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Abstracts for Contributed Presentations

Podium Presentations   Poster Presentations   Invited Talks - Invited Speakers page

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Podium Presentations

Quantum Algorithms from Topological Quantum Field Theories

Key Words: Quantum Computation Quantum Algorithms TQFT Topological invariants

Authors: Gorjan Alagic

Institution(s): Institute for Quantum Computing University of Waterloo

Abstract: Topological Quantum Field Theories (or TQFTs) are abstract constructions from category theory and mathematical physics. Their conception was originally motivated by the search for a physical theory that unifies general relativity and quantum mechanics. At its core, a TQFT is a map from manifolds (e.g., spacetimes) to linear maps (e.g., quantum operations) that satisfies some physically sensible properties. For instance, the disjoint union of two manifolds must be mapped to the tensor product of the two corresponding linear maps. To manifolds without boundary, a TQFT assigns a topologically invariant number called a quantum invariant. This discovery added a beautiful new direction in the study of manifold invariants in pure mathematics. For this reason and many others, this area has seen a tremendous amount of work in the past two decades, from physicists and mathematicians alike.

In this talk, we will discuss how this theory can be applied to design quantum algorithms for approximating certain quantum invariants. The aim of the talk is to give an accessible introduction to some of the ideas in this area, and to motivate quantum computation enthusiasts to study it further. We will begin with the simplest two-dimensional state-sum models. These examples are quite attractive, since they can be described in a combinatorial manner by means of triangulations. We will then define a three-dimensional state-sum TQFT, called the Turaev-Viro theory. Finally, we will discuss a recent result (joint with Stephen Jordan, Robert Koenig, and Ben Reichardt) showing that approximating the Turaev-Viro quantum invariant is a universal problem for quantum computation.

Website: http://www.iqc.ca/~galagic/

Twists in topological codes

Key Words: Topological quantum error-correcting code, anyons, symmetry, topological defect, code deformation

Authors: H. Bombin

Institution(s): Perimeter Institute

Watch Bombin presentation

Abstract: There exists a close relationship between topological quantum error-correcting codes and topological order in condensed matter systems. Indeed, a topological stabilizer code can be regarded as the ground state of a suitable Hamiltonian model, so that "wrong" syndromes correspond to excitations. These excitations are anyons, quasiparticles that carry a topological charge and exhibit exotic statistics.

Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We call these defects twists. Twists give rise to new topological degrees of freedom in the ground state, useful as a quantum memory. Moreover, twists can be braided to perform topologically protected gates on these topological qubits.

Thus, twists provide a new way to encode and compute with topological codes through code deformations. Because the properties of twists depend on the anyon model, codes with different anyon content give rise to different computational capabilities. E.g., in the well-known toric code a process where suitable twists are braided and fused has the same outcome as if they were Ising anyons. These are non-abelian anyons: braiding produces non-trivial gates on encoded qubits.

Adiabatic Quantum Algorithms for the NP-Complete Maximum-Weight Independent Set, Exact Cover and 3SAT Problems

Key Words: Adiabatic Quantum Algorithm, NP-complete, Maximum Independent Set, Exact Cover, 3SAT

Authors: Vicky Choi

Institution(s): Virginia Tech

Watch Choi Video

Abstract: The problem Hamiltonian of the adiabatic quantum algorithm for the maximum-weight independent set problem (MIS) that is based on the reduction to the Ising problem (as described in [Choi08]) has flexible parameters. We show that by choosing the parameters appropriately in the problem Hamiltonian (without changing the problem to be solved) for MIS on CK graphs, we can prevent the first order quantum phase transition and significantly change the minimum spectral gap. We raise the basic question about what the appropriate formulation of adiabatic running time should be. We also describe adiabatic quantum algorithms for Exact Cover and 3SAT in which the problem Hamiltonians are based on the reduction to MIS. We point out that the argument in Altshuler et al.(arXiv:0908.2782 [quant-ph]) that their adiabatic quantum algorithm failed with high probability for randomly generated instances of Exact Cover does not carry over to this new algorithm.

