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Quantum mechanics

Quantum mechanics

Decoding the quantum horizon

05 Jan 2017
Taken from the January 2017 issue of Physics World

Patrick Hayden and Robert Myers describe how the study of “qubits”, quantum bits of information, may hold the key to uniting quantum theory and general relativity into a unified theory of quantum gravity

Scientific reflections
Scientific reflections: Participants of the It from Qubit summer school discuss theories by the Perimeter Institute’s reflecting pool. (Courtesy: JB Park)

In 1990 the distinguished theoretical physicist John Wheeler coined the phrase “it from bit” to encapsulate a radical new view of the universe that he had been developing over the preceding 20 years:

“It from bit symbolizes the idea that every item of the physical world has at bottom…an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes-or-no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin.”

In other words, what Wheeler proposed is that at the most fundamental level, all of physics has a description that can be articulated in terms of information. While Wheeler’s scientific career ran from early work with Niels Bohr on nuclear fission in the 1930s to quantum electrodynamics, general relativity and the foundations of quantum mechanics, this radical idea received little support at the time. However, in hindsight, we can now see that it was truly visionary.

Fast-forward a quarter of a century and a modernized version of Wheeler’s idea is now taking shape. Quantum-information science, which aims to develop new ultrafast computers based on the principles of quantum theory, is forming an exciting confluence with high-energy theory, which studies the elementary subatomic particles and the fundamental forces in nature.

One twist is that in the time since Wheeler’s original work, our understanding of information in quantum mechanics has advanced tremendously. While Wheeler emphasized bits, it appears that intrinsically quantum-mechanical forms of information – now known as “qubits” – are more fundamental. In recent years a growing number of theorists have been exploring whether these curious quanta of information may hold the answer to combining quantum theory and general relativity into a quantum theory of gravity.

Language problem

Despite this exciting convergence, high-energy physics and quantum information theory remain distinct disciplines and communities. Both are mature fields that have grown and developed to study their own problems. A primary challenge to interdisciplinary co-operation, though, is that as knowledge builds, so too does the language that encodes that knowledge. Specialization creates communities with their own dialects and tools. So when a physicist steps from their area of expertise and into another, they can easily get lost in a thicket of unfamiliar terms and descriptions, even before grappling with the new physical principles and phenomena.

In August 2015, with the support of the New York City-based Simons Foundation, the “It from Qubit” collaboration was formed to build new bridges promoting communication and collaboration between the two research communities. We chose the name both as a homage to Wheeler and, by replacing “bit” with “qubit”, to emphasize the crucial role of entirely new ideas and techniques that would surely have surprised and delighted him.

Consisting of 17 senior researchers from the US, Canada, the UK, Japan, Israel and Argentina, plus a growing team of postdoctoral fellows, the collaboration is trying to answer an ambitious list of questions. Is space–time held together by quantum entanglement? Does quantum gravity allow information processing even more powerful than quantum computers? Is there a connection between computational complexity and the principle of least action? The list goes on, but the goal of the collaboration is not just to make progress on these specific problems. Perhaps even more importantly, it aims to motivate, attract and train a broader community of scientists to work at the interface of quantum information and high-energy physics.

Scanning new horizons

A first indication of the new information-based perspective that is a key focus of the It from Qubit collaboration is found by rewinding all the way back to 1972. At that time, Wheeler’s graduate student Jacob Bekenstein used various thought experiments to argue that in a black hole, the area A of the event horizon – the surface of “no return” dividing the interior and the exterior – should be equal to its entropy, S, a quantity whose usual interpretation is purely statistical. This suggestion hinted at some unseen microscopic structure for black holes and hence ran contrary to the common wisdom of the day that black holes were simply elegant classical geometries solving the equations of Albert Einstein’s general relativity. Bekenstein’s idea originally met with strong opposition, but it was vindicated just a few years later when Stephen Hawking showed that the laws of quantum mechanics require black holes to radiate much like black bodies at a finite temperature, and indeed possess entropy just like ordinary thermal systems. The Bekenstein–Hawking formula S = A/4G (where G is the gravitational constant) is now widely regarded as one of the most remarkable discoveries in fundamental physics. With hindsight, we can also see that this elegant formula was the first hint of a connection between information and the structure of space–time, as it encodes information about the statistical-mechanical microstates comprising the black hole in the geometry of the space–time itself.

