# Wouldn’t you like to know what’s going on in my mind?

I suppose most theoretical physicists who (like me) are comfortably past the age of 60 worry about their susceptibility to “crazy-old-guy syndrome.” (Sorry for the sexism, but all the victims of this malady I know are guys.) It can be sad when a formerly great scientist falls far out of the mainstream and seems to be spouting nonsense.

Matthew Fisher is only 55, but reluctance to be seen as a crazy old guy might partially explain why he has kept pretty quiet about his passionate pursuit of neuroscience over the past three years. That changed two months ago when he posted a paper on the arXiv about Quantum Cognition.

Neuroscience has a very seductive pull, because it is at once very accessible and very inaccessible. While a theoretical physicist might think and write about a brane even without having or seeing a brane, everybody’s got a brain (some scarecrows excepted). On the other hand, while it’s not too hard to write down and study the equations that describe a brane, it is not at all easy to write down the equations for a brain, let alone solve them. The brain is fascinating because we know so little about it. And … how can anyone with a healthy appreciation for Gödel’s Theorem not be intrigued by the very idea of a brain that thinks about itself?

(Almost) everybody’s got a brain.

The idea that quantum effects could have an important role in brain function is not new, but is routinely dismissed as wildly implausible. Matthew Fisher begs to differ. And those who read his paper (as I hope many will) are bound to conclude: This old guy’s not so crazy. He may be onto something. At least he’s raising some very interesting questions.

My appreciation for Matthew and his paper was heightened further this Wednesday, when Matthew stopped by Caltech for a lunch-time seminar and one of my interminable dinner-time group meetings. I don’t know whether my brain is performing quantum information processing (and neither does Matthew), but just the thought that it might be is lighting me up like a zebrafish.

Following Matthew, let’s take a deep breath and ask ourselves: What would need to be true for quantum information processing to be important in the brain? Presumably we would need ways to (1) store quantum information for a long time, (2) transport quantum information, (3) create entanglement, and (4) have entanglement influence the firing of neurons. After a three-year quest, Matthew has interesting things to say about all of these issues. For details, you should read the paper.

Matthew argues that the only plausible repositories for quantum information in the brain are the Phosphorus-31 nuclear spins in phosphate ions. Because these nuclei are spin-1/2, they have no electric quadrupole moments and hence corresponding long coherence times — of order a second. That may not be long enough, but phosphate ions can be bound with calcium ions into objects called Posner clusters, each containing six P-31 nuclei. The phosphorus nuclei in Posner clusters might have coherence times greatly enhanced by motional narrowing, perhaps as long as weeks or even longer.

Where energy is being consumed in a cell, ATP sometimes releases diphosphate ions (what biochemists call pyrophosphate), which are later broken into two separate phosphate ions, each with a single P-31 qubit. Matthew argues that the breakup of the diphosphate, catalyzed by a suitable enzyme, will occur at an enhanced rate when these two P-31 qubits are in a spin singlet rather than a spin triplet. The reason is that the enzyme has to grab ahold of the diphosphate molecule and stop its rotation in order to break it apart, which is much easier when the molecule has even rather than odd orbital angular momentum; therefore due to Fermi statistics the spin state of the P-31 nuclei must be antisymmetric. Thus wherever ATP is consumed there is a plentiful source of entangled qubit pairs.

If the phosphate molecules remain unbound, this entanglement will decay in about a second, but it is a different story if the phosphate ions group together quickly enough into Posner clusters, allowing the entanglement to survive for a much longer time. If the two members of an entangled qubit pair are snatched up by different Posner clusters, the clusters may then be transported into different cells, distributing the entanglement over relatively long distances.