Website: http://arxiv.org/abs/1004.2226; http://www.cs.vt.edu/~vchoi

Fast Decoders for Topological Quantum Codes

Key Words: Quantum error-correction, Topological quantum codes, Toric code, Renormalization, Belief propagation, Depolarizing channel, Erasure channel, Color codes

Authors: Guillaume Duclos-Cianci, David Poulin

Institution(s): Université de Sherbrooke

Watch Duclos-Cianci Video

Abstract: Topological quantum computation and topological error correcting codes attracted a lot of interest recently because they require realistic nearest neighbors couplings and, by encoding the information in non-local topological degrees of freedom, they offer a very high resilience to local noise. I will present a family of algorithms, combining real-space renormalization methods and belief propagation, to estimate the free energy of a topologically ordered system in the presence of defects (Phys. Rev. Lett. 104, 050504 (2010)). Such an algorithm is needed to preserve the quantum information stored in the ground space of a topologically ordered system and to decode topological error-correcting codes. For a system of linear size L, our algorithm runs in time log L compared to L^6 needed for the minimum-weight perfect matching algorithm previously used in this context and achieves a higher depolarizing error threshold (16.5% vs 15.5%). I will introduce the intuitions behind the m! ethod and present new developments.

On the relevance of quasi-probability representations to quantum foundations and quantum information theory

Key Words: Quantum foundations, quasi-probability, Wigner functions, phase space, negativity, non-classicality.

Authors: Chris Ferrie, Joseph Emerson

Institution(s): Institute for Quantum Computing University of Waterloo

Abstract: Several quasi-probability representations of quantum states have been proposed to study various problems in quantum information theory and quantum foundations. These representations are often defined only on restricted dimensions and their physical significance in contexts such as drawing quantum-classical comparisons is limited by the non-uniqueness of the particular representation. Here we show how the mathematical theory of frames provides a unified formalism which accommodates all known quasi-probability representations of quantum systems. Moreover, we show that any quasi-probability representation is equivalent to a frame representation and then prove that any such representation of quantum mechanics must exhibit either negativity or a deformed probability calculus.

Website: http://arxiv.org/find/quant-ph/1/au:+Ferrie_C/0/1/0/all/0/1

Closed time-like curves in measurement-based quantum computation

Key Words: closed time-like curves; quantum computation; one-way model

Authors: Raphael Dias da Silva (1) Ernesto F. Galvão (1) Elham Kashefi (2)

Institution(s): (1) Instituto de Física, Univ. Federal Fluminense (Brazil) (2) School of Informatics, Univ. Edinburgh

Watch Dias da Silva video

Abstract: Many results have been recently obtained regarding the power of hypothetical closed time-like curves (CTC’s) in quantum computation. Most of them have been derived using Deutsch’s influential model for quantum CTCs [D. Deutsch, Phys. Rev. D 44, 3197 (1991)]. Deutsch’s model demands self-consistency for the time-travelling system, but in the absence of (hypothetical) physical CTCs, it cannot be tested experimentally.

In this paper we show how the one-way model of measurement-based quantum computation (MBQC) can be used to test Deutsch’s model for CTCs. Using the stabilizer formalism, we identify predictions that MBQC makes about a specific class of CTCs involving travel in time of quantum systems. Using a simple example we show that Deutsch’s formalism leads to predictions conflicting with those of the one-way model.

There exists an alternative, little-discussed model for quantum time-travel due to Bennett and Schumacher (in unpublished work, see http://bit.ly/cjWUT2), which was rediscovered recently by Svetlichny [arXiv:0902.4898v1]. This model uses quantum teleportation to simulate (probabilistically) what would happen if one sends quantum states back in time. We show how the Bennett/ Schumacher/ Svetlichny (BSS) model for CTCs fits in naturally within the formalism of MBQC. We identify a class of CTC’s in this model that can be simulated deterministically using techniques associated with the stabilizer formalism. We also identify the fundamental limitation of Deutsch's model that accounts for its conflict with the predictions of MBQC and the BSS model.

This work was done in collaboration with Raphael Dias da Silva and Elham Kashefi, and has appeared in preprint format (see website).

Website: http://arxiv.org/abs/1003.4971

Fractional scaling of quantum walks on percolation lattices

Key Words: quantum walks, percolation lattices, disordered systems

Authors: Viv Kendon

Institution(s): University of Leeds

Watch Kendon Video

Abstract: Quantum walks have been used as simple models of quantum transport phenomena, applicable to systems as diverse as spin chains and bio-molecules. Here we investigate the properties of quantum walks on percolation lattices, disordered structures appropriate for modelling biological and experimentally realistic systems. Both bond (edge) and site percolation have similar definitions: with independent randomly chosen probability p the bond or site is present in the lattice. In two and higher dimensions, percolation lattices exhibit a phase transition from a set of small disconnected regions to a more highly connected structure ("one giant cluster"). On 2D Cartesian lattices, the critical probability pc = 0.5 (bond) and pc = 0.5927... (site).

