by Seth Lloyd
Another implementation of Loschmidt’s proposal is the spin-echo effect. To understand the spin-echo effect, consider the following macroscopic analogy. Runners line up at the starting line of a racetrack. When the gun goes off, the runners start running laps around the track. Because they run at different speeds and some are on the inside and some on the outside, after a few laps they are spread out all around the track. After ten minutes, a second gun is fired. At the sound of the second gun, the runners turn around and start running in the opposite direction. If each runner runs at the same speed as before, they gradually begin to bunch together as they undo the earlier distance. After ten minutes they all reach the starting line together.
In the spin-echo effect, the runners are nuclear spins. The protons and neutrons that make up the nuclei of atoms spin like little tops, with spin conventionally defined as either “up” or “down,” the difference being determined by the direction of the spin: if you imagine a clock sitting faceup on a table, “spin up” is counterclockwise and “spin down” is clockwise. A convenient way to think of “up” or “down” spin is to curl the fingers of your right hand in the direction that the proton or neutron is spinning. Your thumb will then point along the axis about which it spins, and the direction your thumb is pointing in defines the “direction” of spin—that is, up or down.
Consider a bunch of protons all initially spinning in the same direction. If their spins are known, the entropy of each is zero. Now a microwave pulse sets the spins all precessing at once. (Precession is the wobbling motion a top makes under the force of gravity; nuclear spins are like little tops that wobble under the force of magnetism.) Each spin precesses at a slightly different rate and soon the spins are pointing in all directions, like the runners spread out in all positions on the track. The rate at which each spin precesses is given by its local magnetic field; this rate is “invisible” information, inaccessible to a macroscopic observer. Since the directions the spins are pointing in are now unknown, the spins, on their own, now have a high, almost maximum entropy, equal to the number of bits required to specify their direction of spin (that is, up or down) to the accuracy allowed by quantum mechanics.
The increase in entropy of the individual spins is an example of entropy increase by the spread of information. The precessing spins have become “infected” by the information in the local magnetic field. If we possessed that information, we could infer the directions in which the spins are pointing. As it is, we don’t have that information, and since the spins are becoming correlated with the magnetic field, their individual entropies are increasing.
Now comes the echo. A second microwave pulse inverts the angles that the spins have precessed; for example, an angle of +60° becomes an angle of -60°. Now as each spin precesses, it undoes the angle precessed previously. After the same amount of time it took the spins to become unknown, the spins are once again all pointing in the same direction. Their entropy has decreased back to zero.
The spin-echo effect was first demonstrated experimentally fifty years ago. There are more complicated analogs of Loschmidt’s notion, but all such analogs boil down to the same procedure. If you are a sufficiently skilled experimentalist and Boltzmann says, “Reverse them,” you can reverse them!
Why does the spin-echo effect not constitute a violation of the second law of thermodynamics? The second law says that increases in entropy cannot be reversed. In the case of the spin-echo effect, entropy has only apparently increased. Even though the entropy of the spins taken on their own increases and then decreases during the course of the echo, the underlying entropy of the spins taken together with the magnetic field remains the same.
Exorcising Maxwell’s Demon
There is a second way in which entropy can decrease. Recall that entropy is information that is unknown (that is, invisible). What happens when previously unknown information becomes known, when the invisible becomes visible? What happens when you get information? Then entropy decreases.
This mode of entropy decrease was first identified by James Clerk Maxwell, whose demon operates by getting information about the microscopic state of a volume of gas and is therefore able to decrease its entropy. Maxwell’s demon has provoked many efforts to exorcise it over the years. The full exorcism of the demon was not accomplished until recently. (I played some part in this ceremony myself.) Despite the confusion sown by Maxwell’s demon over the years, the final resolution is surprisingly simple: The underlying laws of physics preserve information. As a result, the total information/entropy of the gas and demon taken together cannot decrease.
In practice, this simple resolution involves considerable subtlety. Later, I’ll present a full quantum-mechanical model of Maxwell’s demon that will explain in detail the way in which the demon gets information and does its work. For the moment, consider a simple bit model, like those already discussed. Take two bits. The first bit, corresponding to the demon, reads 0 initially. The second bit, corresponding to the gas, can read either 0 or 1. The demon has no bits of entropy initially, and the gas has one bit.
The first step in the process of extracting entropy is for the demon’s bit to get information about the gas bit. This can be accomplished by performing a controlled-NOT operation on the demon’s bit with the gas bit as the control. The controlled-NOT flips the demon’s bit if and only if the gas bit is 1. As a result, after the operation the demon’s bit reads the same as the gas bit: they are either both 0 or both 1. That is, the demon’s bit and the gas bit possess one bit of mutual information. The demon’s bit has, in effect, measured the state of the gas bit to obtain that mutual information.
