Entropy, Quantum Information and Entanglement
Professor, Department of Physics and Astronomy
One of the few clues to the connection between quantum mechanics (the physical theory of elementary particles, atoms and radiation) and the theory of general relativity (the theory of gravitation and cosmology), is the remarkable prediction of so-called "Hawking Radiation" from black holes.1 According to the Hawking calculation, black holes – regions of space from which no information can escape – are fundamentally unstable and evaporate, radiating away their mass and energy. And even though black holes have a physical temperature of exactly absolute zero, the radiation emitted by black holes is that of a perfect thermal body at a temperature proportional to 1/M, where M is the mass of the black hole. In fact, the temperature actually increases as the black hole loses mass and evaporates!
Since the radiation is essentially thermal noise, it carries no information about the internal state of the black hole and is therefore consistent with the definition of a black hole. Moreover, it seems that the thermodynamic analogy is quite complete: black holes are believed to possess an entropy, or “degree of disorder,” just as any ordinary body at a nonzero temperature would, even though they are at absolute zero physical temperature and have no internal structure whatsoever.
Horizons and Thermodynamics
It is now appreciated that black hole entropy is a particular example of a more general phenomenon known as horizon entropy. A horizon is the boundary between two regions of the universe that are causally out of contact. Information about events in one region cannot propagate, even in principle, to the other region.
Remarkably, all horizons must radiate random thermal energy and possess entropy proportional to their surface area, even though the horizon itself and its surroundings are simply empty space –a vacuum – at zero temperature!
To appreciate this strange result, it is useful to construct a horizon without the complications of black holes and general relativity. For example, if a driver heading eastbound on the Long Island Expressway accelerates uniformly (suppose the speed of the car increases by 5 mph every second), the driver has actually created a space-time horizon about three light years west of the car. As long as the driver continues to accelerate, the state of affairs in the universe three light years to the west will always be unknown to the driver (see Figure 1). In effect, the driver is disconnected from that part of the universe by a horizon. Looking in the westward direction at the space-time horizon, the driver would detect, in principle, the faint radiant heat and entropy of a thermal body at a temperature less than a billionth of a billionth of a degree above absolute zero – horizon entropy and temperature.
The fact that horizon entropy behaves in most respects just like “textbook” thermodynamic entropy – the entropy accorded molecules moving randomly about – is a deeply mysterious fact. For instance, horizon entropy combined with ordinary thermodynamic entropy obeys a generalized second law of thermodynamics, which states that the total entropy of the universe must increase or stay constant. But the horizon is not a physical object or collection of things, such as a gas of molecules – it is simply an empty piece of space. And it is not even a distinguished empty piece of space in the universe: the other non-accelerating drivers on the Long Island Expressway would observe that particular piece of space as unremarkable. What is the entropy, and exactly where is it, if there is nothing but empty space at the horizon?
Unfortunately, there are no satisfactory answers to these questions. But a striking connection between two very different fields of physics has emerged in the last several years that gives a new perspective to these questions – even if no ready answers are available. The new perspective about black holes, horizons and general relativity surprisingly comes from the condensed matter physics field of quantum computing and quantum information theory.
Quantum Information and Entanglement Entropy
The central property common to these disparate topics is known as quantum entanglement. When the concept of “information” (a computer document, for example) is generalized so that it is consistent with the fundamental physical laws of quantum mechanics, the appearance of long-range correlations or entanglements between bits of information is inevitable. Suppose that the information contained in one “bit” (a 0 or a 1) is represented by the quantum mechanical state of an atom. For example, if the atom is in the lowest energy level (the ground state) then it’s a “0”; if it is in the first excited state, then it’s a “1.” The information is a property of that atom. If we consider two atoms, there seem to be four possible states: 00, 11, 01 and 10. But, according to quantum mechanics, other states are possible as well: the two atoms may be in a superposition of states in which the excited state (and ground state) is “shared” between the two atoms. Such a state might be written as 01+10; the + sign does not mean the addition of 01 and 10, but rather that a measurement of the atoms would reveal one of these two possibilities (01 or 10) with equal probability. It is crucial to realize that the information, in such a quantum state, does not belong to either atom but is held jointly and the individual states of the atoms are said to be entangled. Moreover, the atoms might be separated from one another by a huge distance while their relationship – that one atom holds a 0 and the other a 1 – is rigidly locked. That is, a measurement made on one atom will reveal the state 0 or 1 with equal probability; if the result is 1, then the other atom instantly assumes the definite state of 0 (and vice versa). In this way, information may no longer be associated with a particular point in space, and the information is said to be non-local.
