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Notes: Chapter 15
1. This statement ignores hidden-variable approaches, such as Bohm's. But even in such approaches, we'd want to teleport an object's quantum state (its wavefunction), so a mere measurement of position or velocity would be inadequate. Return to Text
2. Zeilinger's research group also included Dick Bouwmeester, Jian-Wi Pan, Klaus Mattle, Manfred Eibl, and Harald Weinfurter, and De Martini's has included S. Giacomini, G. Milani, F. Sciarrino, and E. Lombardi. Return to Text
3. For the reader who has some familiarity with the formalism of quantum mechanics, here are the essential steps in quantum teleportation. Imagine that the initial state of a photon I have in New York is given by
are the two photon polarization states, and we allow for definite, normalized, but arbitrary values of the coefficients. My goal is to give Nicholas enough information so that he can produce a photon in London in exactly the same quantum state. To "do so, Nicholas and I first
Return to Text
4. In fact, the mathematically inclined reader will note that it is not hard to prove the so-called no-quantum-cloning theorem. Imagine we have a unitary cloning operator U that takes any given state as input and produces two copies of it as output (U maps 1a) —> 1a)1a), for any input state ja)). Note that U acting on a state like (1a) + j13)) yields (1a)1a) + 113)113)), which is not a two-fold copy of the original state (ja) +113))(1a) +113)), and hence no such operator U exists to carry out quantum cloning. (This was first shown by Wootters and Zurek in the early 1980s.) Return to Text
5. Many researchers have been involved in developing both the theory and the experimental realization of quantum teleportation. In addition to those discussed in the text, the work of Sandu Popescu while at Cambridge University played an important part in the Rome experiments, and Jeffrey Kimble's group at the California Institute of Technology has pioneered the teleportation of continuous features of a quantum state, to name a few. Return to Text
6. For extremely interesting progress on entangling many-particle systems, see, for example, B. Julsgaard, A. Kozhekin, and E. S. Polzik, "Experimental long-lived entanglement of two macroscopic objects," Nature 413 (Sept. 2001), 400-403. Return to Text
7. One of the most exciting and active areas of research making use of quantum entanglement and quantum teleportation is the field of quanttrn computing. For recent general-level presentations of quantum computing, see Tom Siegfried, The Bit and the Pendulum (New York: John Wiley, 2000), and George Johnson, A Shortcut Through Time (New York: Knopf, 2003). Return to Text
8. One aspect of the slowing of time at increasing velocity, which we did not discuss in Chapter 3 but will play a role in this chapter, is the so-called twin paradox. The issue is simple to state: if you and I are moving relative to one another at constant velocity, I will think your clock is running slow relative to mine. But since you are as justified as I in claiming to be at rest, you will think that mine is the moving clock and hence is the one that is running slow. That each of us thinks the other's clock is running slow may seem paradoxical, but it's not. At constant velocity, our clocks will continue to get farther apart and hence they don't allow for a direct, face-to-face comparison to determine which is "really" running slow. And all other indirect comparisons (for instance, we compare the times on our clocks by cell phone communication) occur with some elapsed time over some spatial separation, necessarily bringing into play the complications of different observers' notions of now, as in Chapters 3 and 5. I won't go through it here, but when these special relativistic complications are folded into the analysis, there is no contradiction between each of us declaring that the other's clock is running slow (see, e.g., E. Taylor and J. A. Wheeler, Spacetime Physics, for a complete, technical, but elementary discussion). Where things appear to get more puzzling is if, for example, you slow down, stop, turn around, and head back toward me so that we can compare our clocks face to face, eliminating the complications of different notions of now. Upon our meeting, whose clock will be ahead of whose? This is the so-called twin paradox: if you and I are twins, when we meet again, will we be the same age, or will one of us look older? The answer is that my clock will be ahead of yours — if we are twins, I will look older. There are many ways to explain why, but the simplest to note is that when you change your velocity and experience an acceleration, the symmetry between our perspectives is lost — you can definitively claim that you were moving (since, for example, you felt it — or, using the discussion of Chapter 3, unlike mine, your journey through spacetime has not been along a straight line) and hence that your clock ran slow relative to mine. Less time elapsed for you than for me. Return to Text
