Category Archives: Theoretical highlights

Highlighting theoretical progress at IQIM.

Monopoles passing through Flatland!

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Like many mathematically inclined teenagers, I was charmed when I first read the book Flatland by Edwin Abbott Abbott.* It’s a story about a Sphere who visits a two-dimensional world and tries to awaken its inhabitants to the existence of a third dimension. As perceived by Flatlanders, the Sphere is a circle which appears as a point, grows to maximum size, then shrinks and disappears.

My memories of Flatland were aroused as I read a delightful recent paper by Max Metlitski, Charlie Kane, and Matthew Fisher about magnetic monopoles and three-dimensional bosonic topological insulators. To explain why, I’ll need to recall a few elements of the theory of monopoles and of topological insulators, before returning to the connection between the two and why that reminds me of Flatland.

Flatlanders, confined to the surface of a topological insulator, are convinced by a magnetic monopole that there is a third dimension.

Flatlanders, confined to the two-dimensional surface of a topological insulator, are convinced by a magnetic monopole that a third dimension must exist.

Monopoles

Paul Dirac was no ordinary genius. Aside from formulating relativistic electron theory and predicting the existence of antimatter, Dirac launched the quantum theory of magnetic monopoles in a famous 1931 paper. Dirac envisioned a magnetic monopole as a semi-infinitely long, infinitesimally thin string of magnetic flux, such that the end of the string, where the flux spills out, seems to be a magnetic charge. For this picture to make sense, the string should be invisible. Dirac pointed out that an electron with electric charge e, transported around a string carrying flux \Phi, could detect the string (via what later came to be called the Aharonov-Bohm effect) unless the flux is an integer multiple of 2\pi\hbar /e, where \hbar is Planck’s constant. Conversely, in order for the string to be invisible, if a magnetic monopole exists with magnetic charge g_D = 2\pi\hbar /e, then all electric charges must be integer multiples of e. Thus the existence of magnetic monopoles (which have never been observed) could explain quantization of electric charge (which has been observed).

Captivated by the beauty of his own proposal, Dirac concluded his paper by remarking, “One would be surprised if Nature had made no use of it.”

Our understanding of quantized magnetic monopoles advanced again in 1979 when another extraordinary physicist, Edward Witten, discussed a generalization of Dirac’s quantization condition. Witten noted that the Lagrange density of electrodynamics could contain a term of the form

\frac{\theta e^2\hbar}{4\pi^2}~\vec{E}\cdot\vec{B},

where \vec{E} is the electric field and \vec{B} is the magnetic field. This “\theta term” may also be expressed as

\frac{\theta e^2\hbar}{8\pi^2}~ \partial^\mu\left(\epsilon_{\mu\nu\lambda\sigma}A^\nu\partial^\lambda A^\sigma \right),

where A is the vector potential, and hence is a total derivative which makes no contribution to the classical field equations of electrodynamics. But Witten realized that it can have important consequences for the quantum properties of magnetic monopoles. Specifically, the \theta term modifies the field momentum conjugate to the vector potential, which becomes

\vec{E}+\frac{\theta e^2\hbar}{4\pi^2}\vec{B}.

Because the Gauss law condition satisfied by physical quantum states is altered, for a monopole with magnetic charge m g_D , where g_D is Dirac’s minimal charge 2\pi\hbar /e and m is an integer, the allowed values of the electric charge become

q = e\left( n - \frac{\theta m}{2\pi}\right),

where n is an integer. This spectrum of allowed charges remains invariant if \theta advances by 2\pi, suggesting that the parameter \theta is actually an angular variable with period 2\pi. This periodicity of \theta can be readily verified in a theory admitting fermions with the minimal charge e. But if the charged particles are bosons then \theta turns out to be a periodic variable with period 4\pi instead.

That \theta has a different period for a bosonic theory than a fermionic one has an interesting interpretation. As Goldhaber noticed in 1976, dyons carrying both magnetic and electric charge can exhibit statistical transmutation. That is, in a purely bosonic theory, a dyon with magnetic charge g_D= 2\pi\hbar/e and electric charge ne is a fermion if n is an odd integer — when two dyons are exchanged, transport of each dyon’s electric charge in the magnetic field of the other dyon induces a sign change in the wave function. In a fermionic theory the story is different; now we can think of the dyon as a fermionic electric charge bound to a bosonic monopole. There are two canceling contributions to the exchange phase of the dyon, which is therefore a boson for any integer value of n, whether even or odd.

As \theta smoothly increases from 0 to 2\pi, the statistics (whether bosonic or fermionic) of a dyon remains fixed even as the dyon’s electric charge increases by e. For the bosonic theory with \theta = 2\pi, then, dyons with magnetic charge g_D and electric charge ne are bosons for n odd and fermions for n even, the opposite of what happens when \theta=0. For the bosonic theory, unlike the fermionic theory, we need to increase \theta by 4\pi for the physics of dyons to be fully invariant.

