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