The security of quantum cryptography relies on the impossibility to determine the quantum state of a single quantum system. For instance, if we send a photon, polarized along theta on a beam splitter, aligned along x, we get either +1 or -1, but we cannot deduce the direction theta of the polarization of the photon. In contrast, If we have a collection of identical quantum objects all in the same state, for instance, many photons with the same polarization along theta, we can determine that state. The simplest strategy, not the most efficient, consists of performing large enough number of measurements with a polarizer along x: this yields a good estimation of the probabilities to get +1 or -1. If we know a priori that the polarization of the photon is linear, these probabilities allow us to determine plus or minus theta within an additive pi constant, which is irrelevant, since linear polarization is defined by a direction, so that theta and theta plus pi are equivalent. To complete the measurement and lift the degeneracy between the positive and negative solutions, it is sufficient to perform a measurement with the polarizer in another orientation. For instance, we can set the polarizer at the positive value, I am sure you will easily find how you can conclude. More generally, if we do not know a priori that the polarization of the photon is linear, so it could be circular or elliptical, it is possible to determine that polarization by making several different measurements, provided that we have a large enough number of photons, all with the same polarization. Suppose now we have a quantum cloner, that is to say, a device able to produce two perfect copies of a quantum system in states identical to the original state. We can chain a series of such cloners, and produce many copies of the initial system. It is then possible with a series of analyzers in various orientations, to determine the polarization of the input photon. Using such series of cloners, Eve could then intercept a photon sent by Alice, resend an identical copy to Bob, and make a complete tomography of the photon sent by Alice to Bob, without Bob noticing it. Eve could then obtain the polarization of the photons sent by Alice to Bob, and eventually will construct a third copy of the key of Alice and Bob, all that without being noticed. Quantum cloner would be a killer of quantum cryptography. But there is nothing like a quantum cloner. In 1982, William Wootters and Woicech Zurek demonstrated that perfect cloning of a single photon with an arbitrary polarization is impossible. This is the celebrated no-cloning theorem. Its demonstration is based on the basic principles of quantum mechanics. It is extremely important, the security of many quantum cryptography schemes relies on it. I will now demonstrate it for polarized photons, but the demonstration would be the same for any quantum system, whose state belongs in a two dimensional space that is to say for any qubit. It is tempting to consider a laser amplifier as a photon cloner. I'm sure, you heard that laser amplifier makes copies of an input photon and delivers photons identical to the initial one, that is to say with the same frequency, direction and polarization. Could we use such a laser amplifier to realize a perfect photon cloner? That is to say, to produce perfect copies for any polarization epsilon at the input. The answer is no. For each particular kind of laser amplifier either based on population inversion or on other processes, one can show that what you get at the output is not what is shown here. We will come back to these specific cases in the second quantum optics course when we study the interaction between a photon and an atom. For the time being, we do not need to go into the details of the system we would like to use as a cloner. The beauty and the power of the no-cloning theorem is to be general. You can apply it to any particular system. You know enough of quantum physics to understand and appreciate the general demonstration of the theorem which you will discover now. Let us then consider a hypothetical perfect cloner that would yield two perfect copies of the input photon in two different outputs O prime and O double prime. A perfect cloner must work for any polarization of the input photon, not only for x or y. It must also perfectly clone the theta polarization. I use here the yet more simplified notation x or y to designate a one photon state with polarization x or y, and theta, for one photon with polarization along theta. I know that it is annoying not to be fully consistent with the previous notations but, I also know that simplified notations help to understand delicate reasoning. Using the expression of the theta polarized photon in the x, y basis, we can express the output of the would-be cloner in that basis. Let us now write what we expect for a real quantum mechanical device which would be a perfect cloner for polarizations x and y. It should also obey the fundamental rules of quantum mechanics. In particular its action is described by a Hamiltonian and the relation between the input radiation state and the output radiation state is described by an evolution operator which is a linear operator. Linearity of the evolution means that the output associated with the superposition of the input is the superposition of the corresponding outputs. It applies to a theta polarized photon considered as a superposition of a x polarized photon and a y polarized photon. Linearity thus allows us to simply express the output of a real quantum mechanical device. Compare the two results If you expand the tensor product, you find the sum of four terms, an expression obviously different from the result on the left hand side. The difference is yet more visible if we simplify further, the notation for the tensor product of states in O prime and O double prime. Here x,x means that we have state x in O prime, and state x in O double prime. These notations are frequently used to express such tensor product states when there is no ambiguity. Since the two points of view lead to a contradiction, and linearity of the evolution is believed to be a fundamental feature of quantum mechanics, the only possible conclusion is that ideal quantum cloners, which would perfectly duplicate any state belonging to a two dimensional input space, cannot exist. The no-cloning theorem is a very fundamental theorem in quantum physics. It forbids complete determination of the quantum state psi of a single quantum system. It is crucial for the security of all quantum cryptography schemes, but it goes well beyond. For instance, if one could fully determined the quantum state of a single quantum system, one could use a weird property of entangled quantum systems, the so called quantum non-locality, to transmit utilizable information faster than light. I will explain it in our second course on quantum optics, but you understand that if we could transmit utilizable information faster than light, it would mean a formidable change in our conception of the world. The power of the no-cloning theorem stems from its generality. There have been many specific proposals to fully determine the quantum state of a quantum system by duplicating its state, but, all have been proven wrong. This could be expected because of the generality of the no-cloning theorem. In a sense, you can compare the situation to the impossibility of perpetual motion of the first kind. There have been dozens, maybe hundreds of proposals for perpetual motion. Their analysis is sometimes quite involved, often interesting, but it is not logically necessary: we know, a priori, that it is impossible because it would violate the very general law of energy conservation. At this point, however, I want to give you a warning about impossibility theorems. Impossibility theorems are important in physics and in engineering since they prevent one from wasting time in trying to develop skills that would violate them. But there is always the risk to applying an impossibility theorem beyond its domain of validity and then miss a genuine discovery. The history of experimental physics is rich in examples of schemes declared impossible to realize and eventually found correct. It only meant that the envisaged scheme did not fulfil the hypothesis of validity of the impossibility theorem. Remarkable examples can be found with trapping of charged particles, electrons or ions, declared impossible to trap because of the Earnshaw theorem, or with neutral atoms declared impossible to trap with light pressure because of the so called optical Earnshaw theorem. I will explain in our second quantum optics course how it has been possible to circumvent the optical Earnshaw theorem to trap atoms with light. In order to avoid such pitfalls, it is important to understand the domain of validity of the no-cloning theorem. We have demonstrated it for a quantum state belonging to a two dimensional space, and what has been shown is the impossibility to clone all the states of that space. It does not mean that we cannot perfectly clone some particular states. So, if you need a scheme, where only a sub ensemble of the full space of states must be cloned, you should not abandon your idea too fast. So, please remember one of my favorite mottos: impossibility theorems cannot be violated but they may be bypassed.
