Quantum Gravity and Entanglement

Mark Van Raamsdonk

[Every year (since 1949) the Gravity Research Foundation honors best submitted essays in the field of Gravity. This year's prize goes to Mark Van Raamsdonk for his essay "Building Up Spacetime with Quantum Entanglement". The five award-winning essays will be published in the Journal of General Relativity and Gravitation (GRG) and subsequently, in a special issue of the International Journal of Modern Physics D (IJMPD). Today we present here an invited article from Prof. Raamsdonk on his current work.
-- 2Physics.com ]

Author: Mark Van Raamsdonk
Affiliation: Department of Physics and Astronomy,
University of British Columbia, Vancouver, Canada.

Quantum Mechanics and Entanglement :

The development of quantum mechanics in the early 20th century is surely one of the most remarkable achievements of mankind. Quantum mechanics is fundamentally different than the physical theories developed earlier to describe physics on macroscopic scales, yet is absolutely essential in understanding atomic scale physics. At the heart of quantum mechanics is the idea of quantum superposition: in quantum mechanics, objects can in some sense be in two places at once (or more generally two physical configurations at once). Mathematically speaking, every state of a physical system can be associated with a kind of vector, and if A and B are vectors representing two allowed physical configurations, then A + B is also an allowed physical state. In a simple example, A could be a state where an object is in one place, and B could be a state where the same object is in a different place; A + B then represents a state where a single object has no definite location. If a measurement of the object’s location is performed, no definite prediction for the result is possible; we will find it either in one location or the other, and quantum mechanics can at best predict the probability for each possible outcome.

2Physics articles by past winners of the Gravity Research Foundation award:
Alexander Burinskii (2009): "Beam Pulses Perforate Black Hole Horizon"
T. Padmanabhan (2008): "Gravity : An Emergent Perspective"
Steve Carlip (2007): "Symmetries, Horizons, and Black Hole Entropy"

Intimately related to the idea of quantum superposition is the notion of quantum entanglement. If we have a physical system with two parts (e.g. a ball and a box) then in a general quantum state, we cannot say with certainty what is the state of the ball (e.g. whether or not it is in the box) or what is the state of the box (e.g. whether the box is open or closed). But for certain quantum states, this uncertainty can be correlated for the two objects. For example, suppose A represents a quantum state where the ball is in the box and the box is closed, and B represents a quantum state where the ball is not in the box and the box is open. Then in the state A + B, neither the location of the ball nor the state of the box is definite, but a measurement which determines the state of the box effectively also determines the location of the ball: if we measure the state A+B and find the box closed, we can be sure that the ball is in the box; if we find it open, we can be sure the ball is not in the box. In this situation, we say that the ball and the box are entangled, since a measurement of one part of the system influences the quantum state of the other part of the system. In practice, it would be exceedingly difficult to prepare a macroscopic system such as a ball and box in such an entangled state, but such situations are commonplace at the atomic scale. The phenomenon of entanglement is an intrinsically quantum phenomenon; indeed, it can be shown that a computer making use of quantum entanglement can perform certain calculations far faster than any ordinary computer; entanglement is the basic property of quantum systems that allows quantum computation.

Strange as they may seem, the rules of quantum mechanics have now been tested beyond any reasonable doubt and allow us to understand physical processes in nature with incredible precision. For certain properties of elementary particles, predications based on quantum mechanics have been shown to be correct to one part in 100,000,000 or better. We now have a fully quantum mechanical description (known as quantum field theory) for the strong, weak, and electromagnetic forces, that allows us to understand how these interactions operate even at distance scales 100,000,000,000,000 times smaller than we can resolve with our eyes.

Quantum Gravity :

The approach that allowed physicists to develop a quantum mechanical theory for the strong, weak, and electromagnetic forces turns out not to work when applied to the remaining force, the force of gravity. In fact, it fails miserably. As a result, finding the correct quantum mechanical theory of gravity has been a prominent open question for decades; indeed it is one of the greatest challenges in theoretical physics. While Einstein’s Theory of General Relativity is almost entirely adequate for the purposes of describing the observed gravitational dynamics of planets, stars, galaxies, and even the expansion of the universe as a whole, it cannot be the whole story, since it does not incorporate the quantum mechanics principles that are believed to underlie all physics in our universe. Usually, a quantum mechanical description of nature is only necessary at very short distance scales; at macroscopic distance scales, the pre-20th century ``classical’’ physics provides an excellent approximation. But there are certain situations, such as in the interior of a black hole, in the early universe just after the big bang, or in a hypothetical scattering of particles with energies many orders of magnitude larger than we can currently produce in an accelerator, where gravitational effects would be important at distance scales small enough that a quantum mechanical description of the physics is essential. Finding the right theory of quantum gravity is essential if we want to fully understand the workings of nature.

