to a place where the laws of physics as we know them were not yet formed, at a place where the arrow of time was broken and the fabric of space was not yet woven, the powers of such a master of the quantum realm would only be constrained by their ability to come back to the same (or similar) reality from which they departed. All the superheroes of Marvel and DC Comics combined would stand no chance against Ant-Man with a malfunctioning regulator…

The Quantum Realm

The birth of the term itself is an interesting story. Brad Winderbaum, co-producer for the movie, emailed me a couple of weeks after the meeting with the following request: Could I come up with a term describing Ant-Man going to the “microverse”? The term “microverse” carried legal baggage, so something fresh was needed. I offered “going nano”, “going quantum”, “going atomic”, or… “quantum realm”. I didn’t know how small the “microverse” scale was supposed to be in a writer’s mind (in a physicist’s mind it is exactly $10^{-6}$ meters – one thousandth of a millimeter), hence the many options. The reply was quick:

Thanks Spiros! Quantum Realm is a pretty great term.

Et voilà. Ant-Man was going to the quantum realm, a place where time and space dissolve and the only thing strong enough to anchor Scott Lang to reality is… You have to watch the movie to see what that is – it was an off-the-cuff remark I made at the meeting… At the end of the meeting, Paul, Peyton and the others thanked me and asked me if I could fly to San Francisco the next week for the first week of shooting. There, I would have met Michael Douglas and Evangeline Lilly, but I declined the generous offer. It was the week of Innoworks Academy at Caltech, an award-winning summer camp for middle school kids on the free/reduced lunch program. As the camp’s adviser, I resolved to never miss a camp as long as I was in the country and San Francisco is in the same state as Caltech. My mom would be proud of my decision (I hope), though an autograph from Mr. Douglas would have fetched me a really good hug.

The Movie

I just watched the movie yesterday (it is actually good!) and the feeling was surreal. Because I had no idea what to expect. Because I never thought that the people in that room would take what I was saying seriously enough to use it in the movie. I never got a copy of the script and during the official premiere three weeks ago, I was delivering a lecture on the future of quantum computing in a monastery in Madrid, Spain. When I found out that Kevin Feige, president of Marvel Studios, said this at a recent interview, my heart skipped several beats:

But the truth is, there is so much in Ant-Man: introducing a new hero, introducing a very important part of technology in the Marvel universe, the Pym particles. Ant-Man getting on the Avengers’ radar in this film and even – this is the weirdest part, you shouldn’t really talk about it because it won’t be apparent for years – but the whole notion of the quantum realm and the whole notion of going to places that are so out there, they are almost mind-bendingly hard to fathom. It all plays into Phase Three.

The third phase of the Marvel Cinematic Universe is about to go quantum and all I can think of is: I better start reading comic books again. But first, I have to teach a bunch of 11-14 year-old girls quantum physics through Minecraft. It is, after all, the final week of Project Scientist here at Caltech this summer and the theme is coding. With quantum computers at the near horizon, these young women need to learn how to program Asimov’s laws of quantum robotics into our benevolent quantum A.I. overlords. These young women are humanity’s new hope…

# Of Supersoakers and squeezed states

“BBs,” the lecturer said. I was sitting in the center of my row of seats, the two yards between me and the whiteboard empty. But I fancied I hadn’t heard correctly. “You know, like in BB guns?”

I had heard correctly. I nodded.

“Did you play with BB guns when you were a kid?”

I nodded again.

“I had BB guns,” the lecturer ruminated. “I had to defend myself from my brothers.”

I nodded more vigorously. My brother and I love each other, but we’ve crossed toy pistols.

Photons are like BBs, like bullets.”

Light, the lecturer continued, behaves like BBs under certain conditions. Under other conditions, light behaves differently. Different behaviors correspond to different species of light. Some species, we can approximate with classical (nonquantum*) physics. Some species, we can’t.

Kids begged less for BB guns, in my experience, than for water guns. I grew up in Florida, where swimming season stretches from April till September. To reload a BB gun, you have to fetch spent BBs. But, toting a Supersoaker, you swim in ammunition.

Water guns brought to mind water waves, which resemble a species of classical light. If BBs resemble photons, I mused, what about Supersoaker sprays? Water balloons?

I resolved to draw as many parallels as I could between species of light and childhood weapons.

