The Code Book Part 11

In Young's earlier experiment he had shone light at a part.i.tion in which there were two narrow vertical slits, as shown in Figure 71(a) Figure 71(a). On a screen some distance beyond the slits, Young expected to see two bright stripes, projections of the slits. Instead he observed that the light fanned out from the two slits and formed a pattern of several light and dark stripes on the screen. The striped pattern of light on the screen had puzzled him, but now he believed he could explain it wholly in terms of what he had seen on the duck pond.

Young began by a.s.suming that light was a form of wave. If the light emanating from the two slits behaved like waves, then it was just like the ripples behind the two ducks. Furthermore, the light and dark stripes on the screen were caused by the same interactions that caused the water waves to form high peaks, deep troughs and patches of calm. Young could imagine points on the screen where a trough met a peak, resulting in cancelation and a dark stripe, and points on the screen where two peaks (or two troughs) met, resulting in reinforcement and a bright stripe, as shown in Figure 71(b) Figure 71(b). The ducks had provided Young with a deeper insight into the true nature of light, and he eventually published "The Undulatory Theory of Light," an all-time cla.s.sic among physics papers.

Nowadays, we know that light does indeed behave like a wave, but we know that it can also behave like a particle. Whether we perceive light as a wave or as a particle depends on the circ.u.mstances, and this ambiguity of light is known as wave-particle duality. We do not need to discuss this duality any further, except to say that modern physics thinks of a beam of light as consisting of countless individual particles, known as photons, which exhibit wave-like properties. Looked at this way, we can interpret Young's experiment in terms of photons flooding the slits, and then interacting on the other side of the part.i.tion. of light is known as wave-particle duality. We do not need to discuss this duality any further, except to say that modern physics thinks of a beam of light as consisting of countless individual particles, known as photons, which exhibit wave-like properties. Looked at this way, we can interpret Young's experiment in terms of photons flooding the slits, and then interacting on the other side of the part.i.tion.

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Figure 71 Young's slits experiment viewed from above. Diagram (a) shows light fanning out from the two slits in the part.i.tion, interacting and creating a striped pattern on the screen. Diagram (b) shows how individual waves interact. If a trough meets a peak at the screen, the result is a dark stripe. If two troughs (or two peaks) meet at the screen, the result is a bright stripe. Young's slits experiment viewed from above. Diagram (a) shows light fanning out from the two slits in the part.i.tion, interacting and creating a striped pattern on the screen. Diagram (b) shows how individual waves interact. If a trough meets a peak at the screen, the result is a dark stripe. If two troughs (or two peaks) meet at the screen, the result is a bright stripe.

So far, there is nothing particularly strange about Young's experiment. However, modern technology allows physicists to repeat Young's experiment using a filament that is so dim that it emits single photons of light. Photons are produced individually at a rate of, say, one per minute, and each photon travels alone toward the part.i.tion. Sometimes a photon will pa.s.s through one of the two slits, and strike the screen. Although our eyes are not sensitive enough to see the individual photons, they can be observed with the help of a special detector, and over a period of hours we could build up an overall picture of where the photons are striking the screen. With only one photon at a time pa.s.sing through the slits, we would not expect to see the striped pattern observed by Young, because that phenomenon seems to depend on two photons simultaneously traveling through different slits and interacting with each other on the other side. Instead we might expect to see just two light stripes, simply projections of the slits in the part.i.tion. However, for some extraordinary reason, even with single photons the result on the screen is still a pattern of light and dark stripes, just as if photons had been interacting.

This weird result defies common sense. There is no way to explain the phenomenon in terms of the cla.s.sical laws of physics, by which we mean the traditional laws that were developed to explain how everyday objects behave. Cla.s.sical physics can explain the orbits of planets or the trajectory of a cannonball, but cannot fully describe the world of the truly tiny, such as the trajectory of a photon. In order to explain such photon phenomena, physicists resort to quantum theory, an explanation of how objects behave at the microscopic level. However, even quantum theorists cannot agree on how to interpret this experiment. They tend to split into two opposing camps, each with their own interpretation.

The first camp posits an idea known as superposition superposition. The superpositionists begin by stating that we know only two things for certain about the photon-it leaves the filament and it strikes the screen. Everything else is a complete mystery, including whether the photon pa.s.sed through the left slit or the right slit. Because the exact path of the photon is unknown, superpositionists take the peculiar view that the photon somehow pa.s.ses through both slits simultaneously, which would then allow it to interfere with itself and create the striped pattern observed on the screen. But how can one photon pa.s.s through both slits? is a complete mystery, including whether the photon pa.s.sed through the left slit or the right slit. Because the exact path of the photon is unknown, superpositionists take the peculiar view that the photon somehow pa.s.ses through both slits simultaneously, which would then allow it to interfere with itself and create the striped pattern observed on the screen. But how can one photon pa.s.s through both slits?

Superpositionists argue along the following lines. If we do not know what a particle is doing, then it is allowed to do everything possible simultaneously. In the case of the photon, we do not know whether it pa.s.sed through the left slit or the right slit, so we a.s.sume that it pa.s.sed through both slits simultaneously. Each possibility is called a state state, and because the photon fulfills both possibilities it is said to be in a superposition of states superposition of states. We know that one photon left the filament and we know that one photon hit the screen on the other side of the part.i.tion, but in between it somehow split into two "ghost photons" that pa.s.sed through both slits. Superposition might sound silly, but at least it explains the striped pattern that results from Young's experiment performed with individual photons. In comparison, the old-fas.h.i.+oned cla.s.sical view is that the photon must have pa.s.sed through one of the two slits, and we simply do not know which one-this seems much more sensible than the quantum view, but unfortunately it cannot explain the observed result.

Erwin Schrodinger, who won the n.o.bel Prize for Physics in 1933, invented a parable known as "Schrodinger's cat," which is often used to help explain the concept of superposition. Imagine a cat in a box. There are two possible states for the cat, namely dead or alive. Initially, we know that the cat is definitely in one particular state, because we can see that it is alive. At this point, the cat is not in a superposition of states. Next, we place a vial of cyanide in the box along with the cat and close the lid. We now enter a period of ignorance, because we cannot see or measure the state of the cat. Is it still alive, or has it trodden on the vial of cyanide and died? Traditionally we would say that the cat is either dead or alive, we just do not know which. However, quantum theory says that the cat is in a superposition of two states-it is both dead and alive, it satisfies all possibilities. Superposition occurs only when we lose sight of an object, and it is a way of describing an object during a period of ambiguity. When we eventually open the box, we can see whether the cat is alive or dead. The act of looking at the cat forces it to be in one particular state, and at that very moment the superposition disappears. forces it to be in one particular state, and at that very moment the superposition disappears.

