A year ago, the “I’m a little teapot” song kept playing in my head.
I was finishing a collaboration with David Limmer, a theoretical chemist at the University of California Berkeley. David studies quantum and classical systems far from equilibrium, including how these systems exchange energy and information with their environments. Example systems include photoisomers.
Your average-Joe photoisomer spends much of its life in equilibrium, exchanging heat with room-temperature surroundings. The molecule has the shape above, called the cis configuration. Imagine shining a laser or sunlight on the photoisomer. The molecule can absorb a photon, or particle of light, gaining energy. The energized switch has the opportunity to switch: One chemical group can rotate downward. The molecule will occupy its trans configuration.
The molecule now has more energy than it had while equilibrium, albeit less energy than it had right after absorbing the photon. The molecule can remain in this condition for a decent amount of time. (Experts: The molecule occupies a metastable state.) That is, the molecule can store sunlight. For that reason, experimentalists at Harvard and MIT attached photoisomers to graphene nanotubules, improving the nanotubules’ storage of solar fuel.
With what probability does a photoisomer switch upon absorbing a photon? This question has resisted easy answering, because photoisomers prove difficult to model: They’re small, quantum, and far from equilibrium. People have progressed by making assumptions, but such assumptions can lack justifications or violate physical principles. David wanted to derive a simple, general bound—of the sort in which thermodynamicists specialize—on a photoisomer’s switching probability.
He had a hunch as to how he could derive such a bound. I’veblogged,manytimes,about thermodynamic resource theories. Thermodynamic resource theories are simple models, developed in quantum information theory, for exchanges of heat, particles, information, and more. These models involve few assumptions: the conservation of energy, quantum theory, and, to some extent, the existence of a large environment (Markovianity). With such a model, David suspected, he might derive his bound.
I knew nothing about photoisomers when I met David, but I knew about thermodynamic resource theories. I’d contributed to their development, to the theorems that have piled up in the resource-theory corner of quantum information theory. Then, the corner had given me claustrophobia. Those theorems felt so formal, abstract, and idealized. Formal, abstract theory has drawn me ever since I started studying physics in college. But did resource theories model physical reality? Could they impact science beyond our corner of quantum information theory? Did resource theories matter?
I called for connecting thermodynamic resource theories to physical reality four years ago, in a paper that begins with an embarrassing story about me. Resource theorists began designing experiments whose results should agree with our theorems. Theorists also tried to improve the accuracy with which resource theories model experimentalists’ limitations. See David’s and my paper for a list of these achievements. They delighted me, as a step toward the broadening of resource theories’ usefulness.
Like any first step, this step pointed toward opportunities. Experiments designed to test our theorems essentially test quantum mechanics. Scientists have tested quantum mechanics for decades; we needn’t test it much more. Such experimental proposals can push experimentalists to hone their abilities, but I hoped that the community could accomplish more. We should be able to apply resource theories to answer questions cultivated in other fields, such as condensed matter and chemistry. We should be useful to scientists outside our corner of quantum information.
David’s idea lit me up like photons on a solar-fuel-storage device. He taught me about photoisomers, I taught him about resource theories, and we derived his bound. Our proof relies on the “second laws of thermodynamics.” These abstract resource-theory results generalize the second law of thermodynamics, which helps us understand why time flows in only one direction. We checked our bound against numerical simulations (experts: of Lindbladian evolution). Our bound is fairly tight if the photoisomer has a low probability of absorbing a photon, as in the Harvard-MIT experiment.
Experts: We also quantified the photoisomer’s coherences relative to the energy eigenbasis. Coherences can’t boost the switching probability, we concluded. But, en route to this conclusion, we found that the molecule is a natural realization of a quantum clock. Our quantum-clock modeling extends to general dissipative Landau-Zener transitions, prevalent across condensed matter and chemistry.
As I worked on our paper one day, a jingle unfolded in my head. I recognized the tune first: “I’m a little teapot.” I hadn’t sung that much since kindergarten, I realized. Lyrics suggested themselves:
I’m a little isomer with two hands.
Here is my cis pose; here is my trans.
Stand me in the sunlight; watch me spin.
I’ll keep solar energy in!
The song lodged itself in my head for weeks. But if you have to pay an earworm to collaborate with David, do.
In the previous blog post (titled, “On the Coattails of Quantum Supremacy“) we started with Google and ended up with molecules! I also mentioned a recent paper by John Preskill, Jake Covey, and myself (see also this videoed talk) where we assume that, somewhere in the (near?) future, experimentalists will be able to construct quantum superpositions of several orientations of molecules or other rigid bodies. Next, I’d like to cover a few more details on how to construct error-correcting codes for anything from classical bits in your phone to those future quantum computers, molecular or otherwise.
Classical error correction: the basics
Error correction is concerned with the design of an encoding that allows for protection against noise. Let’s say we want to protect one classical bit, which is in either “0” or “1”. If the bit is say in “0”, and the environment (say, the strong magnetic field from a magnet you forgot was laying next to your hard drive) flipped it to “1” without our knowledge, an error would result (e.g., making your phone think you swiped right!)