(a) Two entangled Posner clusters. Each dot is a P-31 nuclear spin, and each dashed line represents a singlet pair. (b) Many entangled Posner clusters. [From Fisher 2015]

What causes a neuron to fire is a complicated story that I won’t attempt to wade into. Suffice it to say that part of the story may involve the chemical binding of a pair of Posner clusters which then melt if the environment is sufficiently acidic, releasing calcium ions and phosphate ions which enhance the firing. The melting rate depends on the spin state of the six P-31 nuclei within the cluster, so that entanglement between clusters in different cells may induce nonlocal correlations among different neurons, which could be quite complex if entanglement is widely distributed.

This scenario raises more questions than it answers, but these are definitely scientific questions inviting further investigation and experimental exploration. One thing that is far from clear at this stage is whether such quantum correlations among neurons (if they exist at all) would be easy to simulate with a classical computer. Even if that turns out to be so, these potential quantum effects involving many neurons could be fabulously interesting. IQIM’s mission is to reach for transformative quantum science, particularly approaches that take advantage of synergies between different fields of study. This topic certainly qualifies.* It’s going to be great fun to see where it leads.

If you are a young and ambitious scientist, you may be contemplating the dilemma: Should I pursue quantum physics or neuroscience? Maybe, just maybe, the right answer is: Both.

*Matthew is the only member of the IQIM faculty who is not a Caltech professor, though he once was.

# Toward physical realizations of thermodynamic resource theories

The thank-you slide of my presentation remained onscreen, and the question-and-answer session had begun. I was presenting a seminar about thermodynamic resource theories (TRTs), models developed by quantum-information theorists for small-scale exchanges of heat and work. The audience consisted of condensed-matter physicists who studied graphene and photonic crystals. I was beginning to regret my topic’s abstractness.

The question-asker pointed at a listener.

“This is an experimentalist,” he continued, “your arch-nemesis. What implications does your theory have for his lab? Does it have any? Why should he care?”

I could have answered better. I apologized that quantum-information theorists, reared on the rarefied air of Dirac bras and kets, had developed TRTs. I recalled the baby steps with which science sometimes migrates from theory to experiment. I could have advocated for bounding, with idealizations, efficiencies achievable in labs. I should have invoked the connections being developed with fluctuation results, statistical mechanical theorems that have withstood experimental tests.

The crowd looked unconvinced, but I scored one point: The experimentalist was not my arch-nemesis.

“My new friend,” I corrected the questioner.

His question has burned in my mind for two years. Experiments have inspired, but not guided, TRTs. TRTs have yet to drive experiments. Can we strengthen the connection between TRTs and the natural world? If so, what tools must resource theorists develop to predict outcomes of experiments? If not, are resource theorists doing physics?

A Q&A more successful than mine.

I explore answers to these questions in a paper released today. Ian Durham and Dean Rickles were kind enough to request a contribution for a book of conference proceedings. The conference, “Information and Interaction: Eddington, Wheeler, and the Limits of Knowledge” took place at the University of Cambridge (including a graveyard thereof), thanks to FQXi (the Foundational Questions Institute).

“Proceedings are a great opportunity to get something off your chest,” John said.

That seminar Q&A had sat on my chest, like a pet cat who half-smothers you while you’re sleeping, for two years. Theorists often justify TRTs with experiments.* Experimentalists, an argument goes, are probing limits of physics. Conventional statistical mechanics describe these regimes poorly. To understand these experiments, and to apply them to technologies, we must explore TRTs.

Does that argument not merit testing? If experimentalists observe the extremes predicted with TRTs, then the justifications for, and the timeliness of, TRT research will grow.

Something to get off your chest. Like the contents of a conference-proceedings paper, according to my advisor.

You’ve read the paper’s introduction, the first eight paragraphs of this blog post. (Who wouldn’t want to begin a paper with a mortifying anecdote?) Later in the paper, I introduce TRTs and their role in one-shot statistical mechanics, the analysis of work, heat, and entropies on small scales. I discuss whether TRTs can be realized and whether physicists should care. I identify eleven opportunities for shifting TRTs toward experiments. Three opportunities concern what merits realizing and how, in principle, we can realize it. Six adjustments to TRTs could improve TRTs’ realism. Two more-out-there opportunities, though less critical to realizations, could diversify the platforms with which we might realize TRTs.