Common tongue

One big puzzle, though, was an apparent mismatch. In standard thermal systems, entropy is proportional to volume but a black hole’s entropy is only proportional to its area. This is one feature that made the Bekenstein–Hawking entropy so unusual. Moreover, when gravity is taken into account, this strange proportionality of entropy with area infects ordinary matter as well. That is, trying to put too much matter and, hence, entropy into a given volume leads to gravitational collapse and the creation of a black hole, whose entropy is, once again, only proportional to its area. As a result, the most entropy, and hence information, that can ever be packed into a given region of space is proportional to the region’s area, not its volume.

In 1984 Rafael Sorkin made headway on this puzzle when studying quantum correlations in quantum field theory. He found that entropy provided a measure of the correlations between degrees of freedom in different regions and that the largest contribution was, in fact, proportional to the area of the boundary separating the two regions, a result very reminiscent of the Bekenstein–Hawking entropy. With modern developments, we can recognize Sorkin’s calculation as evaluating a quantity known in the quantum-information community as “entanglement entropy”. This concept has become central to the discussion of the quantum physics of black holes.

Holographic harbinger

The Bekenstein–Hawking formula motivated other pioneers, such as Gerard ‘t Hooft and Leonard Susskind, to begin advocating for a “holographic” formulation of quantum gravity. As with regular optical holograms, this is the idea that the information in a 3D volume can be encoded on a 2D surface.

In 1997 Juan Maldacena produced a realization of this holographic principle when he found a relationship between two physical theories: quantum gravity in a peculiar kind of space–time called anti-de Sitter space (AdS); and a special kind of quantum field theory called conformal field theory (CFT), in one fewer spatial dimensions. In fact, Maldacena’s “AdS/CFT correspondence” postulates that these two theories provide two different descriptions of the same physical phenomena. AdS is a peculiar space–time geometry where it is possible to stand in the middle and shine a light at the “boundary”. Even though the boundary is infinitely far away, the light beam is reflected and returns in finite time. In the AdS/CFT correspondence, the CFT can be thought of as being defined on the boundary, while the quantum-gravity theory lives on the inside, which is usually called the “bulk”. As bizarre as the correspondence may seem at first sight, it is an idea that has survived the scrutiny of thousands of theoretical physicists over almost 20 years. In recent years, this holographic duality has become the central arena for investigations into the new convergence of high-energy physics and quantum information.

Intriguingly, a generalized version of the Bekenstein– Hawking formula reappeared in a 2006 collaboration of condensed-matter theorist Shinsei Ryu and string theorist Tadashi Takayanagi. They proposed that calculating the entanglement entropy in the boundary CFT could be translated into the gravitational question of evaluating A/4G on certain special surfaces in the bulk AdS space–time. For a given region in the boundary theory, their prescription was essentially to imagine letting gravity pull the region down into the bulk geometry while keeping its edges pinned to the boundary region. The resulting surface should minimize its area in the same way a soap bubble does when pinned within a wire frame. Inserting the area of the resulting bubble into the formula A/4G then yields the entanglement entropy of the region in the boundary CFT. At the time, this provocative idea had appeared like a rabbit pulled from a magician’s hat. Over time, however, Ryu and Takayanagi’s geometric prescription passed increasingly stringent tests of its quantum-information-theoretic properties. Now we can even derive their formula from a careful translation of calculations in the boundary theory to the bulk, and a variety of new insights have emerged from carefully studying this remarkable result.

In particular, the Ryu–Takayanagi formula motivated Mark van Raamsdonk, Brian Swingle and others to start developing the idea that entanglement is key to the emergence of space–time itself. In quantum mechanics, entanglement between different particles joins them into a whole that is fundamentally more than the sum of its parts. Van Raamsdonk and Swingle speculated that the enormous amount of entanglement present in the boundary CFT was effectively stitching together the microscopic degrees of freedom of the bulk quantum-gravity theory to produce the AdS space–time geometry, something very different indeed from the sum of the boundary parts. Those initially vague speculations have quickly given way to more precise statements. Recently, van Raamsdonk and his collaborators have even managed to show that the field equations of general relativity in the bulk emerge from the structure of entanglement in the boundary theory.

