by Seth Lloyd
There is nothing impossible about this kind of oppositeness. It is just that in the classical world the brothers must share a bit of information for each possible question that can be asked. In the quantum version of this story (“Two entangled spins walk into a bar . . .”), the two entangled spins share one and only one quantum bit, and yet they are capable of giving opposite answers to an infinite variety of questions corresponding to the infinite set of possible axes about which they can be measured. Spooky.
The Quantum-Measurement Problem
Quantum measurement is a process during which one quantum system gets information about another. In the case of the particle and the detector in the double-slit experiment, for example, let |left> and |right> be the states (waves) in which the particle goes through either the left or the right slit, respectively, and let |click> and |no click> be the states (waves) in which the detector clicks or does not click. Let |ready> be the state of the detector before the measurement takes place, in which it is ready to detect the particle if the particle goes through the right slit. Just before the measurement, the particle is in the superposition state |left> + |right> and the detector is in the state |ready>. During the measurement, in the |left> part of the superposition the particle goes through the left slit and the detector does not click, while in the |right> part of the superposition the particle goes through the right slit and the detector clicks. Just after the measurement, then, the state of the particle and detector is the superposition |left, no click> + |right, click>. That is, particle and detector are in an entangled state that is a superposition of particle-through-left-slit correlated with no-click-from-detector and particle-through-right-slit correlated with click-from-detector.
Suppose I am in the room while the double-slit experiment is performed, and either I hear a click or I don’t. I, too, am a quantum system, albeit one composed of many pieces. Let |Seth hears click> be the wave that corresponds to my hearing a click and |Seth doesn’t hear click> be the wave that corresponds to my not hearing a click. (Note that these waves are quite complicated waves, corresponding to all the atoms in my body.)
After the sound (if any) has reached my ear, the state of the system comprising the particle, the detector, and me is |left, no click, Seth doesn’t hear click> + |right, click, Seth hears click>. I have become entangled with the particle and the detector. In this entangled state, you can see that my state relative to the particle being on the right and the detector clicking is |Seth hears click>. My state relative to the particle being on the left and the detector not clicking is |Seth doesn’t hear click>. The quantum tree fell in the quantum forest, and there was a quantum someone there to hear it.
This “relative state” picture of the quantum-measurement process illuminates the phenomenon of measurement. The information about which slit the particle went through infects first the detector, then me. If I write you a letter telling you whether or not I heard a click, then when you receive the letter your relative state will reflect what happened: |Seth wrote me to say he heard a click> or |Seth wrote me to say he didn’t hear a click>. Now you have become entangled with the particle, the detector, and me. After the measurement, the information about its outcome spreads to infect whatever it comes into contact with.
Figure 10. Schrödinger’s Cat
Two buttons are substituted for the two slits in the double-slit experiment. If the first button is pressed a cat is fed milk; if the second is pressed, the same cat is fed poison instead. But the quantum-mechanical particle presses both buttons: Schrödinger’s cat is both alive and dead at the same time!
Despite the fact that the relative-state picture of quantum measurement illuminates the phenomenon, there is something disquieting about it. When I hear a click, what happens to the other part of the superposition, in which I didn’t hear the click? Is the person who didn’t hear a click still me? Did I both hear and not hear a click at the same time? The unnerving aspect of this picture was amplified in 1935 by the Austrian physicist Erwin Schrödinger, who imagined that when the detector clicks, it trips a mechanism that kills a cat. In the Schrödinger’s Cat paradox, the state of particle, detector, and cat after the measurement is |left, no click, cat alive> + |right, click, cat dead>. In some weird quantum way, the cat is alive and dead at the same time.
The Schrödinger’s Cat paradox has inspired much confusion. Stephen Hawking got so tired of this paradox that he would often say (in paraphrase of Joseph Goebbels), “Whenever I hear the words ‘Schrödinger’s cat,’ I reach for my gun.” The original solution to the Schrödinger’s Cat paradox, advocated by Bohr, was that when you hear the click and the cat dies, the other part of the superposition—the one in which you don’t hear the click and the cat lives—simply goes away. This disappearance of the parts of the wave that we don’t experience is an instance of the wave-function collapse explained earlier, since the wave in effect collapses into one of its components. In the wave-function-collapse picture of quantum measurement, by the time I write to tell you that I heard a click and when I turned around the cat was dead, the part of the wave in which the cat is still alive has disappeared.
The problem with this solution, as we have seen, is that the dynamical laws of quantum mechanics are reversible. In principle, it is possible to return to the original state, the one before the measurement was made. If the wave function actually collapses, it isn’t possible to perform this reversal; however, in many cases, as in the spin-echo effect and its analogs, you can reverse the dynamical evolution of a many-piece quantum system and return it to its original state. Theory and experiment make wave-function collapse an implausible solution to the measurement problem.