In 1979 Ed Witten was a postdoc at Harvard, where I was a student, though he was visiting CERN for the summer when he wrote his paper about the \theta-dependent monopole charge. I always read Ed’s papers carefully, but I gave special scrutiny to this one because magnetic monopoles were a pet interest of mine. At the time, I wondered whether the Witten effect might clarify how to realize the \theta parameter in a lattice gauge theory. But it certainly did not occur to me that the \theta-dependent electric charge of a magnetic monopole could have important implications for quantum condensed matter physics. Theoretical breakthroughs often have unexpected consequences, which may take decades to emerge.

Symmetry-protected topological phases

Okay, now let’s talk about topological insulators, a very hot topic in condensed matter physics these days. Actually, a topological insulator is a particular instance of a more general concept called a symmetry-protected topological phase of matter (or SPT phase). Consider a d-dimensional hunk of material with a (d-1)-dimensional boundary. If the material is in an SPT phase, then the physics of the d-dimensional bulk is boring — it’s just an insulator with an energy gap, admitting no low-energy propagating excitations. But the physics of the (d-1)-dimensional edge is exotic and exciting — for example the edge might support “gapless” excitations of arbitrarily low energy which can conduct electricity. The exotica exhibited by the edge is a consequence of a symmetry, and is destroyed if the symmetry is broken either explicitly or spontaneously; that is why we say the phase is “symmetry protected.”

The low-energy edge excitations can be described by a (d-1)-dimensional effective field theory. But for a typical SPT phase, this effective field theory is what we call anomalous, which means that for one reason or another the theory does not really make sense. The anomaly tells us something interesting and important, namely that the (d-1)-dimensional theory cannot be really, truly (d-1) dimensional; it can arise only at the edge of a higher-dimensional system.

This phenomenon, in which the edge does not make sense by itself without the bulk, is nicely illustrated by the integer quantum Hall effect, which occurs in a two-dimensional electron system in a high magnetic field and at low temperature, if the sample is sufficiently clean so that the electrons are highly mobile and rarely scattered by impurities. In this case the relevant symmetry is electron number, or equivalently the electric charge. At the one-dimensional edge of a two-dimensional quantum Hall sample, charge carriers move in only one direction — to the right, say, but not to the left. A theory with such chiral electric charges does not really make sense. One problem is that electric charge is not conserved — an electric field along the edge causes charge to be locally created, which makes the theory inconsistent.

The way the theory resolves this conundrum is quite remarkable. A two-dimensional strip of quantum Hall fluid has two edges, one at the top, the other at the bottom. While the top edge has only right-moving excitations, the bottom edge has only left-moving excitations. When electric charge appears on the top edge, it is simultaneously removed from the bottom edge. Rather miraculously, charge can be conveyed across the bulk from one edge to the other, even though the bulk does not have any low-energy excitations at all.

I first learned about this interplay of edge and bulk physics from a beautiful 1985 paper by Curt Callan and Jeff Harvey. They explained very lucidly how an edge theory with an anomaly and a bulk theory with an anomaly can fit together, with each solving the other’s problems. Curiously, the authors did not mention any connection with the quantum Hall effect, which had been discovered five years earlier, and I didn’t appreciate the connection myself until years later.

Topological insulators

In the case of topological insulators, the symmetries which protect the gapless edge excitations are time-reversal invariance and conserved particle number, i.e. U(1) symmetry. Though the particle number might not be coupled to an electromagnetic gauge field, it is instructive for the purpose of understanding the properties of the symmetry-protected phase to imagine that the U(1) symmetry is gauged, and then to consider the potential anomalies that could afflict this gauge symmetry. The first topological insulators conceived by theorists were envisioned as systems of non-interacting electrons whose properties were relatively easy to understand using band theory. But it was not so clear at first how interactions among the electrons might alter their exotic behavior. The wonderful thing about anomalies is that they are robust with respect to interactions. In many cases we can infer the features of anomalies by studying a theory of non-interacting particles, assured that these features survive even when the particles interact.

As have many previous authors, Metlitski et al. argue that when we couple the conserved particle number to a U(1) gauge field, the effective theory describing the bulk physics of a topological insulator in three dimensions may contain a \theta term. But wait … since the electric field is even under time reversal and the magnetic field is odd, the \theta term is T-odd; under T, \theta is mapped to -\theta, so T seems to be violated if \theta has any nonzero value. Except … we have to remember that \theta is really a periodic variable. For a fermionic topological insulator the period is 2\pi; therefore the theory with \theta = \pi is time reversal invariant; \theta = \pi maps to \theta = -\pi under T, which is equivalent to a rotation of \theta by 2\pi. For a bosonic topological insulator the period is 4\pi, which means that \theta = 2\pi is the nontrivial T-invariant value.

If we say that a “trivial” insulator (e.g., the vacuum) has \theta = 0, then we may say that a bulk material with \theta = \pi (fermionic case) or \theta = 2\pi (bosonic case) is a “nontrivial” (a.k.a. topological) insulator. At the edge of the sample, where bulk material meets vacuum, \theta must rotate suddenly by \pi (fermions) or by 2\pi (bosons). The exotic edge physics is a consequence of this abrupt change in \theta.