Under scrutiny, the Supersoaker analogy held little water (sorry). A Supersoaker releases water in a stream, rather than in a coherent wave. By coherent, I mean that the wave has a well-defined wavelength: The distance from the first crest to the second equals the distance from the second to the third, and so on. I can’t even identify crests in the Supersoaker photo below.

Coherent waves vs. Supersoaker not-really-waves

Maybe Supersoaker sprays resemble incoherent light? Incoherent light is a mixture of waves of all different wavelengths. Classical physics approximates incoherent light, examples of which include sunlight. If you tease apart sunlight into coherent components, you’ll find waves with short wavelengths (such as ultraviolet rays), waves with medium (such as light we can see), and waves with long (such as microwaves). You can’t ascribe just one wavelength to incoherent light, just as I seemed unable to ascribe a wavelength to Supersoaker sprays.

But Supersoaker sprays differ from incoherent light in other respects. I’d expect triggers, for instance, to introduce nonlinearity into the spray’s dynamics. Readers who know more than I about fluid mechanics can correct me.

Though far-reaching and forceful, Supersoakers weigh down combatants and are difficult to hide. If you need ammunition small enough for a sneak attack, I recommend water balloons. Water balloons resemble squeezed states, which form a quantum class of light related to the Uncertainty Principle.

Werner Heisenberg proposed that, the more you know about a quantum particle’s position, the less you can know about its momentum, and vice versa. Let’s represent your uncertainty about the position by Δx and your uncertainty about the momentum by Δp. The product of these uncertainties can’t dip below some number, represented by ћ/2:

$\Delta_x \Delta_p \geq \frac{\hbar}{2}$.

Neither uncertainty, for example, can equal zero. Heisenberg’s proposal has evolved into more rigorous, more general forms. But the story remains familiar: The lesser the “spread in the possible values” of some property (like position), the greater the “spread in the possible values” of another property (like momentum).

Imagine plotting the possible positions along a graph’s horizontal axis and the possible momenta along the vertical. The points that could characterize our quantum system form a blob of area ћ/2. Doesn’t the blob resemble a water balloon?

Imagine squeezing a water balloon along one direction. The balloon bulges out along another. Now, imagine squeezing most of the quantum uncertainty along one direction in the diagram. You’ve depicted a squeezed state.

Depiction of a squeezed state

Not all childhood weapons contain water or BBs, and not all states of light contain photons.** A vacuum is a state that consists of zero photons. Classical physics suggests that the vacuum is empty and lacks energy. A sliver of energy, called zero-point energy, pervades each quantum vacuum. The Uncertainty Principle offers one reason why.

The vacuum reminds me of the silent treatment. Silence sounds empty, but it can harbor malevolence as quantum vacua harbor energy. Middle-school outcasts beware zero-point malice.

Retreating up Memory Lane, I ran out of analogies between classes of light and childhood weapons. Children play with lasers (with laser pointers and laser-tag guns), and lasers emit (approximately) coherent light. But laser light’s resemblance to laser light doesn’t count as an analogy. The class of incoherent light includes thermal states. (Non-experts, I’m about to spew jargon. If you have the energy, I recommend Googling the italicized terms. If you haven’t, feel free to skip to the next paragraph.) Physicists model much of the natural world with thermal states. To whichever readers identify childhood weapons that resemble them, I offer ten points. I offer 20 for mimicry of solitons or solitary waves, and 25 for that of parametric down-conversion or photon antibunching.

But if sunshine and Supersoakers lure you away from your laptop, I can’t object. Happy summer.

With thanks to Bassam Helou for corrections and discussions.

*Pardon my simplifying inaccuracy. Some nonquantum physics is nonclassical.

**More precisely, not all Fock states correspond to particle numbers $n > 0$. Alternatively: Not all states of light correspond to positive expectation values $\langle \hat{n} \rangle > 0$ of the particle-number operator $\hat{n}$.

# Holography and the MERA

The AdS/MERA correspondence has been making the rounds of the blogosphere with nice posts by Scott Aaronson and Sean Carroll, so let’s take a look at the topic here at Quantum Frontiers.

The question of how to formulate a quantum theory of gravity is a long-standing open problem in theoretical physics. Somewhat recently, an idea that has gained a lot of traction (and that Spiros has blogged about before) is emergence. This is the idea that space and time may emerge from some more fine-grained quantum objects and their interactions. If we could understand how classical spacetime emerges from an underlying quantum system, then it’s not too much of a stretch to hope that this understanding would give us insight into the full quantum nature of spacetime.