String theory and the AdS/CFT correspondence :

One example of a theory that is fully quantum-mechanical but also includes gravitational physics is provided by string theory. Until the mid 1990s, the mathematical description of string theory was such that it allowed only relatively simple calculations; for example, one could predict the results for scattering of a fixed number of particles (including gravitons) on some fixed spacetime background (e.g. flat spacetime). This was not an entirely satisfactory situation. We recall that in Einstein’s theory of gravity, space itself is a dynamical entity that can be curved or warped by matter and energy; it is the effect of this warping on other objects that gives rise to gravitational ``forces.’’ In a complete theory of quantum gravity, different quantum states should correspond to spacetimes with different geometries (i.e. different warpings); the original formulation of string theory could most readily describe only different types of particles on a fixed geometry.

The situation for string theory changed dramatically between 1995 and 1997 in what is now known as “the Second Superstring Revolution.” (The first revolution was the period in the mid 1980s when it became clear that the original formulation of string theory was mathematically consistent.) This period culminated in a stunning proposal by Juan Maldacena known as the AdS/CFT correspondence, or gauge theory / gravity duality. (This followed an earlier proposal of the same nature by Tom Banks, Willy Fischler, Steve Shenker, and Lenny Susskind). The proposal states that there is an exact equivalence between certain examples of string theory (full-fledged theories of quantum gravity) and certain ordinary quantum mechanical systems without gravity (often quantum field theories). These much simpler ordinary quantum mechanical systems suffer none of the restrictions found in the original formulation of string theory, and thus, via the equivalence, may be used to provide a complete formulation of the corresponding string theory, able to quantum mechanically describe gravity and other forces on a spacetime which can fluctuate dynamically. Remarkably, this much better formulation of string theory turns out to be no more complicated than the quantum mechanical description of the other forces, completely understood almost half a century ago.

Geometry from Entanglement :

According to the AdS/CFT correspondence, there must be a dictionary that allows us to associate to every state of some conventional quantum mechanical system a state of the corresponding equivalent quantum gravity theory. Different states in the quantum mechanics correspond to different spacetime geometries (i.e. different distributions of matter and a different warping of space). For example, the quantum state A might correspond to completely empty space, while the state B corresponds to space with some gravitational waves, and state C corresponds to a space with orbiting black holes. While the dictionary between quantum state and corresponding spacetime is known for very simple states, more generally the correspondence is far from obvious. Ideally, one would like to know the gravity interpretation for an arbitrary quantum state of the conventional system; understanding the general dictionary is a crucial open question for the field.

The central suggestion in my essay [1] is that crucial information about what the spacetime associated to a given quantum state looks like is contained in how the various parts of the ordinary quantum are entangled with each other in the given state. While the arguments rely on some specific results in string theory, it is not difficult to give some sense of where the idea comes from.

To start, suppose that a specific quantum system has a corresponding gravity theory such that each state of the system corresponds to some spacetime. Now consider a second quantum system, which we obtain by taking two copies of the first system (with no physical interactions between the two systems). For the larger system, the simplest states are those with no entanglement between the two parts. That is, we can consider a state A = (A1,A2) in which the first system is in state A1 and the second system is in state A2. Now A1 and A2 each correspond to some particular spacetime according to the AdS/CFT correspondence. Thus, we can interpret the state A of the larger system as corresponding to two completely disconnected spacetimes (imagine our universe and some parallel universe with which there is no possible communication).

More generally, we can consider states which are quantum superpositions such as (A1,A2) + (B1,B2) . For such states, there is entanglement between the two parts. In [1], based on various earlier works, I pointed out that for states with enough entanglement (certain states which are quantum superpositions (A1,A2) + (B1,B2) + (C1,C2) + … with many states in the superposition) the resulting complicated state can be interpreted as a single connected spacetime, in which two distinct parts are connected by something like a wormhole (or a black hole/white hole). Since all the individual states in the superposition had interpretations as disconnected spacetimes, we can say that a quantum superposition of disconnected spacetimes has produced a connected spacetime. Alternately, we can say that by entangling the two parts of our original quantum system, we have managed to connect up two parts of the corresponding spacetime.

Starting from this hint of a connection between entanglement and spacetime geometry, one can argue that more quantitative measures of entanglement in states of a quantum system give direct information about quantitative geometrical quantities in the corresponding spacetimes, such as areas and geodesic distances. The complete picture for how to deduce the spacetime associated with a particular state in the AdS/CFT correspondence is certainly still beyond our reach, but I believe these connections between entanglement and geometry may be an important part of the story. If correct, they suggest a deep connection between quantum gravity and quantum information theory (the natural setting for studies of entanglement in quantum systems) that may be of fundamental importance.

References
[1] Mark Van Raamsdonk, “Building up spacetime with quantum entanglement,” arXiv:1005.3035. Link.
[2] Nielsen, M.A., Chuang, I.L., “Quantum Computation and Quantum Information” (Cambridge University Press, Cambridge, 2000).
[3] Juan Maldacena, “The Illusion of Gravity” -, Scientific American, November 2005. Link.
[4] Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory" (Vintage Series, Random House Inc, February 2000).