Monopoles in Flatland

To understand the edge physics, and in particular to grasp how fermionic and bosonic topological insulators differ, Metlitski et al. invite us to imagine a magnetic monopole with magnetic charge g_D passing through the boundary between the bulk and the surrounding vacuum. To the Flatlanders confined to the surface of the bulk sample, the passing monopole induces a sudden change in the magnetic flux through the surface by a single flux quantum g_D, which could arise due to a quantum tunneling event. What does the Flatlander see?

In a fermionic topological insulator, there is a monopole that carries charge e/2 when inside the sample (where \theta=-\pi) and charge 0 when outside (where \theta=0). Since electric charge is surely conserved in the full three-dimensional theory, the change in the monopole’s charge must be compensated by a corresponding change in the charge residing on the surface. Flatlanders are puzzled to witness a spontaneously arising excitation with charge e/2. This is an anomaly — electric charge conservation is violated, which can only make sense if Flatlanders are confined to a surface in a higher-dimensional world. Though unable to escape their surface world, the Flatlanders can be convinced by the Monopole that an extra dimension must exist.

In a bosonic topological insulator, the story is somewhat different: there is a monopole that carries electric charge 0 when inside the sample (where \theta=-2\pi) and charge –e when outside (where \theta=0). In this case, though, there are bosonic charge-e particles living on the surface. A monopole can pick up a charged particle as it passes through Flatland, so that its charge is 0 both inside the bulk sample and outside in the vacuum. Flatlanders are happy — electric charge is conserved!

But hold on … there’s still something wrong. Inside the bulk (where \theta= -2\pi) a monopole with electric charge 0 is a fermion, while outside in the vacuum (where \theta = 0) it is a boson. In the three-dimensional theory it is not possible for any local process to create an isolated fermion, so if the fermionic monopole becomes a bosonic monople as it passes through Flatland, it must leave a fermion behind. Flatlanders are puzzled to witness a spontaneously arising fermion. This is an anomaly — conservation of fermionic parity is violated, which can only make sense if Flatlanders are confined to a surface in a higher-dimensional world. Once again, the clever residents of Flatland learn from the Monopole about an extra spatial dimension, without ever venturing outside their two-dimensional home.

Topological order gets edgy

This post is already pretty long and I should wrap it up. Before concluding I’ll remark that the theory of symmetry-protected phases has been developing rapidly in recent months.

In particular, a new idea, introduced last fall by Vishwanath and Senthil, has been attracting increasing attention. While in most previously studied SPT phases the unbroken symmetry protects gapless excitations confined to the edge of the sample, Vishwanath and Senthil pointed out another possibility — a gapped edge exhibiting topological order. The surface can support anyons with exotic braiding statistics.

Here, too, anomalies are central to the discussion. While anyons in two-dimensional media are already a much-studied subject, the anyon models that can be realized at the edges of three-dimensional SPT phases are different than anyon models realized in really, truly two-dimensional systems. What’s new are not the braiding properties of the anyons, but rather how the anyons transform under the symmetry. Flatlanders who study the symmetry realization in their gapped two-dimensional world should be able to infer the existence of the three-dimensional bulk.

The pace of discovery picked up this month when four papers appeared simultaneously on the preprint arXiv, by Metlitski-Kane-Fisher, Chen-Fidkowski-Vishwanath, Bonderson-Nayak-Qi, and Wang-Potter-Senthil, all proposing and analyzing models of SPT phases with gapped edges. It remains to be seen, though, whether this physics will be realized in actual materials.

Are we on the edge?

In Flatland, our two-dimensional friend, finally able to perceive the third dimension thanks to the Sphere’s insistent tutelage, begs to enter a world of still higher dimensions, “where thine own intestines, and those of kindred Spheres, will lie exposed to … view.” The Sphere is baffled by the Flatlander’s request, protesting, “There is no such land. The very idea of it is utterly inconceivable.”

Let’s not be so dogmatic as the Sphere. The lessons learned from the quantum Hall effect and the topological insulator have prepared us to take the next step, envisioning our own three-dimensional world as the edge of a higher-dimensional bulk system. The existence of an unseen bulk may be inferred in the future by us edgelings, if experimental explorations of our three-dimensional effective theory reveal anomalies begging for an explanation.

Perhaps we are on the edge … of a great discovery. At least it’s conceivable.

*Disclaimer: The gender politics of Flatland, to put it mildly, is outdated and offensive. I don’t wish to endorse the idea that women are one dimensional! I included the reference to Flatland because the imagery of two-dimensional beings struggling to imagine the third dimension is a perfect fit to the scientific content of this post.

This single-shot life

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The night before defending my Masters thesis, I ran out of shampoo. I ran out late enough that I wouldn’t defend from beneath a mop like Jack Sparrow’s; but, belonging to the Luxuriant Flowing-Hair Club for Scientists (technically, if not officially), I’d have to visit Shopper’s Drug Mart.

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The author’s unofficially Luxuriant Flowing Scientist Hair

Before visiting Shopper’s Drug Mart, I had to defend my thesis. The thesis, as explained elsewhere, concerns epsilons, the mathematical equivalents of seed pearls. The thesis also concerns single-shot information theory.