For readers who feel uncomfortable with superposition, there is the second quantum camp, who favor a different interpretation of Young's experiment. Unfortunately, this alternative view is equally bizarre. The many-worlds interpretation many-worlds interpretation claims that upon leaving the filament the photon has two choices-either it pa.s.ses through the left slit or the right slit-at which point the universe divides into two universes, and in one universe the photon goes through the left slit, and in the other universe the photon goes through the right slit. These two universes somehow interfere with each other, which accounts for the striped pattern. Followers of the many-worlds interpretation believe that whenever an object has the potential to enter one of several possible states, the universe splits into many universes, so that each potential is fulfilled in a different universe. This proliferation of universes is referred to as the claims that upon leaving the filament the photon has two choices-either it pa.s.ses through the left slit or the right slit-at which point the universe divides into two universes, and in one universe the photon goes through the left slit, and in the other universe the photon goes through the right slit. These two universes somehow interfere with each other, which accounts for the striped pattern. Followers of the many-worlds interpretation believe that whenever an object has the potential to enter one of several possible states, the universe splits into many universes, so that each potential is fulfilled in a different universe. This proliferation of universes is referred to as the multiverse multiverse.

Whether we adopt superposition or the many-worlds interpretation, quantum theory is a perplexing philosophy. Nevertheless, it has shown itself to be the most successful and practical scientific theory ever conceived. Besides its unique capacity to explain the result of Young's experiment, quantum theory successfully explains many other phenomena. Only quantum theory allows physicists to calculate the consequences of nuclear reactions in power stations; only quantum theory can explain the wonders of DNA; only quantum theory explains how the sun s.h.i.+nes; only quantum theory can be used to design the laser that reads the CDs in your stereo. Thus, like it or not, we live in a quantum world.

Of all the consequences of quantum theory, the most technologically important is potentially the quantum computer. As well as destroying the security of all modern ciphers, the quantum computer would herald a new era of computing power. One of the pioneers of quantum computing is David Deutsch, a British physicist who began working on the concept in 1984, when he attended a conference on the theory of computation. While listening to a lecture at the conference, Deutsch spotted something that had previously been overlooked. The tacit a.s.sumption was that all computers essentially operated according to the laws of cla.s.sical physics, but Deutsch was convinced that computers ought to obey the laws of quantum physics instead, because quantum laws are more fundamental.

Ordinary computers operate at a relatively macroscopic level, and at that level quantum laws and cla.s.sical laws are almost indistinguishable. It did not therefore matter that scientists had generally thought of ordinary computers in terms of cla.s.sical physics. However, at the microscopic level the two sets of laws diverge, and at this level only the laws of quantum physics hold true. At the microscopic level, quantum laws reveal their true weirdness, and a computer constructed to exploit these laws would behave in a drastically new way. After the conference, Deutsch returned home and began to recast the theory of computers in the light of quantum physics. In a paper published in 1985 he described his vision of a quantum computer operating according to the laws of quantum physics. In particular, he explained how his quantum computer differed from an ordinary computer.

Imagine that you have two versions of a question. To answer both questions using an ordinary computer, you would have to input the first version and wait for the answer, then input the second version and wait for the answer. In other words, an ordinary computer can address only one question at a time, and if there are several questions it has to address them sequentially. However, with a quantum computer, the two questions could be combined as a superposition of two states and inputted simultaneously-the machine itself would then enter a superposition of two states, one for each question. Or, according to the many-worlds interpretation, the machine would enter two different universes, and answer each version of the question in a different universe. Regardless of the interpretation, the quantum computer can address two questions at the same time by exploiting the laws of quantum physics. version and wait for the answer, then input the second version and wait for the answer. In other words, an ordinary computer can address only one question at a time, and if there are several questions it has to address them sequentially. However, with a quantum computer, the two questions could be combined as a superposition of two states and inputted simultaneously-the machine itself would then enter a superposition of two states, one for each question. Or, according to the many-worlds interpretation, the machine would enter two different universes, and answer each version of the question in a different universe. Regardless of the interpretation, the quantum computer can address two questions at the same time by exploiting the laws of quantum physics.

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Figure 72 David Deutsch. ( David Deutsch. (photo credit 8.1) To get some idea of the power of a quantum computer, we can compare its performance with that of a traditional computer by seeing what happens when each is used to tackle a particular problem. For example, the two types of computer could tackle the problem of finding a number whose square and cube together use all the digits from 0 to 9 once and only once. If we test the number 19, we find that 192 = 361 and 19 = 361 and 193 = 6,859. The number 19 does not fit the requirement because its square and cube include only the digits: 1, 3, 5, 6, 6, 8, 9, i.e., the digits 0, 2, 4, 7 are missing and the digit 6 is repeated. = 6,859. The number 19 does not fit the requirement because its square and cube include only the digits: 1, 3, 5, 6, 6, 8, 9, i.e., the digits 0, 2, 4, 7 are missing and the digit 6 is repeated.

To solve this problem with a traditional computer, the operator would have to adopt the following approach. The operator inputs the number 1 and then allows the computer to test it. Once the computer has done the necessary calculations, it declares whether or not the number fulfills the criterion. The number 1 does not fulfill the criterion, so the operator inputs the number 2 and allows the computer to carry out another test, and so on, until the appropriate number is eventually found. It turns out that the answer is 69, because 692 = 4,761 and 69 = 4,761 and 693 = 328,509, and these numbers do indeed include each of the ten digits once and only once. In fact, 69 is the only number that satisfies this requirement. It is clear that this process is time-consuming, because a traditional computer can test only one number at a time. If the computer takes one second to test each number, then it would have taken 69 seconds to find the answer. In contrast, a quantum computer would find the answer in just 1 second. = 328,509, and these numbers do indeed include each of the ten digits once and only once. In fact, 69 is the only number that satisfies this requirement. It is clear that this process is time-consuming, because a traditional computer can test only one number at a time. If the computer takes one second to test each number, then it would have taken 69 seconds to find the answer. In contrast, a quantum computer would find the answer in just 1 second.

