Quantum Information

As a means of exactly derandomizing certain quantum information processing tasks, unitary designs have become an important concept in quantum information theory. A unitary design is a collection of unitary matrices that approximates the entire unitary group, much like a spherical design approximates the entire unit sphere. We use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. The tightness of these bounds is then considered, where specific unitary 2-designs are introduced that are analogous to SIC-POVMs and complete sets of MUBs in the complex projective case. Additionally, we catalogue the known constructions of unitary t-designs and give an upper bound on the size of the smallest weighted unitary t-design in U(d). This is joint work with Aidan Roy (Calgary): \'Unitary designs and codes,\' arXiv:0809.3813.

Abstract: Adapting the concept of Wigner functions to finite dimensional systems is no simple matter. Basic issues with existing proposals are that they are either over-complete in the sense that the degrees of freedom in the Wigner function does not match the degrees of freedom in the original density matrix, or that they work only for restricted dimensions, namely for prime powers. We propose a new way to define a Wigner function and associated quantum phase space for some non-prime power dimensions. This allows to perform quantum state tomography from the measurement statistics of a minimal number of yes/no measurements.

By definition, SIC-POVMs are symmetric in the sense that the magnitude of the inner product between any pair of vectors is constant. All known constructions are based on additional symmetries, mainly with respect to the Weyl-Heisenberg group. Analyzing solutions for small dimensions, Zauner has identified an additional symmetry of order three and conjectured that these symmetries can be used to construct SIC-POVMs for all dimensions. Appleby has confirmed that all numerical solutions of Renes et al. indeed have that additional symmetry. This leads to the main questions addressed in the talk: Do all SIC-POVMs necessarily possess these symmetries, or can we construct SIC-POVMs without or with other symmetries?

We study the possibility of a self-correcting quantum memory based on stabilizer codes with geometrically-local stabilizer generators. We prove that the distance of such stabilizer codes in D dimensions is bounded by O(L^{D-1}) where L is the linear size of the D-dimensional lattice. In addition, we prove that in D=1 and D=2, the energy barrier separating different logical states is upper-bounded by a constant independent of L. This shows that in such systems there is no natural energy dissipation mechanism which prevents errors from accumulating. Our results are in contrast with the existence of a classical 2D self-correcting memory, the 2D Ising ferromagnet.

In the closed system setting I will show how to obtain extremely accurate adiabatic QC by proper choice of the interpolation between the initial and final Hamiltonians. Namely, given an analytic interpolation whose first N initial and final time derivatives vanish, the error can be made to be smaller than 1/N^N, with an evolution time which scales as N and the square of the norm of the time-derivative of the Hamiltonian, divided by the cube of the gap (joint work with Ali Rezakhani and Alioscia Hamma). In the open system setting I will describe a method for protecting adiabatic QC by use of a hybrid encoding-dynamical decoupling scheme. This strategy can be used to protect spin-based universal adiabatic QC against arbitrary 1-local noise using only global magnetic fields. By combining error bounds for the closed and open system settings, I will show that in principle the method is scalable to arbitrarily large computations. References: Closed system case: arXiv:0808.2697 Open system case: Phys. Rev. Lett. 100, 160506 (2008)

The rise of quantum information science has been paralleled by the development of a vigorous research program aimed at obtaining an informational characterization or reconstruction of the quantum formalism, in a broad framework for stochastic theories that encompasses quantum and classical theory, but also a wide variety of other theories that can serve as foils to them. Such a reconstruction, at its most ambitious, is envisioned as playing a role in quantum physics similar to Einstein\'s reconstruction of the dynamics and kinetics of macroscopic bodies, and later of their gravitational interactions, on the basis of simple principles with clear operational meanings and experimental consequences. But short of such an ambitious goal, it could still lead to a principled understanding of the features of quantum mechanics that account for its greater-than-classical information-processing power, an understanding which could help guide the search for new quantum algorithms and protocols. I will summarize a convex operational framework for possible physical theories, and present results from a project to characterize quantum mechanics in terms of principles tightly linked to the possibility or impossibility of various information processing protocols. Previous results identified properties, like the existence of information-disturbance tradeoffs and restrictions on cloning and broadcasting, common to all nonclassical theories. In this talk I will focus on recent results involving protocols that are less generic. These are: the existence of exponentially secure bit commitment in non-classical theories without entanglement, the consequences for theories of the existence of a conclusive teleportation scheme, and sufficient conditions for the existence of a deterministic teleportation scheme. I\'ll also discuss sufficient conditions for \'remote steering\' of ensembles using entanglement, rendering insecure bit commitment protocols of the form shown to be secure in the unentangled case. Connections to the category-theoretic approach of Coecke and Abramsky, Selinger, Baez, and collaborators may be touched on if time permits. Joint work with various groups of collaborators including Jonathan Barrett, Matthew Leifer, Alexander Wilce, Oscar Dahlsten, and Ben Toner.

I would like to provide a short, possibly elementary, introduction to the problem of computing string amplitudes at higher genus for superstrings. Essentially, I will recall which is the mathematical problem in defining the path integral measure (which has a well defined algebraic geometry realization for bosonic strings) and the solution proposed by d~@~YHocker and Phong for the genus 2 case. Their main results are the chiral splitted form of the measure, and its explicit expression in genus two. They proposed the splitting form to work at any genus and assumed some restriction for the explicit form which however did not permitted them to find a solution for genera higher then 2. I will tell something about the technology which permitted us to find explicit solutions for genus 3 and four. Indeed, we showed that the restrictions imposed by d~@~YHocker and Phong have no solution whereas the most general form compatible with modular invariance and clustering provide a unique solution, at least for genus 3 and 4. I will try to be as less technical as possible.

In this talk I will give an introduction to the simulation of quantum many-body systems using the so-called tensor networks. After a brief historical review, I will introduce the basics on tensor network representations of quantum states, and will explain some recent developments. In particular, in the last part of my talk I will focus on recent results obtained in the simulation of 2-dimensional quantum lattice systems of infinite size.

A quantum channel models a physical process in which noise is added to a quantum system via interaction with its environment. Protecting quantum systems from such noise can be viewed as an extension of the classical communication problem introduced by Shannon sixty years ago. A fundamental quantity of interest is the quantum capacity of a given channel, which measures the amount of quantum information which can be protected, in the limit of many transmissions over the channel. In this talk, I will show that certain pairs of channels, each with a capacity of zero, can have a strictly positive capacity when used together, implying that the quantum capacity does not completely characterize a channel\'s ability to transmit quantum information. As a corollary, I will show that a commonly used lower bound on the quantum capacity - the coherent information, or hashing bound - is an overly pessimistic benchmark against which to measure the performance of quantum error correction because the gap between this bound and the capacity can be arbitrarily large.

There are many results showing that the probability of entanglement is high for large dimensions. Recently, Arveson showed that the probability of entanglement is zero when the rank of a bipartite state is no larger than half the dimension of the smaller space. We show that that the probability of entanglement is zero when the rank of a bipartite state is no larger than half the maximum of the rank of its two reduced density matrices. Our approach is quite different from that of Arveson and uses a different measure. But both approaches show that the separable states lie in a lower dimensional manifold given a reasonable parameterization of the separable states. This is joint work with Elisabeth Werner, using on characterizations of the extreme points of qubit channels given by Ruskai, Szarek and Werner and the extreme points of entanglement breaking channels given by M. Horodecki, Shor and Ruskai.