Now let’s encode our single logical bit into three physical bits, whose possible states are represented by the eight corners of the cube below. Let’s encode the logical bit as “0” —> 000 and “1” —> 111, corresponding to the corners of the cube marked by the black and white ball, respectively. For our (local) noise model, we assume that flips of only one of the three physical bits are more likely to occur than flips of two or three at the same time.
Error correction is, like many Hollywood movies, an origin story. If, say, the first bit flips in our above code, the 000 state is mapped to 100, and 111 is mapped to 011. Since we have assumed that the most likely error is a flip of one of the bits, we know upon observing that 100 must have come from the clean 000, and 011 from 111. Thus, in either case of the logical bit being “0” or “1”, we can recover the information by simply observing which state the majority of the bits are in. The same things happen when the second or third bits flip. In all three cases, the logical “0” state is mapped to one of its three neighboring points (above, in blue) while the logical “1” is mapped to its own three points, which, crucially, are distinct from the neighbors of “0”. The set of points that are closer to 000 than to 111 is called a Voronoi tile.
Now, let’s adapt these ideas to molecules. Consider the rotational states of a dumb-bell molecule consisting of two different atoms. (Let’s assume that we have frozen this molecule to the point that the vibration of the inter-atomic bond is limited, essentially creating a fixed distance between the two atoms.) This molecule can orient itself in any direction, and each such orientation can be represented as a point on the surface of a sphere. Now let us encode a classical bit using the north and south poles of this sphere (represented in the picture below as a black and a white ball, respectively). The north pole of the sphere corresponds to the molecule being parallel to the z-axis, while the south pole corresponds to the molecule being anti-parallel.
This time, the noise consists of small shifts in the molecule’s orientation. Clearly, if such shifts are small, the molecule just wiggles a bit around the z-axis. Such wiggles still allow us to infer that the molecule is (mostly) parallel and anti-parallel to the axis, as long as they do not rotate the molecule all the way past the equator. Upon such correctable rotations, the logical “0” state — the north pole — is mapped to a point in the northern hemisphere, while logical “1” — the south pole — is mapped to a point in the southern hemisphere. The northern hemisphere forms a Voronoi tile of the logical “0” state (blue in the picture), which, along with the corresponding tile of the logical “1” state (the southern hemisphere), tiles the entire sphere.
Quantum error correction
To upgrade these ideas to the quantum realm, recall that this time we have to protect superpositions. This means that, in addition to shifting our quantum logical state to other states as before, noise can also affect the terms in the superposition itself. Namely, if, say, the superposition is equal — with an amplitude of in “0” and in “1” — noise can change the relative sign of the superposition and map one of the amplitudes to . We didn’t have to worry about such sign errors before, because our classical information would always be the definite state of “0” or “1”. Now, there are two effects of noise to worry about, so our task has become twice as hard!
Not to worry though. In order to protect against both sources of noise, all we need to do is effectively stagger the above constructions. Now we will need to design a logical “0” state which is itself a superposition of different points, with each point separated from all of the points that are superimposed to make the logical “1” state.
Diatomic molecules: For the diatomic molecule example, consider superpositions of all four corners of two antipodal tetrahedra for the two respective logical states.
The logical “0” state for the quantum code is now itself a quantum superposition of orientations of our diatomic molecule corresponding to the four black points on the sphere to the left (the sphere to the right is a top-down view). Similarly, the logical “1” quantum state is a superposition of all orientations corresponding to the white points.
Each orientation (black or white point) present in our logical states rotates under fluctuations in the position of the molecule. However, the entire set of orientations for say logical “0” — the tetrahedron — rotates rigidly under such rotations. Therefore, the region from which we can successfully recover after rotations is fully determined by the Voronoi tile of any one of the corners of the tetrahedron. (Above, we plot the tile for the point at the north pole.) This cell is clearly smaller than the one for classical north-south-pole encoding we used before. However, the tetrahedral code now provides some protection against phase errors — the other type of noise that we need to worry about if we are to protect quantum information. This is an example of the trade-off we must make in order to protect against both types of noise; a licensed quantum mechanic has to live with such trade-offs every day.
Oscillators: Another example of a quantum encoding is the GKP encoding in the phase space of the harmonic oscillator. Here, we have at our disposal the entire two-dimensional plane indexing different values of position and momentum. In this case, we can use a checkerboard approach, superimposing all points at the centers of the black squares for the logical “0” state, and similarly all points at the centers of the white squares for the logical “1”. The region depicting correctable momentum and position shifts is then the Voronoi cell of the point at the origin: if a shift takes our central black point to somewhere inside the blue square, we know (most likely) where that point came from! In solid state circles, the blue square is none other than the primitive or unit cell of the lattice consisting of points making up both of the logical states.
