### July 6-10, 2009

Workshop on Operator Structures in Quantum Information

### Contributed Talks/Posters

**Quantum
error correction on infinite-dimensional Hilbert spaces**

**by**

Cedric Beny Centre for Quantum Technologies, National University of
Singapore

Coauthors: Achim Kempf, David W. Kribs
We present a generalization of quantum error correction to infinite-dimensional
Hilbert spaces. The generalization yields new classes of quantum
error correcting codes that have no finite-dimensional counterparts.
The error correction theory we develop begins with a shift of focus
from states to algebras of observables. Standard subspace codes
and subsystem codes are seen as the special case of algebras of
observables given by finite-dimensional von Neumann factors of type
I. Our generalization allows for the correction of codes characterized
by any von Neumann algebra, depending on that nature of the noise.

Paper reference: arXiv:0811.0421

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*The Stinespring Theorem made difficult*

by

Man-Duen Choi Math Dpartment , University of Toronto

For more than three decades, we thought we had known everything
about completely positive linear maps. Because of the sudden arrival
of the era of quantum computers, we were awakened of many unknown
aspects of the the standard structure theorems. Here, I will show
off the magic of the difficult Stinespring Theorem in the finite
dimensinal case.

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*Multi-particle decoherence free subspaces and incoherently
generated coherences*

by Raisa Karasik Applied Science & Technology and Berkeley Quantum
Information Center, University of California, Berkeley

Coauthors: K.-P. Marzlin, B. C. Sanders and K. B. Whaley

Decoherence often arises from the interaction between a quantum
system and its environment and constitutes a major obstacle for
quantum computation and information as it leads to destruction of
fragile quantum states. Encoding into decoherence-free subspaces
is one strategy to protect quantum states. The bad news is that
we have found that decoherence-free subspaces do not exist for extended
systems in more than one dimension for a broad class of realistic
reservoirs, but the good news is that we have discovered that in
some cases the dynamics of a quantum system can connive together
with the environmental interactions to reduce decoherence. This
property relies on the counterintuitive phenomenon of “incoherently
generated coherences” and allows for identification of a new
class of decoherence-free states. Examples of such systems are given
from cavity quantum electrodynamics and squeezed light.

Paper reference: Phys. Rev. A 77(5): 052301 ( 2008), Phys. Rev.
A 76(1): 012331 (2007)

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*Dilation properties of a bipartite quantum state and quantum
analogs of Bell-type inequalities*

by

Elena R. Loubenets. Moscow State Institute of Electronics and Mathematics

We introduce the specifically constructed dilations of a bipartite
quantum state; prove the existence of such dilations for any bipartite
state, possibly infinitely dimensional, and present, in terms of
these dilations, the quantum analogs of bipartite Bell-type inequalities.

Paper reference: JPA 41_ 445303 (2008), JPA 41_ 445304 (2008),
Banach Center Publ 73_ 325 (2006), JPA 38_ L653 (2005)

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*Random quantum channels: graphical
calculus*

by Ion Nechita, University of Ottawa and Université Lyon
1

Coauthors: Benoît Collins (University of Ottawa and CNRS)

With the aim of studying random constructions arising in quantum
information theory, we introduce a diagrammatic notation for tensors,
inspired by ideas of Penrose and Coecke. Then, interpreting Weingarten
calculus in our formalism, we describe a method for computing expectation
values of diagrams which contain Haar-distributed random unitary
matrices. This is done by the means of a graph-expansion of the
original diagram.

As a first set of applications of the above methods, we compute
eigenvalue statistics for outputs of tensor products of independent
and conjugate random quantum channels. We obtain the almost sure
behavior of the eigenvalues. In the case of conjugate channels,
our results improve on known bounds for the largest eigenvalue obtained
by Hayden and Winter.

Paper reference: arXiv:0905.2313

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*Upper bounds on the rate of nondegenerate p-ary stabilizer
codes.*

by

Yingkai Ouyang, University of Waterloo

For prime p and q = p2, by viewing the normalizer of a stabilizer
code as a q-ary classical code, we use upper bounds on the rate
of q-ary codes to obtain upper bounds on the rate of nondegenerate
stabilizer codes for fixed stabilizer code distance d and code length
n. This is because the distance of any nondegenerate stabilizer
code is the distance of its normalizer seen as a classical q-ary
code. This substantially improves the quantum Hamming bound for
nondegenerate stabilizer codes. The best upper bounds obtained via
this method follow from Aaltonen's generalization of the McEcliece,
Rumsey, Rodemich and Welch bounds to q-ary codes and the Elias-Bassalygo
bound. Previously Rains showed that the maximum distance of an arbitrary
quantum code is [(3-v3)/4]n. Using Aaltonen's bound, we show that
as n goes goes to infinity, the maximum distance of a nondegenerate
stabilizer code is 0.3161n, beating Rain's bound.

