Linear Matrix Inequalities, Semidefinite Programming and Quantum Information Theory

Transcription

1 Linear Matrix Inequalities, Semidefinite Programming and Quantum Information Theory Toulouse, January Antonio Acín (ICFO-The Institute of Photonic Sciences, Barcelona) Non-commutative polynomial optimisation problems in quantum information theory We consider optimisation problems with polynomial inequality constraints in non-commuting variables and present a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution. We discuss several applications of the method to problems in quantum physics. 2. Guillaume Aubrun (Université Lyon 1) Dvoretzky s theorem and the complexity of entanglement detection The well-known Horodecki criterion asserts that a state ρ on C d C d is entangled if and only if there exists a positive map Φ : M d M d such that the operator (Φ I)(ρ) is not positive semidefinite. We show that that the number of such maps needed to detect all the robustly entangled states (i.e. states ρ which remain entangled even in the presence of randomizing noise) exceeds exp(cd 3 / log d). The proof is based on a study of the approximability of the set of states (resp. of separable states) by polytopes with few vertices or few faces, and ultimately relies on the Dvoretzky Milman theorem about the dimension of almost spherical sections of convex bodies. The result can be interpreted as a geometrical manifestation of the complexity of entanglement detection. 3. Howard Barnum (Leibniz Universität Hannover and University of New Mexico) Hyperbolicity Cones, Convex Analysis and Semidefinite Programming Hyperbolicity cones, associated to hyperbolic polynomials, are a broad class of cones that initially arose in the theory of partial differential equations. It includes most of the cones supporting efficient convex optimization algorithms, for example polyhedral, positive semidefinite matrix, symmetric, and homogeneous cones, and hyperbolicity itself directly supports efficient optimization. Generalizations of a conjecture of Lax state that such cones are spectrahedral (i.e. slices of the PSD cone), or at least are spectrahedral shadows, i.e. semidefinite representable. This review talk will cover important properties and examples of hyperbolicity cones, relaxations of them obtained by taking derivatives, and some of what is known about the generalized Lax conjectures. Hyperbolicity cones share many properties of an information-theoretic flavor with the cone of quantum states and effects (PSD Hermitian cone). If time permits I ll discuss an important subclass of these cones, those that are isometric and have full spectrum, shown by Bauschke, Güler, Lewis and Sendov to share even more such properties with the quantum one. 4. Sabine Burgdorf (CWI, Amsterdam) Conic optimization approach to quantum graph parameters The completely positive semidefinite cone is a new matrix cone, which consists of all the symmetric matrices (of a given size) that admit a Gram representation by positive semidefinite matrices of any size. This cone can be used to model quantum graph parameters or, more generally, to characterize the set of bipartite quantum correlations as projection of an affine section of it. We will give an overview of the known structure of the completely positive semidefinite cone and of its use to model quantum graph parameters. These parameters are analogs of classical graph parameters, like the chromatic and stability numbers of a graph, which arise naturally in the context of nonlocal games and the study of entanglement in zero-error communication. In particular, we will discuss a hierarchy of polyhedral inner approximations that can be used to express some quantum graph parameters by way of linear programs. We will also mention results about