Asymmetric molecules (a.k.a. rigid rotors): Now let’s briefly return to molecules. Above, we considered diatomic molecules that had a symmetry axis, i.e., that were left unchanged under rotations about the axis that connects the two atoms. There are of course more general molecules out there, including ones that are completely asymmetric under any possible (proper) 3D rotation (see figure below for an example).
BONUS: There is a subtle mistake relating to the geometry of the rotation group in the labeling of this figure. Let me know if you can find it in the comments!
All of the orientations of the asymmetric molecule, and more generally a rigid body, can no longer be parameterized by the sphere. They can be parameterized by the 3D rotation group : each orientation of an asymmetric molecule is labeled by the 3D rotation necessary to obtain said orientation from a reference state. Such rotations, and in turn the orientations themselves, are parameterized by an axis (around which to rotate) and an angle (by which one rotates). The rotation group luckily can still be viewed by humans on a sheet of paper. Namely, can be thought of as a ball of radius with opposite points identified. The direction of each vector lying inside the ball corresponds to the axis of rotation, while the length corresponds to the angle. This may take some time to digest, but it’s not crucial to the story.
So far we’ve looked at codes defined on cubes of bits, spheres, and phase-space lattices. Turns out that even can house similar encodings! In other words, can also be cut up into different Voronoi tiles, which in turn can be staggered to create logical “0” and “1” states consisting of different molecular orientations. There are many ways to pick such states, corresponding to various subgroups of . Below, we sketch two sets of black/white points, along with the Voronoi tile corresponding to the rotations that are corrected by each encoding.
Voronoi tiles of the black point at the center of the ball representing the 3D rotation group, for two different molecular codes. This and the Voronoi cells corresponding to the other points tile together to make up the entire ball. 3D printing all of these tiles would make for cool puzzles!
Achieving supremacy was a big first step towards making quantum computing a practical and universal tool. However, the largest obstacles still await, namely handling superposition-poisoning noise coming from the ever-curious environment. As quantum technologies advance, other possible routes for error correction are by encoding qubits in harmonic oscillators and molecules, alongside the “traditional” approach of using arrays of physical qubits. Oscillator and molecular qubits possess their own mechanisms for error correction, and could prove useful (granted that the large high-energy space required for the procedures to work can be accessed and controlled). Even though molecular qubits are not yet mature enough to be used in quantum computers, we have at least outlined a blueprint for how some of the required pieces can be built. We are by no means done however: besides an engineering barrier, we need to further develop how to run robust computations on these exotic spaces.
Author’s note: I’d like to acknowledge Jose Gonzalez for helping me immensely with the writing of this post, as well as for drawing the comic panels in the previous post. The figures above were made possible by Mathematica 12.
Most readers have by now heard that Google has “achieved” quantum “supremacy”. Notice the only word not in quotes is “quantum”, because unlike previous proposals that have also made some waves, quantumness is mostly not under review here. (Well, neither really are the other two words, but that story has already been covered quite eloquently by John, Scott, and Toby.) The Google team has managed to engineer a device that, although noisy, can do the right thing a large-enough fraction of the time for people to be able to “quantify its quantumness”.
However, the Google device, while less so than previous incarnations, is still noisy. Future devices like it will continue to be noisy. Noise is what makes quantum computers so darn difficult to build; it is what destroys the fragile quantum superpositions that we are trying so hard to protect (remember, unlike a classical computer, we are not protecting things we actually observe, but their superposition).
Protecting quantum information is like taking your home-schooled date (who has lived their entire life in a bunker) to the prom for the first time. It is a fun and necessary part of a healthy relationship to spend time in public, but the price you pay is the possibility that your date will hit it off with someone else. This will leave you abandoned, dancing alone to Taylor Swift’s “You Belong With Me” while crying into your (spiked?) punch.
The high school sweetheart/would-be dance partner in the above provocative example is the quantum superposition — the resource we need for a working quantum computer. You want it all to yourself, but your adversary — the environment — wants it too. No matter how much you try to protect it, you’ll have to observe it eventually (after all, you want to know the answer to your computation). And when you do (take your date out onto the crowded dance floor), you run the risk of the environment collapsing the information before you do, leaving you with nothing.
Protecting quantum information is also like (modern!) medicine. The fussy patient is the quantum information, stored in delicate superposition, while quantumists are the doctors aiming to prevent the patient from getting sick (or “corrupted”). If our patient incurs say “quasiparticle poisoning”, we first diagnose the patient’s syndromes, and, based on this diagnosis, apply procedures like “lattice surgery” and “state injection” to help our patient successfully recover.
Error correction with qubits
Error correction sounds hard, and it should! Not to fear: plenty of very smart people have thought hard about this problem, and have come up with a plan — to redundantly encode the quantum superposition in a way that allows protection from errors caused by noise. Such quantum error-correction is an expansion of the techniques we currently use to protect classical bits in your phone and computer, but now the aim is to protect, not the definitive bit states 0 or 1, but their quantum superpositions. Things are even harder now, as the protection machinery has to do its magic without disturbing the superposition itself (after all, we want our quantum calculation to run to its conclusion and hack your bank).
