We are interested in the problem of characterizing the correlations that arise when performing local measurements on separate quantum systems. In a previous work [Phys. Rev. Lett. 98, 010401 (2007)], we introduced an infinite hierarchy of conditions necessarily satisfied by any set of quantum correlations. Each of these conditions could be tested using semidefinite programming. We present here new results concerning this hierarchy. We prove in particular that it is complete, in the sense that any set of correlations satisfying every condition in the hierarchy has a quantum representation in terms of commuting measurements. Although our tests are conceived to rule out nonquantum correlations, and can in principle certify that a set of correlations is quantum only in the asymptotic limit where all tests are satisfied, we show that in some cases it is possible to conclude that a given set of correlations is quantum after performing only a finite number of tests. We provide a criterion to detect when such a situation arises, and we explain how to reconstruct the quantum states and measurement operators reproducing the given correlations. Finally, we present several applications of our approach. We use it in particular to bound the quantum violation of various Bell inequalities.

We introduce a hierarchy of conditions necessarily satisfied by any distribution P_{alphabeta} representing the probabilities for two separate observers to obtain outcomes alpha and beta when making local measurements on a shared quantum state. Each condition in this hierarchy is formulated as a semidefinite program. Among other applications, our approach can be used to obtain upper bounds on the quantum violation of an arbitrary Bell inequality. It yields, for instance, tight bounds for the violations of the Collins et al. inequalities.

Einstein-Podolsky-Rosen (EPR) steering is a form of bipartite quantum correlation that is intermediate between entanglement and Bell nonlocality. It allows for entanglement certification when the measurements performed by one of the parties are not characterised (or are untrusted) and has applications in quantum key distribution. Despite its foundational and applied importance, EPR steering lacks a quantitative assessment. Here we propose a way of quantifying this phenomenon and use it to study the steerability of several quantum states. In particular we show that every pure entangled state is maximally steerable, the projector onto the anti-symmetric subspace is maximally steerable for all dimensions, we provide a new example of one-way steering, and give strong support that states with positive-partial-transposition are not steerable.Introduction.-Quantum systems display correlations that do not have a counterpart in classical physics. In the early days of quantum theory E. Schrodinger noticed a consequence of these stronger-than-classical correlations and named it EPR steering [1]. EPR steering refers to the following phenomenon: two parties, Alice and Bob, share an entangled state |ψ AB . By measuring her subsystem, Alice can remotely change (i.e. steer) the state of Bob's subsystem in such a way that would be impossible if their systems were only classically correlated. The simplest example of steering is given by the maximally entangled state of two qubits |φ + = (|00 +|11 )/ √ 2. Alice can project Bob's system into the basis {|a , |a ⊥ } by making a measurement of her subsystem in the conjugate basis {|a * , |a ⊥ * }. As such, she can remotely prepare any state on Bob's subsystem, a feature that is impossible if they share only separable states.EPR steering was recently given an operational interpretation as the distribution of entanglement by an untrusted party [2]: Alice wants to convince Bob, who does not trust her, that they share an entangled state. Bob, in order to be convinced, asks Alice to remotely prepare a collection of states of his subsystems. Alice performs her measurements (which are unknown to Bob) and communicates the results to him. By looking at the conditional states prepared by Alice, Bob is able to certify if they must have come from measurements on an entangled state. Interestingly, EPR steering is a form of quantum correlation that lies in between entanglement [3] and Bell nonlocality [4] since, on the one hand not every entangled state is steerable, and on the other hand some steerable states do not violate a Bell inequality [2]. Furthermore, like nonlocality, steering can be demonstrated in simple 'tests', for example it is sufficient to consider only two measurements with two outcomes for Alice, preparing a collection of four states for Bob. As such, steering can be certified experimentally through the violation of steering inequalities, the analogue of Bell inequalities [5]. In fact several steering tests have been reported [6,7], including a recent loophole-free experiment [8]...

We analyze the asymptotic security of the family of Gaussian modulated Quantum Key Distribution protocols for Continuous Variables systems. We prove that the Gaussian unitary attack is optimal for all the considered bounds on the key rate when the first and second momenta of the canonical variables involved are known by the honest parties.PACS numbers: 03.67. Dd, 03.67.Hk In 1984 Bennet and Brassard introduced the concept of Quantum Cryptography and presented the first Quantum Key Distribution (QKD) protocol: BB84 [1]. The original idea was that in Quantum Mechanics, and contrary to Classical Physics, the observation of a system invariably perturbs the system under observation. Therefore, if two honest parties, Alice and Bob, establish a quantum channel and use it to send information, an eavesdropper's presence could be detected by analyzing how the noise-free channel has changed. It was then shown that QKD protocols are completely secure against any eavesdropping attacks as long as the bit error rates do not exceed a certain value (see for instance [2] and references therein). In the meantime, new applications of Quantum Mechanics to certain information tasks started to develop: coin tossing, dense coding, teleportation...All these results first appeared in the context of discrete systems, but many of them were later translated into the language of Continuous Variables (CV) systems. This is per se an interesting theoretical problem. However, the main motivation for dealing with these systems comes from a practical point of view: although the set of feasible operations is reduced, the so-called Gaussian operations are easy to implement and amazingly precise. Quantum cryptography with continuous variables systems [3,4,5,6,7,8] was the most immediate result: the transmission of coherent or squeezed pulses of light, together with homodyne measurements, allows performing QKD with very high key rates [9].The security analysis of these new protocols is not straightforward. First of all, the commonly used reconciliation and privacy amplification protocols are designed to correct and distill secret bits from binary random variables, although some have been adapted to continuous variables [10,11]. Second, the dimension of the Hilbert space on which the CV systems are defined is infinite in theory, which makes a complete tomography impossible in principle, thus preventing Alice and Bob to know precisely the state they are actually sharing. Therefore, security proofs for CV protocols have to consider the optimal attack by Eve when Alice and Bob know their state is in some set, usually defined by the momenta of the quadratures up to second order [12]. In her search for information, Eve's possible attacks can be classified in three different types [13]: individual attacks, where Eve interacts individually with the sent states and measures them individually before public reconciliation; collective attacks, where Eve applies the same unitary individual attack over the sent states, but performs her (possibly collective) meas...

We consider optimization problems with polynomial inequality constraints in noncommuting variables. These non-commuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints. Such problems arise naturally in quantum theory and quantum information science. To solve them, we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution. We also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem.