Alice communicates with words drawn uniformly amongst $\{\ket{j}\}_{j=1..n}$, the canonical orthonormal basis. Sometimes however Alice interleaves quantum decoys $\{\frac{\ket{j}+i\ket{k}}{\sqrt{2}}\}$ between her messages. Such pairwise superpositions of possible words cannot be distinguished from the message words. Thus as malevolent Eve observes the quantum channel, she runs the risk of damaging the superpositions (by causing a collapse). At the receiving end honest Bob, whom we assume is warned of the quantum decoys' distribution, checks upon their integrity with a measurement. The present work establishes, in the case of individual attacks, the tradeoff between Eve's information gain (her chances, if a message word was sent, of guessing which) and the disturbance she induces (Bob's chances, if a quantum decoy was sent, to detect tampering). Besides secure channel protocols, quantum decoys seem a powerful primitive for constructing n-dimensional quantum cryptographic applications. Moreover the methods employed in this article should be of strong interest to anyone concerned with the old but fundamental problem of how much information may be gained about a system, versus how much this will disturb the system, in quantum mechanics. Keywords: d-level systems cryptography
Alice communicates with words drawn uniformly amongst {∣j⟩}j=1..n, the canonical orthonormal basis. Sometimes however Alice interleaves quantum decoys {2∣j⟩+i∣k⟩} between her messages. Such pairwise superpositions of possible words cannot be distinguished from the message words. Thus as malevolent Eve observes the quantum channel, she runs the risk of damaging the superpositions (by causing a collapse). At the receiving end honest Bob, whom we assume is warned of the quantum decoys' distribution, checks upon their integrity with a measurement. The present work establishes, in the case of individual attacks, the tradeoff between Eve's information gain (her chances, if a message word was sent, of guessing which) and the disturbance she induces (Bob's chances, if a quantum decoy was sent, to detect tampering). Besides secure channel protocols, quantum decoys seem a powerful primitive for constructing n-dimensional quantum cryptographic applications. Moreover the methods employed in this article should be of strong interest to anyone concerned with the old but fundamental problem of how much information may be gained about a system, versus how much this will disturb the system, in quantum mechanics. Keywords: d-level systems cryptography
Alice communicates with words drawn uniformly amongst {|j>}j=1…n, the canonical orthonormal basis. Sometimes however, Alice interleaves quantum decoys [Formula: see text] between her messages. Such pairwise superpositions of possible words cannot be distinguished from the message words. Thus as malevolent Eve observes the quantum channel, she runs the risk of damaging the superpositions (by causing a collapse). At the receiving end honest Bob, whom we assume is warned of the quantum decoys' distribution, checks upon their integrity with a measurement. The present work establishes, in the case of individual attacks, the tradeoff between Eve's information gain (her chances, if a message word was sent, of guessing which) and the disturbance she induces (Bob's chances, if a quantum decoy was sent, to detect tampering). Whilst not directly applicable to secure channel protocols, quantum decoys seem a powerful primitive for constructing other n-dimensional quantum cryptographic applications. Moreover, the methods employed in this article should be of strong interest to anyone concerned with the old but fundamental problem of how much information may be gained about a system, versus how much this will disturb the system, in quantum mechanics.