Ordinary information theory emerged in 1948, midwifed by American engineer Claude E. Shannon. Shannon calculated how efficiently we can pack information into symbols when encoding long messages. Consider encoding this article in the fewest possible symbols. Because “the” appears many times, you might represent “the” by one symbol. Longer strings of symbols suit misfits like “luxuriant” and “oobleck.” The longer the article, the fewer encoding symbols you need per encoded word. The encoding-to-encoded ratio decreases, toward a number called the Shannon entropy, as the message grows infinitely long.

Claude Shannon

We don’t send infinitely long messages, excepting teenagers during phone conversations. How efficiently can we encode just one article or sentence? The answer involves single-shot information theory, or—to those stuffing long messages into the shortest possible emails to busy colleagues—“1-shot info.” Pioneered within the past few years, single-shot theory concerns short messages and single trials, the Twitter to Shannon’s epic. Like articles, quantum states can form messages. Hence single-shot theory blended with quantum information in my thesis.

Continue reading →

Entanglement = Wormholes

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One of the most enjoyable and inspiring physics papers I have read in recent years is this one by Mark Van Raamsdonk. Building on earlier observations by Maldacena and by Ryu and Takayanagi. Van Raamsdonk proposed that quantum entanglement is the fundamental ingredient underlying spacetime geometry.* Since my first encounter with this provocative paper, I have often mused that it might be a Good Thing for someone to take Van Raamsdonk’s idea really seriously.

Now someone has.

I love wormholes. (Who doesn’t?) Picture two balls, one here on earth, the other in the Andromeda galaxy. It’s a long trip from one ball to the other on the background space, but there’s a shortcut:You can walk into the ball on earth and moments later walk out of the ball in Andromeda. That’s a wormhole.

I’ve mentioned before that John Wheeler was one of my heros during my formative years. Back in the 1950s, Wheeler held a passionate belief that “everything is geometry,” and one particularly intriguing idea he called “charge without charge.” There are no pointlike electric charges, Wheeler proclaimed; rather, electric field lines can thread the mouth of a wormhole. What looks to you like an electron is actually a tiny wormhole mouth. If you were small enough, you could dive inside the electron and emerge from a positron far away. In my undergraduate daydreams, I wished this idea could be true.

But later I found out more about wormholes, and learned about “topological censorship.” It turns out that if energy is nonnegative, Einstein’s gravitational field equations prevent you from traversing a wormhole — the throat always pinches off (or becomes infinitely long) before you get to the other side. It has sometimes been suggested that quantum effects might help to hold the throat open (which sounds like a good idea for a movie), but today we’ll assume that wormholes are never traversable no matter what you do.

Alice and Bob are in different galaxies, but each lives near a black hole, and their black holes are connected by a wormhole.

Love in a wormhole throat: Alice and Bob are in different galaxies, but each lives near a black hole, and their black holes are connected by a wormhole. If both jump into their black holes, they can enjoy each other’s company for a while before meeting a tragic end.

Are wormholes any fun if we can never traverse them? The answer might be yes if two black holes are connected by a wormhole. Then Alice on earth and Bob in Andromeda can get together quickly if each jumps into a nearby black hole. For solar mass black holes Alice and Bob will have only 10 microseconds to get acquainted before meeting their doom at the singularity. But if the black holes are big enough, Alice and Bob might have a fulfilling relationship before their tragic end.

This observation is exploited in a recent paper by Juan Maldacena and Lenny Susskind (MS) in which they reconsider the AMPS puzzle (named for Almheiri, Marolf, Polchinski, and Sully). I wrote about this puzzle before, so I won’t go through the whole story again. Here’s the short version: while classical correlations can easily be shared by many parties, quantum correlations are harder to share. If Bob is highly entangled with Alice, that limits his ability to entangle with Carrie, and if he entangles with Carrie instead he can’t entangle with Alice. Hence we say that entanglement is “monogamous.” Now, if, as most of us are inclined to believe, information is “scrambled” but not destroyed by an evaporating black hole, then the radiation emitted by an old black hole today should be highly entangled with radiation emitted a long time ago. And if, as most of us are inclined to believe, nothing unusual happens (at least not right away) to an observer who crosses the event horizon of a black hole, then the radiation emitted today should be highly entangled with stuff that is still inside the black hole. But we can’t have it both ways without violating the monogamy of entanglement!

The AMPS puzzle invites audacious reponses, and AMPS were suitably audacious. They proposed that an old black hole has no interior — a freely falling observer meets her doom right at the horizon rather than at a singularity deep inside.

