June 17, 2024

July 6-10, 2009
Workshop on Operator Structures in Quantum Information

Abstracts Invited Talks

Variance bounds and commutators
by Koenraad Audenaert, Maths dept., Royal Holloway, University of London

Murthy and Sethi gave a sharp upper bound on the variance of a real random variable in terms of the range of values of that variable. We generalise this bound to the complex case and also to the quantum case. In doing so, we make contact with several geometrical and matrix analytical concepts, such as the numerical range. Based on the quantum bound, we give a new and simplified proof for an upper bound on the Frobenius norm of commutators recently proven by Bottcher and Wenzel (B W). We also make some headway to generalise the B W bound to other norms. This is ongoing work and our proofs may still contain some pleasant gaps at the time of the presentation.

Universal quantum channel coding
by Igor Bjelakovic, Technische Universitaet Berlin
Coauthors: Holger Boche and Janis Noetzel

We determine the optimal rates of universal quantum codes for entanglement transmission and generation under channel uncertainty. In the simplest scenario the sender and receiver are provided merely with the information that the channel they use belongs to a given set of channels, so that they are forced to use quantum codes that are reliable for the whole set of channels. This is precisely the quantum analog of the compound channel coding problem. We determine the entanglement transmission and entanglement-generating capacities of compound quantum channels and show that they are equal.
Finally, we show how the results on quantum compound capacities imply those for random quantum capacity of arbitrarily varying quantum channels (AVQC) via Ahlswede's robustification technique.
A final derandomization step leads to a variant of famous Ahlswede dichotomy determining the entanglement transmission capacity of AVQC.

Paper reference: arXiv:0811.4588

Quantum hypothesis testing of non-i.i.d. states and its connection to reversible resource theories
by Fernando Brandao, Imperial College London
Coauthors: Martin Plenio

In the first part of the talk I will present extensions of quantum hypothesis testing to the case of non-indenpendent and identically distributed states; I will consider the setting where one would like to discriminate many copies of a given quantum state from a family of non-i.i.d. states. We say such a family of states has the exponential distinguishability (ED) property if the discrimination can be performed with exponential acuracy, in the number of copies of the first state. I will present certain conditions on sets of states under which we can prove the ED property. The proof combines recent developments on the characterization of permutation-symmetric quantum states, such as the exponential de Finetti theorem, and concepts from entanglement theory, such as the idea of non-lockability in entanglement measures.

In the second part of the talk, I will consider a new approach to the study of resource theories. These theories analyse the implications of restrictions on the physical processes available to the convertability of a physical state into another. A well-known example of a resource theory is entanglement theory, which emerges when distant parties only have access to local operations and classical communication. I will argue that whenever the set of non-resource states satisfies the ED property, then one can achieve reversible trasformations of the resource states in the framework where all operations not capable of generating resource can be used. Moreover, I will show that the unique measure fully charaterizing the rates of convertability is given by the optimal rate of distinguishability of a resource state to non-resource states. I will end up showing two applications of this result to entanglement theory.

Random quantum channels: almost sure confinement of the eigenvalues.
by Benoit Collins, University of Ottawa
Coauthors: Ion Nechita

Given a Hilbert space V of dimension k and a sequence of spaces W_n of dimension n, we consider a sequence of random quantum channels obtained by the random inclusion in V W_n of a subspace H_n of dimension approximately tkn (t in (0, 1)). As n tends to infinity, we confine almost surely the singular values of the outputs of all states in H_n. As an application, we obtain new bounds for minimum output entropy entropies and give new examples of random channels violating the additivity of the Renyi entropy (for all parameters p>1). Our techniques rely on free probability type norm estimates for random matrices.

Paper reference: arXiv:0906.1877

Superactivation of the Zero-Error Classical Capacity of a Quantum Channel
by Toby Cubitt (University of Bristol)

The zero-error classical capacity of a quantum channel is the asymptotic rate at which it can be used to send classical bits perfectly, so that they can be decoded with zero probability of error. We show that there exist pairs of quantum channels, neither of which individually have any zero-error capacity whatsoever (even if arbitrarily many uses of the channels are available), but such that access to even a single copy of both channels allows classical information to be sent perfectly reliably. In other words, we prove that the zero-error classical capacity can be superactivated. This result is the first example of superactivation of a classical capacity of a quantum channel.


A liberation process for quantum mutual information
by Patrick Hayden, McGill University
Coauthors: Nima Lashkari and Tobias Osborne

In free probability theory, the mutual free information is defined as the amount of time required for a stochastic time evolution called the liberation process to render two noncommutative random variables free. In this talk, I'll describe a quantum mechanical version of the liberation process and find that the time it takes to decouple two quantum systems is precisely the familiar quantum mutual information defined in terms of the von Neumann entropies of the state. In contrast, no formula for the mutual free information in terms of the free entropy is known.

Comments on Hastings' additivity counterexamples
by Christopher King, Northeastern University
Coauthors: M. Fukuda, D. Moser

Matt Hastings recently provided a proof of the existence of channels which violate the additivity conjecture for minimal output entropy. In this talk I will describe the main ideas of Hastings' proof. This will lead to some bounds for the minimal dimensions needed to obtain a counterexample.