La distribution quantique de clés est une technique cryptographique permettant l'échange de clés secrètes dont la confidentialité est garantie par les lois de la mécanique quantique. Le comportement particulier des particules élémentaires est exploité. En effet, en mécanique quantique, toute mesure sur l'état d'une particule modifie irrémédiablement cet état. En jouant sur cette propriété, deux parties, souvent appelées Alice et Bob, peuvent encoder une clé secrète dans des porteurs quantiques tels que des photons uniques. Toute tentative d'espionnage demande à l'espion, Eve, une mesure de l'état du photon qui transmet un bit de clé et donc se traduit par une perturbation de l'état. Alice et Bob peuvent alors se rendre compte de la présence d'Eve par un nombre inhabituel d'erreurs de transmission. L'information échangée par la distribution quantique n'est pas directement utilisable mais doit être d'abord traitée. Les erreurs de transmissions, qu'elles soient dues à un espion ou simplement à du bruit dans le canal de communication, doivent être corrigées grâce à une technique appelée réconciliation. Ensuite, la connaissance partielle d'un espion qui n'aurait perturbé qu'une partie des porteurs doit être supprimée de la clé finale grâce à une technique dite d'amplification de confidentialité. Cette thèse s'inscrit dans le contexte de la distribution quantique de clé où les porteurs sont des états continus de la lumière. En particulier, une partie importante de ce travail est consacrée au traitement de l'information continue échangée par un protocole particulier de distribution quantique de clés, où les porteurs sont des états cohérents de la lumière. La nature continue de cette information implique des aménagements particuliers des techniques de réconciliation, qui ont surtout été développées pour traiter l'information binaire. Nous proposons une technique dite de réconciliation en tranches qui permet de traiter efficacement l'information continue. L'ensemble des techniques développées a été utilisé en collaboration avec l'Institut d'Optique à Orsay, France, pour produire la première expérience de distribution quantique de clés au moyen d'états cohérents de la lumière modulés continuement. D'autres aspects importants sont également traités dans cette thèse, tels que la mise en perspective de la distribution quantique de clés dans un contexte cryptographique, la spécification d'un protocole complet, la création de nouvelles techniques d'amplification de confidentialité plus rapides à mettre en œuvre ou l'étude théorique et pratique d'algorithmes alternatifs de réconciliation. Enfin, nous étudions la sécurité du protocole à états cohérents en établissant son équivalence à un protocole de purification d'intrication. Sans entrer dans les détails, cette équivalence, formelle, permet de valider la robustesse du protocole contre tout type d'espionnage, même le plus compliqué possible, permis par les lois de la mécanique quantique. En particulier, nous généralisons l'algorithme de réconciliation en tranches pour le transformer en un protocole de purification et nous établissons ainsi un protocole de distribution quantique sûr contre toute stratégie d'espionnage. Quantum key distribution is a cryptographic technique, which allows to exchange secret keys whose confidentiality is guaranteed by the laws of quantum mechanics. The strange behavior of elementary particles is exploited. In quantum mechnics, any measurement of the state of a particle irreversibly modifies this state. By taking advantage of this property, two parties, often called Alice and bob, can encode a secret key into quatum information carriers such as single photons. Any attempt at eavesdropping requires the spy, Eve, to measure the state of the photon and thus to perturb this state. Alice and Bob can then be aware of Eve's presence by a unusually high number of transmission errors. The information exchanged by quantum key distribution is not directly usable but must first be processed. Transmission errors, whether they are caused by an eavesdropper or simply by noise in the transmission channel, must be corrected with a technique called reconciliation. Then, the partial knowledge of an eavesdropper, who would perturb only a fraction of the carriers, must be wiped out from the final key thanks to a technique called privacy amplification. The context of this thesis is the quantum key distribution with continuous states of light as carriers. An important part of this work deals with the processing of continuous information exchanged by a particular protocol, where the carriers are coherent states of light. The continuous nature of information in this case implies peculiar changes to the reconciliation techniques, which have mostly been developed to process binary information. We propose a technique called sliced error correction, which allows to efficiently process continuous information. The set of the developed techniques was used in collaboration with the Institut d'Optique, Orsay, France, to set up the first experiment of quantum key distribution with continuously-modulated coherent states of light. Other important aspects are also treated in this thesis, such as placing quantum key distribution in the context of a cryptosystem, the specification of a complete protocol, the creation of new techniques for faster privacy amplification or the theoretical and practical study of alternate reconciliation algorithms. Finally, we study the security of the coherent state protocol by analyzing its equivalence with an entanglement purification protocol. Without going into the details, this formal equivalence allows to validate the robustness of the protocol against any kind of eavesdropping, even the most intricate one allowed by the laws of quantum mechanics. In particular, we generalize the sliced error correction algorithm so as to transform it into a purification protocol and we thus establish a quantum key distribution protocol secure against any eavesdropping strategy.