MS are also audacious, but in a different way. They helpfully summarize their key point succinctly in a simple equation:

ER = EPR

Here, EPR means Einstein-Podolsky-Rosen, whose famous paper highlighted the weirdness of quantum correlations, while ER means Einstein-Rosen (sorry, Podolsky), who discovered wormhole solutions to the Einstein equations. (Both papers were published in 1935.) MS (taking Van Raamsdonk very seriously) propose that whenever any two quantum subsystems are entangled they are connected by a wormhole. In many cases, these wormholes are highly quantum mechanical, but in some cases (where the quantum system under consideration has a weakly coupled “gravitational dual”), the wormhole can have a smooth geometry like the one ER described. That wormholes are not traversable is important for the consistency of ER = EPR: just as Alice cannot use their shared entanglement to send a message to Bob instantaneously, so she is unable to send Bob a message through their shared wormhole.

AMPS imagined that Alice could distill qubit C from the black hole’s early radiation and carry it back to the black hole, successfully verifying its entanglement with another qubit B distilled from the recent radiation. Monogamy then ensures that qubit B cannot be entangled with qubit A behind the horizon. Hence when Alice falls through the horizon she will not observe the quiescent vacuum state in which A and B are entangled; instead she encounters a high-energy particle. MS agree with this conclusion.

AMPS go on to say that Alice’s actions before entering the black hole could not have created that energetic particle; it must have been there all along, one of many such particles constituting a seething firewall.

Here MS disagree. They argue that the excitation encountered by Alice as she crosses the horizon was actually created by Alice herself when she interacted with qubit C. How could Alice’s actions, executed far, far away from the black hole, dramatically affect the state of the black hole’s interior? Because C and A are connected by a wormhole!

The ER = EPR conjecture seems to allow us to view the early radiation with which the black hole is entangled as a complementary description of the black hole interior. It’s not clear yet whether this picture works in detail, and even if it does there could still be firewalls; maybe in some sense the early radiation is connected to the black hole via a wormhole, yet this wormhole is wildly fluctuating rather than a smooth geometry. Still, MS provide a promising new perspective on a deep problem.

As physicists we often rely on our sense of smell in judging scientific ideas, and earlier proposed resolutions of the AMPS puzzle (like firewalls) did not smell right. At first whiff, ER = EPR may smell fresh and sweet, but it will have to ripen on the shelf for a while. If this idea is on the right track, there should be much more to say about it. For now, wormhole lovers can relish the possibilities.

Eventually, Wheeler discarded “everything is geometry” in favor of an ostensibly deeper idea: “everything is information.” It would be a fitting vindication of Wheeler’s vision if everything in the universe, including wormholes, is made of quantum correlations.

*Update: Commenter JM reminded me to mention Brian Swingle’s beautiful 2009 paper, which preceded Van Raamsdonk’s and proposed a far-reaching connection between quantum entanglement and spacetime geometry.

“Nature, you instruct me.”

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“Settle thy studies.”

Alone in his workroom, a student contemplates his future. Piles of books teeter next to him. Boxes line the walls; and glass vials, the boxes. Sunbeams that struggle through the stained-glass window illuminate dust.

The student’s name is Faust. I met him during my last winter in college, while complementing Physics 42: Introductory Quantum Mechanics with German 44: The Faust Tradition. A medieval German alchemist, Faust has inspired plays, novels, operas, the short story “The Devil and Daniel Webster” about an American Congressman, and the film “Bedazzled” starring Brendan Frasier.

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The Faust tradition in popular culture. Mephistopheles is the demon who buys Faust’s soul. (http://xkcd.com/501/)

As I wondered what to pursue a PhD in, so (roughly speaking) did Faust. In plays by Christopher Marlowe and Johann Wolfgang von Goethe, Faust wavers among law, theology, philosophy, and medicine. You’ve probably heard what happens next: Faust chooses sorcery, conjures a demon, and bargains away his soul. Hardly the role model for a college student. I preferred to keep my soul, though Maxwell’s demon had stolen my heart.

A few decades after Goethe penned Faust, English physicist James Maxwell proposed a thought experiment. Consider a box divided into two rooms, he wrote, and a demon controlling the door between the rooms. Since others have explained Maxwell’s paradox, I won’t parrot them. Suffice to say, the demon helps clarify why time flows, what knowledge is, and how information relates to matter. Quantum-information physicists, I learned in a seminar after German 44, study Maxwell’s demon. Via the demon, experiment, and math, QI physicists study the whole world. I wanted to contemplate the whole world, like Goethe’s Faust. By studying QI, I might approximate my goal. Faust, almost as much as my QI seminar, convinced me to pursue a PhD in physics.

Fast forward two years. Someone must have misread my application, because Caltech let me sign my soul to its PhD program. I am the newest Preskillite. Or Preskillnik. Whichever term, if either, irks my supervisor more.

For five years, I will haunt this blog. (Spiros will haunt me if I don’t haunt it.) I’ll try to post one article per month. Pure quantum information occupies me usually: abstract math that encodes physical effects, like entropy (a key to why time flows), decoherence (a system’s transformation from quantum to ordinary), and entanglement (one particle’s ability to affect another, instantaneously, from across a room).

In case I wax poetic about algebra, I apologize in advance. Apologies if I write too many stories about particles in boxes. In addition to training a scientist’s lens on atoms, I enjoy training it on science, culture, and communities. Tune in for scientists’ uses (and abuses) of language, why physics captivates us, and the bittersweetness of representing half our species in a roomful of male physicists (advantage: I rarely wait in line to use a physics department’s bathroom).