Fortunately, there is a simple and elegant alternative to the wave-function collapse explanation. The measurement problem stems from the presence of those parts of the wave function corresponding to alternatives that do not actually happen. It would be nice to be able to ignore them. That is, when the detector has clicked and I have written to tell you the cat is dead, I would like to put the matter to rest and cease to worry about the parts of the wave function in which the cat is still alive. Let bygones be bygones. When are we justified in ignoring these other parts of the wave function? The answer to this question was supplied by Robert Griffiths and Roland Omnes and further elaborated by Murray Gell-Mann and James Hartle: We can ignore the other parts of the wave function at exactly the moment when they have no further effect on us.
This solution to the measurement problem depends not only on the present but on the future. If the other parts of the wave function will never again interfere with ours, then we say that the future history of the wave function decoheres. This “decoherent-histories” approach to quantum mechanics neatly resolves most of the troubling aspects of the measurement problem.
In the case of the double-slit experiment, for example, there are two possible histories. In one of them, the particle goes through the left slit and lands on the wall. In the other, the particle goes through the right slit and lands on the wall. These histories are coherent, not decoherent: they interfere with each other to create the pattern of bands on the wall.
Now add the detector to the right-hand slit. There are still two possible histories. In one of them, the particle goes through the left slit and lands on the wall. In the other, the particle goes through the right slit, trips the detector, and lands on the wall. Because of the detector, the interference pattern goes away. These histories are decoherent: they do not interfere with each other. Similarly, in the Schrödinger’s Cat paradox, once the detector has clicked and the cat is dead, looking at the cat again to see if it is still dead makes no difference for the future: the cat stays dead. The histories of that experiment are thus decoherent. In this case, we can say that the cat is either dead or alive, but not both.
There is a simple criterion for deciding whether a set of histories is coherent or decoherent. Think of what happens when you make a measurement. Measurement destroys coherence. But it can’t destroy coherence
if there is no coherence there to destroy. If making a sequence of measurements on a quantum system changes its future behavior, then the histories corresponding to the possible sequences of outcomes of the measurements are coherent. If the sequence of measurements has no effect on the system’s future behavior, the histories are decoherent. In the double-slit experiment, measurement destroys the interference pattern and changes the behavior of the system: The histories of the double-slit experiment are thus coherent.
Many Worlds
The decoherent-histories picture of quantum mechanics provides an intuitively satisfying resolution to the measurement problem. During measurement, the particle and detector become entangled and the wave function is a superposition of two states. One of these states corresponds to what “actually happens.” As long as the future history of particle and detector (and the cat and me and you) is decoherent, then the other state has no further effect.
The other state—the other part of the wave function—is in some sense still there, however, even though we can safely ignore it. This feature has led some people to advocate the so-called Many Worlds interpretation of quantum mechanics, according to which this other part of the wave function corresponds to another world, in which the cat is happily alive. The cat, say Many Worlds advocates, is really dead and really alive at the same time.
There is some division in the physics community about the Many Worlds interpretation of quantum mechanics. In 1997, I debated the issue with the Oxford physicist David Deutsch, who is a strong advocate of the Many Worlds picture. I’m not sure who won the debate in this particular world.9 For the rest of this book, I will use the “many histories” interpretation of quantum mechanics advocated by Gell-Mann and Hartle. In this interpretation, quantum mechanics supports sets of decoherent histories as described above. Out of this set, one history really happens. The remainder of the histories correspond to inaccessible pieces of the wave function. These histories correspond to possible events that didn’t really happen. (Or, as Deutsch would urge us to say, that didn’t really happen in this world.)
In my opinion, the Many Worlds picture does injustice to the word “really.” Normally, people use the word “really” to refer to things that are actually the case: I really wrote these words; you are really reading them. There are other parts of the wave function in which I wrote something else and you are watching TV. But those parts of the wave function don’t correspond to what really happened. They are like the forking paths in Borges’s story: even if they are there, they have no effect on reality.
CHAPTER 6
Atoms at Work
Talking to Atoms
New York City is full of people wandering the streets talking to the empty air. When asked what they are doing, they claim the air is talking back in voices only they can hear. One morning, when I was a graduate student in New York, I was having breakfast at the counter of the Polish coffee shop just around the corner from my apartment on Second Street at Avenue B. As I dug into my kielbasa and eggs, the man sitting next to me grabbed my arm, gazed into my eyes, and said, “They took Einstein’s brain and transplanted it into my head.”
“Really?” I said. “Then there’s some questions I need to ask you.” I proceeded to ask him his opinions on quantum mechanics and general relativity. Unfortunately, in light of his answers, the transplant seems to have been less than successful.
