results on the universal scaling behavior of the single electron box in the strong tunneling limit.

3.12.3 Breaking Integrability

Giuseppe Mussardo

SISSA, Trieste mussardo@he.sissa.it

Thanks to integrable quantum field theories, there has been in recent year new understanding on a large number of models of interest in statistical mechanics or in condensed matter physics, e.g. Ising model in a magnetic field or Sine-Gordon model. Integrability has permitted to determine, for instance, the exact spectrum of many systems, the explicit determination of the correlation functions of their order parameters, as well as their thermodynamical properties. In the seminar there will be discussed two methods which permits to extend this analysis also to non-integrable models: the first one is based on the Form Factor Perturbation Theory while the second is based on semi-classical techniques. Both approaches will be then illustrated by studying in details the non-integrable features of theories with kink excitations, like for instance the Ising model or Double Sine Gordon. We present the evolution of the spectrum of the stable particles and the computation of the decay width of the unstable ones.

3.13 Quantum Information

Organizers A. Holevo (Moscow), M. B. Ruskai (Medford)

3.13.1 One-and-a-Half Quantum de Finetti Theorems

Matthias Christandl

Cambridge mc380@cam.ac.uk

When n-k systems of an n-party permutation invariant density matrix are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. Here we show that an upper bound on the trace distance of this approximation is given by 2kd2/n, where d is the dimension of the system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for states in representations of the unitary group by coherent states.

For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our “half” the-

orem). It arises from a combinatorial formula for the distance of symmetric Werner states to product Werner states, making a connection to the recently defined shifted Schur functions. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. This is joint work with Robert Koenig, Graeme Mitchison and Renato Renner.

3.13.2 Catalytic Quantum Error Correction

Igor Devetak

University of Southern California devetak@usc.edu

We exhibit a natural generalization of the stabilizer formalism for entanglementassisted quantum error correction. Conventional stabilizer codes for quantum channels without entanglement assistance are equivalent to isotropic (or self-orthogonal T) symplectic codes. When entanglement assistance is included, the isotropicity condition is no longer necessary. A catalytic quantum code is one which borrows the use of a perfect quantum channel and returns it at the end of the protocol. One of the consequences of the above result is that any classical code over GF(4) can be made into a catalytic quantum code. In particular, classical codes over GF(4) attaining the Shannon limit correspond to catalytic quantum codes attaining the hashing bound.

3.13.3 Quantum State Transformations and the Schubert Calculus

Patrick Hayden

McGill University patrick@cs.mcgill.ca

The problem of relating the eigenvalues of a density operator to those of its reductions, known as the quantum marginal problem, is closely connected to determining the amount of communication required to convert one entangled quantum state into another. I’ll develop this connection and show how the solution to a restricted version of the marginal problem can be used to extract simple conditions governing the existence of transformations in particular cases.

3.13.4 The Local Hamiltonian Problem

Julia Kempe

CNRS & University of Paris kempe@lri.fr

Most physical systems are described by a sum of local Hamiltonians, i.e. Hamiltonians that act on a few particles each. Computing the ground state energy of these

systems is notoriously hard in general and has been studied in many settings, most importantly on square lattices. Kitaev was the first to cast the problem in complexity theoretic terms; he showed that the 5-Local Hamiltonian problem is as hard as any problem in QMA, the quantum analogue of NP. We will review the status of the problem since then with some new rigorous perturbation theory techniques on the way and also give a connection to adiabatic quantum computing.

3.13.5The Information-Disturbance Tradeoff and the Continuity of Stinespring’s Representation

Dennis Kretschmann

TU Braunschweig d.kretschmann@tu-bs.de

Stinespring’s famous dilation theorem is the basic structure theorem for quantum channels: it states that every quantum channel (i.e., completely positive and trace preserving map) arises from a unitary evolution on a larger system. The theorem not only provides a neat characterization of the set of permissible quantum operations, but is also a most useful tool in quantum information science.

Here I will present a continuity theorem for Stinespring’s dilation: if two quantum channels are close in cb-norm, then we can always find unitary implementations which are close in operator norm, with dimension-independent bounds. This result can be seen as a generalization of Uhlmann’s theorem from states to channels and allows to derive a formulation of the information-disturbance tradeoff in terms of quantum channels, as well as a continuity estimate for the no-broadcasting theorem. Other applications include a strengthened proof of the no-go theorem for quantum bit commitment.

Joint work with D. Schlingemann and R. F. Werner.

3.13.6 Locality Estimates for Quantum Spin Systems

Robert Sims

University of Vienna robert.sims@univie.ac.at

We review some recent results that express or rely on the locality properties of the dynamics of quantum spin systems. In particular, we present a slightly sharper version of the recently obtained Lieb-Robinson bound on the group velocity for such systems on a large class of metric graphs. Using this bound we provide expressions of the quasi-locality of the dynamics in various forms, present a proof of the Exponential Clustering Theorem, and discuss a multi-dimensional Lieb-Schultz-Mattis Theorem.