As I prepare to move to Caltech, a Faust line keeps replaying in my mind. It encapsulates my impression of a PhD, though written 200 years ago: “Nothing I had; and yet, enough for youth—/ delight in fiction, and the thirst for truth.”

Pleasure to meet you, Quantum Frontiers. Drink with me.

A Public Lecture on Quantum Information

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Sooner or later, most scientists are asked to deliver a public lecture about their research specialties. When successful, lecturing about science to the lay public can give one a feeling of deep satisfaction. But preparing the lecture is a lot of work!

Caltech sponsors the Earnest C. Watson lecture series (named after the same Earnest Watson mentioned in my post about Jane Werner Watson), which attracts very enthusiastic audiences to Beckman Auditorium nine times a year. I gave a Watson lecture on April 3 about Quantum Entanglement and Quantum Computing, which is now available from iTunes U and also on YouTube:

I did a Watson lecture once before, in 1997. That occasion precipitated some big changes in my presentation style. To prepare for the lecture, I acquired my first laptop computer and learned to use PowerPoint. This was still the era when a typical physics talk was handwritten on transparencies and displayed using an overhead projector, so I was sort of a pioneer. And I had many anxious moments in the late 1990s worrying about whether my laptop would be able to communicate with the projector — that can still be a problem even today, but was a more common problem then.

I invested an enormous amount of time in preparing that 1997 lecture, an investment still yielding dividends today. Aside from figuring out what computer to buy (an IBM ThinkPad) and how to do animation in PowerPoint, I also learned to draw using Adobe Illustrator under the tutelage of Caltech’s digital media expert Wayne Waller. And apart from all that technical preparation, I had to figure out the content of the lecture!

That was when I first decided to represent a qubit as a box with two doors, which contains a ball that can be either red or green, and I still use some of the drawings I made then.

Entanglement, illustrated with balls in boxes.

Entanglement, illustrated with balls in boxes.

This choice of colors was unfortunate, because people with red-green color blindness cannot tell the difference. I still feel bad about that, but I don’t have editable versions of the drawings anymore, so fixing it would be a big job …

I also asked my nephew Ben Preskill (then 10 years old, now a math PhD candidate at UC Berkeley), to make a drawing for me illustrating weirdness.

The desire to put weirdness to work has driven the emergence of quantum information science.

The desire to put weirdness to work has driven the emergence of quantum information science.

I still use that, for sentimental reasons, even though it would be easier to update.

The turnout at the lecture was gratifying (you can’t really see the audience with the spotlight shining in your eyes, but I sensed that the main floor of the Auditorium was mostly full), and I have gotten a lot of positive feedback (including from the people who came up to ask questions afterward — we might have been there all night if the audio-visual staff had not forced us to go home).

I did make a few decisions about which I have had second thoughts. I was told I had the option of giving a 45 minute talk with a public question period following, or a 55 minute talk with only a private question period, and I opted for the longer talk. Maybe I should have pushed back and insisted on allowing some public questions even after the longer talk — I like answering questions. And I was told that I should stay in the spotlight, to ensure good video quality, so I decided to stand behind the podium the whole time to curb my tendency to pace across the stage. But maybe I would have seemed more dynamic if I had done some pacing.

I got some gentle criticism from my wife, Roberta, who suggested I could modulate my voice more. I have heard that before, particularly in teaching evaluations that complain about my “soporific” tone. I recall that Mike Freedman once commented after watching a video of a public lecture I did at the KITP in Santa Barbara — he praised its professionalism and “newscaster quality”. But that cuts two ways, doesn’t it? Paul Ginsparg listened to a podcast of that same lecture while doing yardwork, and then sent me a compliment by email, with a characteristic Ginspargian twist. Noting that my sentences were clear, precise, and grammatical, Paul asked: “is this something that just came naturally at some early age, or something that you were able to acquire at some later stage by conscious design (perhaps out of necessity, talks on quantum computing might not go over as well without the reassuring smoothness)?”

Another criticism stung more. To illustrate the monogamy of entanglement, I used a slide describing the frustration of Bob, who wants to entangle with both Alice and Carrie, but finds that he can increase his entanglement with Carrie only my sacrificing some of his entanglement with Alice.

Entanglement is monogamous. Bob is frustrated to find that he cannot be fully entangled with both Alice and Carrie.

Entanglement is monogamous. Bob is frustrated to find that he cannot be fully entangled with both Alice and Carrie.

This got a big laugh. But I used the same slide in a talk at the APS Denver meeting the following week (at a session celebrating the 100th anniversary of Niels Bohr’s atomic model), and a young woman came up to me after that talk to complain. She suggested that my monogamy metaphor was offensive and might discourage women from entering the field!

After discussing the issue with Roberta, I decided to address the problem by swapping the gender roles. The next day, during the question period following Stephen Hawking’s Public Lecture, I spoke about Betty’s frustration over her inability to entangle fully with both Adam and Charlie. But is that really an improvement, or does it reflect negatively on Betty’s morals? I would appreciate advice about this quandary in the comments.