Another peculiar feature of information emerges when it is made to conform to the laws of quantum mechanics: according to quantum mechanics, information cannot be copied with perfect fidelity. However, information can be moved by destroying it in one place and creating it in another – a process now called “teleportation.” Teleportation first involves the sender (Alice) and receiver (Bob) sharing an entangled quantum state (Figure 2). For instance, using the example of Alice and Bob, the state S = 01+10 is created, and Alice and Bob each take one of the atoms. To send information to Bob, Alice must entangle the information state she wishes to send with her part of the entangled resource she shares with Bob. Then, after making a particular measurement on the combined quantum system she now holds, the information disappears from Alice’s atoms and instantaneously appears in Bob’s atoms.2 Classical computers do not have the features I have described above because their information bits are being continuously reset into their determined states (this is, in part, what runs down the batteries on your laptop). Entanglement is the quantum counterpart of a wire, or channel, connecting sender to receiver, through which information is sent. Like all wires, it has an intrinsic capacity to carry information; for instance, the CAT 5 patch cord that connects your computer to the wall jack can carry about 100 million bits of information every second. Trying to send more information would result in the corruption of some of the information that you are trying to send. Unsurprisingly, the capacity of a quantum channel is simply the number of pairs of bits, entangled in the manner of state S, that are shared between sender and receiver.
Even though entropy has the connotation of disorder and randomness, entropy and quantum information are intimately connected. Before the state S is used to teleport information, a measurement of either shared atom would yield a 0 or 1 with equal probability. Suppose a new batch of states, ready to teleport information, was cooked up and Alice decides to test some of them (for quality control) by measuring her atoms. If they all yielded the same answers (all 1s for instance), she would conclude that this batch must be uncorrelated with Bob’s counterpart atoms and it would therefore be flawed. Entanglements between quantum objects manifest themselves as fluctuations in the measured properties of those objects, and the fluctuations, in turn, are perceived as randomness or entropy when measurements are made. This entanglement entropy is a measure of the information capacity of a quantum channel. Such “man-made” entanglements have been demonstrated recently to teleport information over distances more than 100 kilometers.
Quantum Mechanics of the Vacuum: Quantum Fields
A naturally occurring kind of entanglement seems to be responsible for the fluctuations we perceive as horizon entropy. But what, exactly, is entangled in the empty space surrounding a horizon? In an old-fashioned television picture tube, electrons are accelerated through a vacuum where they strike a fluorescent screen and make a picture. When the picture tube is on, electrons move, say, to the left because every successive point in space to the left of the electron is at a higher voltage. In setting the voltage at each point in space, the vacuum has been “imprinted” with some information that tells the electron how to move. But the laws of quantum mechanics govern information, even if that information exists in empty space! Suppose we encode the instructions for the motion of an electron by a series of bits, which in turn, correspond to the voltages at each point in empty space. The voltage at each point may be thought of as an atom in either an excited state (1) or the ground state (0). A 01 combination of adjacent “voltage bits” tells the electron to move to the right; a 10 combination tells it to move to the left; and the electron does nothing at all if it encounters a 00 or 11. One possible quantum state is shown in Figure 3.