The operator begins by representing the numbers in a special way so as to exploit the power of the quantum computer. One way to represent the numbers is in terms of spinning particles-many fundamental particles possess an inherent spin, and they can either spin eastward or westward, rather like a basketball spinning on the end of a finger. When a particle is spinning eastward it represents 1, and when it is spinning westward it represents 0. Hence, a sequence of spinning particles represents a sequence of 1's and 0's, or a binary number. For example, seven particles, spinning east, east, west, east, west, west, west respectively, together represent the binary number 1101000, which is equivalent to the decimal number 104. Depending on their spins, a combination of seven particles can represent any number between 0 and 127. numbers is in terms of spinning particles-many fundamental particles possess an inherent spin, and they can either spin eastward or westward, rather like a basketball spinning on the end of a finger. When a particle is spinning eastward it represents 1, and when it is spinning westward it represents 0. Hence, a sequence of spinning particles represents a sequence of 1's and 0's, or a binary number. For example, seven particles, spinning east, east, west, east, west, west, west respectively, together represent the binary number 1101000, which is equivalent to the decimal number 104. Depending on their spins, a combination of seven particles can represent any number between 0 and 127.

With a traditional computer, the operator would then input one particular sequence of spins, such as west, west, west, west, west, west, east, which represents 0000001, which is simply the decimal number 1. The operator would then wait for the computer to test the number to see whether it fits the criterion mentioned earlier. Next the operator would input 0000010, which would be a sequence of spinning particles representing 2, and so on. As before, the numbers would have to be entered one at a time, which we know to be time-consuming. However, if we are dealing with a quantum computer, the operator has an alternative way of inputting numbers which is much faster. Because each particle is fundamental, it obeys the laws of quantum physics. Hence, when a particle is not being observed it can enter a superposition of states, which means that it is spinning in both directions at the same time, and so is representing both 0 and 1 at the same time. Alternatively, we can think of the particle entering two different universes: in one universe it spins eastward and represents 1, while in the other it spins westward and represents 0.

The superposition is achieved as follows. Imagine that we can observe one of the particles, and it is spinning westward. To change its spin, we would fire a sufficiently powerful pulse of energy, enough to kick the particle into spinning eastward. If we were to fire a weaker pulse, then sometimes we would be lucky and the particle would change its spin, and sometimes we would be unlucky and the particle would keep its westward spin. So far the particle has been in clear view all along, and we have been able to follow its progress. However, if the particle is spinning westward and put in a box out of our view, and we fire a weak pulse of energy at it, then we have no idea whether its spin has been changed. The particle enters a superposition of eastward and westward spins, just as the cat entered a superposition of being dead and alive. By taking seven westward-spinning particles, placing them in a box, and firing seven weak pulses of energy at them, then all seven particles enter a superposition. then we have no idea whether its spin has been changed. The particle enters a superposition of eastward and westward spins, just as the cat entered a superposition of being dead and alive. By taking seven westward-spinning particles, placing them in a box, and firing seven weak pulses of energy at them, then all seven particles enter a superposition.

With all seven particles in a superposition, they effectively represent all possible combinations of eastward and westward spins. The seven particles simultaneously represent 128 different states, or 128 different numbers. The operator inputs the seven particles, while they are still in a superposition of states, into the quantum computer, which then performs its calculations as if it were testing all 128 numbers simultaneously. After 1 second the computer outputs the number, 69, which fulfills the requested criterion. The operator gets 128 computations for the price of one.

A quantum computer defies common sense. Ignoring the details for a moment, a quantum computer can be thought of in two different ways, depending on which quantum interpretation you prefer. Some physicists view the quantum computer as a single ent.i.ty that performs the same calculation simultaneously on 128 numbers. Others view it as 128 ent.i.ties, each in a separate universe, each performing just one calculation. Quantum computing is Twilight Zone Twilight Zone technology. technology.

When traditional computers operate on 1's and 0's, the 1's and 0's are called bits, which is short for binary digits. Because a quantum computer deals with 1's and 0's that are in a quantum superposition, they are called quantum bits, or qubits qubits (p.r.o.nounced "cubits"). The advantage of qubits becomes even clearer when we consider more particles. With 250 spinning particles, or 250 qubits, it is possible to represent roughly 10 (p.r.o.nounced "cubits"). The advantage of qubits becomes even clearer when we consider more particles. With 250 spinning particles, or 250 qubits, it is possible to represent roughly 1075 combinations, which is greater than the number of atoms in the universe. If it were possible to achieve the appropriate superposition with 250 particles, then a quantum computer could perform 10 combinations, which is greater than the number of atoms in the universe. If it were possible to achieve the appropriate superposition with 250 particles, then a quantum computer could perform 1075 simultaneous computations, completing them all in just one second. simultaneous computations, completing them all in just one second.

The exploitation of quantum effects could give rise to quantum computers of unimaginable power. Unfortunately, when Deutsch created his vision of a quantum computer in the mid-1980s, n.o.body could quite envisage how to create a solid, practical machine. For example, scientists could not actually build anything that could calculate with spinning particles in a superposition of states. One of the greatest hurdles was maintaining a superposition of states throughout the calculation. A superposition exists only while it is not being observed, but an observation in the most general sense includes any interaction with anything external to the superposition. A single stray atom interacting with one of the spinning particles would cause the superposition to collapse into a single state and cause the quantum calculation to fail. maintaining a superposition of states throughout the calculation. A superposition exists only while it is not being observed, but an observation in the most general sense includes any interaction with anything external to the superposition. A single stray atom interacting with one of the spinning particles would cause the superposition to collapse into a single state and cause the quantum calculation to fail.

Another problem was that scientists could not work out how to program a quantum computer, and were therefore not sure what sort of computations it might be capable of doing. However, in 1994 Peter Shor of AT&T Bell Laboratories in New Jersey did succeed in defining a useful program for a quantum computer. The remarkable news for crypta.n.a.lysts was that Shor's program defined a series of steps that could be used by a quantum computer to factor a giant number-just what was required to crack the RSA cipher. When Martin Gardner set his RSA challenge in Scientific American Scientific American, it took six hundred computers several months to factor a 129- digit number. In comparison, Shor's program could factor a number a million times bigger in one-millionth of the time. Unfortunately, Shor could not demonstrate his factorization program, because there was still no such thing as a quantum computer.