2 the closure of the completely positive semidefinite cone and SDP-based outer approximations. This talk is based on joint work with Monique Laurent and Teresa Piovesan. 5. Matthias Christandl (University of Copenhagen) Membership in moment polytopes is in NP and conp We show that the problem of deciding membership in the moment polytope associated with a finite-dimensional unitary representation of a compact, connected Lie group is in NP and conp. This is the first non-trivial result on the computational complexity of this problem, which naively amounts to a quadratically-constrained program. Our result applies in particular to the Kronecker polytopes, and therefore to the problem of deciding positivity of the stretched Kronecker coefficients. In contrast, it has recently been shown that deciding positivity of a single Kronecker coefficient is NP-hard in general [Ikenmeyer, Mulmuley and Walter, arxiv: ]. We discuss the consequences of our work in the context of complexity theory and the quantum marginal problem. 6. Douglas Farenick (University of Regina) Spectra and variation of quantum random variables In this lecture I will discuss the study of essentially bounded matrix-valued (or quantum) random variables from the point of view of operator and measure theory. The Gelfand spectrum of such a quantum random variable coincides with the hypoconvex hull of its essential range. Moreover, a notion of operator-valued variance is introduced, leading to a formulation of the moment problem in the context of quantum probability spaces in terms of operator-theoretic properties involving semi- invariant subspaces and spectral theory. As an application of quantum variance, new measures of random and inherent quantum noise are introduced for measurements of quantum systems, modifying some recent ideas of Polterovich. This lecture is based on joint work with Sarah Plosker and Michael Kozdron. 7. Aram Harrow (MIT) SDP hierarchies for entangled states and games Many questions in quantum information can be expressed as polynomial optimization problems. For example: is a given quantum state entangled? Which nonlocal correlations can be achieved by local measurements of entangled states? In these talks I will explain what is known about the hardness of these problems. In some cases they are NP-complete and in some cases they are related to the Unique Games Conjecture. In recent work we can show that these computational hardness results can also be turned into unconditional no-go theorems for SDPs; i.e. showing that no small SDP can give a good approximation of certain quantum problems. I will describe how quantum and classical information theory can be used to prove approximation guarantees for SDP hierarchies in some cases of these problems. 8. Didier Henrion (LAAS - CNRS Toulouse) Invariant set approximations for polynomial dynamical systems with the Lasserre hierarchy We consider the dynamical system x t+1 = f(x t ), t = 0, 1,... with polynomial vector field f : R n R n and state x t R n. Given a compact semialgebraic set X of R n, its maximal invariant set X I is defined as the set of initial states x 0 in X such that x t+1 = f(x t ) remains in X for all t = 0, 1,.... The maximal invariant set can have a complicated geometry (think e.g. of the Mandelbrot or Julia fractal sets corresponding to n = 2 and f quadratic). We show that the Lasserre hierarchy of moment-sum-of-squares semidefinite programs can generate a family of semialgebraic outer approximations X k of X I converging in Lebesgue measure, that is, lim k volumex k = volumex I. Joint work with Milan Korda and Colin Jones (EPFL). 2

3 9. Michael Kech (TU München) Semidefinite Programming in Quantum State Tomography Quantum state tomography is an integral task in quantum information science, its implementation, however, is expensive. Though, in many tomography tasks there is some prior information about the quantum state, allowing for a more resourceful tomography procedure. Most prominently, the quantum state is assumed to be pure, or more generally low-rank. Based on semidefinite programming, low-rank quantum states can be reconstructed from linear measurements using considerably less measurements than a full tomography would (This originates from the work of Recht, Fazel and Parillo in 2007.). To date, all approaches following this path were based on probabilistic measurements. I will present explicit constructions of measurements that allow for the reconstruction of low-rank quantum states via the same semidefinite programs, requiring a close to optimal number of measurements. 10. Cécilia Lancien (Universite Claude Bernard Lyon 1 & Universitat Autonoma de Barcelona) Relaxations of separability in multipartite quantum systems Certifying that a bipartite state is not separable can always be done by exhibiting an entanglement witness constructed from a positive map. In the multipartite case, the picture becomes more intricate. Indeed, even asserting that a state is not biseparable (i.e. a convex combination of states which are separable across a given bipartition) may be a delicate task. We show however that it is always possible to construct such a genuine multipartite entanglement witness by lifting entanglement witnesses which only reveal bipartite entanglement. In small dimensions, this approach is quite versatile since it allows for a formulation of the problem as a semi-definite program, and can therefore be solved efficiently. Nevertheless, as one could have expected, any state-independent construction is condemned to become weaker and weaker as the dimensions grow. We back up this affirmation by focussing on one specific positive map relaxation of separability, namely positivity under partial transposition. We thus prove that the set of states which have a positive partial transpose across every cut is much bigger than the set of biseparable states. This substantiates that a universal scheme can only detect a small fraction of genuinely multipartite entangled states. (arxiv: , joint work with O. Gühne, M. Huber and R. Sengupta) 11. Jean Bernard Lasserre (LAAS - CNRS Toulouse) The moment-lp and moment-sos approaches in polynomial optimization and some other applications In a first part we provide an introduction to the basics of the moment-lp and moment-sos approaches to global polynomial optimization. In particular, we describe the hierarchy of LP and semidenite programs to approximate the optimal value of such problems. In fact, the same methodology also applies to solve (or approximate) Generalized Moment Problems (GMP) whose data are described by basic semi-algebraic sets and polynomials (or even semi-algebraic functions). Indeed, Polynomial optimization is a particular (and even the simplest) instance of the GMP. In a second part, we describe how to use this methodology for solving several problems (outside optimization) viewed as particular instances of the GMP. This includes: Approximating compact basic semi-algebraic sets defined by quantiers. Computing convex polynomials underestimators of polynomials on a box. Bounds on measures satisfying some moment conditions. Approximating the volume of compact basic semi-algebraic sets. Approximating the Gaussian measure of non-compact basic semi-algebraic sets. Approximating the Lebesgue decomposition of a measure µ w.r.t. another measure ν, based only on the moments of µ and ν. 12. Alexander Müller-Hermes (University of Copenhagen) Some questions about positive maps arising in quantum information theory 3