One type of emergence is exhibited in holography, which is the idea that certain (D+1)-dimensional systems with gravity are exactly equivalent to D-dimensional quantum theories without gravity. (Note that we’re calling time a dimension here. For example, you would say that on a day-to-day basis we experience D = 4 dimensions.) In this case, that extra +1 dimension and the concomitant gravitational dynamics are emergent phenomena.

A nice aspect of holography is that it is explicitly realized by the AdS/CFT correspondence. This correspondence proposes that a particular class of spacetimes—ones that asymptotically look like anti-de Sitter space, or AdS—are equivalent to states of a particular type of quantum system—a conformal field theory, or CFT. A convenient visualization is to draw the AdS spacetime as a cylinder, where time marches forward as you move up the cylinder and different slices of the cylinder correspond to snapshots of space at different instants of time. Conveniently, in this picture you can think of the corresponding CFT as living on the boundary of the cylinder, which, you should note, has one less dimension than the “bulk” inside the cylinder.

Even within this nice picture of holography that we get from the AdS/CFT correspondence, there is a question of how exactly do CFT, or boundary quantities map onto quantities in the AdS bulk. This is where a certain tool from quantum information theory called tensor networks has recently shown a lot of promise.

A tensor network is a way to efficiently represent certain states of a quantum system. Moreover, they have nice graphical representations which look something like this:

Beni discussed one type of tensor network in his post on holographic codes. In this post, let’s discuss the tensor network shown above, which is known as the Multiscale Entanglement Renormalization Ansatz, or MERA.

The MERA was initially developed by Guifre Vidal and Glen Evenbly as an efficient approximation to the ground state of a CFT. Roughly speaking, in the picture of a MERA above, one starts with a simple state at the centre, and as you move outward through the network, the MERA tells you how to build up a CFT state which lives on the legs at the boundary. The MERA caught the eye of Brian Swingle, who noticed that it looks an awfully lot like a discretization of a slice of the AdS cylinder shown above. As such, it wasn’t a preposterously big leap to suggest a possible “AdS/MERA correspondence.” Namely, perhaps it’s more than a simple coincidence that a MERA both encodes a CFT state and resembles a slice of AdS. Perhaps the MERA gives us the tools that are required to construct a map between the boundary and the bulk!

So, how seriously should one take the possibility of an AdS/MERA correspondence? That’s the question that my colleagues and I addressed in a recent paper. Essentially, there are several properties that a consistent holographic theory should satisfy in both the bulk and the boundary. We asked whether these properties are still simultaneously satisfied in a correspondence where the bulk and boundary are related by a MERA.

What we found was that you invariably run into inconsistencies between bulk and boundary physics, at least in the simplest construals of what an AdS/MERA correspondence might be. This doesn’t mean that there is no hope for an AdS/MERA correspondence. Rather, it says that the simplest approach will not work. For a good correspondence, you would need to augment the MERA with some additional structure, or perhaps consider different tensor networks altogether. For instance, the holographic code features a tensor network which hints at a possible bulk/boundary correspondence, and the consistency conditions that we proposed are a good list of checks for Beni and company as they work out the extent to which the code can describe holographic CFTs. Indeed, a good way to summarize how our work fits into the picture of quantum gravity alongside holography and tensors networks is by saying that it’s nice to have good signposts on the road when you don’t have a map.

# Hello, my name is QUANTUM MASTER EQUATION

“Why does it have that name?”

“I don’t know.” Lecturers have shrugged. “It’s just a name.”

This spring, I asked about master equations. I thought of them as tools used in statistical mechanics, the study of vast numbers of particles. We can’t measure vast numbers of particles, so we can’t learn about stat-mech systems everything one might want to know. The magma beneath Santorini, for example, consists of about 1024 molecules. Good luck measuring every one.

Imagine, as another example, using a quantum computer to solve a problem. We load information by initializing the computer to a certain state: We orient the computer’s particles in certain directions. We run a program, then read out the output.

Suppose the computer sits on a tabletop, exposed to the air like leftover casserole no one wants to save for tomorrow. Air molecules bounce off the computer, becoming entangled with the hardware. This entanglement, or quantum correlation, alters the computer’s state, just as flies alter a casserole.* To understand the computer’s output—which depends on the state, which depends on the air—we must have a description of the air. But we can’t measure all those air molecules, just as we can’t measure all the molecules in Santorini’s magma.

