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Yet, How can we distinguish ? Their the Symmetrization Postulate is rarely evoked. possibly different internal states are not a good cri- terium, as the dynamics can in general affect the internal degrees of freedom of the particles. The same is valid A. The Spin-Statistics Connection for their momentum or other dynamical variables. But their spatial location can actually be used to distinguish To determine whether a given particle is a or them. Let us imagine we have two identical particles, a , we need to investigate its statistical behaviour one in Alice’s possession and the other with Bob. If in the presence of (at least one) other identical particles, these two parties are kept distant enough so that the when they are all indistinguishable, and this behaviour wave functions of the particles practically never overlap will be very different for the two types of particles. (during the time we consider this system), then it is Indirect methods could also help us reach a conclusion, possible to keep track of the particles just by keeping but before any of that a simple and intriguing property track of the classical parties. This situation is not un- can actually come to our rescue: the spin-statistics common in quantum mechanics. If, on the other hand, connection. the wave functions do overlap at some point, then we no longer know which particle is with which party. And if Spin-Statistics Theorem — Particles with integer we just do not or cannot involve these classical parties spin are . Particles with half-integer spin are at all, then it is in general also impossible to keep track . of identical particles. In both these cases, the particles become completely indistinguishable, they are identified This is not only a widely known empirical rule in by completely arbitrary labels, with no physical meaning Physics, but in fact a theorem (originally proved by Pauli (as opposed to Alice and Bob). In these situations the [3]), even if its proofs are not all completely clear and free description of our system becomes ambiguous and the from controversy. Thanks to it, it is very easy to deter- so-called exchange degeneracy appears. The problem mine whether some particle is either a fermion or a boson. of finding the correct and unambiguous description for In particular, this criterion works also for composite par- such systems is very general and requires the introduc- ticles. It is quite surprising to find such a connection tion of a new postulate for quantum mechanics: the between the spin of a particle and its statistical nature, Symmetrization Postulate. a connection whose origins I believe are still not well un- derstood. Symmetrization Postulate — In a system contain- ing indistinguishable particles, the only possible states of the system are the ones described by vectors that are, with III. QUANTUM INFORMATION respect to permutations of the labels of those particles:

• either completely symmetrical — in which case the The use of quantum systems and their unique proper- particles are called bosons; ties to encode, transmit, process and store information offers a completely new way to deal with information, • either completely antisymmetrical — in which case representing a revolution for Information Sciences, and the particles are called fermions. possibly for our Information Society as well. It is con- ceivable that one day we will have a more fundamental This is in fact a generalization of Pauli’s Exclusion description of Nature than Quantum Physics and this Principle, in two ways. First, it extends it to a whole class may well represent yet again a revolution in the way we of particles which suffer the same restrictions: fermions. deal with information. But before trying to reach that But if goes even further and introduces a new class of far, we should ask ourselves if we have already explored particles, bosons, which have a very different behaviour, all the properties of the quantum world in terms of their almost the opposite, as they are forced to share the same relevance for information processing. I think not. There quantum numbers. To decide which particles should be is still at least one other property, as fundamental as the associated to a particular symmetry is something that ones already mentioned, that should be considered: par- must ultimately be determined by observation. The Sym- ticle statistics [15], or the apparent fact that every parti- metrization Postulate matches the study of such symme- cle is either a fermion or a boson and that their collective tries with our empirical knowledge: as far as we know behaviour obeys precise rules. Now, can the effects of today, there are two classes of particles in Nature ac- particle statistics play any role in quantum information cording to their collective behaviour in indistinguishable processing? Can they be used to perform useful quantum situations. These are, of course, bosons and fermions: information tasks? And in an efficient way? no particles have been found so far that under the same For the last couple of years we have been exploring the circumstances could be described by vectors that are nei- role of indistinguishable particles and quantum statistics ther symmetrical nor antisymmetrical. It is important to in quantum information processing, both for fermions 3 and bosons [16]. We have proved that, using only the two qubits (encoded in the internal degree of freedom of effects of particle statistics, it is possible to perform a two electrons or two photons) can be achieved using only quantum information task — such as transfer of entangle- the properties of fermions and bosons. Furthermore, this ment [11], to do useful quantum information processing method offers interesting applications to the detection of — such as entanglement concentration [12], and do it in entanglement and the purification of mixed states. a optimal way — in particular, in a state discrimination Two main features emerge from the above results: par- protocol [13]. All these results make use the antibunching ticle statistics appears as a resource that can replace con- of indistinguishable electrons impinging in a beam split- trolled operations (conditional interactions) in a natural ter, as well as of the bunching of photons in a similar way, and information processing using indistinguishable situation, both a clear signature of their statistics [14]. particles is different for fermions and bosons. The ob- Using two pairs of entangled particles, it was shown tained results can also be tested with current technology. for both fermions (electrons) and bosons (photons) that Moreover, they establish that indistinguishable particles indistinguishability enforces a transfer of entanglement and quantum statistics can play a new and important role from the internal to the spatial degrees of freedom with- in quantum information and that this connection should out any interaction between these degrees of freedom be further explored. [11]. Furthermore, sub-ensembles selected by local mea- surements of the path will in general have different amounts of entanglement in the internal degrees of free- dom depending on the statistics of the particles involved. Acknowledgements Then, an entanglement concentration scheme was pro- posed which uses only the effects of particle statistics [12]. Although its efficiency is the same for both fermions and This article is based on a talk delivered at the In- bosons, the protocol itself is slightly different depend- ternational Meeting on Quantum Information Science: ing on the nature of the particles. Moreover, no explicit Foundations of Quantum Information, held in Camerino, controlled operation is required at any stage. Finally, Italy, in April 2004. I would like to thank the support particle statistics is applied to the problem of optimal from Funda¸c˜ao para a Ciˆencia e a Tecnologia (Portugal) ambiguous discrimination of quantum states [13]. It was and the 3rd Community Support Framework of the Euro- shown that the Helstrom optimal single-shot discrimina- pean Social Fund, as well as acknowledge the QuantLog tion probability to distinguish non-orthogonal states of initiative.

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