4 We study some problems on positive maps between matrix algebras arising in quantum information theory. First we consider linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with n copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every positive integer n there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all n. In general we show that an affirmative answer to the existence question of such non-trivial tensor-stable positive maps would imply the existence of NPPT bound entanglement. A possible approach to check that a given trace-preserving positive map is not tensor-stable positive is to show that some Renyi-divergence evaluated at some pair of quantum states increases under the action of the map. Motivated by this we study the contraction of Renyi-divergences and similar distance measures under positive maps. By using some recent results from entanglement theory and the SDP for finding a completely positive extension by Heinosaari et al. we show that in two-dimensions any tracepreserving positive map restricted to the span of two quantum states extends to a quantum channel. Therefore, any Renyi-divergence (and some similar distance measures) must contract under the action of such a map. 13. David Reeb (Leibniz University Hannover) Extending Quantum Operations For a given set of input-output pairs of quantum states or observables, we ask whether there exists a physical (i.e. completely positive) operation connecting them. This question - as well as approximations and variations thereof - can be formulated as a semidefinite program. The SDP duality connects the problem in a quantitative way to a natural notion of complete positivity on a subspace that we define, even though the Choi matrix is not available in this context. Although strong duality holds always, the optima are not always attained in the problems considered here. The SDP duality is also used to prove versions of Aveson s extension theorem in finite dimensions, and to provide a counterexample to a conjecture by Alberti and Uhlmann. [talk based on: J. Math. Phys. 53, (2012), arxiv: ] 14. Daniel Reitzner & Michal Sedlak (Institute of Physics, Slovak Academy of Sciences) Incompatibility of quantum testers Every protocol that manipulates information can be described by a sequence of consecutive operations. For quantum systems such sequences are called quantum circuits. They contain many details that do not affect the output of the protocol which makes them cumbersome for optimization of various tasks. On the other hand, quantum combs parametrize only input-output relations of the protocol and, hence, allow us to handle just the relevant parameters. In this contribution, we introduce the quantum comb framework and after briefly mentioning some of its applications we will address the problem of quantum tester incompatibility. Quantum testers are specific types of quantum combs that perform measurements on channels (state transformations). We will focus on the question of whether two testers can be obtained from a single one by postprocessing. This question can be cast as a semi-definite program. This formulation together with the numerical solution offers valuable insight for the analytical approach in the case of incompatibility of factorized qubit testers with two outcomes. 15. Peter Wittek (ICFO-The Institute of Photonic Sciences, Barcelona) Towards Solving Bilevel Optimization Problems in Quantum Information Theory In bilevel optimization, an optimization problem is nested within another. The simplest case is when the nested, lower level problem has an identical objective function, but with the opposite sign as the higher level problem this is a case of min-max optimization. In the more general case, the lower level objective is unrelated to the higher level objective, and either can be nonconvex. Problems of the simpler and the generic type naturally arise in quantum information theory. Examples include bilevel SDPs, but also bilevel polynomial optimization problems that can relaxed as SDPs. The former case is relatively straightforward, we give an example problem of this by simulating POVMs with projective measurements. The latter problem may ask for the generalization of the relaxation of bilevel polynomial optimization problem of commuting 4

5 variables to noncommuting operators, which in turn requires the generalization of the Navascués- Pironio-Acín (NPA) hierarchy to more general measures. A possible application of this approach would be finding more noise-tolerant measurement settings in randomness extraction. In this talk, we report on work in progress towards a generic solution. 5