Below pc, the quantum walk will not be able to spread. Approaching pc from above, the spreading slows down completely, as the number of long-distance connected paths reduces to zero. For p = 1, the lattice is fully connected, and the standard quantum walk spreading applies (linear in T). In between, we find the quantum walks show fractional scaling of the spreading, i.e., proportional to T to the power alpha (0.5 < alpha < 1).

Our (numerical) results are skewed by finite size effects: the increase in alpha from zero begins before p = pc. It then flattens toward the classical random walk spreading rate of alpha = 0.5 around p = 0.85, followed by a steep rise to the quantum value of alpha = 1 at p = 1. At this stage, we do not have enough data to predict the large T behaviour, but think the steep rise will becomes a "step" function at p = 1 as T -> infinity. The randomness in the percolation lattice would thus act as decoherence in the large T limit. However, such a limit could only be approached for quite large values of T, and from the point of view of models for disordered systems on smaller scales (tens of sites), the faster-than-classical fractional scaling is very much the dominant feature.

Quantum computation on the edge of a symmetry-protected topological order

Key Words: quantum computation, topological order

Authors: Akimasa Miyake

Institution(s): Perimeter Institute for Theoretical Physics

Abstract: We elaborate the idea of quantum computation through measuring the correlation of a gapped ground state, while the bulk Hamiltonian is utilized to stabilize the resource. A simple computational primitive, by pulling out a single spin adiabatically from the bulk followed by its measurement, is shown to make any ground state of the one-dimensional isotropic Haldane phase useful ubiquitously as a quantum logical wire. The primitive is compatible with certain discrete symmetries that are crucial to protect this topological order, and the antiferromagnetic Heisenberg spin-1 chain of a finite length is practically a sufficient resource. Our approach manifests a holographic principle in that the logical information of a universal quantum computer can be written and processed perfectly on the edge state (i.e. boundary) of the system, supported by the persistent entanglement from the bulk even when the ground state and its evolution cannot be exactly analyzed. Reference: arXiv:1003.4! 662

For How Long Is It Possible To Quantum Compute?

Key Words: quantum error correction, fault-tolerant quantum computation, critical environments

Authors: Eduardo R. Mucciolo, E. Novais, Harold U. Baranger

Institution(s): University of Central Florida, USA; ABC Federal University, Brazil; Duke University, USA

Abstract: One of the key problems in quantum information processing is to understand the physical limits to quantum computation. Several strategies have been proposed to attenuate errors caused by the interaction of the computer with its surrounding environment and quantum error correction (QEC) is likely the most versatile. A large effort has been devoted to proving that resilience can be achieved by concatenating QEC codes in logical structures. In our work we look into this question at a different angle: we provide an upper bound on the time available to computation given a certain computer, a QEC code, and a decohering environment. We consider a broad class of environments, including those where correlation effects can be induced by gapless modes.

Our approach is based on a Hamiltonian formulation where we use coarse graining in time to derive an explicit quantum evolution operator for the logical qubits, taking into account the QEC code. We show that this evolution operator has the same form as that for the original physical qubits, except for a reduced coupling to the environment which can be evaluated systematically for a given geometry and QEC code structure.

To quantify the effectiveness of QEC, we compute the trace distance between the real and ideal states of a logical qubit after an arbitrary number of QEC cycles. We derive expressions for the long-time trace distance for several for super-ohmic-, ohmic-, and sub-ohmic-like baths. Given a confidence threshold for the trace distance, we establish the maximum time available for computation in those three cases. This maximum time is controlled by an exponent related to the spatial dimensions and other characteristics of the computer and the environment. For the super-ohmic regime, we find that computation can continue indefinitely, while in the other regimes the maximum time depends strongly on the QEC code, on the number of logical qubits, and on the original environment-computer strength interaction.