One opportunity is the physical realization of thermal embezzlement. TRTs, like thermodynamic laws, dictate how systems can and cannot evolve. Suppose that a state $R$ cannot transform into a state $S$: $R \not\mapsto S$. An ancilla $C$, called a catalyst, might facilitate the transformation: $R + C \mapsto S + C$. Catalysts act like engines used to extract work from a pair of heat baths.

Engines degrade, so a realistic transformation might yield $S + \tilde{C}$, wherein $\tilde{C}$ resembles $C$. For certain definitions of “resembles,”** TRTs imply, one can extract arbitrary amounts of work by negligibly degrading $C$. Detecting the degradation—the work extraction’s cost—is difficult. Extracting arbitrary amounts of work at a difficult-to-detect cost contradicts the spirit of thermodynamic law.

The spirit, not the letter. Embezzlement seems physically realizable, in principle. Detecting embezzlement could push experimentalists’ abilities to distinguish between close-together states $C$ and $\tilde{C}$. I hope that that challenge, and the chance to violate the spirit of thermodynamic law, attracts researchers. Alternatively, theorists could redefine “resembles” so that $C$ doesn’t rub the law the wrong way.

The paper’s broadness evokes a caveat of Arthur Eddington’s. In 1927, Eddington presented Gifford Lectures entitled The Nature of the Physical World. Being a physicist, he admitted, “I have much to fear from the expert philosophical critic.” Specializing in TRTs, I have much to fear from the expert experimental critic. The paper is intended to point out, and to initiate responses to, the lack of physical realizations of TRTs. Some concerns are practical; some, philosophical. I expect and hope that the discussion will continue…preferably with more cooperation and charity than during that Q&A.

If you want to continue the discussion, drop me a line.

*So do theorists-in-training. I have.

**A definition that involves the trace distance.

# Bits, bears, and beyond in Banff

Another conference about entropy. Another graveyard.

Last year, I blogged about the University of Cambridge cemetery visited by participants in the conference “Eddington and Wheeler: Information and Interaction.” We’d lectured each other about entropy–a quantification of decay, of the march of time. Then we marched to an overgrown graveyard, where scientists who’d lectured about entropy decades earlier were decaying.

This July, I attended the conference “Beyond i.i.d. in information theory.” The acronym “i.i.d.” stands for “independent and identically distributed,” which requires its own explanation. The conference took place at BIRS, the Banff International Research Station, in Canada. Locals pronounce “BIRS” as “burrs,” the spiky plant bits that stick to your socks when you hike. (I had thought that one pronounces “BIRS” as “beers,” over which participants in quantum conferences debate about the Measurement Problem.) Conversations at “Beyond i.i.d.” dinner tables ranged from mathematical identities to the hiking for which most tourists visit Banff to the bears we’d been advised to avoid while hiking. So let me explain the meaning of “i.i.d.” in terms of bear attacks.

The BIRS conference center. Beyond here, there be bears.

Suppose that, every day, exactly one bear attacks you as you hike in Banff. Every day, you have a probability p1 of facing down a black bear, a probability p2 of facing down a grizzly, and so on. These probabilities form a distribution {pi} over the set of possible events (of possible attacks). We call the type of attack that occurs on a given day a random variable. The distribution associated with each day equals the distribution associated with each other day. Hence the variables are identically distributed. The Monday distribution doesn’t affect the Tuesday distribution and so on, so the distributions are independent.

Information theorists quantify efficiencies with which i.i.d. tasks can be performed. Suppose that your mother expresses concern about your hiking. She asks you to report which bear harassed you on which day. You compress your report into the fewest possible bits, or units of information. Consider the limit as the number of days approaches infinity, called the asymptotic limit. The number of bits required per day approaches a function, called the Shannon entropy HS, of the distribution:

Number of bits required per day → HS({pi}).