Given that the picture tube has the capacity to be used in this way – voltage bits set in a particular way to move electrons around as needed – what is the natural state of the voltage bits when the picture tube is “off ”? That is, what is the state of the voltage bits when we haven’t applied any voltages and set them to any particular values? This is what physicists refer to as the quantum state of the vacuum (and it is not as ridiculous as it sounds). What we call empty space – space devoid of matter or energy – corresponds to the voltage bits being in their lowest energy state. One might think that the lowest energy quantum state for the set of voltages (called a quantum field) is all 0s (or all 1s) in Figure 3 – but nothing could be further from the truth! The lowest energy state has enormous complexity and behaves as if each voltage bit is entangled with every other. Just like atoms, voltage bits are able to lower their energy by sharing their excited states with other voltage bits and producing entangled states like S. This feature follows from the Heisenberg uncertainty principle, which states that an excited state can lower its momentum and energy by making its position highly uncertain. If I denote by Sij the state in which the voltage bit at point i is entangled with the voltage bit at point j (that is, Sij=0i1j+1i0 j), the lowest energy state, F, of the quantum field is a superposition of every different way that the voltage bits can be paired into states like S. An example for four voltage bits is shown in Figure 4a. Multiplying out all such products reveals that every possible combination of 0s and 1s appear with some probability (Figure 4b).3
This is the secret life of voltages inside the dark, unplugged television tube sitting in your attic and, for that matter, the empty space all around us. Why, then, doesn’t an electron see these voltages and spontaneously begin to move randomly about in empty space? The reason is that for the electron to gain some energy and move about, the quantum field must lose some energy – but it is already in the lowest energy state so it can’t lose any more. Suppose we make an imaginary partition in the middle of an empty box, with Alice in possession of the voltage bits on the left and Bob in possession of those on the right (see Figure 4b). If Bob measures and determines the values of his bits, Alice’s bits instantaneously assume their correlated values. The resulting energy of the part of the quantum field belonging to Alice will sometimes have a large energy and sometimes a small energy depending on the random outcomes of Bob’s measurements. An electron placed on Alice’s side would move randomly, exchanging energy with the random energy changes of the quantum field. The random motion is effected only through those entangled pairs of voltage bits straddling the partition, and the number of such pairs is proportional to the surface area of the partition. From Alice’s perspective, the random energy and motion is ascribed to the electron being in thermal contact with a body possessing temperature and entropy – the imaginary partition of the box.
If the word “horizon” replaces “partition,” this argument explains horizon entropy! Unfortunately, such a jump has little physical basis until the connections between quantum mechanics and general relativity are better understood. However, this connection has an interesting consequence as far as information and computation is concerned: each entangled pair of voltage bits straddling a boundary is capable of teleporting one bit of information through the boundary. Since the number of such bit pairs is proportional to the area of the boundary, the maximum amount of information you can store in your computer is limited by the surface area (not the volume) of your computer! For a laptop computer, the theoretical maximum is about 1069 bits.
It is clear that entanglement entropy is a highly non-local quantity as it involves the knowledge of the quantum state of fields at every point in space. In contrast, more familiar local quantities such as energy and momentum are properties of individual particles or points in space. My research in the past several years has been to develop a local picture for entanglement entropy. Even though computing entanglement entropy across a horizon depends on knowing the quantum fields everywhere on either side of the horizon, the answer, paradoxically, only depends upon the surface area of the horizon. The area factor comes from enumerating all of the entangled pairs in the ground state of the quantum field that straddle the horizon. A few years ago, I realized that the kind of quantum operator that counts these pairs is closely related to what is known as a boundary changing operator. One might picture the action of such an operator as the creation of a twist in a fabric ribbon that is held at both ends. To create a twist, one end of the ribbon must be released and twisted; then the twist may be pushed to any location. Although the location of the twist is seemingly well localized in one place, its creation must have involved manipulation of the field all the way out to the boundary. However, in a two-dimensional space-time, an exact transformation (known as a duality) exists that can turn a non-local, boundary-changing operator into a local one. Through this transformation, the entanglement entropy of a gas of photons (the “voltage” bits) in a two-dimensional space-time is turned into the conventional thermal entropy of a one dimensional gas of “boundary-changing particles.” The original entangled photons do not interact with one another, but the boundary-changing particles interact very strongly with one another, giving rise to a very strange “boundary” thermodynamics from which I was able to derive the entanglement entropy.4
My results were confirmed computationally (about a year after I published them) by Ingo Peschel and colleagues in Berlin.5 However, it seems that the duality trick only works in a two-dimensional space-time, precisely because the infinite “ribbon” can only represent one spatial dimension. Even with this limitation, this work has given a technical meaning to horizon entropy as a property of the boundary between two regions rather than the regions themselves.
2. Alice must additionally send some conventional information to Bob for him to unambiguously decode the message. Without such a feature, teleportation would allow communication faster than the speed of light.
3. The quantum state of the vacuum, as described, has been confirmed by many experiments, including the Casimir effect: the weak attraction between two metal plates in a perfect vacuum.
4. Levine, G.C. (2004). “Entanglement Entropy in a Boundary Impurity Model,” Phys. Rev. Lett., 93, 266402.
5. Peschel, I. (2005). “Entanglement Entropy with Interface Defects,” J. Phys. A: Math. Gen., 38, 4327. Zhou, J., Peschel, I., and Wang, X. (2006). “Critical Entanglement of XXZ Heisenberg Chains with Defects,” Phys. Rev., B73, 24417. Phys. Rev., D7, 2333 (1973).