We can package our knowledge about the computer’s state into a mathematical object, called a density operator, labeled by ρ(t). A quantum master equation describes how ρ(t) changes. I had no idea, till this spring, why we call master equations “master equations.” Had someone named “John Master” invented them? Had the inspiration for the Russell Crowe movie Master and Commander? Or the Igor who lisps, “Yeth, mathter” in adaptations of Frankenstein?

Jenia Mozgunov, a fellow student and Preskillite, proposed an answer: Using master equations, we can calculate how averages of observable properties change. Imagine describing a laser, a cavity that spews out light. A master equation reveals how the average number of photons (particles of light) in the cavity changes. We want to predict these averages because experimentalists measure them. Because master equations spawn many predictions—many equations—they merit the label “master.”

Jenia’s hypothesis appealed to me, but I wanted certainty. I wanted Truth. I opened my laptop and navigated to Facebook.

“Does anyone know,” I wrote in my status, “why master equations are called ‘master equations’?”

Ian Durham, a physicist at St. Anselm College, cited Tom Moore’s Six Ideas that Shaped Physics. Most physics problems, Ian wrote, involve “some overarching principle.” Example principles include energy conservation and invariance under discrete translations (the system looks the same after you step in some direction). A master equation encapsulates this principle.

Ian’s explanation sounded sensible. But fewer people “liked” his reply on Facebook than “liked” a quip by a college friend: Master equations deserve their name because “[t]hey didn’t complete all the requirements for the doctorate.”

My advisor, John Preskill, dug through two to three books, one set of lecture notes, one German Wikipedia page, one to two articles, and Google Scholar. He concluded that Nordsieck, Lamb, and Uhlenbeck coined “master equation.” According to a 1940 paper of theirs,** “When the probabilities of the elementary processes are known, one can write down a continuity equation for W [a set of probabilities], from which all other equations can be derived and which we will call therefore the ‘master’ equation.”

“Are you sure you were meant to be a physicist,” I asked John, “rather than a historian?”

“Procrastination is a powerful motivator,” he replied.

Lecturers have shrugged at questions about names. Then they’ve paused, pondered, and begun, “I guess because…” Theorems and identities derive their names from symmetries, proof techniques, geometric illustrations, and applications to problems I’d thought unrelated. A name taught me about uses for master equations. Names reveal physics I wouldn’t learn without asking about names. Names aren’t just names. They’re lamps and guides.

Pity about the origin of “master equation,” though. I wish an Igor had invented them.

*Apologies if I’ve spoiled your appetite.

**A. Nordsieck, W. E. Lamb, and G. E. Uhlenbeck, “On the theory of cosmic-ray showers I,” Physica 7, 344-60 (1940), p. 353.

# 20 years of qubits: the arXiv data

Editor’s Note: The preceding post on Quantum Frontiers inspired the always curious Paul Ginsparg to do some homework on usage of the word “qubit” in papers posted on the arXiv. Rather than paraphrase Paul’s observations I will quote his email verbatim, so you can experience its Ginspargian style.

fig has total # uses of qubit in arxiv (divided by 10) per month, and
total # docs per month:
an impressive 669394 total in 29587 docs.

the graph starts at 9412 (dec '94), but that is illusory since qubit
only shows up in v2 of hep-th/9412048, posted in 2004.
the actual first was quant-ph/9503016 by bennett/divicenzo/shor et al
(posted 23 Mar '95) where they carefully attribute the term to
schumacher ("PRA, to appear '95") and jozsa/schumacher ("J. Mod Optics
'94"), followed immediately by quant-ph/9503017 by deutsch/jozsa et al
(which no longer finds it necessary to attribute term)

[neither of schumacher's first two articles is on arxiv, but otherwise
probably have on arxiv near 100% coverage of its usage and growth, so
permits a viral epidemic analysis along the lines of kaiser's "drawing
theories apart"  of use of Feynman diagrams in post wwII period].

ever late to the party, the first use by j.preskill was
quant-ph/9602016, posted 21 Feb 1996

