Security of Quantum Key Distribution with entangled quNits

We consider a generalisation of Ekert's entanglement-based quantum cryptographic protocol where qubits are replaced by qu$N$its (i.e., N-dimensional systems). In order to study its robustness against optimal incoherent attacks, we derive the information gained by a potential eavesdropper during a cloning-based individual attack. In doing so, we generalize Cerf's formalism for cloning machines and establish the form of the most general cloning machine that respects all the symmetries of the problem. We obtain an upper bound on the error rate that guarantees the confidentiality of quNit generalisations of the Ekert's protocol for qubits.Comment: 15 pages, equation 15 and conclusions corrected the 14th of April 2003, new results adde

Inequalities that test locality in quantum mechanics

Quantum theory violates Bell's inequality, but not to the maximum extent that is logically possible. We derive inequalities (generalizations of Cirel'son's inequality) that quantify the upper bound of the violation, both for the standard formalism and the formalism of generalized observables (POVMs). These inequalities are quantum analogues of Bell inequalities, and they can be used to test the quantum version of locality. We discuss the nature of this kind of locality. We also go into the relation of our results to an argument by Popescu and Rohrlich (Found. Phys. 24, 379 (1994)) that there is no general connection between the existence of Cirel'son's bound and locality.Comment: 5 pages, 1 figure; the argument has been made clearer in the revised version; 1 reference adde

Non-local correlations as an information theoretic resource

It is well known that measurements performed on spatially separated entangled quantum systems can give rise to correlations that are non-local, in the sense that a Bell inequality is violated. They cannot, however, be used for super-luminal signalling. It is also known that it is possible to write down sets of ``super-quantum'' correlations that are more non-local than is allowed by quantum mechanics, yet are still non-signalling. Viewed as an information theoretic resource, super-quantum correlations are very powerful at reducing the amount of communication needed for distributed computational tasks. An intriguing question is why quantum mechanics does not allow these more powerful correlations. We aim to shed light on the range of quantum possibilities by placing them within a wider context. With this in mind, we investigate the set of correlations that are constrained only by the no-signalling principle. These correlations form a polytope, which contains the quantum correlations as a (proper) subset. We determine the vertices of the no-signalling polytope in the case that two observers each choose from two possible measurements with d outcomes. We then consider how interconversions between different sorts of correlations may be achieved. Finally, we consider some multipartite examples.Comment: Revtex. 12 pages, 6 figure

Robust Randomness Amplifiers: Upper and Lower Bounds

A recent sequence of works, initially motivated by the study of the nonlocal properties of entanglement, demonstrate that a source of information-theoretically certified randomness can be constructed based only on two simple assumptions: the prior existence of a short random seed and the ability to ensure that two black-box devices do not communicate (i.e. are non-signaling). We call protocols achieving such certified amplification of a short random seed randomness amplifiers. We introduce a simple framework in which we initiate the systematic study of the possibilities and limitations of randomness amplifiers. Our main results include a new, improved analysis of a robust randomness amplifier with exponential expansion, as well as the first upper bounds on the maximum expansion achievable by a broad class of randomness amplifiers. In particular, we show that non-adaptive randomness amplifiers that are robust to noise cannot achieve more than doubly exponential expansion. Finally, we show that a wide class of protocols based on the use of the CHSH game can only lead to (singly) exponential expansion if adversarial devices are allowed the full power of non-signaling strategies. Our upper bound results apply to all known non-adaptive randomness amplifier constructions to date.Comment: 28 pages. Comments welcom

Quantum entanglement can be simulated without communication

It has recently been shown that all causal correlations between two parties which output each one bit, a and b, when receiving each one bit, x and y, can be expressed as convex combinations of local correlations (i.e., correlations that can be simulated with local random variables) and non-local correlations of the form a+b=xy mod 2. We show that a single instance of the latter elementary non-local correlation suffices to simulate exactly all possible projective measurements that can be performed on the singlet state of two qubits, with no communication needed at all. This elementary non-local correlation thus defines some unit of non-locality, which we call a nl-bit.Comment: 4 pages RevTex, 3 eps figure

Classifying N-qubit Entanglement via Bell's Inequalities

All the states of N qubits can be classified into N-1 entanglement classes from 2-entangled to N-entangled (fully entangled) states. Each class of entangled states is characterized by an entanglement index that depends on the partition of N. The larger the entanglement index of an state, the more entangled or the less separable is the state in the sense that a larger maximal violation of Bell's inequality is attainable for this class of state.Comment: 4 pages, 3 figure

Testing the bounds on quantum probabilities

Bounds on quantum probabilities and expectation values are derived for experimental setups associated with Bell-type inequalities. In analogy to the classical bounds, the quantum limits are experimentally testable and therefore serve as criteria for the validity of quantum mechanics.Comment: 9 pages, Revte

Secrecy extraction from no-signalling correlations

Quantum cryptography shows that one can guarantee the secrecy of correlation on the sole basis of the laws of physics, that is without limiting the computational power of the eavesdropper. The usual security proofs suppose that the authorized partners, Alice and Bob, have a perfect knowledge and control of their quantum systems and devices; for instance, they must be sure that the logical bits have been encoded in true qubits, and not in higher-dimensional systems. In this paper, we present an approach that circumvents this strong assumption. We define protocols, both for the case of bits and for generic $d$-dimensional outcomes, in which the security is guaranteed by the very structure of the Alice-Bob correlations, under the no-signalling condition. The idea is that, if the correlations cannot be produced by shared randomness, then Eve has poor knowledge of Alice's and Bob's symbols. The present study assumes, on the one hand that the eavesdropper Eve performs only individual attacks (this is a limitation to be removed in further work), on the other hand that Eve can distribute any correlation compatible with the no-signalling condition (in this sense her power is greater than what quantum physics allows). Under these assumptions, we prove that the protocols defined here allow extracting secrecy from noisy correlations, when these correlations violate a Bell-type inequality by a sufficiently large amount. The region, in which secrecy extraction is possible, extends within the region of correlations achievable by measurements on entangled quantum states.Comment: 23 pages, 4 figure

Sufficient conditions for three-particle entanglement and their tests in recent experiments

We point out a loophole problem in some recent experimental claims to produce three-particle entanglement. The problem consists in the question whether mixtures of two-particle entangled states might suffice to explain the experimental data. In an attempt to close this loophole, we review two sufficient conditions that distinguish between N-particle states in which all N particles are entangled to each other and states in which only M particles are entangled (with M<N). It is shown that three recent experiments to obtain three-particle entangled states (Bouwmeester et al., Pan et al., and Rauschenbeutel et al.) do not meet these conditions. We conclude that the question whether these experiments provide confirmation of three-particle entanglement remains unresolved. We also propose modifications of the experiments that would make such confirmation feasible.Comment: 16 page

A toy model for quantum mechanics

The toy model used by Spekkens [R. Spekkens, Phys. Rev. A 75, 032110 (2007)] to argue in favor of an epistemic view of quantum mechanics is extended by generalizing his definition of pure states (i.e. states of maximal knowledge) and by associating measurements with all pure states. The new toy model does not allow signaling but, in contrast to the Spekkens model, does violate Bell-CHSH inequalities. Negative probabilities are found to arise naturally within the model, and can be used to explain the Bell-CHSH inequality violations.Comment: in which the author breaks his vow to never use the words "ontic" and "epistemic" in publi