You are probably surprised to hear of teleportation in a serious course. Do we, quantum scientists, claim that we can do what you can see in Star Trek, teleporting a human being? The answer is clearly no. What we can do is teleport the quantum state of a single quantum object. It is not a small achievement, and it should be quite surprising for anybody who has studied quantum mechanics. Remember indeed that one cannot determine by measurement the full quantum state of a single object. You get only a projection of its quantum state. To get the full state, you need many measurements on many copies. But you cannot make copies of a single quantum object because of the no-cloning theorem. You may remember from quantum optics one, the Wootters Zurek no-cloning theorem. It is impossible to duplicate a quantum system with a 100 percent fidelity. Since you cannot know the state, you must thus think that it is impossible to teleport a quantum state. In fact, quantum teleportation exists and does not contradict the no-cloning theorem, because when you go from the initial situation to the final situation, the quantum state is erased from the initial place, so there is never more than one sample of the quantum state. In a sense, it achieves the same operation as the cut and paste of classical computers. What is forbidden in the quantum world is copy and paste. We want to teleport the polarization state of photon nu_0 phi(nu_0), which is expressed with its two x and y polarization components lambda and mu. As usual, polarization y is perpendicular to the figure and polarization x is in the figure and perpendicular to the direction of propagation. We adopt the same convention for all legs. To teleport the state of nu_0, we dispose of a pair of photons nu_1 nu_2 in the entangled Bell state Psi minus. Photons nu_0 and nu_1 will be combined on a beam splitter followed by a joint detection system. Initially, the three photons nu_0, nu_1, and nu_2 are described by the state Upsilon. The first operation consists of making a partial Bell measurement on nu_0 and nu_1. You have just seen that a joint detection in the two outputs of the beam splitter corresponds to the Bell state Psi minus of nu_0, nu_1. If the result is positive, a signal will be sent to the shutter V which will be open. Otherwise, the shutter V remains closed. It means that photon nu_2 will be transmitted only if the result of the Bell measurement of Psi minus is positive. But what is then the state of nu_2? At this point, you are going to encounter one of the most surprising features of quantum mechanics. To describe the state of nu_2, we must take into account the result of the measurement on nu_0 and nu_1, although this measurement happens at a distance. This results from the standard formalism of quantum mechanics, which allows you to determine the state of the three photons system just after a positive result for the measurement of Psi minus on two photons only. Go into your favorite textbook on quantum mechanics, and you will find how to get the state of the three photons just after the measurement. It is obtained by projection of the initial state onto the eigenspace associated with the result of the measurement. That space is spanned by the two state vectors that are tensor products of Psi minus of nu_0, nu_1 by x_2 or y_2 respectively. Let us effect that projection, projecting successively on each of these eigenvectors. Projection on Psi minus of nu_0, nu_1 times x_2 has one term only associated with the term x_0,y_1, x_2 in Upsilon, and the coefficient is minus lambda over two. The state resulting from the projection is Psi minus of nu_0, nu_1 times x_2 with a coefficient minus lambda over 2. Similarly, the projection on Psi minus of nu_0, nu_1 times y_2 has one term only associated with y_0, x_1, y_2 in Upsilon. The projection yields the state Psi minus of nu_0, nu_1 times y_2 with the coefficient minus mu over 2. The total projection of Upsilon returns the sum of these two partial projections. It is, within an irrelevant global minus factor, the product of Psi minus of nu_0, nu_1 by a state of nu_2 which is identical to the initial state of nu_0 with its two components lambda and mu. We have thus copied the state of nu_0 onto nu_2 and nu_0 has disappeared. This is quantum teleportation. You should not be surprised by the term minus 1 over 2 in factor which makes the norm of the vector less than one. Projection does not conserve the norm in general which you should find obvious, if you think of a geometrical projection. You may wonder why I have used an entangled state of nu_1 and nu_2 different from the one considered in previous sections. It turns out that it yields the simplest teleportation protocol. I have already told you that it is possible to obtain any Bell state with adequate crystalline plates. The scheme you have just seen is remarkable and its experimental demonstration in 1997 was a landmark in the emergence of ideas about quantum networks. You have noticed, however, that it works only in a limited fraction of the cases, the ones when the result of the measurement on nu_0, nu_1 is Psi minus. The probability to find the system in that state is the square modulus of the projection of the initial state onto Psi minus, that is to say, one over four. It means that our method will work only in one fourth of the cases. Is it possible to generalize the method in order to make it 100 percent efficient? The answer is yes, in principle, when we know how to make an efficient complete Bell measurement, which is not the case at the moment. With current technology, however, we can do better. As we have seen, it is possible to identify not only the result associated with the Bell state Psi minus, but also the one associated with Psi plus, provided that one has two polarizers and four detectors. You can rewind to section three if this is not clear. Reasoning as in the case of Psi minus, you can calculate the result of the projection of the initial state Upsilon on the eigenspace associated with Psi plus. The probability is one fourth. The state resulting from the projection is found equal to Psi plus of nu_0, nu_1 times a state of nu_2 which is not exactly the initial state of nu_0. But