For example, consider a qubit — the fundamental quantum unit represented by two shelves (which, e.g., could be the ground and excited states of an atom, the absence or presence of a photon in a box, or the zeroth and first quanta of a really cold LC circuit). This qubit can be in any quantum superposition of the two shelves, described by 2 probability amplitudes, one corresponding to each shelf. Observing this qubit will collapse its state onto either one of the shelves, changing the values of the 2 amplitudes. Since the resource we use for our computation is precisely this superposition, we definitely do not want to observe this qubit during our computation. However, we are not the only ones looking: the environment (other people at the prom: the trapping potential of our atom, the jiggling atoms of our metal box, nearby circuit elements) is also observing this system, thereby potentially manipulating the stored quantum state without our knowledge and ruining our computation.
Now consider 50 such qubits. Such a space allows for a superposition with different amplitudes (instead of just for the case of a single qubit). We are once again plagued by noise coming from the environment. But what if we now, less ambitiously, want to store only one qubit’s worth of information in this 50-qubit system? Now there is room to play with! A clever choice of how to do this (a.k.a. the encoding) helps protect from the bad environment.
The entire prospect of building a bona-fide quantum computer rests on this extra overhead or quantum redundancy of using a larger system to encode a smaller one. It sounds daunting at first: if we need 50 physical qubits for each robust logical qubit, then we’d need “I-love-you-3000” physical qubits for 60 logical ones? Yes, this is a fact we all have to live with. But granted we can scale up our devices to that many qubits, there is no fundamental obstacle that prevents us from then using error correction to make next-level computers.
To what extent do we need to protect our quantum superposition from the environment? It would be too ambitious to protect it from a meteor shower. Or a power outage (although that would be quite useful here in California). So what then can we protect against?
Our working answer is local noise — noise that affects only a few qubits that are located near each other in the device. We can never be truly certain if this type of noise is all that our quantum computers will encounter. However, our belief that this is the noise we should focus on is grounded in solid physical principles — that nature respects locality, that affecting things far away from you is harder than making an impact nearby. (So far Google has not reported otherwise, although much more work needs to be done to verify this intuition.)
The harmonic oscillator
In what other ways can we embed our two-shelf qubit into a larger space? Instead of scaling up using many physical qubits, we can utilize a fact that we have so far swept under the rug: in any physical system, our two shelves are already part of an entire bookcase! Atoms have more than one excited state, there can be more than one photon in a box, and there can be more than one quantum in a cold LC circuit. Why don’t we use some of that higher-energy space for our redundant encoding?
The noise in our bookcase will certainly be different, since the structure of the space, and therefore the notion of locality, is different. How to cope with this? The good news is that such a space — the space of the harmonic oscillator — also has a(t least one) natural notion of locality!
Whatever the incarnation, the oscillator has associated with it a position and momentum (different jargon for these quantities may be used, depending on the context, but you can just think of a child on a swing, just quantized). Anyone who knows the joke about Heisenberg getting pulled over, will know that these two quantities cannot be set simultaneously.
Nevertheless, local errors can be thought of as small shifts in position or momentum, while nonlocal errors are ones that suddenly shift our bewildered swinging quantized child from one side of the swing to the other.
Armed with a local noise model, we can extend our know-how from multi-qubit land to the oscillator. One of the first such oscillator codes were developed by Gottesman, Kitaev, and Preskill (GKP). Proposed in 2001, GKP encodings posed a difficult engineering challenge: some believed that GKP states could never be realized, that they “did not exist”. In the past few years however, GKP states have been realized nearly simultaneously in two experimentalplatforms. (Food for thought for the non-believers!)
Parallel to GKP codes, another promising oscillator encoding using cat states is also being developed. This encoding has historically been far easier to create experimentally. It is so far the only experimental procedure achieving the break-even point, at which the actively protected logical information has the same lifetime as the system’s best unprotected degree of freedom.
Can we mix and match all of these different systems? Why yes! While Google is currently trying to build the surface code out of qubits, using oscillators (instead of qubits) for the surface code and encoding said oscillators either in GKP (see related IBM post) [1,2,3] or cat [4,5] codes is something people are seriously considering. There is even more overhead, but the extra information one gets from the correction procedure might make for a more fault-tolerant machine. With all of these different options being explored, it’s an exciting time to be into quantum!
It turns out there are still other systems we can consider, although because they are sufficiently more “out there” at the moment, I should first say “bear with me!” as I explain. Forget about atoms, photons in a box, and really cold LC circuits. Instead, consider a rigid 3-dimensional object whose center of mass has been pinned in such a way that the object can rotate any way it wants. Now, “quantize” it! In other words, consider the possibility of having quantum superpositions of different orientations of this object. Just like superpositions of a dead and alive cat, of a photon and no photon, the object can be in quantum superposition of oriented up, sideways, and down, for example. Superpositions of all possible orientations then make up our new configuration space (read: playground), and we are lucky that it too inherits many of the properties we know and love from its multi-qubit and oscillator cousins.