Then, in 1996, Lov Grover, also at Bell Labs, discovered another powerful program. Graver's program is a way of searching a list at incredibly high speed, which might not sound particularly interesting until you realize that this is exactly what is required to crack a DES cipher. To crack a DES cipher it is necessary to search a list of all possible keys in order to find the correct one. If a conventional computer can check a million keys a second, it would take over a thousand years to crack a DES cipher, whereas a quantum computer using Grover's program could find the key in less than four minutes.

It is purely coincidental that the first two quantum computer programs to be invented have been exactly what crypta.n.a.lysts would have put at the top of their wish lists. Although Shor's and Grover's programs generated tremendous optimism among codebreakers, there was also immense frustration, because there was still no such thing as a working quantum computer that could run these programs. Not surprisingly, the potential of the ultimate weapon in decryption technology has whetted the appet.i.te of organizations such as America's Defense Advanced Research Projects Agency (DARPA) and the Los Alamos National Laboratory, who are desperately trying to build devices that can handle qubits, in the same way that silicon chips handle bits. Although a number of recent breakthroughs have boosted morale among researchers, it is fair to say that the technology remains remarkably primitive. In 1998, Serge Haroche at the University of Paris VI put the hype surrounding the breakthroughs into perspective when he dispelled claims that a real quantum computer is only a few years away. He said this was like painstakingly a.s.sembling the first layer of a house of cards, then boasting that the next 15,000 layers were a mere formality. who are desperately trying to build devices that can handle qubits, in the same way that silicon chips handle bits. Although a number of recent breakthroughs have boosted morale among researchers, it is fair to say that the technology remains remarkably primitive. In 1998, Serge Haroche at the University of Paris VI put the hype surrounding the breakthroughs into perspective when he dispelled claims that a real quantum computer is only a few years away. He said this was like painstakingly a.s.sembling the first layer of a house of cards, then boasting that the next 15,000 layers were a mere formality.

Only time will tell if and when the problems of building a quantum computer can be overcome. In the meantime, we can merely speculate as to what impact it would have on the world of cryptography. Ever since the 1970s, codemakers have had a clear lead in the race against codebreakers, thanks to ciphers such as DES and RSA. These sorts of ciphers are a precious resource, because we have come to trust them to encrypt our e-mails and guard our privacy. Similarly, as we enter the twenty-first century more and more commerce will be conducted on the Internet, and the electronic marketplace will rely on strong ciphers to protect and verify financial transactions. As information becomes the world's most valuable commodity, the economic, political and military fate of nations will depend on the strength of ciphers.

Consequently, the development of a fully operational quantum computer would imperil our personal privacy, destroy electronic commerce and demolish the concept of national security. A quantum computer would jeopardize the stability of the world. Whichever country gets there first will have the ability to monitor the communications of its citizens, read the minds of its commercial rivals and eavesdrop on the plans of its enemies. Although it is still in its infancy, quantum computing presents a potential threat to the individual, to international business and to global security.

Quantum Cryptography While crypta.n.a.lysts antic.i.p.ate the arrival of quantum computers, cryptographers are working on their own technological miracle-an encryption system that would reestablish privacy, even when confronted with the might of a quantum computer. This new form of encryption is fundamentally different from any that we have previously encountered in that it offers the hope of perfect privacy. In other words, this system would be flawless and would guarantee absolute security for eternity. Furthermore, it is based on quantum theory, the same theory that is the foundation for quantum computers. So while quantum theory is the inspiration for a computer that could crack all current ciphers, it is also at the heart of a new unbreakable cipher called with the might of a quantum computer. This new form of encryption is fundamentally different from any that we have previously encountered in that it offers the hope of perfect privacy. In other words, this system would be flawless and would guarantee absolute security for eternity. Furthermore, it is based on quantum theory, the same theory that is the foundation for quantum computers. So while quantum theory is the inspiration for a computer that could crack all current ciphers, it is also at the heart of a new unbreakable cipher called quantum cryptography quantum cryptography.

The story of quantum cryptography dates back to a curious idea developed in the late 1960s by Stephen Wiesner, then a graduate student at Columbia University. Sadly, it was Wiesner's misfortune to invent an idea so ahead of its time that n.o.body took it seriously. He still recalls the reaction of his seniors: "I didn't get any support from my thesis adviser-he showed no interest in it at all. I showed it to several other people, and they all pulled a strange face, and went straight back to what they were already doing." Wiesner was proposing the bizarre concept of quantum money, which had the great advantage of being impossible to counterfeit.

Wiesner's quantum money relied heavily on the physics of photons. When a photon travels through s.p.a.ce it vibrates, as shown in Figure 73(a) Figure 73(a). All four photons are traveling in the same direction, but the angle of vibration is different in each case. The angle of vibration is known as the polarization of the photon, and a lightbulb generates photons of all polarizations, which means that some photons will vibrate up and down, some from side to side, and others at all angles in between. To simplify matters, we shall a.s.sume that photons have only four possible polarizations, which we label[image] and and[image] . .

By placing a filter known as a Polaroid in the path of the photons, it is possible to ensure that the emerging beam of light consists of photons that vibrate in one particular direction; in other words, the photons all have the same polarization. To some extent, we can think of the Polaroid filter as a grating, and photons as matchsticks randomly scattered onto the grating. The matchsticks will slip through the grating only if they are at the correct angle. Any photon that is already polarized in the same direction as the Polaroid filter will automatically pa.s.s through it unchanged, and photons that are polarized perpendicular to the filter will be blocked.

Unfortunately, the matchstick a.n.a.logy breaks down when we think about diagonally polarized photons approaching a vertical Polaroid filter. Although matchsticks oriented diagonally would be blocked by a vertical grating, this is not necessarily the case with diagonally polarized photons approaching a vertical Polaroid filter. In fact, diagonally polarized photons are in a quantum quandary when confronted by a vertical Polaroid filter. What happens is that, half of them at random will be blocked, and half will pa.s.s through, and those that do pa.s.s through will be reoriented with a vertical polarization. Figure 73(b) Figure 73(b) shows eight photons approaching a vertical Polaroid filter, and shows eight photons approaching a vertical Polaroid filter, and Figure 73(c) Figure 73(c) shows that only four of them successfully pa.s.s through it. All the vertically polarized photons have pa.s.sed through, all the horizontally polarized photons have been blocked, and half of the diagonally polarized photons have pa.s.sed through. shows that only four of them successfully pa.s.s through it. All the vertically polarized photons have pa.s.sed through, all the horizontally polarized photons have been blocked, and half of the diagonally polarized photons have pa.s.sed through.