Website: http://arxiv.org/abs/1004.3247

Testing Contextuality on Quantum Ensembles with One Clean Qubit

Key Words: contextuality, DQC1, NMR, solid state

Authors: Osama Moussa (1), Colm A. Ryan (1), David G. Cory (2,3), and Raymond Laflamme (1,3)

Institution(s): (1) Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L3G1, Canada (2) Department of Nuclear Science and Engineering, MIT, Cambridge, Massachusetts 02139, USA (3) Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2J2W9, Canada

Abstract: The main question at hand is whether nature is fundamentally contextual. A number of recent experimental results have tackled this question in different settings, and this work examines an important, hitherto unexplored, piece of the puzzle; proposing and demonstrating a protocol to experimentally test contextuality without recourse to the isolation of individual quantum systems nor strong measurement.

Website: http://link.aps.org/doi/10.1103/PhysRevLett.104.160501

A Numerical Quantum and Classical Adversary

Key Words: adversary semidefinite clock decoherence algorithm

Authors: Michael Mullan, Emanuel Knill

Institution(s): University of Colorado at Boulder, National Institute of Standards and Technology

Abstract: The Quantum Adversary Method has proven to be a successful technique for deriving lower bounds on a wide variety of problems. However, it assumes perfect quantum computation, which in most modern devices, is unrealistic. Here, we develop a generalization of this technique without this assumption, which can be applied to arbitrary small problems automatically. To do this, we start by reformulating the objective value of the semidefinite program of the spectral adversary method. By relating the final measurement stage of a quantum computation to remote state preparation, we prove that the optimal value of the new objective corresponds to the probability that the quantum computer will output the correct value after a specified number of queries. Once in this framework, the addition of decoherence is natural. In particular, the optimum probability of success can be determined for any probability of phase error. In the limit of complete phase decoherence, we recover the optimal p! robability of success for a classical computation. Our semidefinite programming formulation is suitably general, and so has application outside that of algorithms. In particular, we apply it to the optimization of quantum clocks.

Factoring numbers with periodic interferograms

Key Words: factorization algorithm, interferometers, Gauss sums, exponential sums, Shor's algorithm, quantum computation

Authors: Vincenzo Tamma (1)(2), Heyi Zhang (1), Xuehua He (1), Augusto Garuccio (2), Wolfgang P. Schleich (3), and Yanhua Shih (1)

Institution(s): (1) Department of Physics, University of Maryland, Baltimore County, Baltimore, Maryland 21250, (2) Dipartimento Interateneo di Fisica, Università degli Studi di Bari, 70100 Bari, Italy, (3) Institut für Quantenphysik, Universität Ulm, Albert-Einstein-Allee 11, D-89081 Ulm, Germany

Abstract: Download pdf abstract here

Poster Presentations

Non-equilibrium Thermo Field Dynamics and its Application to Quantum Information

Key Words: non-equilibrium thermo field dynamics, quantum Brownian motion, quantum stochastic differential equations, martingale operator, qunatum error correction, quantum teleportation of continuous variables

Authors: Toshihico Arimitsu

Institution(s): Graduate School of Pure and Applied Sciences, University of Tsukuba

Abstract: We present our original framework of the canonical operator formalism for dissipative and/or stochastic quantum systems named Non-Equilibrium Thermo Field Dynamics (NETFD). NETFD is a canonical operator formalism providing us with a full set of methods to tackle problems in dissipative quantum systems in a similar way as quantum mechanics and quantum field theory. Within NETFD, the dynamics of dissipative quantum systems is described by the stochastic and/or dissipative “Schroedinger equation ” for an unstable vacuum that has the same amount of information contained in the statistical operator (the density operator). A consistent and unified system of stochastic differential equations in the operator formalism, together with a theory of quantum Brownian motion, allows one to analyze time-evolution of noisy quantum systems in a practical manner.

As an attractive application of NETFD, we will treat the system of qubits under the influence of spatially correlated noises. By the introduction of tilde degrees of freedom, completely positive maps (CP maps) describing the error or the error-correction procedure are represented within NETFD by operators acting on both bra- and ket-thermal vacuums for qubits, which makes analyses of error-correction procedures transparent; this makes a contrast to the case within the density operator formalism where CP maps are super-operator acting on density operator. It is shown that errors due to the correlated noises can be corrected by the quantum error-correction code and error-correction procedure that is prepared for spatially independent noises. This result is valid generally for the stabilizer code, which is quite a large class of quantum error-correction codes.

If time permits, we will present, as another application of NETFD, the study on the influence of imperfect generation of squeezed states and of imperfect measurements to quantum teleportation of continuous variables by investigating the fidelity of the quantum channel.

