The conference will be held at the Institut Henri Poincaré, 2327 October 2017
Organizers
Confirmed Speakers
Talks  Antonio Acín  Noncommutative polynomial optimisation problems in quantum information theory
We discuss questions in quantum physics that can be cast as noncommutative polynomial optimisation problems and discuss their solution in terms of semidefinite programming. This ranges from new approaches to detect entangled states to the computation of ground state energies of interacting systems. We argue that these methods can play an important role for the certification of quantum effects.
 Radosław Adamczak  Entropic and metric uncertainty relations for random unitary matrices
I will discuss recent results concerning almost optimal entropic and metric (totalvariation and Hellinger) uncertainty relations which hold with high probability for measurements given by i.i.d. Haar distributed random unitary matrices. If time permits I will also discuss some applications to information locking and data hiding. The talk will be partially based on joint work with Rafal Latala, Zbigniew Puchala and Karol Zyczkowski.
 Benoit Collins  MOE estimates for quantum channels arising from random isometries and free probability
We consider the model of random quantum channels where the Stinespring isometries are chosen uniformly at random. We show that under a suitable choice of parameters for the dimensions of the input, output and ancilla, the output space is almost deterministic and conveniently described with a new free probability tool, the free compression norm. Then we study the problem of maximizing the Lp norm on this output space. Surprisingly, the output maximizing the Lp norm does not depend on p and has a very simple form. As a consequence we obtain the best bounds known so far for the violation of the additivity of the minimum output entropy. This talk is based on two joint works with Belinschi and Nechita.
 Nilanjana Datta  Concentration of quantum states from quantum functional and transportation cost inequalities
Quantum functional inequalities (e.g. the logarithmic Sobolev and Poincaré inequalities) have found widespread application in the study of the behavior of primitive quantum Markov semigroups. The classical counterparts of these inequalities are related to each other via a socalled transportation cost inequality of order 2 (T2). The latter inequality relies on the notion of a metric on the set of probability distributions called the Wasserstein distance of order 2. (T2) in turn implies a transportation cost inequality of order 1 (T1). In this paper, we introduce quantum generalizations of the inequalities (T1) and (T2), making use of appropriate quantum versions of the Wasserstein distances, one recently defined by Carlen and Maas and the other defined by us. We establish that these inequalities are related to each other, and to the quantum modified logarithmic Sobolev and Poincaré inequalities, as in the classical case. We also show that these inequalities imply certain concentrationtype results for the invariant state of the underlying semigroup. We consider the example of the generalized depolarizing semigroup to derive concentration inequalities for any finite dimensional fullrank quantum state. These inequalities are then applied to derive upper bounds on the error probabilities occurring in the setting of finite blocklength quantum parameter estimation. This is joint work with Cambyse Rouzé.
 Omar Fawzi  Entropy accumulation
We ask the question whether entropy accumulates, in the sense that the operationally relevant total uncertainty about an npartite system A=(A1,…An) corresponds to the sum of the entropies of its parts Ai. The Asymptotic Equipartition Property implies that this is indeed the case to first order in n, under the assumption that the parts Ai are identical and independent of each other. Here, we show that entropy accumulation occurs more generally, i.e., without an independence assumption, provided one quantifies the uncertainty about the individual systems Ai by the von Neumann entropy of suitably chosen conditional states. In this talk, I will introduce the more general framework and describe the main steps of the proof. Based on joint work with Frederic Dupuis and Renato Renner, most of it available in https://arxiv.org/abs/1607.01796
 Motohisa Fukuda  Additivity questions and tensor powers of random quantum channels
Perhaps considering minimum output entropy of high tensor powers of quantum channels is one of best ways to understand capacity of quantum channels. However, if addivity violation is a local phenomena in terms of tensor powers, we may be able to obtain a simple formula for regularized minimum output entropy. In this talk, we discuss this problem with random quantum channels generated by the orthogonal group.
 Carlos GonzálezGuillén  Spectral gap of random quantum channels
We consider random quantum channels arising form uniform random isometries via the Stinespring representation. We will show that these channels are generically gapped, that is, there is a separation between the first and second singular values of the channels. We will see that, in a broad sense, this result can be seen as a generalisation of Hasting's result for a set of random unital channels. Moreover, we will discuss different applications of this result to study entropy and correlations in 1D tensor networks.