Examples of rigid bodies include airplanes (which can roll, pitch and yaw, even while “fixed” on a particular trajectory vector) and robot arms (which can rotate about multiple joints). Given that we’re not quantizing those (yet?), what rigid body should we have in mind as a serious candidate? Well, in parallel to the impressive engineering successes of the multi-qubit and oscillator paradigms, physicists and chemists have made substantial progress in trapping and cooling molecules. If a trapped molecule is cold enough, it’s vibrational and electronic states can be neglected, and its rotational states form exactly the rigid body we are interested in. Such rotational states, as far as we can tell, are not in the realm of Avengers-style science fiction.
The idea to use molecules for quantum computing dates all the way back to a 2001 paper by Dave DeMille, but in a recent paper by Jacob Covey, John Preskill, and myself, we propose a framework of how to utilize the large space of molecular orientations to protect against (you guessed it!) a type of local noise. In the second part of the story, called “Quantum Error Correction with Molecules“, I will cover a particular concept that is not only useful for a proper error-correcting code (classical and quantum), but also one that is quite fun to try and understand. The concept is based on a certain kind of tiling, called Voronoi tiles or Thiessen polygons, which can be used to tile anything from your bathroom floor to the space of molecular orientations. Stay tuned!
Every spring, a portal opens between Waltham, Massachusetts and another universe.
The other universe has a Watch City dual to Waltham, known for its watch factories. The cities throw a festival to which explorers, inventors, and tourists flock. Top hats, goggles, leather vests, bustles, and lace-up boots dot the crowds. You can find pet octopodes, human-machine hybrids, and devices for bending space and time. Steam powers everything.
Watch City Steampunk Festival
So I learned thanks to Maxim Olshanyi, a professor of physics at the University of Massachusetts Boston. He hosted my colloquium, “Quantumsteampunk: Quantum information meets thermodynamics,” earlier this month. Maxim, I discovered, has more steampunk experience than I. He digs up century-old designs for radios, builds the radios, and improves upon the designs. He exhibits his creations at the Watch City Steampunk Festival.
I never would have guessed that Maxim moonlights with steampunkers. But his hobby makes sense: Maxim has transformed our understanding of quantum integrability.
Integrability is to thermalization as Watch City is to Waltham. A bowl of baked beans thermalizes when taken outside in Boston in October: Heat dissipates into the air. After half-an-hour, large-scale properties bear little imprint of their initial conditions: The beans could have begun at 112ºF or 99º or 120º. Either way, the beans have cooled.
Integrable systems avoid thermalizing; more of their late-time properties reflect early times. Why? We can understand through an example, an integrable system whose particles don’t interact with each other (whose particles are noninteracting fermions). The dynamics conserve the particles’ momenta. Consider growing the system by adding particles. The number of conserved quantities grows as the system size. The conserved quantities retain memories of the initial conditions.
Imagine preparing an integrable system, analogously to preparing a bowl of baked beans, and letting it sit for a long time. Will the system equilibrate, or settle down to, a state predictable with a simple rule? We might expect not. Obeying the same simple rule would cause different integrable systems to come to resemble each other. Integrable systems seem unlikely to homogenize, since each system retains much information about its initial conditions.
Maxim and collaborators exploded this expectation. Integrable systems do relax to simple equilibrium states, which the physicists called the generalized Gibbs ensemble (GGE). Josiah Willard Gibbs cofounded statistical mechanics during the 1800s. He predicted the state to which nonintegrable systems, like baked beans in autumnal Boston, equilibrate. Gibbs’s theory governs classical systems, like baked beans, as does the GGE theory. But also quantum systems equilibrate to the GGE, and Gibbs’s conclusions translate into quantum theory with few adjustments. So I’ll explain in quantum terms.
Consider quantum baked beans that exchange heat with a temperature- environment. Let denote the system’s Hamiltonian, which basically represents the beans’ energy. The beans equilibrate to a quantum Gibbs state, . The denotes Boltzmann’s constant, a fundamental constant of nature. The partition function enables the quantum state to obey probability theory (normalizes the state).
Maxim and friends modeled their generalized Gibbs ensemble on the Gibbs state. Let denote a quantum integrable system’s conserved quantity. This system equilibrates to . The normalizes the state. The intensive parameters ’s serve analogously to temperature and depend on the conserved quantities’ values. Maxim and friends predicted this state using information theory formalized by Ed Jaynes. Inventing the GGE, they unlocked a slew of predictions about integrable quantum systems.
A radio built by Maxim. According to him, “The invention was to replace a diode with a diode bridge, in a crystal radio, thus gaining a factor of two in the output power.”
I define quantum steampunk as the intersection of quantum theory, especially quantum information theory, with thermodynamics, and the application of this intersection across science. Maxim has used information theory to cofound a branch of quantum statistical mechanics. Little wonder that he exhibits homemade radios at the Watch City Steampunk Festival. He also holds a license to drive steam engines and used to have my postdoc position. I appreciate having older cousins to look up to. Here’s hoping that I become half the quantum steampunker that I found by Massachusetts Bay.