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Figure 73 (a) Although photons of light vibrate in all directions, we a.s.sume for simplicity that there are just four distinct directions, as shown in this diagram. (b) The lamp has emitted eight photons, which are vibrating in various directions. Each photon is said to have a polarization. The photons are heading toward a vertical Polaroid filter. (c) On the other side of the filter, only half the photons have survived. The vertically polarized photons have pa.s.sed through, and the horizontally polarized photons have not. Half the diagonally polarized photons have pa.s.sed through, and are thereafter vertically polarized. (a) Although photons of light vibrate in all directions, we a.s.sume for simplicity that there are just four distinct directions, as shown in this diagram. (b) The lamp has emitted eight photons, which are vibrating in various directions. Each photon is said to have a polarization. The photons are heading toward a vertical Polaroid filter. (c) On the other side of the filter, only half the photons have survived. The vertically polarized photons have pa.s.sed through, and the horizontally polarized photons have not. Half the diagonally polarized photons have pa.s.sed through, and are thereafter vertically polarized.

It is this ability to block certain photons that explains how Polaroid sungla.s.ses work. In fact, you can demonstrate the effect of Polaroid filters by experimenting with a pair of Polaroid sungla.s.ses. First remove one lens, and close that eye so that you are looking with just the other eye through the remaining lens. Not surprisingly, the world looks quite dark because the lens blocks many of the photons that would otherwise have reached your eye. At this point, all the photons reaching your eye have the same polarization. Next, hold the other lens in front of the lens you are looking through, and rotate it slowly. At one point in the rotation, the loose lens will have no effect on the amount of light reaching your eye because its orientation is the same as the fixed lens-all the photons that get through the loose lens also pa.s.s through the fixed lens. If you now rotate the loose lens through 90, it will turn completely black. In this configuration, the polarization of the loose lens is perpendicular to the polarization of the fixed lens, so that any photons that get through the loose lens are blocked by the fixed lens. If you now rotate the loose lens by 45, then you reach an intermediate stage in which the lenses are partially misaligned, and half of the photons that pa.s.s through the loose lens manage to get through the fixed lens.

Wiesner planned to use the polarization of photons as a way of creating dollar bills that can never be forged. His idea was that dollar bills should each contain 20 light traps, tiny devices that are capable of capturing and retaining a photon. He suggested that banks could use four Polaroid filters oriented in four different ways ([image]) to fill the 20 light traps with a sequence of 20 polarized photons, using a different sequence for each dollar bill. For example, Figure 74 Figure 74 shows a bill with the polarization sequence ( shows a bill with the polarization sequence ([image]) Although the polarizations are explicitly shown in Figure 74 Figure 74, in reality they would be hidden from view. Each note also carries a traditional serial number, which is B2801695E for the dollar bill shown. The issuing bank can identify each dollar bill according to its polarization sequence and its printed serial number, and would keep a master list of serial numbers and the corresponding polarization sequences.

A counterfeiter is now faced with a problem-he cannot merely forge a dollar bill which carries an arbitrary serial number and a random polarization sequence in the light traps, because this pairing will not appear on the bank's master list, and the bank will spot that the dollar bill is a fake. To create an effective forgery, the counterfeiter must use a genuine bill as a sample, somehow measure its 20 polarizations, and then make a duplicate dollar bill, copying across the serial number and loading the light traps in the appropriate way. However, measuring photon polarizations is a notoriously tricky task, and if the counterfeiter cannot accurately measure them in the genuine sample bill, then he cannot hope to make a duplicate. bank's master list, and the bank will spot that the dollar bill is a fake. To create an effective forgery, the counterfeiter must use a genuine bill as a sample, somehow measure its 20 polarizations, and then make a duplicate dollar bill, copying across the serial number and loading the light traps in the appropriate way. However, measuring photon polarizations is a notoriously tricky task, and if the counterfeiter cannot accurately measure them in the genuine sample bill, then he cannot hope to make a duplicate.

To understand the difficulty of measuring the polarization of photons, we need to consider how we would go about trying to perform such a measurement. The only way to learn anything about the polarization of a photon is by using a Polaroid filter. To measure the polarization of the photon in a particular light trap, the counterfeiter selects a Polaroid filter and orients it in a particular way, say vertically,[image] . If the photon emerging from the light trap happens to be vertically polarized, it will pa.s.s through the vertical Polaroid filter and the counterfeiter will correctly a.s.sume that . If the photon emerging from the light trap happens to be vertically polarized, it will pa.s.s through the vertical Polaroid filter and the counterfeiter will correctly a.s.sume that it is a vertically polarized photon. If the emerging photon is horizontally polarized, it will not pa.s.s through the vertical Polaroid filter, and the counterfeiter will correctly a.s.sume that it is a horizontally polarized photon. However, if the emerging photon happens to be diagonally polarized ( it is a vertically polarized photon. If the emerging photon is horizontally polarized, it will not pa.s.s through the vertical Polaroid filter, and the counterfeiter will correctly a.s.sume that it is a horizontally polarized photon. However, if the emerging photon happens to be diagonally polarized ([image] or or[image] ), it might or might not pa.s.s through the filter, and in either case the counterfeiter will fail to identify its true nature. A ), it might or might not pa.s.s through the filter, and in either case the counterfeiter will fail to identify its true nature. A[image] photon might pa.s.s through the vertical Polaroid filter, in which case the counterfeiter will wrongly a.s.sume that it is a vertically polarized photon, or the same photon might not pa.s.s through the filter, in which case he will wrongly a.s.sume that it is a horizontally polarized photon. Alternatively, if the counterfeiter chooses to measure the photon in another light trap by orientating the filter diagonally, say photon might pa.s.s through the vertical Polaroid filter, in which case the counterfeiter will wrongly a.s.sume that it is a vertically polarized photon, or the same photon might not pa.s.s through the filter, in which case he will wrongly a.s.sume that it is a horizontally polarized photon. Alternatively, if the counterfeiter chooses to measure the photon in another light trap by orientating the filter diagonally, say[image] , then this would correctly identify the nature of a diagonally polarized photon, but it would fail to accurately identify a vertically or horizontally polarized photon. , then this would correctly identify the nature of a diagonally polarized photon, but it would fail to accurately identify a vertically or horizontally polarized photon.