In case you watch the video, there are a couple of things you should know. First, in his introduction, Tom Soifer quotes from a poem about me, but neglects to name the poet. It is former Caltech postdoc Patrick Hayden. And second, toward the end of the lecture I talk about some IQIM outreach activities, but neglect to name our Outreach Director Spiros Michalakis, without whose visionary leadership these things would not have happened.

The most touching feedback I received came from my Caltech colleague Oskar Painter. I joked in the lecture about how mild mannered IQIM scientists can unleash the superpower of quantum information at a moment’s notice.

Mild mannered professor unleashes the super power of quantum information.

Mild mannered professor unleashes the superpower of quantum information.

After watching the video, Oskar shot me an email:

“I sent a link to my son [Ewan, age 11] and daughter [Quinn, age 9], and they each watched it from beginning to end on their iPads, without interruption.  Afterwards, they had a huge number of questions for me, and were dreaming of all sorts of “quantum super powers” they imagined for the future.”

Largest prime number found?

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Over the past few months, I have been inundated with tweets about the largest prime number ever found. That number, according to Nature News, is 2^{57,885,161}-1. This is certainly a very large prime number and one would think that we would need a supercomputer to find a prime number larger than this one. In fact, Nature mentions that there are infinitely many prime numbers, but the powerful prime number theorem doesn’t tell us how to find them!
nature_news_highlightWell, I am here to tell you of the discovery of the new largest prime number ever found, which I will call P_{euclid}. Here it is:

P_{euclid} = 2\cdot 3\cdot 5\cdot 7\cdot 11 \cdot \cdots \cdot (2^{57,885,161}-1) +1.

This number, the product of all prime numbers known so far plus one, is so large that I can’t even write it down on this blog post. But it is certainly (proof left as an exercise…!) a prime number (see Problem 4 in The allure of elegance) and definitely larger than the one getting all the hype. Finally, I will be getting published in Nature!

In the meantime, if you are looking for a real challenge, calculate how many digits my prime number has in base 10. Whoever gets it right (within an order of magnitude), will be my co-author in the shortest Nature paper ever written.

Update 2: I read somewhere that in order to get attention to your blog posts, you should sprinkle them with grammatical errors and let the commenters do the rest for you. I wish I was mastermind-y enough to engineer this post in this fashion. Instead, I get the feeling that someone will run a primality test on P_{euclid} just to prove me wrong. Well, what are you waiting for? In the meantime, another challenge: What is the smallest number (ballpark it using Prime Number Theorem) of primes we need to multiply together before adding one, in order to have a number with a larger prime factor than 2^{57,885,161}-1?

Update: The number P_{euclid} given above may not be prime itself, as pointed out quickly by Steve Flammia, Georg and Graeme Smith. But, it does contain within it the new largest prime number ever known, which may be the number itself. Now, if only we had a quantum computer to factor numbers quickly…Wait, wasn’t there a polynomial time primality test?

Note: The number mentioned is the largest known Mersenne prime. That Mersenne primes are crazy hard to find is an awesome problem in number theory.

Post-Quantum Cryptography

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As an undergraduate, I took Introduction to Algorithms from Ron Rivest. One of the topics he taught was the RSA public-key cryptosystem which he had created with Adi Shamir and Leonard Adleman. At the time, RSA was only about a decade old, yet already one of the banner creations of computer science. Today many of us rely on it routinely for the security of banking transactions. The internet would not be the same without it and its successors (such as elliptic curve cryptography, ECC). However, as you may have heard, quantum computation spells change for cryptography. Today I’ll tell a little of this story and talk about prospects for the future.

Ron Rivest

Ron Rivest

What is public-key cryptography (PKC)? The basic notion is due to Ralph Merkle in 1974 and (in a stronger form) to Whitfield Diffie and Martin Hellman in 1976. Their remarkable proposal was that two parties, “Alice” and “Bob”, could cooperate in cryptographic protocols, even if they had never met before. All prior cryptography, from the ancients up through and after the cryptographic adventures of WWII, had relied on the cooperating parties sharing in advance some “secret key” that gave them an edge over any eavesdropper “Eve”.
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QIP 2013 from the perspective of a greenhorn (grad student)

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caltechCrew_qip2013

Most of Caltech’s contingent during QIP’s banquet. Not pictured: sword dancers, jug balancers and Gorjan.

A couple of weeks ago, about half of IQI (now part of IQIM) flew from Pasadena to Beijing in order to attend QIP 2013, the 16th annual workshop on quantum information processing. I wish I could report that the quantum information community solved the world’s problems over the past year, or at least built a 2^10 qubit universal quantum computer, but unfortunately, we’re not quite there yet. As a substitute, I’ll mention a few of the talks that I particularly enjoyed and the really hard open problems that they left us with.