With thanks to Maxim and the rest of the University of Massachusetts Boston Department of Physics for their hospitality.
The next Watch City Steampunk Festival takes place on May 9, 2020. Contact me if you’d attend a quantum-steampunk meetup!
I was working with Tony Bartolotta, a PhD student in theoretical physics at Caltech, and Jason Pollack, a postdoc in cosmology at the University of British Columbia. They acted as the souls of consideration. We missed out on dozens of opportunities to bicker—about the paper’s focus, who undertook which tasks, which journal to submit to, and more. Bickering would have spiced up the story behind our paper, because the paper concerns disagreement.
Quantum observables can disagree. Observables are measurable properties, such as position and momentum. Suppose that you’ve measured a quantum particle’s position and obtained an outcome . If you measure the position immediately afterward, you’ll obtain again. Suppose that, instead of measuring the position again, you measure the momentum. All the possible outcomes have equal probabilities of obtaining. You can’t predict the outcome.
The particle’s position can have a well-defined value, or the momentum can have a well-defined value, but the observables can’t have well-defined values simultaneously. Furthermore, if you measure the position, you randomize the outcome of a momentum measurement. Position and momentum disagree.
How should we quantify the disagreement of two quantum observables, and ? The question splits physicists into two camps. Pure quantum information (QI) theorists use uncertainty relations, whereas condensed-matter and high-energy physicists prefer out-of-time-ordered correlators. Let’s meet the camps in turn.
Heisenberg intuited an uncertainty relation that Robertson formalized during the 1920s,
Imagine preparing a quantum state and measuring , then repeating this protocol in many trials. Each trial has some probability of yielding the outcome . Different trials will yield different ’s. We quantify the spread in values with the standard deviation . We define analogously. denotes Planck’s constant, a number that characterizes our universe as the electron’s mass does.
denotes the observables’ commutator. The numbers that we use in daily life commute: . Quantum numbers, or operators, represent and . Operators don’t necessarily commute. The commutator represents how little and resemble 7 and 5.
Robertson’s uncertainty relation means, “If you can predict an measurement’s outcome precisely, you can’t predict a measurement’s outcome precisely, and vice versa. The uncertainties must multiply to at least some number. The number depends on how much fails to commute with .” The higher an uncertainty bound (the greater the inequality’s right-hand side), the more the operators disagree.
Heisenberg and Robertson explored operator disagreement during the 1920s. They wouldn’t have seen eye to eye with today’s QI theorists. For instance, QI theorists consider how we can apply quantum phenomena, such as operator disagreement, to information processing. Information processing includes cryptography. Quantum cryptography benefits from operator disagreement: An eavesdropper must observe, or measure, a message. The eavesdropper’s measurement of one observable can “disturb” a disagreeing observable. The message’s sender and intended recipient can detect the disturbance and so detect the eavesdropper.
How efficiently can one perform an information-processing task? The answer usually depends on an entropy , a property of quantum states and of probability distributions.Uncertainty relations cry out for recasting in terms of entropies. So QI theorists have devised entropic uncertainty relations, such as
The entropy quantifies the difficulty of predicting the outcome of an measurement. is defined analogously. is called the overlap. It quantifies your ability to predict what happens if you prepare your system with a well-defined value, then measure . For further analysis, check out this paper. Entropic uncertainty relations have blossomed within QI theory over the past few years.
Pure QI theorists, we’ve seen, quantify operator disagreement with entropic uncertainty relations. Physicists at the intersection of condensed matter and high-energy physics prefer out-of-time-ordered correlators (OTOCs). I’ve blogged about OTOCs so many times, Quantum Frontiers regulars will be able to guess the next two paragraphs.
Consider a quantum many-body system, such as a chain of qubits. Imagine poking one end of the system, such as by flipping the first qubit upside-down. Let the operator represent the poke. Suppose that the system evolves chaotically for a time afterward, the qubits interacting.Information about the poke spreads through many-body entanglement, or scrambles.
Imagine measuring an observable of a few qubits far from the qubits. A little information about migrates into the qubits. But measuring reveals almost nothing about , because most of the information about has spread across the system. disagrees with , in a sense. Actually, disagrees with . The represents the time evolution.
The OTOC’s smallness reflects how much disagrees with at any instant . At early times , the operators agree, and the OTOC . At late times, the operators disagree loads, and the OTOC .
Different camps of physicists, we’ve seen, quantify operator disagreement with different measures: Today’s pure QI theorists use entropic uncertainty relations. Condensed-matter and high-energy physicists use OTOCs. Trust physicists to disagree about what “quantum operator disagreement” means.
I want peace on Earth. I conjectured, in 2016 or so, that one could reconcile the two notions of quantum operator disagreement. One must be able to prove an entropic uncertainty relation for scrambling, wouldn’t you think?