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Figure 74 Stephen Wiesner's quantum money. Each note is unique because of its serial number, which can be seen easily, and the 20 light traps, whose contents are a mystery. The light traps contain photons of various polarizations. The bank knows the sequence of polarizations corresponding to each serial number, but a counterfeiter does not. Stephen Wiesner's quantum money. Each note is unique because of its serial number, which can be seen easily, and the 20 light traps, whose contents are a mystery. The light traps contain photons of various polarizations. The bank knows the sequence of polarizations corresponding to each serial number, but a counterfeiter does not.

The counterfeiter's problem is that he must use the correct orientation of Polaroid filter to identify a photon's polarization, but he does not know which orientation to use because he does not know the polarization of the photon. This catch-22 is an inherent part of the physics of photons. Imagine that the counterfeiter chooses a[image] -filter to measure the photon emerging from the second light trap, and the photon does not pa.s.s through the filter. The counterfeiter can be sure that the photon was not -filter to measure the photon emerging from the second light trap, and the photon does not pa.s.s through the filter. The counterfeiter can be sure that the photon was not[image] -polarized, because that type of photon would have pa.s.sed through. However, the counterfeiter cannot tell whether the photon was -polarized, because that type of photon would have pa.s.sed through. However, the counterfeiter cannot tell whether the photon was[image] -polarized, which would certainly not have pa.s.sed through the filter, or whether it was -polarized, which would certainly not have pa.s.sed through the filter, or whether it was[image] or or[image] -polarized, either of which stood a fifty-fifty chance of being blocked. -polarized, either of which stood a fifty-fifty chance of being blocked.

This difficulty in measuring photons is one aspect of the uncertainty principle, developed in the 1920s by the German physicist Werner Heisenberg. He translated his highly technical proposition into a simple statement: "We cannot cannot know, as a matter of principle, the present in all its details." This does not mean that we cannot know everything because we do not have enough measuring equipment, or because our equipment is poorly designed. Instead, Heisenberg was stating that it is logically impossible to measure every aspect of a particular object with perfect accuracy. In this particular case, we cannot measure every aspect of the photons within the light traps with perfect accuracy. The uncertainty principle is another weird consequence of quantum theory. know, as a matter of principle, the present in all its details." This does not mean that we cannot know everything because we do not have enough measuring equipment, or because our equipment is poorly designed. Instead, Heisenberg was stating that it is logically impossible to measure every aspect of a particular object with perfect accuracy. In this particular case, we cannot measure every aspect of the photons within the light traps with perfect accuracy. The uncertainty principle is another weird consequence of quantum theory.

Wiesner's quantum money relied on the fact that counterfeiting is a two-stage process: first the counterfeiter needs to measure the original note with great accuracy, and then he has to replicate it. By incorporating photons in the design of the dollar bill, Wiesner was making the bill impossible to measure accurately, and hence creating a barrier to counterfeiting.

A naive counterfeiter might think that if he cannot measure the polarizations of the photons in the light traps, then neither can the bank. He might try manufacturing dollar bills by filling the light traps with an arbitrary sequence of polarizations. However, the bank is able to verify which bills are genuine. The bank looks at the serial number, then consults its confidential master list to see which photons should be in which light traps. Because the bank knows which polarizations to expect in each light trap, it can correctly orient the Polaroid filter for each light trap and perform an accurate measurement. If the bill is counterfeit, the counterfeiter's arbitrary polarizations will lead to incorrect measurements and the bill will stand out as a forgery. For example, if the bank uses a -filter to measure what should be a -polarized photon, but finds that the filter blocks the photon, then it knows that a counterfeiter has filled the trap with the wrong photon. If, however, the bill turns out to be genuine, then the bank refills the light traps with the appropriate photons and puts it back into circulation.

In short, the counterfeiter cannot measure the polarizations in a genuine bill because he does not know which type of photon is in each light trap, and cannot therefore know how to orient the Polaroid filter in order to measure it correctly. On the other hand, the bank is able to check the polarizations in a genuine bill, because it originally chose the polarizations, and so knows how to orient the Polaroid filter for each one.

Quantum money is a brilliant idea. It is also wholly impractical. To start with, engineers have not yet developed the technology for trapping photons in a particular polarized state for a sufficiently long period of time. Even if the technology did exist, it would be too expensive to implement it. It might cost in the region of $1 million to protect each dollar bill. Despite its impracticality, quantum money applied quantum theory in an intriguing and imaginative way, so despite the lack of encouragement from his thesis adviser, Wiesner submitted a paper to a scientific journal. It was rejected. He submitted it to three other journals, and it was rejected three more times. Wiesner claims that they simply did not understand the physics. more times. Wiesner claims that they simply did not understand the physics.

It seemed that only one person shared Wiesner's excitement for the concept of quantum money. This was an old friend by the name of Charles Bennett, who several years earlier had been an undergraduate with him at Brandeis University. Bennett's curiosity about every aspect of science is one of the most remarkable things about his personality. He says he knew at the age of three that he wanted to be a scientist, and his childhood enthusiasm for the subject was not lost on his mother. One day she returned home to find a pan containing a weird stew bubbling on the cooker. Fortunately she was not tempted to taste it, as it turned out to be the remains of a turtle that the young Bennett was boiling in alkali in order to strip the flesh from the bones, thereby obtaining a perfect specimen of a turtle skeleton. During his teenage years, Bennett's curiosity moved from biology to biochemistry, and by the time he got to Brandeis he had decided to major in chemistry. At graduate school he concentrated on physical chemistry, then went on to do research in physics, mathematics, logic and, finally, computer science. moved from biology to biochemistry, and by the time he got to Brandeis he had decided to major in chemistry. At graduate school he concentrated on physical chemistry, then went on to do research in physics, mathematics, logic and, finally, computer science.

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Figure 75 Charles Bennett. ( Charles Bennett. (photo credit 8.2) Aware of Bennett's broad range of interests, Wiesner hoped that he would appreciate quantum money, and handed him a copy of his rejected ma.n.u.script. Bennett was immediately fascinated by the concept, and considered it one of most beautiful ideas he had ever seen. Over the next decade he would occasionally reread the ma.n.u.script, wondering if there was a way to turn something so ingenious into something that was also useful. Even when he became a research fellow at IBM's Thomas J. Watson Laboratories in the early 1980s, Bennett still could not stop thinking about Wiesner's idea. The journals might not want to publish it, but Bennett was obsessed by it.