The second step of the process asks the demon to reduce the entropy of the gas. The demon can do this by performing a controlled-NOT operation on the gas bit using his own bit as control. Since the two bits are the same, the second controlled-NOT restores the gas bit to 0. If the demon’s bit is 0, he leaves the gas bit in the state 0. If the demon’s bit is 1, he flips the gas bit from 1 to 0. In either case, the gas bit is now in the state 0 and has zero bits of entropy. The demon has reduced the entropy of the gas by one bit.
The final situation is as follows. The gas bit is in the state 0. The demon’s bit is 0 if the gas bit was initially in the state 0, and is 1 if the gas bit was initially in the state 1. The two controlled-NOT operations have in effect swapped the initial bit of the demon with the initial bit of the gas. Even though the entropy of the gas has decreased by a bit, the total amount of information in the gas and the demon taken together remains constant. The demon does not violate the second law of thermodynamics.
Note that the transfer of information from the gas to the demon takes place in compliance with Landauer’s principle, discussed earlier. The demon’s goal is to “erase” the gas bit by restoring it to 0. But because the underlying laws of physics preserve information, he can restore the gas bit to 0 only by transferring the information in the gas bit to his own bit. The total information remains constant.
In a Scientific American article on Maxwell’s demon, Charles Bennett of IBM showed how Landauer’s principle prevents the demon from violating the second law of thermodynamics in extracting work from a one-particle gas.7 In a paper published in Physical Review, I showed that this argument applies not only to systems of bits but to all physical systems—heat engines, hurricanes, what have you.8 Physical dynamics can be used to get information, and that information can be used to decrease the entropy of a particular element of a system, but the total amount of information/entropy does not decrease. (The reader who is interested in demonology is directed to Harvey Leff and Andrew Rex’s two compendia of papers on Maxwell’s demon.)
If the resolution of the Maxwell’s demon problem rests simply on the foundations of the information-preserving character of physical law, why has this problem caused so much confusion over the last century and a half? The problem lies with the distinction between information and entropy. Recall that entropy is invisible information, or ignorance—informat
ion that is unavailable. But the distinction between “visible” and “invisible” depends on who is looking. It is in fact possible to decrease entropy by looking at something.
To see how the distinction between visible and invisible information plays out in the case of Maxwell’s demon, compare the perspective of a demon with that of an outsider. Like the demon, the outsider knows that the demon’s bit is originally 0 but does not know the initial value of the gas bit. Unlike the demon, the outsider cannot follow the results of the sequence of controlled-NOT operations. She knows only that this sequence of controlled-NOT operations is taking place. That is, the outsider and the demon agree on the dynamics of the interaction between the demon’s bit and the gas bit but draw the line between visible and invisible in different places. The demon’s bit after the first op is visible to the demon but not to the outsider.
Before the first controlled-NOT, the demon and the outsider agree that the entropy of the demon’s bit is zero and the entropy of the gas bit is one. After the first controlled-NOT, the demon’s bit is perfectly correlated with the state of the gas bit. That is, the demon now “knows” the value of the gas bit. More precisely, as far as the demon is concerned, the entropy of the gas bit is zero, because it is conditioned on the state of the demon’s bit after the operation, a state the demon knows. The information in the gas bit has gone from being invisible to the demon to being visible. As far as the demon is concerned, entropy has decreased by one bit, and visible information has increased by one bit.
Now consider the perspective of the outsider. After the first controlled-NOT, the outsider knows that the demon’s bit and the gas bit are perfectly correlated. They read either 00 or 11, but the outsider does not know which. Accordingly, the outsider considers the demon’s bit and the gas bit taken together to have one bit of entropy. Since the information in the gas and the demon remains invisible to the outsider, the outsider considers the entropy to have remained constant, with a value of one bit.
After the second controlled-NOT, the bit that was originally in the gas has been transferred to the demon. Both demon and outsider agree that the gas has been restored to the state 0. From the demon’s perspective, his bit is visible and registers one bit of information: the entropy is zero. From the outsider’s perspective, the demon’s bit is invisible: the entropy is one bit. Both demon and outsider agree that the total amount of information is one bit. The second law of thermodynamics applies to the total amount of information, both visible and invisible.
Maxwell’s demon completes the discussion of entropy increase and entropy decrease for the moment. At bottom, as the statistical mechanicians of the late nineteenth century showed, the world is made up of bits. The second law of thermodynamics is a statement about information processing: the underlying physical dynamics of the universe preserve bits and prevent their number from decreasing. To fully understand these physical dynamics requires us to look at quantum mechanics, which describes how physical systems behave at their most fundamental level. Before turning to quantum mechanics, however, let’s look briefly at the information-processing capacity of classical systems, such as atoms in a gas or snooker balls on a table.
Atomic Computation
The position and velocity of an atom in a gas register information. Indeed, positions and velocities of atoms were the very first quantities to which the basic formulas of information were applied. Atoms register bits.