You might try substituting for the in Ineq. , and for the . You’d expect the uncertainty bound to tighten—the inequality’s right-hand side to grow—when the system scrambles. Scrambling—the condensed-matter and high-energy-physics notion of disagreement—would coincide with a high uncertainty bound—the pure-QI-theory notion of disagreement. The two notions of operator disagreement would agree. But the bound I’ve described doesn’t reflect scrambling. Nor do similar bounds that I tried constructing. I banged my head against the problem for about a year.
The sky brightened when Jason and Tony developed an interest in the conjecture. Their energy and conversation enabled us to prove an entropic uncertainty relation for scrambling, published this month.1 We tested the relation in computer simulations of a qubit chain. Our bound tightens when the system scrambles, as expected: The uncertainty relation reflects the same operator disagreement as the OTOC. We reconciled two notions of quantum operator disagreement.
As Quantum Frontiers regulars will anticipate, our uncertainty relation involves weak measurements and quasiprobability distributions: I’ve been studying their roles in scrambling over the pastthreeyears,with colleagues for whose collaborations I have the utmost gratitude. I’m grateful to have collaborated with Tony and Jason. Harmony helps when you’re tackling (quantum operator) disagreement—even if squabbling would spice up your paper’s backstory.
Yoram Alhassid asked the question at the end of my Yale Quantum Institute colloquium last February. I knew two facts about Yoram: (1) He belongs to Yale’s theoretical-physics faculty. (2) His PhD thesis’s title—“On the Information Theoretic Approach to Nuclear Reactions”—ranks among my three favorites.1
Over the past few months, I’ve grown to know Yoram better. He had reason to ask about quantum statistical mechanics, because his research stands up to its ears in the field. If forced to synopsize quantum statistical mechanics in five words, I’d say, “study of many-particle quantum systems.” Examples include gases of ultracold atoms. If given another five words, I’d add, “Calculate and use partition functions.” A partition function is a measure of the number of states, or configurations, accessible to the system. Calculate a system’s partition function, and you can calculate the system’s average energy, the average number of particles in the system, how the system responds to magnetic fields, etc.
My colloquium concerned quantum thermodynamics, which I’ve bloggedabout many times.So I should have been able to distinguish quantum thermodynamics from its neighbors. But the answer I gave Yoram didn’t satisfy me. I mulled over the exchange for a few weeks, then emailed Yoram a 502-word essay. The exercise grew my appreciation for the question and my understanding of my field.
An adaptation of the email appears below. The adaptation should suit readers who’ve majored in physics, but don’t worry if you haven’t. Bits of what distinguishes quantum thermodynamics from quantum statistical mechanics should come across to everyone—as should, I hope, the value of question-and-answer sessions:
One distinction is a return to the operational approach of 19th-century thermodynamics. Thermodynamicists such as Sadi Carnot wanted to know how effectively engines could operate. Their practical questions led to fundamental insights, such as the Carnot bound on an engine’s efficiency. Similarly, quantum thermodynamicists often ask, “How can this state serve as a resource in thermodynamic tasks?” This approach helps us identify what distinguishes quantum theory from classical mechanics.
Asking, “How can this state serve as a resource?” leads quantum thermodynamicists to design quantum engines, ratchets, batteries, etc. We analyze how these devices can outperform classical analogues, identifying which aspects of quantum theory power the outperformance. This question and these tasks contrast with the questions and tasks of many non-quantum-thermodynamicists who use statistical mechanics. They often calculate response functions and (e.g., ground-state) properties of Hamiltonians.
These goals of characterizing what nonclassicality is and what it can achieve in thermodynamic contexts resemble upshots of quantum computing and cryptography. As a 21st-century quantum information scientist, I understand what makes quantum theory quantum partially by understanding which problems quantum computers can solve efficiently and classical computers can’t. Similarly, I understand what makes quantum theory quantum partially by understanding how much more work you can extract from a singlet (a maximally entangled state of two qubits) than from a product state in which the reduced states have the same forms as in the singlet, .
As quantum thermodynamics shares its operational approach with quantum information theory, quantum thermodynamicists use mathematical tools developed in quantum information theory. An example consists of generalizedentropies. Entropies quantify the optimal efficiency with which we can perform information-processing and thermodynamic tasks, such as data compression and work extraction.
Most statistical-mechanics researchers use just the Shannon and von Neumann entropies, and , and perhaps the occasional relative entropy. These entropies quantify optimal efficiencies in large-system limits, e.g., as the number of messages compressed approaches infinity and in the thermodynamic limit.
Other entropic quantities have been defined and explored over the past two decades, in quantum and classical information theory. These entropies quantify the optimal efficiencies with which tasks can be performed (i) if the number of systems processed or the number of trials is arbitrary, (ii) if the systems processed share correlations, (iii) in the presence of “quantum side information” (if the system being used as a resource is entangled with another system, to which an agent has access), or (iv) if you can tolerate some probability that you fail to accomplish your task. Instead of limiting ourselves to and , we use also “-smoothed entropies,” Rényi divergences, hypothesis-testing entropies, conditional entropies, etc.