The Shannon entropy describes many asymptotic properties of i.i.d. variables. Similarly, the von Neumann entropy HvN describes many asymptotic properties of i.i.d. quantum states.

But you don’t hike for infinitely many days. The rate of black-bear attacks ebbs and flows. If you stumbled into grizzly land on Friday, you’ll probably avoid it, and have a lower grizzly-attack probability, on Saturday. Into how few bits can you compress a set of nonasymptotic, non-i.i.d. variables?

We answer such questions in terms of ɛ-smooth α-Rényi entropies, the sandwiched Rényi relative entropy, the hypothesis-testing entropy, and related beasts. These beasts form a zoo diagrammed by conference participant Philippe Faist. I wish I had his diagram on a placemat.

“Beyond i.i.d.” participants define these entropies, generalize the entropies, probe the entropies’ properties, and apply the entropies to physics. Want to quantify the efficiency with which you can perform an information-processing task or a thermodynamic task? An entropy might hold the key.

Many highlights distinguished the conference; I’ll mention a handful.  If the jargon upsets your stomach, skip three paragraphs to Thermodynamic Thursday.

Aram Harrow introduced a resource theory that resembles entanglement theory but whose agents pay to communicate classically. Why, I interrupted him, define such a theory? The backstory involves a wager against quantum-information pioneer Charlie Bennett (more precisely, against an opinion of Bennett’s). For details, and for a quantum version of The Princess and the Pea, watch Aram’s talk.

Graeme Smith and colleagues “remove[d] the . . . creativity” from proofs that certain entropic quantities satisfy subadditivity. Subadditivity is a property that facilitates proofs and that offers physical insights into applications. Graeme & co. designed an algorithm for checking whether entropic quantity Q satisfies subadditivity. Just add water; no innovation required. How appropriate, conference co-organizer Mark Wilde observed. BIRS has the slogan “Inspiring creativity.”

Patrick Hayden applied one-shot entropies to AdS/CFT and emergent spacetime, enthused about elsewhere on this blog. Debbie Leung discussed approximations to Haar-random unitaries. Gilad Gour compared resource theories.

Conference participants graciously tolerated my talk about thermodynamic resource theories. I closed my eyes to symbolize the ignorance quantified by entropy. Not really; the photo didn’t turn out as well as hoped, despite the photographer’s goodwill. But I could have closed my eyes to symbolize entropic ignorance.

Thermodynamics and resource theories dominated Thursday. Thermodynamics is the physics of heat, work, entropy, and stasis. Resource theories are simple models for transformations, like from a charged battery and a Tesla car at the bottom of a hill to an empty battery and a Tesla atop a hill.

My advisor’s Tesla. No wonder I study thermodynamic resource theories.

Philippe Faist, diagrammer of the Entropy Zoo, compared two models for thermodynamic operations. I introduced a generalization of resource theories for thermodynamics. Last year, Joe Renes of ETH and I broadened thermo resource theories to model exchanges of not only heat, but also particles, angular momentum, and other quantities. We calculated work in terms of the hypothesis-testing entropy. Though our generalization won’t surprise Quantum Frontiers diehards, the magic tricks in my presentation might.

At twilight on Thermodynamic Thursday, I meandered down the mountain from the conference center. Entropies hummed in my mind like the mosquitoes I slapped from my calves. Rising from scratching a bite, I confronted the Banff Cemetery. Half-wild greenery framed the headstones that bordered the gravel path I was following. Thermodynamicists have associated entropy with the passage of time, with deterioration, with a fate we can’t escape. I seem unable to escape from brushing past cemeteries at entropy conferences.

Not that I mind, I thought while scratching the bite in Pasadena. At least I escaped attacks by Banff’s bears.

With thanks to the conference organizers and to BIRS for the opportunity to participate in “Beyond i.i.d. 2015.”