 Razvan Gurau  Invitation to Random Tensors
Random matrices are ubiquitous in modern theoretical physics and provide insights on a wealth of phenomena, from the spectra of heavy nuclei to the theory of strong interactions or random two dimensional surfaces. The backbone of all the analytical results in matrix models is their 1/N expansion (where N is the size of the matrix). Despite early attempts in the '90, the generalization of this 1/N expansion to higher dimensional random tensor models has proven very challenging. This changed with the discovery of the 1/N expansion (originally for colored and subsequently for arbitrary invariant) tensor models in 2010. I this talk I will present a short introduction to the modern theory of random tensors and its connections to conformal field theory and random higher dimensional geometry.
 Madalin Guta  Identification and information geometry of quantum inputoutput systems
This talk deals with the problem of identifying and estimating dynamical parameters of continuoustime quantum open systems, in the inputoutput formalism. I will discuss several aspects of this problem: The first aspect concerns the structure of the space of identifiable parameters for ergodic dynamics, assuming full access to the output state for arbitrarily long times. I will show that the equivalence classes of undistinguishable parameters are orbits of a Lie group acting on the space of dynamical parameters. The second aspect concerns the information geometric structure on this space. I will show that the space of identifiable parameters is the base space of a principal bundle given by the action of the group, and carries a Riemannian metric based on the quantum Fisher information of the output. The metric can be computed explicitly in terms of the Markov covariance of certain fluctuation operators, and relate it to the horizontal bundle of the connection. The third direction concerns the identification of linear inputoutput systems in the timedependent and stationary setups. The fourth direction concerns the design of measurements to achieve the quantum Fisher information.
 Teiko Heinosaari  Maximally incompatible quantum observables
I will discuss a way to quantify the degrees of incompatibility of two observables in a probabilistic physical theory and, based on this, a global measure of the degree of incompatibility inherent in such theories, across all observable pairs. This makes it possible to compare probabilistic theories with respect to the nonclassical feature of incompatibility, raising many interesting questions. I will show that quantum theory contains observables that are as incompatible as any probabilistic physical theory can have. In particular, two of the most common pairs of complementary observables (position and momentum; number and phase) are maximally incompatible.
 Anthony Leverrier  A Gaussian de Finetti theorem and application to truncations of random Haar matrices
de Finetti theorems are pervasive in finitedimensional quantum information theory as they state that permutation invariant quantum systems are in some sense close to convex mixtures of i.i.d. states. In this work, I’ll consider infinitedimensional quantum systems that are invariant under a larger symmetry group, namely the unitary group U(n), and show that such states are similarly well approximated by convex mixtures of Gaussian i.i.d. states. I’ll then discuss how to apply this result to study truncations of random Haar unitary matrices. This talk is based on arXiv:1612.05080.
 Ke Li  Discriminating quantum states: the multiple Chernoff distance
Suppose we are given n copies of one of the quantum states {rho_1,..., rho_r}, with an arbitrary prior distribution that is independent of n. The multiple hypothesis Chernoff bound problem concerns the minimal average error probability P_e in detecting the true state. It is known that P_e=exp{En+o(n)} decays exponentially to zero. However, this error exponent E is generally unknown, except for the case r=2. In this talk, I will give a solution to the longstanding open problem of identifying the above error exponent, by proving Nussbaum and Szkola's conjecture that E=min_{i neq j} C(rho_i, rho_j). The righthand side of this equality is called the multiple quantum Chernoff distance, and C(rho_i,rho_j):= max_{0 <= s <= 1} {log Tr (rho_i^s rho_j^(1s))} has been previously identified as the optimal error exponent for testing two hypotheses, rho_i versus rho_j. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finitedimensional, but otherwise general, quantum states. This upper bound, up to a statesdependent factor, matches the multiplestate generalization of Nussbaum and Szkola's lower bound. Specialized to the case r=2, we give an alternative proof to the achievability of the binaryhypothesis Chernoff distance, which was originally proved by Audenaert et al.
 Satya Majumdar  Number variance and entanglement entropy of trapped fermions via random matrix theory
Consider N free Fermions in a one dimensional harmonic trap. How many Fermions are there at zero temperature in an interval [L,L]? The ground state quantum fluctuations of the number of Fermions in [L,L] can be mapped to the classical fluctutions of the number of eigenvalues in [L,L] of a Gaussian random Hermitian matrix with complex entries. This mapping allows us to compute exactly for large N, using a Coulomb gas approach, the variance of number of Fermions in the quantum system at T=0, as a function of L. The variance exhibits, as a function of L, a very interesting nonmonotonic behaviour. I'll then discuss how these results can be used to compute also the ground state entanglement entropy of the interval [L,L] with the rest of the system.