Another hallmark of quantum thermodynamics is results’ generality and simplicity. Thermodynamics characterizes a system with a few macroscopic observables, such as temperature, volume, and particle number. The simplicity of some quantum thermodynamics served a chemist collaborator and me, as explained in the introduction of https://arxiv.org/abs/1811.06551.
Yoram’s question reminded me of one reason why, as an undergrad, I adored studying physics in a liberal-arts college. I ate dinner and took walks with students majoring in economics, German studies, and Middle Eastern languages. They described their challenges, which I analyzed with the physics mindset that I was acquiring. We then compared our approaches. Encountering other disciplines’ perspectives helped me recognize what tools I was developing as a budding physicist. How can we know our corner of the world without stepping outside it and viewing it as part of a landscape?
1The title epitomizes clarity and simplicity. And I have trouble resisting anything advertised as “the information-theoretic approach to such-and-such.”
Merchandise spilled outside shops onto the streets, restaurateurs parked diners under trees, and ice-cream cones begged to be eaten on park benches. People thronged the streets, markets filled public squares, and the scents of flowers wafted from vendors’ stalls. I couldn’t blame the city. Its sunshine could have drawn Merlin out of his crystal cave. Insofar as a city lives, Barcelona epitomized a quotation by thermodynamicist Ilya Prigogine: “The main character of any living system is openness.”
Prigogine (1917–2003), who won the Nobel Prize for chemistry, had brought me to Barcelona. I was honored to receive, at the Joint European Thermodynamics Conference (JETC) there, the Ilya Prigogine Prize for a thermodynamics PhD thesis. The JETC convenes and awards the prize biennially;the last conference had taken place in Budapest. Barcelona suited the legacy of a thermodynamicist who illuminated open systems.
The conference center. Not bad, eh?
Ilya Prigogine began his life in Russia, grew up partially in Germany, settled in Brussels, and worked at American universities. His nobelprize.orgbiography reveals a mind open to many influences and disciplines: Before entering university, his “interest was more focused on history and archaeology, not to mention music, especially piano.” Yet Prigogine pursued chemistry.
He helped extend thermodynamics outside equilibrium. Thermodynamics is the study of energy, order, and time’s arrow in terms of large-scale properties, such as temperature, pressure, and volume. Many physicists think that thermodynamics describes only equilibrium. Equilibrium is a state of matter in which (1) large-scale properties remain mostly constant and (2) stuff (matter, energy, electric charge, etc.) doesn’t flow in any particular direction much. Apple pies reach equilibrium upon cooling on a countertop. When I’ve described my research as involving nonequilibrium thermodynamics, some colleagues have asked whether I’ve used an oxymoron. But “nonequilibrium thermodynamics” appears in Prigogine’s Nobel Lecture.
Another Nobel laureate, Lars Onsager, helped extend thermodynamics a little outside equilibrium. He imagined poking a system gently, as by putting a pie on a lukewarm stovetop or a magnet in a weak magnetic field. (Experts: Onsager studied the linear-response regime.) You can read about his work in my blog post “Long live Yale’s cemetery.” Systems poked slightly out of equilibrium tend to return to equilibrium: Equilibrium is stable. Systems flung far from equilibrium, as Prigogine showed, can behave differently.
A system can stay far from equilibrium by interacting with other systems. Imagine placing an apple pie atop a blistering stove. Heat will flow from the stove through the pie into the air. The pie will stay out of equilibrium due to interactions with what we call a “hot reservoir” (the stove) and a “cold reservoir” (the air). Systems (like pies) that interact with other systems (like stoves and air), we call “open.”
You and I are open: We inhale air, ingest food and drink, expel waste, and radiate heat. Matter and energy flow through us; we remain far from equilibrium. A bumper sticker in my high-school chemistry classroom encapsulated our status: “Old chemists don’t die. They come to equilibrium.” We remain far from equilibrium—alive—because our environment provides food and absorbs heat. If I’m an apple pie, the yogurt that I ate at breakfast serves as my stovetop, and the living room in which I breakfasted serves as the air above the stove. We live because of our interactions with our environments, because we’re open. Hence Prigogine’s claim, “The main character of any living system is openness.”
JETC 2019 fostered openness. The conference sessions spanned length scales and mass scales, from quantum thermodynamics to biophysics to gravitation. One could arrive as an expert in cell membranes and learn about astrophysics.
I remain grateful for the prize-selection committee’s openness. The topics of earlier winning theses include desalination, colloidal suspensions, and falling liquid films. If you tipped those topics into a tube, swirled them around, and capped the tube with a kaleidoscope glass, you might glimpse my thesis’s topic, quantum steampunk. Also, of the nine foregoing Prigogine Prize winners, only one had earned his PhD in the US. I’m grateful for the JETC’s consideration of something completely different.
When Prigogine said, “openness,” he referred to exchanges of energy and mass. Humans can exhibit openness also to ideas. The JETC honored Prigogine’s legacy in more ways than one. Here’s hoping I live up to their example.