9. John Wheeler, among others, has suggested a possible central role for observers in a quantum universe, summed up in one of his famous aphorisms: "No elementary phenomenon is a phenomenon until it is an observed phenomenon." You can read more about Wheeler's fascinating life in physics in John Archibald Wheeler and Kenneth Ford, Geons, Black Holes, and Quantum Foam: A Life in Physics (New York: Norton, 1998). Roger Penrose has also studied the relation between quantum physics and the mind in his The Emperor's New Mind, and also in Shadows of the Mind: A Search for the Missing Science of Consciousness (Oxford: Oxford University Press, 1994). Return to Text
10. See, for example, "Reply to Criticisms" in Albert Einstein, vol. 7 of The Library of Living Philosophers, P. A. Schilpp, ed. (New York: MJF Books, 2001). Return to Text
11. W. J. van Stockum, Proc. R. Soc. Edin. A 57 (1937), 135. Return to Text
12. The expert reader will recognize that I am simplifying. In 1966, Robert Geroch, who was a student of John Wheeler, showed that it is at least possible, in principle, to construct a wormhole without ripping space. But unlike the more intuitive, space-tearing approach to building wormholes in which the mere existence of the wormhole does not entail time travel, in Geroch's approach the construction phase itself would necessarily require that time become so distorted that one could freely travel backward and forward in time (but no farther back than the initiation of the construction itself). Return to Text
13. Roughly speaking, if you passed through a region containing such exotic matter at nearly the speed of light and took the average of all your measurements of the energy density you detected, the answer you'd find would be negative. Physicists say that such exotic matter violates the so-called averaged weak energy condition. Return to Text
14. The simplest realization of exotic matter comes from the vacuum fluctuations of the electromagnetic field between the parallel plates in the Casimir experiment, discussed in Chapter 12. Calculations show that the decrease in quantum fluctuations between the plates, relative to empty space, entails negative averaged energy density (as well as negative pressure). Return to Text
15. For a pedagogical but technical account of wormholes, see Matt Visser, Lorentzian Wormholes: From Einstein to Hawking (New York: American Institute of Physics Press, 1996). Return to Text
2. Zeilinger's research group also included Dick Bouwmeester, Jian-Wi Pan, Klaus Mattle, Manfred Eibl, and Harald Weinfurter, and De Martini's has included S. Giacomini, G. Milani, F. Sciarrino, and E. Lombardi. Return to Text
3. For the reader who has some familiarity with the formalism of quantum mechanics, here are the essential steps in quantum teleportation. Imagine that the initial state of a photon I have in New York is given by
4. In fact, the mathematically inclined reader will note that it is not hard to prove the so-called no-quantum-cloning theorem. Imagine we have a unitary cloning operator U that takes any given state as input and produces two copies of it as output (U maps 1a) —> 1a)1a), for any input state ja)). Note that U acting on a state like (1a) + j13)) yields (1a)1a) + 113)113)), which is not a two-fold copy of the original state (ja) +113))(1a) +113)), and hence no such operator U exists to carry out quantum cloning. (This was first shown by Wootters and Zurek in the early 1980s.) Return to Text
5. Many researchers have been involved in developing both the theory and the experimental realization of quantum teleportation. In addition to those discussed in the text, the work of Sandu Popescu while at Cambridge University played an important part in the Rome experiments, and Jeffrey Kimble's group at the California Institute of Technology has pioneered the teleportation of continuous features of a quantum state, to name a few. Return to Text
6. For extremely interesting progress on entangling many-particle systems, see, for example, B. Julsgaard, A. Kozhekin, and E. S. Polzik, "Experimental long-lived entanglement of two macroscopic objects," Nature 413 (Sept. 2001), 400-403. Return to Text
7. One of the most exciting and active areas of research making use of quantum entanglement and quantum teleportation is the field of quanttrn computing. For recent general-level presentations of quantum computing, see Tom Siegfried, The Bit and the Pendulum (New York: John Wiley, 2000), and George Johnson, A Shortcut Through Time (New York: Knopf, 2003). Return to Text