 Camille Male  Distributional symmetries and non commutative notions of independence
The properties of the limiting non commutative distribution of random matrices can be usually understood thanks to the symmetry of the model, e.g. Voiculescu's asymptotic free independence occurs for random matrices invariant in law by conjugation by unitary matrices. This talk presents an approach for the study of random matrices invariant in law by conjugation by permutation matrices, the theory of traffics. A traffic is a non commutative random variable in a space with more structure than a general non commutative probability space, so that the notion of traffic distribution is richer than the notion of non commutative distribution. It comes with a notion of independence which is able to encode the different notions of non commutative independence.
 James Mingo  The Role of the Transpose in Free Probability: the partial transpose of Rcyclic operators
Like tensor independence, free independence gives us rules for doing calculations. With random matrix models, we usually need tensor independence of the entries and some kind of group invariance of the joint distribution of the entries to get the (asymptotic) freeness necessary to apply the tools of free probability. A few years ago Mihai Popa and I found that the transpose also produces asymptotic freeness, i.e. a matrix could be asymptotically free from its own transpose. Since that we have expanded this work to the case of the partial transposes that arise in quantum information theory. In this talk, I will explain what happens when one transposes certain Rcyclic operators.
 Ashley Montanaro  Probabilistic techniques for simulating quantum computational supremacy experiments
The fast pace of recent experimental developments has led to the hope that quantum computers will soon demonstrate computational performance far beyond classical computers: a milestone known as quantum (computational) supremacy. However, the relative simplicity of nearterm experiments leaves open the possibility that they could be simulated classically more easily than full quantum computers could. In this talk, I will discuss two such classical simulation results for proposed architectures for quantum computational supremacy experiments. The first is a classical algorithm for simulating certain noisy commuting quantum computations ("IQP circuits") in polynomial time, based on Fourier analysis over Z_2^n. The second is experimental work demonstrating that simple probabilistic methods can be used to simulate boson sampling experiments significantly more efficiently than previously thought. This implies that quantum computational supremacy is unlikely to be achieved via boson sampling in the near future. The talk is based on joint work with Michael Bremner and Dan Shepherd (Quantum 1, 8 (2017); arXiv:1610.01808), and joint work with Alex Neville, Chris Sparrow, Raphael Clifford, Eric Johnston, Patrick Birchall and Anthony Laing (arXiv:1705.00686).
 Ramis Movassagh  Generic Local Hamiltonians are Gapless
We prove that quantum local Hamiltonians with generic interactions are gapless. In fact, we prove that there is a continuous density of states arbitrary above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for herein may include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. We calculate the scaling of the gap with the system's size in the case that the local terms are distributed according to gaussian β−orthogonal random matrix ensemble. As a corollary there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. In addition to the lack of an energy gap, we prove that the ground state is degenerate when the local eigenvalue distribution is discrete. Time permitting we will present another very new result on the eigenvalue distribution of sums of matrices from the knowledge of the summands. This theory and techniques utilize modern free probability theory and other ideas from random matrix theory. References: RM "Generic Local Hamiltonians are Gapless", Phys. Rev. Lett (2017) (RM, Alan Edelman) Phys. Rev. Lett. 107, 097205 (2011)
 Anand Natarajan  Algorithms and lower bounds for entangled XOR games
In this talk I will consider the complexity of the problem of determining, given a nonlocal XOR game with k >= 3 players, whether there exists an entangled strategy for the game using commuting operators that succeeds with probability 1. Our first result is a polynomialtime algorithm to solve this problem for any k; previously, this problem was not known to be decidable, and indeed, it was shown recently by Slofstra that the same question for a different class of games is undecidable. Our second result is a characterization of random symmetric XOR games: we show that there exists a constant C_k depending only on k, such that a random symmetric kplayer XOR game with question alphabet size n has no perfect entangled strategy with high probability when the number of clauses is above C_k *n. Moreover, for random (not necessarily symmetric) games in the regime where no perfect entangled strategy exists, we show a lower bound on the number of levels in the NPA (NavascuesPironioAcin) hierarchy of semidefinite programs needed to witness this fact. Joint work with Adam Bene Watts and Aram Harrow (both at MIT).