#articles by primary subject area as follows (hep-th is surprisingly
low given the firewall connection...):

quant-ph 22096
cond-mat.mes-hall 3350
cond-mat.supr-con 880
cond-mat.str-el 376
cond-mat.mtrl-sci 250
math-ph 244
hep-th 228
physics.atom-ph 224
cond-mat.stat-mech 213
cond-mat.other 200
physics.optics 177
cond-mat.quant-gas 152
physics.gen-ph 120
gr-qc 105
cond-mat 91
cs.CC 85
cs.IT 67
cond-mat.dis-nn 55
cs.LO 49
cs.CR 43
physics.chem-ph 33
cs.ET 25
physics.ins-det 21
math.CO,nlin.CD 20
physics.hist-ph,physics.bio-ph,math.OC 19
hep-ph 18
cond-mat.soft,cs.DS,math.OA 17
cs.NE,cs.PL,math.QA 13
cs.AR,cs.OH 12
physics.comp-ph 11
math.LO 10
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# Who named the qubit?

Perhaps because my 40th wedding anniversary is just a few weeks away, I have been thinking about anniversaries lately, which reminded me that we are celebrating the 20th anniversary of a number of milestones in quantum information science. In 1995 Cirac and Zoller proposed, and Wineland’s group first demonstrated, the ion trap quantum computer. Quantum error-correcting codes were invented by Shor and Steane, entanglement concentration and purification were described by Bennett et al., and there were many other fast-breaking developments. It was an exciting year.

But the event that moved me to write a blog post is the 1995 appearance of the word “qubit” in an American Physical Society journal. When I was a boy, two-level quantum systems were called “two-level quantum systems.” Which is a descriptive name, but a mouthful and far from euphonious. Think of all the time I’ve saved in the past 20 years by saying “qubit” instead of “two-level quantum system.” And saying “qubit” not only saves time, it also conveys the powerful insight that a quantum state encodes a novel type of information. (Alas, the spelling was bound to stir controversy, with the estimable David Mermin a passionate holdout for “qbit”. Give it up, David, you lost.)

Ben Schumacher. Thanks for the qubits, Ben!

For the word “qubit” we know whom to thank: Ben Schumacher. He introduced the word in his paper “Quantum Coding” which appeared in the April 1995 issue of Physical Review A. (History is complicated, and in this case the paper was actually submitted in 1993, which allowed another paper by Jozsa and Schumacher to be published earlier even though it was written and submitted later. But I’m celebrating the 20th anniversary of the qubit now, because otherwise how will I justify this blog post?)

In the acknowledgments of the paper, Ben provided some helpful background on the origin of the word:

The term “qubit” was coined in jest during one of the author’s many intriguing and valuable conversations with W. K. Wootters, and became the initial impetus for this work.

I met Ben (and other luminaries of quantum information theory) for the first time at a summer school in Torino, Italy in 1996. After reading his papers my expectations were high, all the more so after Sam Braunstein warned me that I would be impressed: “Ben’s a good talker,” Sam assured me. I was not disappointed.

(I also met Asher Peres at that Torino meeting. When I introduced myself Asher asked, “Isn’t there someone with a similar name in particle theory?” I had no choice but to come clean. I particularly remember that conversation because Asher told me his secret motivation for studying quantum entanglement: it might be important in quantum gravity!)

A few years later Ben spent his sabbatical year at Caltech, which gave me an opportunity to compose a poem for the introduction to Ben’s (characteristically brilliant) talk at our physics colloquium. This poem does homage to that famous 1995 paper in which Ben not only introduced the word “qubit” but also explained how to compress a quantum state to the minimal number of qubits from which the original state can be recovered with a negligible loss of fidelity, thus formulating and proving the quantum version of Shannon’s famous source coding theorem, and laying the foundation for many subsequent developments in quantum information theory.

Sometimes when I recite a poem I can sense the audience’s appreciation. But in this case there were only a few nervous titters. I was going for edgy but might have crossed the line into bizarre.. Since then I’ve (usually) tried to be more careful.

(For reading the poem, it helps to know that the quantum state appears to be random when it has been compressed as much as possible.)

On Quantum Compression (in honor of Ben Schumacher)

Ben.
He rocks.
I remember
When
He showed me how to fit
A qubit
In a small box.

I wonder how it feels
To be compressed.
And then to pass
A fidelity test.

Or does it feel
At all, and if it does
Would I squeal
Or be just as I was?

If not undone
I’d become as I’d begun
And write a memorandum
On being random.
Had it felt like a belt
Of rum?

And might it be predicted