One day, Bennett explained the concept of quantum money to Gilles Bra.s.sard, a computer scientist at the University of Montreal. Bennett and Bra.s.sard, who had collaborated on various research projects, discussed the intricacies of Wiesner's paper over and over again. Gradually they began to see that Wiesner's idea might have an application in cryptography. For Eve to decipher an encrypted message between Alice and Bob, she must first intercept it, which means that she must somehow accurately perceive the contents of the transmission. Wiesner's quantum money was secure because it was impossible to accurately perceive the polarizations of the photons trapped in the dollar bill. Bennett and Bra.s.sard wondered what would happen if an encrypted message was represented and transmitted by a series of polarized photons. In theory, it seemed that Eve would be unable to accurately read the encrypted message, and if she could not read the encrypted message, then she could not decipher it.

Bennett and Bra.s.sard began to concoct a system based on the following principle. Imagine that Alice wants to send Bob an encrypted message, which consists of a series of 1's and 0's. She represents the 1's and 0's by sending photons with certain polarizations. Alice has two possible schemes for a.s.sociating photon polarizations with 1 or 0. In the first scheme, called the rectilinear rectilinear or +-scheme, she sends or +-scheme, she sends[image] to represent 1, and to represent 1, and[image] to represent 0. In the other scheme, called the to represent 0. In the other scheme, called the diagonal diagonal or -scheme, she sends or -scheme, she sends[image] to represent 1, and to represent 1, and[image] to represent 0. To send a binary to represent 0. To send a binary message, she switches between these two schemes in an unpredictable way. Hence, the binary message 1101101001 could be transmitted as follows: message, she switches between these two schemes in an unpredictable way. Hence, the binary message 1101101001 could be transmitted as follows: [image]

Alice transmits the first 1 using the +-scheme, and the second 1 using the -scheme. Hence, 1 is being transmitted in both cases, but it is represented by differently polarized photons each time.

If Eve wants to intercept this message, she needs to identify the polarization of each photon, just as the counterfeiter needs to identify the polarization of each photon in the dollar bill's light traps. To measure the polarization of each photon Eve must decide how to orient her Polaroid filter as each one approaches. She cannot know for sure which scheme Alice will be using for each photon, so her choice of Polaroid filter will be haphazard and wrong half the time. Hence, she cannot have complete knowledge of the transmission.

An easier way to think of Eve's dilemma is to pretend that she has two types of Polaroid detector at her disposal. The +-detector is capable of measuring horizontally and vertically polarized photons with perfect accuracy, but is not capable of measuring diagonally polarized photons with certainty, and merely misinterprets them as vertically or horizontally polarized photons. On the other hand, the -detector can measure diagonally polarized photons with perfect accuracy, but cannot measure horizontally and vertically polarized photons with certainty, misinterpreting them as diagonally polarized photons. For example, if she uses the -detector to measure the first photon, which is[image] , she will misinterpret it as , she will misinterpret it as[image] or or[image] . If she misinterprets it as . If she misinterprets it as[image] , then she does not have a problem, because this also represents 1, but if she misinterprets it as , then she does not have a problem, because this also represents 1, but if she misinterprets it as[image] then she is in trouble, because this represents 0. To make matters worse for Eve, she only gets one chance to measure the photon accurately. A photon is indivisible, and so she cannot split it into two photons and measure it using both schemes. then she is in trouble, because this represents 0. To make matters worse for Eve, she only gets one chance to measure the photon accurately. A photon is indivisible, and so she cannot split it into two photons and measure it using both schemes.

This system seems to have some pleasant features. Eve cannot be sure of accurately intercepting the encrypted message, so she has no hope of deciphering it. However, the system suffers from a severe and apparently insurmountable problem-Bob is in the same position as Eve, inasmuch as he has no way of knowing which polarization scheme Alice is using for each photon, so he too will misinterpret the message. The obvious solution to the problem is for Alice and Bob to agree on which polarization scheme they will use for each photon. For the example above, Alice and Bob would share a list, or key, that reads + + + + . However, we are now back to the same old problem of key distribution-somehow Alice has to get the list of polarization schemes securely to Bob. deciphering it. However, the system suffers from a severe and apparently insurmountable problem-Bob is in the same position as Eve, inasmuch as he has no way of knowing which polarization scheme Alice is using for each photon, so he too will misinterpret the message. The obvious solution to the problem is for Alice and Bob to agree on which polarization scheme they will use for each photon. For the example above, Alice and Bob would share a list, or key, that reads + + + + . However, we are now back to the same old problem of key distribution-somehow Alice has to get the list of polarization schemes securely to Bob.

Of course, Alice could encrypt the list of schemes by employing a public key cipher such as RSA, and then transmit it to Bob. However, imagine that we are now in an era when RSA has been broken, perhaps following the development of powerful quantum computers. Bennett and Bra.s.sard's system has to be self-sufficient and not rely on RSA. For months, Bennett and Bra.s.sard tried to think of a way around the key distribution problem. Then, in 1984, the two found themselves standing on the platform at Croton-Harmon station, near IBM's Thomas J. Watson Laboratories. They were waiting for the train that would take Bra.s.sard back to Montreal, and pa.s.sed the time by chatting about the trials and tribulations of Alice, Bob and Eve. Had the train arrived a few minutes early, they would have waved each other goodbye, having made no progress on the problem of key distribution. Instead, in a eureka! eureka! moment, they created quantum cryptography, the most secure form of cryptography ever devised. moment, they created quantum cryptography, the most secure form of cryptography ever devised.

Their recipe for quantum cryptography requires three preparatory stages. Although these stages do not involve sending an encrypted message, they do allow the secure exchange of a key which can later be used to encrypt a message.

Stage 1. Alice begins by transmitting a random sequence of 1's and 0's (bits), using a random choice of rectilinear (horizontal and vertical) and diagonal polarization schemes. Figure 76 Figure 76 shows such a sequence of photons on their way to Bob. shows such a sequence of photons on their way to Bob.