 Yan Pautrat  Invariant measure for quantum trajectories
Quantum trajectories represent the state of a quantum system undergoing repeated indirect measurements. A quantum trajectory is therefore a sequence of density matrices, and a natural question is to describe its asymptotic behaviour as the number of measurements goes to infinity. In generic situations, this behaviour can at best be a convergence in distribution and is therefore related to the existence of an invariant measure for the evolution. We give conditions for the existence and uniqueness of an invariant measure, and show the convergence in distribution. This is a joint work with Tristan Benoist, Martin Fraas, and Clément Pellegrini.
 Zbigniew Puchała  Asymptotic properties of random quantum states and channels
Properties of random mixed states of dimension N distributed uniformly with respect to the HilbertSchmidt measure are investigated. We show that for large N, due to the concentration of measure phenomenon, the trace distance between two random states tends to a fixed number 1/4+1/π, which yields the Helstrom bound on their distinguishability. To arrive at this result we apply free random calculus and derive the symmetrized MarchenkoPastur distribution. We consider also random quantum channels, especially the limiting behaviour of the diamond norm for two independent random quantum operations. In the case of flat measure on random channels, the limiting value of the diamond norm is equal to 1/2+2/π.
 Roland Speicher  Random Matrices and Their Limits
The free probability perspective on random matrices is that the large size limit of random matrices is given by some (usually interesting) operators on Hilbert spaces and corresponding operator algebras. The prototypical example for this is that independent GUE random matrices converge to free semicircular operators, which generate the free group von Neumann algebra. The usual convergence in distribution has been strengthened in recent years to a strong convergence, also taking operator norms into account. All this is on the level of polynomials. In my talk I will recall this and then go over from polynomials to rational functions (in noncommuting variables). Unbounded operators will also play a role.
 Daniel Stilck Franca  Dimensionality reduction of SDPs through sketching
We show how to sketch semidefinite programs (SDPs) using positive maps in order to reduce their dimension. More precisely, we use JohnsonLindenstrauss transforms to produce a smaller SDP whose solution preserves feasibility or approximates the value of the original problem with high probability. These techniques allow us to improve both complexity and storage space requirements necessary to solve SDPs. They apply to problems in which the Schatten 1norm of the matrices specifying the SDP and of a solution to the problem is constant in the problem size. Furthermore, we provide some nogo results which clarify the limitations of positive, linear sketches in this setting. Finally, we discuss an application to uncertainty relations.
 Yoshimichi Ueda  Matrix liberation process
We introduce a natural random matrix counterpart of the socalled liberation process introduced by Voiculescu in the framework of free probability, and consider its possible LDP in the large N limit.
 Stephan Weis  Stability of the set of quantum states
A convex set C is stable if the midpoint map (x,y) > (x+y)/2 is open. For compact C the Vesterstrøm–O’Brien theorem asserts that C is stable if and only if the barycentric map from the set of all Borel probability measures to C is open. Equivalently, the convex hull of any continuous function on C is continuous. We briefly discuss aspects of an extension of this theorem to certain noncompact sets [1]. An example is the set of quantum states (density operators on a separable Hilbert space) where continuity properties of entropic characteristics have been obtained from the stability. In the main part of the talk we present consequences of the stability for a finitedimensional quantum system. 1) One result is a sufficient condition for the discontinuity of a maximumentropy inference map under linear constraints (MaxEnt map) in term of the geometry of the linear image of the set of quantum states [2]. A corollary is that the irreducible threeparty correlation of three qubits is discontinuous at the GHZstate. 2) A second result is a continuity characterization of a MaxEnt map defined by a twodimensional family of linear constraints [3]. A corollary is that the nonanalyticity of the ground state energy of a oneparameter Hamiltonian is witnessed by a discontinuity of the MaxEnt map constrained on expectation values of the two energy terms. [1] Shirokov, M.E., 2012. Stability of convex sets and applications. Izvestiya: Mathematics 76, 840856. [2] Rodman, L., Spitkovsky, I.M., Szkoła, A., Weis, S., 2016. Continuity of the maximumentropy inference: Convex geometry and numerical ranges approach. Journal of Mathematical Physics 57, 015204. [3] Weis, S., 2016. Maximumentropy inference and inverse continuity of the numerical range. Reports on Mathematical Physics 77, 251–263.
 Ping Zhong  Some noncommutative probability aspects of meandric systems
The talk will consider a family of diagrammatic objects (wellknown to combinatorialists and mathematical physicists) which go under the names of ”meandric systems” or ”semimeandric systems”. I will review some connections which these objects are known to have with free probability, and I will show in particular how the socalled ”semimeandric polynomials” can be retrieved from a natural consideration of operators on the qFock space. This is joint work with Alexandru Nica.