Einstein was somewhat confused about quantum mechanics. As noted, he never fully believed or trusted in the theory, objecting to its intrinsically chancy nature. Quantum mechanics went against his powerful intuition, just as it goes against pretty much everyone else’s, and Einstein had more right than most to trust his instincts. But Einstein’s intuition here led him astray. Quantum mechanics is inherently chancy, and there are literally millions of experiments that confirm the accuracy of the theory.
To see how quantum mechanics injects chance into the universe, it’s useful to contemplate a simple example of a throw of the quantum dice (in my job as an atomic masseur, I have some practice in the quantum crap game): Take an atom. Zap it with a laser. Now zap the atom with another laser and look to see if it emits light. If it emits light, call that 0. If it doesn’t emit light, call that 1. Half the time, the atom will emit light (a 0), and half the time it will remain dark (a 1). Thus, a brand-new bit is born.
Let’s look more closely at atom zapping, a process that allows us to talk to atoms, and to hear them talk back. Unlike my conversation with the New York street person, a real answer emerges: the atom replies to your question by emitting light, or not. To understand the significance of the atom’s answer, we must learn some of the language of atoms.
You are to an atom as Earth is to an ant: very large. Atoms are typically a few ten-billionths of a meter across—tiny, bouncy spheres held together by electricity. An atom consists of a compact nucleus (Latin for “nut”) 100,000 times smaller still, made up of protons (which are positively charged) and neutrons (lacking a charge). Most of the mass of the atom lies in its nucleus, which is surrounded by a cloud of electrons, whose masses are a couple of thousand times smaller than those of protons or neutrons. Electrons are negatively charged, so they are attracted to the positively charged nucleus; there are the same number of electrons in the surrounding cloud as there are protons in the core, so the atom as a whole is electrically neutral.
The electric force binds electrons to the nucleus. When an atom is in its normal state, called the “ground state,” the electrons are clustered as close as they can be to the nucleus. (The nucleus itself is bound together by the so-called strong force, which is 1,000 times stronger than the electromagnetic force.) What does “as close as they can be” mean? Why don’t the electrons just fall into the nucleus, bonking the atom on its nut? In fact, classical mechanics predicts just this bonking process. If classical mechanics were correct, atoms would survive for only a tiny fraction of a second before disintegrating in a burst of light. But the correct picture of atoms is given by quantum mechanics, not classical mechanics. Quantum mechanics guarantees the stability of atoms, and the stability of atoms, in turn, is one of the most concrete confirmations of quantum mechanics. Without quantum mechanics, the life of an atom would be exciting but short.
But how does quantum mechanics guarantee the stability of atoms? Recall that each electron has a wave associated with its position and velocity. The places where an electron’s wave is big are where the electron is likely to be found. The shorter the length of the wave, the faster the electron is moving. Finally, the rate at which the wave wiggles up and down is proportional to the electron’s energy.
Suppose we want to fit an electron’s wave around an atom’s nucleus. The simplest wave that can fit around a nucleus is a sphere: the wave wraps smoothly all the way around. The next simplest wave has one peak as it wraps; then comes a wave with two peaks, and so on. Each of these types of waves corresponds to an electron in a definite energy state. The simplest wave is the spherical one with no peaks; in this state, the ground state, the electron has the lowest energy. The second wave wiggles; the electron has more energy. The more peaks in an electron’s wave, the faster it wiggles and the greater its energy.
Attach a rock to a rubber band and whirl it around your head. The faster the rock moves, the more energy it has, and the farther away from your head it whirls, because the rubber band has to stretch to compensate for the additional speed of the rock. The same holds for the electron: the more energy it has, the farther away from the nucleus it orbits. The electron can snuggle up closest to the nucleus when it has its minimum energy, which occurs when it is in the simple, spherical wave, or it can orbit farther away. Wave-particle duality implies that an atom’s electrons consist of a set of discrete waves, so there are only so many orbits they can take. They never fall into the nucleus, and we can count the possible options (no peaks, one peak, two peaks, and so on).
When an electron jumps from a higher energy state to a lower one, it emits a chunk, or quantum, of light—
a photon—whose energy is equal to the difference between the energies of the two states. Atoms of different kinds—a phosphorus atom, for example, with fifteen electrons, or an iron atom, with twenty-six—emit photons with characteristic energies. Because of the correspondence between energy and the rate at which the emitted photons wiggle up and down, these photons correspond to light of a characteristic frequency. These frequencies are called an atom’s “spectrum.”
The fact that atoms emit light with characteristic spectra was observed in the first half of the nineteenth century. Because they did not know about quanta or photons, classical physicists were unable to account for these spectra. The explanation of atomic spectra was the first great triumph of quantum mechanics. By using the simple relationships between a wavelength and the speed of electrons, and between the frequency with which a wave wiggles and its energy, Niels Bohr was able to calculate the spectrum of the hydrogen atom, showing that this quantum-mechanical model agreed closely with findings observed by experiment.