What about processing that information? When two atoms in a gas collide, the information they register is transformed and processed. How does the information processing performed during atomic collision relate to the information processing performed by the logic gates described in the first part of this book? In fact, as pointed out by Edward Fredkin of Carnegie Mellon University and Tommaso Toffoli of Boston University, atomic collisions naturally perform AND, OR, NOT, and COPY logic operations. In the language of information processing, atomic collisions are computationally universal.
In Fredkin and Toffoli’s model, each possible atomic collision performs AND, OR, NOT, or COPY operations on suitably defined input and output bits. By assigning the proper initial positions and velocities to atoms in a gas, it is a straightforward matter to “wire up” any desired logic circuit. Atoms bouncing in a gas are, in principle, capable of universal digital computation.
In practice, of course, it is rather difficult to make a gas of atoms perform a computation. Even if we did have control over the position and velocity of individual atoms, quantum mechanics limits the accuracy to which position and velocity can be simultaneously specified. Moreover, the collisions between atoms in a gas is intrinsically chaotic; that is, a slight error in the specification of the initial positions and velocities of atoms will typically grow in time, via the butterfly effect, until it contaminates the entire computation. As will be seen in the following chapters, though, both of these limitations can be overcome by using more suitable quantum-mechanical systems to perform computation.
Although practical limitations prevent using collisions between atoms in a gas to compute, the fact that atomic collisions in principle allow computation implies that the long-term future of a gas of atoms is intrinsically unpredictable. The halting problem (see chapter 2) foils not only conventional digital computers but any system capable of performing digital logic. Since colliding atoms intrinsically perform digital logic, their long-term future behavior is uncomputable.
This computational capacity of colliding spheres throws light on the possibility of a third demon, this one evoked by the Marquis Pierre-Simon de Laplace. In an essay on using Newtonian mechanics to predict the future behavior of heavenly bodies, Laplace wrote:
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.
A being capable of performing this prodigious task of prediction is called Laplace’s demon.
Even if the underlying laws of physics were fully deterministic, however, the computational ability of simple systems such as colliding spheres implies that to perform the type of simulation Laplace envisaged, the calculating demon would have to have at least as much computational power as the universe as a whole. Since, as we shall see, computational power requires physical resources, Laplace’s demon would have to use at least as much space, time, and energy as the universe itself.
A second problem with Laplace’s demon is that the laws of quantum mechanics are not deterministic in the sense required by Laplace. In quantum mechanics, what happens in the future is predictable only in a probabilistic way. In fact, the motions of heavenly bodies are intrinsically chaotic and thus are constantly pumping information up from microscopic to macroscopic scales. As will be shown in the next chapter, the result of this celestial chaos is that even Laplace’s heavenly bodies move in a probabilistic fashion that cannot be predicted, even by a demon.
CHAPTER 5
Quantum Mechanics
In the Garden
I was standing in the garden of the Master’s Lodge at Emmanuel College, Cambridge, sipping a glass of champagne. It was the spring of 1983. My fellow graduate students and I were talking about the usual stuff of Cambridge life: boat races, May balls, and the upcoming mathematical tripos examinations that would determine our future. A stunning older woman interrupted us. “You fools!” she exclaimed, in a pronounced Spanish accent. “Don’t you see that the world’s greatest author is sitting over there with no one to talk to?” I looked where she was pointing and saw an old blind man in a white suit sitting quietly on a bench. It was Jorge Luis Borges, and the woman was his companion, Maria Kodama. She shepherded us o
ver to him.
There was a question I had always wanted to ask Borges, and at last I had the opportunity. In his story “The Garden of Forking Paths,” Borges envisions a world in which all possibilities actually happen. At each decision point, each fork in the path, the world takes not one alternative but both at once. So, Borges writes:
In the work of Ts’ui Pên, all possible outcomes occur; each one is the point of departure for other forkings. Sometimes, the paths of this labyrinth converge: for example, you arrive at this house, but in one of the possible pasts you are my enemy, in another, my friend. . . . [Ts’ui Pên] did not believe in a uniform, absolute time. He believed in an infinite series of times, in a growing, dizzying net of divergent, convergent and parallel times. This network of times which approached one another, forked, broke off, or were unaware of one another for centuries, embraces all possibilities of time. We do not exist in the majority of these times; in some you exist, and not I; in others I, and not you; in others, both of us.
“Dr. Borges,” I said, “when you wrote your story, were you aware that it mirrors the so-called Many Worlds interpretation of quantum mechanics? In this interpretation, whenever anyone makes a measurement that reveals information about the world being one way or another, the world splits in two and takes both paths. In the conventional interpretation of quantum mechanics, the Copenhagen interpretation, if I ask a nuclear particle whether it is spinning clockwise or counterclockwise, it picks one spin or the other with equal probability. But in the Many Worlds interpretation, at the moment of measurement the world’s path forks and it takes not one fork or the other but both at once.”