# 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…

# 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.

# Mingling stat mech with quantum info in Maryland

I felt like a yoyo.

I was standing in a hallway at the University of Maryland. On one side stood quantum-information theorists. On the other side stood statistical-mechanics scientists.* The groups eyed each other, like Jets and Sharks in West Side Story, except without fighting or dancing.

This March, the groups were generous enough to host me for a visit. I parked first at QuICS, the Joint Center for Quantum Information and Computer Science. Established in October 2014, QuICS had moved into renovated offices the previous month. QuICSland boasts bright colors, sprawling armchairs, and the scent of novelty. So recently had QuICS arrived that the restroom had not acquired toilet paper (as I learned later than I’d have preferred).

Photo credit: QuICS

From QuICS, I yoyo-ed to the chemistry building, where Chris Jarzynski’s group studies fluctuation relations. Fluctuation relations, introduced elsewhere on this blog, describe out-of-equilibrium systems. A system is out of equilibrium if large-scale properties of it change. Many systems operate out of equilibrium—boiling soup, combustion engines, hurricanes, and living creatures, for instance. Physicists want to describe nonequilibrium processes but have trouble: Living creatures are complicated. Hence the buzz about fluctuation relations.

My first Friday in Maryland, I presented a seminar about quantum voting for QuICS. The next Tuesday, I was to present about one-shot information theory for stat-mech enthusiasts. Each week, the stat-mech crowd invites its speaker to lunch. Chris Jarzynski recommended I invite QuICS. Hence the Jets-and-Sharks tableau.

“Have you interacted before?” I asked the hallway.

“No,” said a voice. QuICS hadn’t existed till last fall, and some QuICSers hadn’t had offices till the previous month.**

Silence.

“We’re QuICS,” volunteered Stephen Jordan, a quantum-computation theorist, “the Joint Center for Quantum Information and Computer Science.”

So began the mingling. It continued at lunch, which we shared at three circular tables we’d dragged into a chain. The mingling continued during the seminar, as QuICSers sat with chemists, materials scientists, and control theorists. The mingling continued the next day, when QuICSer Alexey Gorshkov joined my discussion with the Jarzynski group. Back and forth we yoyo-ed, between buildings and topics.

“Mingled,” said Yigit Subasi. Yigit, a postdoc of Chris’s, specialized in quantum physics as a PhD student. I’d asked how he thinks about quantum fluctuation relations. Since Chris and colleagues ignited fluctuation-relation research, theorems have proliferated like vines in a jungle. Everyone and his aunty seems to have invented a fluctuation theorem. I canvassed Marylanders for bushwhacking tips.

Imagine, said Yigit, a system whose state you know. Imagine a gas, whose temperature you’ve measured, at equilibrium in a box. Or imagine a trapped ion. Begin with a state about which you have information.

Imagine performing work on the system “violently.” Compress the gas quickly, so the particles roil. Shine light on the ion. The system will leave equilibrium. “The information,” said Yigit, “gets mingled.”

Imagine halting the compression. Imagine switching off the light. Combine your information about the initial state with assumptions and physical laws.*** Manipulate equations in the right way, and the information might “unmingle.” You might capture properties of the violence in a fluctuation relation.

With Zhiyue Lu and Andrew Maven Smith of Chris Jarzynski’s group (left) and with QuICSers (right)

I’m grateful to have exchanged information in Maryland, to have yoyo-ed between groups. We have work to perform together. I have transformations to undergo.**** Let the unmingling begin.

With gratitude to Alexey Gorshkov and QuICS, and to Chris Jarzynski and the University of Maryland Department of Chemistry, for their hospitality, conversation, and camaraderie.

*Statistical mechanics is the study of systems that contain vast numbers of particles, like the air we breathe and white dwarf stars. I harp on about statistical mechanics often.