if it was possible to change the sign of the x component of that state, then one would get exactly the initial state phi teleported on nu_2. Changing that sign is possible if the device V is equivalent to a half wave-plate conveniently oriented. Using an electro-optic device, such an operation can be implemented on-demand by application of a suitable voltage and the state phi initially on nu_0 has been now teleported on nu_2. So one can generalize the teleportation operation by taking into account the results of a Bell measurement on nu zero and nu one. If the result corresponds to Psi minus, the device V is simply open. If the result correspond to Psi plus, a voltage is applied to change the sign of the x component. In other cases, V is left closed. It means that teleportation works only half the times. At this point, I'm sure that many of you have realized that the process could be generalized to the cases when nu zero, nu one would be found in Phi plus or Phi minus. Simple operations at V would complete the teleportation operation. Instructions to make the right operation at V would be sent through the channel represented as a black wire. Since there are four different cases, it would suffice to use two classical control bits to choose the right configuration of V. Because of the lack of a complete Bell measurement for single photons, teleportation can be realized in only half the cases: the ones associated with Psi minus or Psi plus. But there is no fundamental reason preventing one to realize a complete Bell measurement. You can think of it. Maybe one of you will find a solution. A remarkable feature of quantum teleportation is the necessity to use two communication channels, a classical channel and a quantum channel. Can you identify these two channels? To answer the question, it suffices to think of the succession of the events after the measurement on nu zero, nu one is done. For the sake of the discussion, we assume that it is possible to make a complete Bell measurement. Immediately after that measurement, nu two is now in a specific state and no longer a member of an entangled pair. But its state depends on the result of the measurement. That state is different from the state to teleport, except if the result is Psi minus. In order to get the state we want to teleport, the device V is set according to the result of the measurement on nu zero, nu one. Since there are four possible results, two classical bits suffice to select the right setting of V. Then, photon nu two passes V and is transformed into the state we wanted to teleport. Two channels have been involved to complete the teleportation. The projection of the state of nu two on one specific state happens through the quantum channel, which is constituted by the two entangled photons, nu one, nu two. Remember that the two entangled photons constitute a single quantum entity even if the two photons are at a distance of each other. That global system is the quantum channel. The classical channel is obviously the line through which the information is sent to V. Now, don't you think that the way I've described the succession of operations raises questions about timing? Take a moment to think about it before continuing. This is one of the surprising features of quantum teleportation. Consider again the description of the process of teleportation. In a symmetric situation for photons nu one and nu two, as in the figure, photon nu two arrives at the apparatus V at about the time when the measurement is performed on nu zero, nu one. At that moment, nu two is projected onto a state that depends on the result of the measurement. Then, V is set according to the result of the measurement on nu zero, nu one. It will demand some time before the classical information, which cannot travel faster than light, arrives at V. It means that one needs to delay the arrival of nu two at V in order to wait for the two classical bits arriving at V. A simple solution is to implement an optical delay line with mirrors or an optical fiber to lengthen the path of nu two. A more elegant solution would be to store the state of photon nu two in a quantum memory and re-emit after a delay a photon with the stored state. Such a memory would be very useful in many operations of quantum communication and networks, and an active research is going on about quantum memories. To my knowledge, there is not yet good enough quantum memories able to store and re-emit efficiently single photons after a significant storage time. The discussion about timing allows me to emphasize the extraordinary character of quantum teleportation, in fact, closely related to the extraordinary character of entanglement. On the fundamental point of view, the fact that the quantum channel is instantaneous is amazing. Remember that Einstein could not accept such a feature! But it is not possible, however, to use it for instantaneous transmission of utilizable information because the teleportation is completed only after the classical channel has been used. It means that quantum non-locality, mysterious as it is, does not contradict relativity if we restrict the impossibility of faster than light propagation to utilizable information. But the mere fact that the description in our ordinary space suggests non-locality is a clue that we have the possibility of new kinds of technologies. Quantum teleportation is one of the fascinating quantum technologies. Who would have thought in the early 1980s at the time when classical communication networks were just appearing that it would be possible one day to cut and paste a quantum state? Quantum teleportation is a key technology for developing quantum networks: the would be quantum Internet. It opens the possibility of long-distance transport of a quantum state. Indeed, if it is possible to provide in advance two entangled qubits at two widely separated places, the state teleportation will happen whatever the distance, without attenuation. This is dramatically different from sending a qubit along a path, for instance, an optical fiber, where there is attenuation. Fascinating, isn't it? Many laboratories do work on such a possibility which will become a reality with good enough quantum memories.