Stage 2. Bob has to measure the polarization of these photons. Since he has no idea what polarization scheme Alice has used for each one, he randomly swaps between his +-detector and his -detector. Sometimes Bob picks the correct detector, and sometimes he picks the wrong one. If Bob uses the wrong detector he may well misinterpret Alice's photon. If Bob uses the wrong detector he may well misinterpret Alice's photon. Table 27 Table 27 covers all the possibilities. For example, in the top line, Alice uses the rectilinear scheme to send 1, and thus transmits ; then Bob uses the correct detector, so he detects , and correctly notes down 1 as the first bit of the sequence. In the next line, Alice does the same thing, but Bob uses the incorrect detector, so he might detect covers all the possibilities. For example, in the top line, Alice uses the rectilinear scheme to send 1, and thus transmits ; then Bob uses the correct detector, so he detects , and correctly notes down 1 as the first bit of the sequence. In the next line, Alice does the same thing, but Bob uses the incorrect detector, so he might detect[image] or or[image] which means that he might correctly note down 1 or incorrectly note down 0. which means that he might correctly note down 1 or incorrectly note down 0.

Stage 3. At this point, Alice has sent a series of 1's and 0's and Bob has detected some of them correctly and some of them incorrectly. To clarify the situation, Alice then telephones Bob on an ordinary insecure line, and tells Bob which polarization scheme she used for each photon-but not how she polarized each photon. So she might say that the first photon was sent using the rectilinear scheme, but she will not say whether she sent say whether she sent[image] or or[image] . Bob then tells Alice on which occasions he guessed the correct polarization scheme. On these occasions he definitely measured the correct polarization and correctly noted down 1 or 0. Finally, Alice and Bob ignore all the photons for which Bob used the wrong scheme, and concentrate only on those for which he guessed the right scheme. In effect, they have generated a new shorter sequence of bits, consisting only of Bob's correct measurements. This whole stage is ill.u.s.trated in the table at the bottom of . Bob then tells Alice on which occasions he guessed the correct polarization scheme. On these occasions he definitely measured the correct polarization and correctly noted down 1 or 0. Finally, Alice and Bob ignore all the photons for which Bob used the wrong scheme, and concentrate only on those for which he guessed the right scheme. In effect, they have generated a new shorter sequence of bits, consisting only of Bob's correct measurements. This whole stage is ill.u.s.trated in the table at the bottom of Figure 76 Figure 76.

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Figure 76 Alice transmits a series of 1's and 0's to Bob. Each 1 and each 0 is represented by a polarized photon, according to either the rectilinear (horizontal/vertical) or diagonal polarization scheme. Bob measures each photon using either his rectilinear or his diagonal detector. He chooses the correct detector for the leftmost photon and correctly interprets it as 1. However, he chooses the incorrect detector for the next photon. He happens to interpret it correctly as 0, but this bit is nevertheless later discarded because Bob cannot be sure that he has measured it correctly. Alice transmits a series of 1's and 0's to Bob. Each 1 and each 0 is represented by a polarized photon, according to either the rectilinear (horizontal/vertical) or diagonal polarization scheme. Bob measures each photon using either his rectilinear or his diagonal detector. He chooses the correct detector for the leftmost photon and correctly interprets it as 1. However, he chooses the incorrect detector for the next photon. He happens to interpret it correctly as 0, but this bit is nevertheless later discarded because Bob cannot be sure that he has measured it correctly.

These three stages have allowed Alice and Bob to establish a common series of digits, such as the sequence 11001001 agreed in Figure 76 Figure 76. The crucial property of this sequence is that it is random, because it is derived from Alice's initial sequence, which was itself random. Furthermore, the occasions when Bob uses the correct detector are also random. The agreed sequence does not therefore const.i.tute a message, but it could act as a random key. At last, the actual process of secure encryption can begin. from Alice's initial sequence, which was itself random. Furthermore, the occasions when Bob uses the correct detector are also random. The agreed sequence does not therefore const.i.tute a message, but it could act as a random key. At last, the actual process of secure encryption can begin.

Table 27 The various possibilities in stage 2 of photon exchange between Alice and Bob. The various possibilities in stage 2 of photon exchange between Alice and Bob.

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This agreed random sequence can be used as the key for a onetime pad cipher. Chapter 3 Chapter 3 described how a random series of letters or numbers, the onetime pad, can give rise to an unbreakable cipher-not just practically unbreakable, but absolutely unbreakable. Previously, the only problem with the onetime pad cipher was the difficulty of securely distributing the random series, but Bennett and Bra.s.sard's arrangement overcomes this problem. Alice and Bob have agreed on a onetime pad, and the laws of quantum physics actually forbid Eve from successfully intercepting it. It is now time to put ourselves in Eve's position, and then we will see why she is unable to intercept the key. described how a random series of letters or numbers, the onetime pad, can give rise to an unbreakable cipher-not just practically unbreakable, but absolutely unbreakable. Previously, the only problem with the onetime pad cipher was the difficulty of securely distributing the random series, but Bennett and Bra.s.sard's arrangement overcomes this problem. Alice and Bob have agreed on a onetime pad, and the laws of quantum physics actually forbid Eve from successfully intercepting it. It is now time to put ourselves in Eve's position, and then we will see why she is unable to intercept the key.

As Alice transmits the polarized photons, Eve attempts to measure them, but she does not know whether to use the +-detector or the -detector. On half the occasions she will choose the wrong detector. This is exactly the same position that Bob is in, because he too picks the wrong detector half the time. However, after the transmission Alice tells Bob which scheme he should have used for each photon and they agree to use only the photons which were measured when Bob used the right detector. However, this does not help Eve, because for half these photons she will have measured them using the incorrect detector, and so will have misinterpreted some of the photons that make up the final key.

Another way to think about quantum cryptography is in terms of a pack of cards rather than polarized photons. Every playing card has a value and a suit, such as the jack of hearts or the six of clubs, and usually we can look at a card and see both the value and the suit at the same time. However, imagine that it is only possible to measure either the value or the suit, but not both. Alice picks a card from the pack, and must decide whether to measure the value or the suit. Suppose that she chooses to measure the suit, which is "spades," which she notes. The card happens to be the four of spades, but Alice knows only that it is a spade. Then she transmits the card down a phone line to Bob. While this is happening, Eve tries to measure the card, but unfortunately she chooses to measure its value, which is "four." When the card reaches Bob he decides to measure its suit, which is still "spades," and he notes this down. Afterward, Alice calls Bob and asks him if he measured the suit, which he did, so Alice and Bob now know that they share some common knowledge-they both have "spades" written on their notepads. However, Eve has "four" written on her notepad, which is of no use at all. measure its suit, which is still "spades," and he notes this down. Afterward, Alice calls Bob and asks him if he measured the suit, which he did, so Alice and Bob now know that they share some common knowledge-they both have "spades" written on their notepads. However, Eve has "four" written on her notepa