**Before QuICS’s birth, a future QuICSer had collaborated with a postdoc of Chris’s on combining quantum information with fluctuation relations.

***Yes, physical laws are assumptions. But they’re glorified assumptions.

****Hopefully nonviolent transformations.

# Generally speaking

My high-school calculus teacher had a mustache like a walrus’s and shoulders like a rower’s. At 8:05 AM, he would demand my class’s questions about our homework. Students would yawn, and someone’s hand would drift into the air.

“I have a general question,” the hand’s owner would begin.

“Only private questions from you,” my teacher would snap. “You’ll be a general someday, but you’re not a colonel, or even a captain, yet.”

Then his eyes would twinkle; his voice would soften; and, after the student asked the question, his answer would epitomize why I’ve chosen a life in which I use calculus more often than laundry detergent.

Many times though I witnessed the “general” trap, I fell into it once. Little wonder: I relish generalization as other people relish hiking or painting or Michelin-worthy relish. When inferring general principles from examples, I abstract away details as though they’re tomato stains. My veneration of generalization led me to quantum information (QI) theory. One abstract theory can model many physical systems: electrons, superconductors, ion traps, etc.

Little wonder that generalizing a QI model swallowed my summer.

QI has shed light on statistical mechanics and thermodynamics, which describe energy, information, and efficiency. Models called resource theories describe small systems’ energies, information, and efficiencies. Resource theories help us calculate a quantum system’s value—what you can and can’t create from a quantum system—if you can manipulate systems in only certain ways.

Suppose you can perform only operations that preserve energy. According to the Second Law of Thermodynamics, systems evolve toward equilibrium. Equilibrium amounts roughly to stasis: Averages of properties like energy remain constant.

Out-of-equilibrium systems have value because you can suck energy from them to power laundry machines. How much energy can you draw, on average, from a system in a constant-temperature environment? Technically: How much “work” can you draw? We denote this average work by < W >. According to thermodynamics, < W > equals the change ∆F in the system’s Helmholtz free energy. The Helmholtz free energy is a thermodynamic property similar to the energy stored in a coiled spring.

One reason to study thermodynamics?

Suppose you want to calculate more than the average extractable work. How much work will you probably extract during some particular trial? Though statistical physics offers no answer, resource theories do. One answer derived from resource theories resembles ∆F mathematically but involves one-shot information theory, which I’ve discussed elsewhere.

If you average this one-shot extractable work, you recover < W > = ∆F. “Helmholtz” resource theories recapitulate statistical-physics results while offering new insights about single trials.

Helmholtz resource theories sit atop a silver-tasseled pillow in my heart. Why not, I thought, spread the joy to the rest of statistical physics? Why not generalize thermodynamic resource theories?

The average work <W > extractable equals ∆F if heat can leak into your system. If heat and particles can leak, <W > equals the change in your system’s grand potential. The grand potential, like the Helmholtz free energy, is a free energy that resembles the energy in a coiled spring. The grand potential characterizes Bose-Einstein condensates, low-energy quantum systems that may have applications to metrology and quantum computation. If your system responds to a magnetic field, or has mass and occupies a gravitational field, or has other properties, <W > equals the change in another free energy.

A collaborator and I designed resource theories that describe heat-and-particle exchanges. In our paper “Beyond heat baths: Generalized resource theories for small-scale thermodynamics,” we propose that different thermodynamic resource theories correspond to different interactions, environments, and free energies. I detailed the proposal in “Beyond heat baths II: Framework for generalized thermodynamic resource theories.”

“II” generalizes enough to satisfy my craving for patterns and universals. “II” generalizes enough to merit a hand-slap of a pun from my calculus teacher. We can test abstract theories only by applying them to specific systems. If thermodynamic resource theories describe situations as diverse as heat-and-particle exchanges, magnetic fields, and polymers, some specific system should shed light on resource theories’ accuracy.

If you find such a system, let me know. Much as generalization pleases aesthetically, the detergent is in the details.