Representations of quantum operations – We start by introducing a geometrical representation (real vector space) of quantum states and quantum operations. To do so we exploit an isomorphism from positive matrices to a subcone of the Minkowski future light-cone. Pure states map onto certain light-like vectors, whilst the axis of revolution encodes the overall probability of occurrence for the state. This extension of the Generalized Bloch Sphere enables us to cater for non-trace-preserving quantum operations, and in particular to view the per-outcome effects of generalized measurements. We show that these consist of the product of an orthogonal transform about the axis of the cone of revolution and a positive real symmetric linear transform. In the case of a qubit the representation becomes all the more interesting since it elegantly associates, to each measurement element of a generalized measurement, a Lorentz transformation in Minkowski space. We formalize explicitly this correspondence between ‘observation of a quantum system’ and ‘special relativistic change of inertial frame’. To end this part we review the state-operator correspondence, which was successfully exploited by Choi to derive the operator-sum representation of quantum operations. We go further and show that all of the important theorems concerning quantum operations can in fact be derived as simple corollaries of those concerning quantum states. Using this methodology we derive novel composition laws upon quantum states and quantum operations, Schmidt-type decompositions for bipartite pure states and some powerful formulae relating to the correspondence. Quantum cryptography – The key principle of quantum cryptography could be summarized as follows. Honest parties communicate using quantum states. To the eavesdropper these states are random and non-orthogonal. In order to gather information she must measure them, but this may cause irreversible damage. Honest parties seek to detect her mischief by checking whether certain quantum states are left intact. Thus tradeoff between the eavesdropper’s information gain, and the disturbance she necessarily induces, can be viewed as the power engine behind quantum cryptographic protocols. We begin by quantifying this tradeoff in the case of a measure distinguishing two non-orthogonal equiprobable pure states. A formula for this tradeoff was first obtained by Fuchs and Peres, but we provide a shorter, geometrical derivation (within the framework of the above mentioned conal representation). Next we proceed to analyze the Information gain versus disturbance tradeoff in a scenario where Alice and Bob interleave, at random, pairwise superpositions of two message words within their otherwise classical communications. This work constitutes one of the few results currently available regarding d-level systems quantum cryptography, and seems to provide a good general primitive for building such protocols. The proof crucially relies on the state-operator correspondence formulae derived in the first part, together with some methods by Banaszek. Finally we make use of this analysis to prove the security of a ‘blind quantum computation’ protocol, whereby Alice gets Bob to perform some quantum algorithm for her, but prevents him from learning her input to this quantum algorithm.
Alice and Bob wish to communicate without the archvillainess Eve eavesdropping on their conversation. Alice, decides to take two college courses, one in cryptography, the other in quantum mechanics. During the courses, she discovers she can use what she has just learned to devise a cryptographic communication system that automatically detects whether or not Eve is up to her villainous eavesdropping. Some of the topics discussed are Heisenberg's Uncertainty Principle, the Vernam cipher, the BB84 and B92 cryptographic protocols. The talk ends with a discussion of some of Eve's possible eavesdropping strategies, opaque eavesdropping, translucent eavesdropping, and translucent eavesdropping with entanglement.
The security of two-state quantum-key distribution against individual attack is estimated when the channel has losses and noises. We assume that Alice and Bob use two nonorthogonal single-photon polarization states. To make our analysis simple, we propose a modified B92 protocol in which Alice and Bob make use of inconclusive results, and Bob performs a kind of symmetrization of received states. Using this protocol, Alice and Bob can estimate Eve's information gain as a function of a few parameters that reflect the imperfections of devices, or Eve's disturbance. In some parameter regions, Eve's maximum information gain shows counterintuitive behavior, namely, it decreases as the amount of disturbances increases. For a small noise rate, Eve can extract perfect information in the case where the angle between Alice's two states is small or large, while she cannot extract perfect information for intermediate angles. We also estimate the secret key gain, which is the net growth of the secret key per pulse. We show the region where the modified B92 protocol over a realistic channel is secure against individual attack.