While entanglement entropy remains at the forefront of the studies of quantum information and quantum gravity, a growing list of other concepts including Rényi entropy, relative entropy, quantum error correction and circuit complexity are each finding a place in this discussion.

Meeting of minds

In July last year, the Perimeter Institute for Theoretical Physics (PI) in Canada hosted the It from Qubit collaboration’s opening gambit in trying to broach some of the divides between fields. As one of our first tasks was to improve our own fluency in both languages, we decided to organize this meeting as a combination of both a workshop and a summer school.

Interest in the summer school far exceeded our expectations, to put it mildly. After doubling the originally planned enrolment and filling PI to the gills, we still had to turn away more than 200 applicants. In the end, 180 researchers from around the world converged on PI. Many of those students and postdocs who couldn’t attend physically due to space limitations did so virtually through live webcasts of the lectures and seminars.

Participants represented a wide array of fields, including quantum gravity, particle theory, condensed-matter physics, foundations of quantum theory, quantum information and computer science. They also represented a broad range of experience, from graduate students to senior professors. But every­one had a common denominator: they were there to learn.

To accommodate the wide range in backgrounds of the students and experts alike, we knew from the start that this meeting needed to be rather different from a standard conference or summer school. After lengthy discussions, we came up with a programme that offered a broad range of activities, from introductory lectures to cutting-edge research seminars. There were often two or three events running in parallel, so it was up to each “student” to decide how to participate at their own level and make the most of the meeting. In problem-solving sessions, junior graduate students could find themselves working alongside world-leading researchers, both diving into unfamiliar waters. Animated conversations involving both senior researchers and students ran the gamut from hashing out basic concepts from the morning’s lecture over lunch, to chalkboard discussions of people’s latest research ideas in front of PI’s reflecting pool.

There was a remarkable enthusiasm and energy animating the entire two weeks of the meeting. For us as organizers, the two weeks passed in something of a blur, but certain moments stand out: getting to know every nook and cranny of PI (and its temporary occupants) in a cheerfully nerdy scavenger hunt, watching our colleagues’ beautiful introductory lectures, which finally put to rest some of our most egregious confusions about each other’s fields, and some exhilarating seminars, such as the one by postdoc Daniel Harlow on AdS/CFT as a quantum error correcting code. We’ve lost track of the number of attendees who enthusiastically informed us that they expect their attendance to result in new projects and research.

One of the goals of the It from Qubit collaboration is to train a new generation of researchers to be fluent in both quantum-information science and fundamental physics, because that’s where the real progress will come from. They represent the future of the field, unquestionably. At the meeting, Ted Jacobson, a quantum-gravity expert from the University of Maryland, US, observed that much of the progress in the field is already being driven by younger researchers. “Of the 10 people who are producing the most interesting new ideas right now, sparking the field, probably eight of them are young people,” he said. “It’s fantastic, and it gives me great hope for the field…They have the benefit of all the hard work that came before and the revelations of string theory and AdS/CFT duality. And now they’re charging forward with it and really making sense of it.”

Found in translation

Is space–time built from entanglement? Are black holes nature’s most powerful computers? We don’t know yet, but, regardless, it is genuinely exciting to see ideas that were originally formulated for completely different reasons having resonance and utility; a huge number of new ideas get found in translation. It’s given us new perspectives on our own fields. Things that we thought were routine and uninteresting are revealed to be much more profound, while things that we thought were crucially important recede a little into the background. For example, nothing is more routine to an information theorist than the fact that adding noise smudges distinguishability. But in translation, that routine fact becomes an energy constraint on the space–times that can emerge from quantum mechanics. The routine becomes profound. The process of translation is still in its infancy but the growing community of bilingual researchers is rapidly accelerating the pace of progress.