The emphases of the talks mainly bifurcated towards computer science versus physics. I was better prepared to understand the talks emphasizing the latter, so my comments will mainly describe those talks. Patrick Hayden’s talk: “summoning information in spacetime: or where and when can a qubit be?” was one of my favorites. To the extent that I understood things, the goal of this work is to better understand how quantum information can propagate forward in time. If a qubit were created at spacetime location S, and then if it were forced to remain localized, the no-cloning theorem would give strict bounds regarding how it could move forward in time. The qubit would follow a worldline and that would be the end of things. However, qubits don’t need to remain localized, as teleportation experiments have pretty clearly demonstrated, and it therefore seems like qubits can propagate into the future in more subtle ways–ways that at face value appear to violate the no-cloning theorem. Patrick and the undergraduate that he worked with on this project, Alex May, came up with a pictorial approach to better understand these subtleties. The really hard open problems that these ideas could potentially be applied to include: firewalls, position-dependent quantum cryptography and to paradoxes concerning the apparent no-cloning violations near black hole event horizons.
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Ignacio Cirac and Peter Zoller get what they deserve

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Ignacio Cirac, Dave Wineland, and Peter Zoller receiving the 2010 Franklin medal.

Ignacio Cirac, Dave Wineland, and Peter Zoller receiving the 2010 Franklin medal.

A good thing about a blog is that when my friends win prizes I have the opportunity to say nice things about them. This seems to be happening a lot lately (Kitaev, Wineland, Kimble, Hawking, Polchinski, …).

Today’s very exciting news is that Ignacio Cirac and Peter Zoller have won the 2013 Wolf Prize in Physics “for groundbreaking theoretical contributions to quantum information processing, quantum optics, and the physics of quantum gases.”
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Introduction to Quantum Information

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First slide, viewed on my laptop.

First slide, viewed on my laptop.

I’m lazy. The only reason I ever do anything is that sometimes in a weak moment I agree to do something, and after that I don’t have the nerve to back out. And that’s how I happened to give the introductory lectures leading off the 12th Canadian Summer School on Quantum Information last June.

The video of the lectures recently became available on YouTube in two one-hour segments, which is my reason for posting about them now:

Here are the slides I used. The school is pitched at beginning graduate students who have a solid background in quantum mechanics but may not be very familiar with quantum information concepts.

Andrew Childs, who knows my character flaws well, invited me to lecture at the school nearly a year in advance. Undaunted by my silence, he kept resending the invitation at regular intervals to improve his chances of catching me on a weak day. Sure enough, feeling a twinge of guilt over blowing off David Poulin when he made the same request the year before, and with a haunting sense that I had refused to do something Andrew had asked me to do on an earlier occasion (though I can’t recall what), one day in September I said yes, feeling the inevitable stab of regret just seconds after pushing the Send button. I consoled myself with the thought that this could be a Valuable Service to the Community.

Actually, it was fun to think about what to include in my lectures. The job was easier because I knew that the other lecturers who would follow me, all of them excellent, would be able to dig more deeply into some of the topics I would introduce. I decided that my first responsibility should be to convey what makes the topic important and exciting, without getting too bogged down in technicalities which were likely to be addressed later in the school. That meant emphasizing the essence of what makes quantum information different from ordinary “classical” information, and expounding on the theme that classical systems cannot in general simulate quantum systems efficiently.

The conditions under which I delivered the lectures were not quite ideal. Preparing PowerPoint slides is incredibly time consuming, and I believe in the principle that such a task can fill however much time is allotted for it. Therefore, as a matter of policy, I try to delay starting on the slides until the last moment, which has sometimes gotten me into hot water. In this case it meant working on the slides during the flight from LA to Toronto, in the car from Toronto to Waterloo, and then for a few more hours in my hotel room until I went to bed about midnight, with my alarm set for 6 am so I could finish my preparations in the morning.

It seemed like a good plan. But around 2 am I was awakened by an incredibly loud pounding, which sounded like a heavy mallet hammering on the ceiling below me. As I discovered when I complained to the front desk, this was literally true — they were repairing the air-conditioning ducts in the restaurant underneath my room. I was told that the hotel could not do anything about the noise, because the restaurant is under different ownership. I went back to bed, but lost patience around 3:30 am and demanded a different room, on the other side of the hotel. I was settled in my (perfectly quiet) new room by 4 am, but I was too keyed up to sleep, and read a book on my iPad until it was 6 am and time to get up.

I worked in my room as late as I could, then grabbed a taxi, showing the driver a map with the location of the summer school marked on it. Soon after he dropped me off, I discovered I was on the wrong side of the University of Waterloo campus, about a 20 minute walk from where I was supposed to be. It was about 8:15, and the school was to begin at 8:30, so I started jogging, though not, as it turned out, in the right direction. After twice asking passersby for help, I got to the lecture hall just in time, my heart pounding and my shirt soaked with sweat. Not in the best of moods, I barked at Andrew that I needed coffee, which he dutifully fetched for me.

Though my head was pounding and my legs felt rubbery, adrenalin kicked in as I started lecturing. I felt like I was performing in a lower gear than usual, but I wasn’t sure whether the audience could tell.

And as often happens when I reluctantly agree to do something, when it was all over I was glad I had done it.