Website: http://www.jstage.jst.go.jp/article/iis/15/3/441/_pdf

Detecting Majorana bound states

Key Words: Non-abelian anyons, Aharonov-Bohm effect

Authors: Colin Benjamin 1, Jiannis K. Pachos 2

Institution(s): 1. Center for Simulational Physics, Dept. of Physics & Astronomy, Univ. of Georgia, Athens, GA 30602, USA. 2. Quantum Information group, School of Physics & Astronomy, Univ. of Leeds, UK

Abstract: We propose a set of interferometric methods on how to detect Majorana bound states induced by a topological insulator. The existence of these states can be easily determined by the conductance oscillations as function of magnetic flux and/or electric voltage. We study the system in the presence and absence of Majorana bound states and observe strikingly different behaviors. Importantly, we show that the presence of coupled Majorana bound states can induce a persistent current in absence of any external magnetic field.

Minor-Embedding in Adiabatic Quantum Optimization

Key Words: Adiabatic Quantum Computation, Graph Minor, Embedding, Quantum Architecture Design, Universal Graph

Authors: Vicky Choi

Institution(s): Virginia Tech

Abstract: We introduce the notion of minor-embedding in adiabatic quantum optimization. A minor-embedding of a graph G in a quantum hardware graph U is a subgraph of U such that G can be obtained from it by contracting edges. We show that the NP-hard quadratic unconstrained binary optimization (QUBO) problem on a graph G can be solved using an adiabatic quantum computer that implements an Ising spin-1/2 Hamiltonian, by reduction through minor-embedding of G in the quantum hardware graph U. There are two components to this reduction: embedding and parameter setting. The embedding problem is to find a minor-embedding of a graph G in U. The parameter setting problem is to determine the corresponding parameters, qubit biases and coupler strengths, of the embedded Ising Hamiltonian. We will also describe the intertwined adiabatic quantum architecture design problem, which is to construct a hardware graph U that satisfies all known physical constraints and, at the same time, permits an efficient minor-embedding algorithm. We illustrate an optimal complete-graph-minor hardware graph. We will also discuss several related algorithmic problems that need to be investigated in order to facilitate the design of adiabatic algorithms and AQC architectures.

Algorithmic Approach to Adiabatic Quantum Optimization

Key Words: Adiabatic Quantum Optimization, Algorithm, QPT, Visualization, Perturbation, Local Minima, Degeneracy

Authors: Neil G. Dickson, M. H. S. Amin

Institution(s): D-Wave Systems, Inc.

Abstract: Recent papers have suggested that brute force Adiabatic Quantum Optimization (AQO) fails on random hard problem instances because of the presence of 1st-order Quantum Phase Transitions (QPTs). However, by removing several assumptions, we present a simple, heuristic algorithm for eliminating 1st-order QPTs, based on 2nd-order perturbation. Hard random and structured instances with ~10^3 to ~10^18 highly-degenerate local minima and a unique global minimum were generated, and the algorithm was simulated using Quantum Monte Carlo (QMC) and special analysis techniques. A new visualization provides a very detailed look into QPTs and the operation of this algorithm. The algorithm was found to eliminate or reduce the severity of many of the QPTs. This suggests a possible path forward for development of non-brute-force, practical AQO algorithms.

Homodyne detection in view of joint probability and quantum state of laser

Key Words: homodyne detection joint probability quantum state laser

Authors: Toru Kawakubo

Institution(s): Kyoto University

Abstract: A long-standing problem concerning the quantum state of a laser field is resolved. Considering the photon-number superselection rule, it turns out that the phase of only one laser in the system can be arbitrary fixed. Although the relative phase between independent laser fields are initially random, localization of the relative phase almost always occurs in an interference experiment. Using the concept of conditional independence, it is shown to be valid to assign a coherent state to a laser field, although its phase may be a priori unknown.

Efficient Direct Tomography for Matrix Product States

Key Words: Tomography, Matrix Product states, Tensor network states, entanglement, state certification

Authors: Olivier Landon-Cardinal (1), Yi-Kai Liu (2), David Poulin (1)

Institution(s): 1) D épartement de Physique, Universit é de Sherbrooke, Sherbrooke, Qu ébec, Canada 2) Institute for Quantum Information, California Institute of Technology, Pasadena, CA, USA

Abstract: Matrix product states (MPS) are a variational class of states that can be specified by a small number of parameters. Their importance in quantum many-body physics and quantum information science stems from the fact that they seem to capture the low energy physics of a wide range of one-dimensional systems. We adress the following question: given a state, is it possible to efficiently perform quantum state tomography in order to extract its MPS tensor representation ?