  • Video recordings of the lectures at the It from Qubit summer school, along with problem sets and solutions, can be found online

Quantum gravity: Einstein meets Schrödinger

Abstract blue, purple, white and black image consisting of small squares and wiggly lines

Quantum theory provides us with a description of physical phenomena on the scale of molecules, atoms and smaller in terms of wavefunctions and probabilities. In contrast, Einstein’s general theory of relativity gives an elegant geometric description of gravity, which dominates the physics of our universe at very long scales. Together, quantum theory and gravity provide us with two remarkably successful and yet exceptionally dissimilar descriptions of the universe.

When one tries to merge quantum theory with gravity, a puzzling new fundamental length scale appears: l2P = ħG/c3, as observed by Max Planck in his seminal 1899 paper (Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin 5 440), where ħ is Planck’s constant divided by 2π, G is the gravitational constant and c is the speed of light in a vacuum. This “Planck length” takes an incredibly small value: approximately 10–35 m – some 25 orders of magnitude smaller than the size of an atom, and some 17 orders of magnitude smaller than the smallest distances probed in today’s best experiments. Hence, the study of quantum gravity is usually seen as an exotic regime within the domain of high-energy physics, which aims to understand the fundamental building blocks of our physical universe at the smallest scales. Here the Planck length becomes a central challenge to formulating a consistent theory of quantum gravity as space–time itself experiences dramatic fluctuations at these distances.

The challenge of combining quantum theory and gravity into a single consistent framework has eluded theoretical physicists for over 80 years now. However, in recent years, high-energy theorists have been making exciting progress by borrowing techniques and concepts originally developed in the study of quantum information. The latter is primarily interested in harnessing the weirdness of quantum theory to develop new systems for ultrafast computation and ultra-secure communications. However, it has become clear that quantum information theory is also a powerful new lens through which to examine the conundrums of quantum gravity.

How entanglement is key to the quantum world

Image of three concentric circles overlapping with another set of three concentric circles

No matter how many internal degrees of freedom are in a quantum system, it is described by just one global wavefunction or state. Hence, it is generally impossible to assign a precise state to the individual parts of a quantum system. Even when the global state is certain, the individual parts will be correlated and uncertain, a phenomenon known as quantum entanglement. This feature was first highlighted in a famous thought experiment by Albert Einstein, Boris Podolsky and Nathan Rosen (1935 Phys. Rev. 47 777). Their experiment involved measuring the polarization of a pair of photons in what we now call an entangled state, and finding that the results were correlated no matter how far apart the measurements were made.

While the aim of their paper was to expose a flaw in quantum mechanics, we now understand that entanglement is one of the key features distinguishing quantum physics from classical physics. Entanglement has been found to be a powerful resource for cryptography and a necessary prerequisite for ultrafast quantum computations. Moreover, it is an essential characteristic underlying many new exotic phases and phenomena in condensed-matter physics. And, as we hope our article conveys, entanglement has taken a role of increasing importance in quantum field theory and quantum gravity in recent years.

Because of the connection between entanglement and uncertainty, the amount of entanglement can be measured in terms of a kind of entropy. In statistical thermodynamics, one is generally uncertain about the global state of the system, which might be represented by a microcanonical or canonical ensemble. The associated entropy is then S = –kB Σpi log pi where pi is the probability that the system will be found in the i’th microstate, and kB is Boltzmann’s constant.

In John von Neumann’s extension of this formula to quantum theory, the probabilities pi are replaced by the eigenvalues of the corresponding density matrix. There is no reason to restrict this quantum formula to the standard ensembles of statistical mechanics, however. In an entangled quantum system, the density matrix of a subsystem can have a non-zero entropy even when the whole has no entropy at all. That subsystem entropy is known as entanglement entropy. In a many-body system that can be partitioned in many different ways, the associated collection of entanglement entropies gives a detailed map of the correlations between subsystems. Rafael Sorkin was the first to examine partitioning the degrees of freedom in a quantum field theory by considering different spatial regions and as described in the main text, this led him to suggest that entanglement entropy may be the origin of the Bekenstein–Hawking formula.

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