Entanglement in Quantum Spin Chains
by Vladimir Korepin, Yang Institute for Theoretical Physics

We consider models of interacting spins [Heisenberg, AKLT ...] with unique ground state.We are interested in reduced density matrix in the ground state.We calculate the spectrum. This helps to evaluate von Neumann entropy and Renyi entropy of the block. Main examples are XY model, for VBS we can also calculate eigenvectors of the density matrix.

Paper reference: arXiv:0805.3542 , arXiv:0804.1741 , arXiv:0711.3882 , arXiv:0707.2534 , arXiv:quant-ph/0609098


The stability property of the set of quantum states and its applications
by M.E. Shirokov, Steklov Mathematical Institute

In this talk I will consider the (convex) stability property of the set of quantum states and its stronger version, providing useful tools in study of infinite dimensional quantum systems. I will briefly describe applications of the stability property to analysis of continuity of the important characteristics related to the classical capacity of a quantum channel and to the notion of entanglement of a state of a composite system.
The stronger version of stability makes possible to develop the special approximation approach to study of concave (convex) functions on the set of quantum states, which can be applied to many characteristics used in the quantum information theory (the von Neumann entropy, the output entropy of a quantum channel, the mutual information, etc.).

Paper reference: arXiv:0804.1515, arXiv:0904.1963

Discriminating correlated states of quantum lattice systems
by Milan Mosonyi, Centre for Quantum Technologies, National Uiversity of Singapore
Coauthors: F. Hiai, T. Ogawa, M. Fannes, M. Hayashi

Asymptotic hypothesis testing in its simplest form is about discriminating two states of a lattice system, based on measurements on finite blocks that asymptotically cover the whole lattice. In general, it is not possible to discriminate the local states with certainty, and one's aim is to minimize the probability of error, subject to certain constraints. Hypothesis testing results show that, in various settings, the error probabilities vanish with an exponential speed, and the decay rates coincide with certain relative-entropy like quantities. In this talk, I present a general method, based on the analysis of the asymptotic Renyi entropies, to obtain the exact error exponents for various classes of correlated states on cubic lattices. The examples include the temperature states of quasi-free fermionic and bosonic lattices, finitely correlated states, and the discrimination problem of i.i.d. states with group symmetric measurements. The discrimination problem of temperature states of a spin chain with translation-invariant and finite-range interaction is also treated with a different method.

Paper reference: arXiv:0706.2141, arXiv:0707.2020, arXiv:0802.0567, arXiv:0808.1450, arXiv:0904.0704

Majorization, entanglement catalysis, stochastic domination and $\ell_p$ norms
by Ion Nechita, University of Ottawa and Université Lyon 1
Co-author: Guillaume Aubrun


Operator Spaces: A natural language for Bell Inequalities
by Carlos Palazuelos, Universidad Complutense de Madrid

Coauthors: David Pérez-García (Universidad Complutense de Madrid) Michael M. Wolf (Niels Bohr Institute) Ignacio Villanueva (Universidad Complutense de Madrid) Marius Junge (University of Illinois at Urbana-Champaign)

In this talk we will show how Operator Space Theory appears in the study of Bell Inequalities. We will show the power of this theory by using it to prove the existence of tripartite quantum states which can lead to arbitrarily large violations of Bell Inequalities for dichotomic observables. We will also comment some other results in the framework of general Bell Inequalities, as well as some physical consequences.

Mapping cones of positive maps of B(H) into itself with H finite dimensional
by Erling Stormer, University of Oslo

I´ll discuss mapping cones of positive maps of B(H) into itself with H finite dimensional. They are cones of positive maps such that composition with completely positive maps are still in the cone. As applications we obtain characterizations of linear functionals with strong positivity properties with respect to a given cone and obtain several new and old characterizations of separable and PPT-states.


Semidefinite programs for completely bounded norms
by John Watrous, University of Waterloo

In this talk I will explain how the completely bounded trace and spectral norms, for finite dimensional mappings, can be efficiently expressed in terms of semidefinite programs. This provides an efficient method by which these norms may be both calculated and verified, and gives alternate proofs of some known facts about them.

Paper reference: arXiv:0901.4709


A strong converse for classical channel coding using entangled inputs
by Stephanie Wehner (California Institute of Technology)

A fully general strong converse for channel coding states that when the rate of sending classical information exceeds the capacity of a quantum channel, the probability of correctly decoding goes to zero exponentially in the number of channel uses, even when we allow code states which are entangled across several uses of the channel. Such a statement was previously only known for classical channels and the quantum identity channel. By relating the problem to the additivity of minimum output entropies, we show that a strong converse holds for a large class of channels, including all unital qubit channels, the d-dimensional depolarizing channel and the Werner-Holevo channel. This further justifies the interpretation of the classical capacity as a sharp threshold for information-transmission.

Joint work with Robert Koenig.


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