8. One aspect of the slowing of time at increasing velocity, which we did not discuss in Chapter 3 but will play a role in this chapter, is the so-called twin paradox. The issue is simple to state: if you and I are moving relative to one another at constant velocity, I will think your clock is running slow relative to mine. But since you are as justified as I in claiming to be at rest, you will think that mine is the moving clock and hence is the one that is running slow. That each of us thinks the other's clock is running slow may seem paradoxical, but it's not. At constant velocity, our clocks will continue to get farther apart and hence they don't allow for a direct, face-to-face comparison to determine which is "really" running slow. And all other indirect comparisons (for instance, we compare the times on our clocks by cell phone communication) occur with some elapsed time over some spatial separation, necessarily bringing into play the complications of different observers' notions of now, as in Chapters 3 and 5. I won't go through it here, but when these special relativistic complications are folded into the analysis, there is no contradiction between each of us declaring that the other's clock is running slow (see, e.g., E. Taylor and J. A. Wheeler, Spacetime Physics, for a complete, technical, but elementary discussion). Where things appear to get more puzzling is if, for example, you slow down, stop, turn around, and head back toward me so that we can compare our clocks face to face, eliminating the complications of different notions of now. Upon our meeting, whose clock will be ahead of whose? This is the so-called twin paradox: if you and I are twins, when we meet again, will we be the same age, or will one of us look older? The answer is that my clock will be ahead of yours — if we are twins, I will look older. There are many ways to explain why, but the simplest to note is that when you change your velocity and experience an acceleration, the symmetry between our perspectives is lost — you can definitively claim that you were moving (since, for example, you felt it — or, using the discussion of Chapter 3, unlike mine, your journey through spacetime has not been along a straight line) and hence that your clock ran slow relative to mine. Less time elapsed for you than for me. Return to Text
9. John Wheeler, among others, has suggested a possible central role for observers in a quantum universe, summed up in one of his famous aphorisms: "No elementary phenomenon is a phenomenon until it is an observed phenomenon." You can read more about Wheeler's fascinating life in physics in John Archibald Wheeler and Kenneth Ford, Geons, Black Holes, and Quantum Foam: A Life in Physics (New York: Norton, 1998). Roger Penrose has also studied the relation between quantum physics and the mind in his The Emperor's New Mind, and also in Shadows of the Mind: A Search for the Missing Science of Consciousness (Oxford: Oxford University Press, 1994). Return to Text
10. See, for example, "Reply to Criticisms" in Albert Einstein, vol. 7 of The Library of Living Philosophers, P. A. Schilpp, ed. (New York: MJF Books, 2001). Return to Text
11. W. J. van Stockum, Proc. R. Soc. Edin. A 57 (1937), 135. Return to Text
12. The expert reader will recognize that I am simplifying. In 1966, Robert Geroch, who was a student of John Wheeler, showed that it is at least possible, in principle, to construct a wormhole without ripping space. But unlike the more intuitive, space-tearing approach to building wormholes in which the mere existence of the wormhole does not entail time travel, in Geroch's approach the construction phase itself would necessarily require that time become so distorted that one could freely travel backward and forward in time (but no farther back than the initiation of the construction itself). Return to Text
13. Roughly speaking, if you passed through a region containing such exotic matter at nearly the speed of light and took the average of all your measurements of the energy density you detected, the answer you'd find would be negative. Physicists say that such exotic matter violates the so-called averaged weak energy condition. Return to Text
14. The simplest realization of exotic matter comes from the vacuum fluctuations of the electromagnetic field between the parallel plates in the Casimir experiment, discussed in Chapter 12. Calculations show that the decrease in quantum fluctuations between the plates, relative to empty space, entails negative averaged energy density (as well as negative pressure). Return to Text
15. For a pedagogical but technical account of wormholes, see Matt Visser, Lorentzian Wormholes: From Einstein to Hawking (New York: American Institute of Physics Press, 1996). Return to Text