We describe a method for reconstructing these states from a small number of efficiently-implementable measurements. Our method is exponentially faster than standard tomography, and can be used to certify that the unknown state is an MPS. The basic idea is to use local unitary operations to disentangle parts of the system, giving direct access to the tensor representation. This compares favorably with recently and independently proposed methods that recover the MPS tensors by performing a variational minimization, which requires significantly more elaborate computations. Our method also has the advantage of recovering any MPS, while other approaches exclude important examples such as GHZ. There is ongoing work to extend this method to other tensor network states.

Website: http://arxiv.org/abs/1002.4632

Mixed state entanglement in relativistic frames

Key Words: Mixed states; Wigner rotation; Seperability; accelerated observers.

Authors: Shahpoor Moradi

Institution(s): University of Razi

Abstract: In recent years, much interest has been focused in quantum entanglement in relativistic regime. For inertial observers entanglement has no invariant meaning. The reason is that under a Lorentz boost the spin undergoes a Wigner rotation whose direction and magnitude depend on the momentum of the particle. Even if the initial state is a direct product of a function of momentum and a function of spin, the transformed state is not a direct product. Spin and momentum appear to be entangled. A state which is maximally entangled in an inertial frame becomes less entangled if the observers are relatively accelerated. This phenomenon shows that entanglement is an observer-dependent quantity in non inertial frames. The entanglement is degraded by the Unruh effect and asymptotically reaches a nonvanishing minimum value in the infinite acceleration limit. This means that the state always remains entangled to a degree and can be used in quantum information tasks, such as teleportation, b! etween parties in relative uniform acceleration. As a further step along these lines, we investigate the entanglement of mixture of a maximally entangled state and a separable state orthogonal to it in the inertial and non inertial frames. In an inertial frame we study entanglement under Wigner rotations induced by Lorentz transformations. In non inertial frame we investigate the entanglement of scalar and Dirac fields as seen by two accelerated observers. In both cases we show that there are states that will change from entangled into separable for a certain value of velocity or acceleration.

Website: http://pra.aps.org/abstract/PRA/v79/i6/e064301

Inequalities for the quantum set

Authors: Tzyh Haur YANG (1), Miguel NAVASCUES (2), Lana SHERIDAN (1), Valerio SCARANI (1,3)

Institution(s): (1) Centre for Quantum Technologies, National University of Singapore, Singapore (2) Universidad Complutense de Madrid, Madrid, Spain (3) Department of Physics, National University of Singapore, Singapore

Abstract: The set of probabilities distributions that can arise from measurements on composite quantum systems is not trivial. It contains the set of “local” probabilities, i.e. those that can be generated by shared randomness; and is contained in the set of “no-signaling” probabilities; but is strictly different from both. A characterization in terms of an operational hierarchy of conditions has been given recently [1,2]. For the simplest case of 2 parties, 2 inputs and 2 outputs (“CHSH scenario”), analytical conditions that approximate quite well the quantum set are known [3,4,5] and have been improved in [1,2]. But this is the only case in which such a “quantum inequality” is known. In this work, we exploit the techniques of “macroscopic locality” [6] and obtain a recipe to construct quantum inequalities from Bell’s inequalities. The recipe applies to scenarios with 2 parties and 2 outputs but any number of inputs. Since a large number of Bell inequalities are known for these scenarios, this is an important step in the analytical characterization of the quantum set. We study the tightness of the conditions that are obtained through our method.

[1] M. Navascues, S. Pironio, A. Acin, Phys. Rev. Lett. 98, 010401 (2007)
[2] M. Navascues, S. Pironio, A. Acin, New J. Phys. 10, 073013 (2008)
[3] B. Tsirelson, Had. J. Suppl. 8, 329 (1993)
[4] L.J. Landau, Found. Phys. 18, 449 (1988)
[5] L. Masanes, arXiv:quant-ph/0512100v1
[6] M. Navascues, H. Wunderlich, Proc.Roy.Soc.Lond.A 466, 881 (2009)

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