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*All entangled states are useful for channel discrimination*

by

Marco Piani, Institute for Quantum Computing, University of Waterloo

Coauthors: John Watrous

We prove that every entangled state is useful as a resource for
the problem of minimum-error channel discrimination. More specifically,
given a single copy of an arbitrary bipartite entangled state, it
holds that there is an instance of a quantum channel discrimination
task for which this state allows for a correct discrimination with
strictly higher probability than every separable state.

Paper reference: arXiv:0901.2118

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*Tsirelson's problem*

by

Volkher Scholz, LUH

Coauthors: Reinhard F. Werner

The situation of two independent observers conducting measurements
on a joint quantum system is usually modelled using a Hilbert space
of tensor product form, each factor associated to one observer.
Correspondingly, the operators describing the observables are then
acting non-trivially only on one of the tensor factors. However,
the same situation can also be modelled by just using one joint
Hilbert space, and requiring that all operators associated to different
observers commute, i.e. are jointly measurable without causing disturbance.
The problem of Tsirelson is now to decide the question whether all
quantum correlation functions between two independent observers
derived from commuting observables can also be expressed using observables
defined on a Hilbert space of tensor product form. Tsirelson showed
already that the distinction is irrelevant in the case that the
ambient Hilbert space is of finite dimension. We show here that
the problem is equivalent to the question whether all quantum correlation
functions can be approximated by correlation function derived from
finite-dimensional systems. We also discuss some physical examples
which fulfill this requirement.

Paper reference: arXiv:0812.4305

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**Two-sided bounds on minimum-error quantum measurement, quantum
conditional min-entropy, and on the reversibility of quantum dynamics
using the first Jezek-Rehacek-Fiurasek-Hradil iterate**

**by**

Jon Tyson Harvard University

Using a unified framework, we obtain two-sided bounds on the following
quantities of interest in quantum information theory:

1. The minimum-error quantum distinguishability of arbitrary ensembles
of mixed quantum states.

2. The approximate reversibility of quantum dynamics in terms of
entanglement fidelity.

3. The conditional min-entropy of arbitrary bipartite quantum states.

Our primary tool is an abstract generalization of Jezek, Rehacek,
and Fiurasek's [Phys. Rev. A 65, 060301] successive approximation
scheme for computing optimal measurements and Jezek, Fiurasek, and
Hradil's [Phys. Rev. A 68, 012305] scheme for maximum-likelihood
reconstruction of quantum channels.

In the case of measurements, we prove a concise factor-of-two estimate
for the failure-rate of optimally distinguishing an arbitrary ensemble
of mixed quantum states, generalizing work of Holevo [Theor. Probab.
Appl. 23, 411 (1978)] and Curlander [Ph.D. Thesis, MIT, 1979]. A
modification of the minimal principle of Concha and Poor [Proceedings
of the 6th International Conference on Quantum Communication, Measurement,
and Computing (Rinton, Princeton, NJ, 2003)] is used to derive a
sub-optimal measurement which has an error rate within a factor
of two of the optimal by construction. This measurement is quadratically
weighted, and has appeared as the first iterate of a sequence of
measurements proposed by Jezek, Rehacek, and Fiurasek [Phys. Rev.
A 65, 060301]. Unlike the so-called "pretty good" measurement,
it coincides with Holevo's asymptotically-optimal measurement in
the case of non-equiprobable pure states. A quadratically-weighted
version of the measurement bound by Barnum and Knill [J. Math. Phys.
43, 2097 (2002)] is proven. Bounds on the distinguishability of
syndromes in the sense of Schumacher and Westmoreland [Phys. Rev.
A 56, 131 (1997)] appear as a corollary, allowing one to convert
pure-state distinguishability bounds into mixed-state bounds with
only a factor-of-two degradation in the failure rate.

Our reversibility bounds for quantum channels are obtained using
a quadratically-weighted version of Barnum and Knill's reversal
channel. The quadratic weighting of our map is interpreted using
a state-dependent functional calculus for quantum channels. Some
advantages of our reversal are:

1. We obtain relatively simple reversibility (and distinguishability)
estimates, without negative matrix powers.

2. Our channel reversal reduces to Holevo's asymptotically-optimal
measurement when used to reverse a pure-state-ensemble-preparation
channel.

3. One can replace the output target state with any other state
for free.

These advantages are obtained at no cost in tightness of our bounds.

If time permits, I will show how to use matrix monotonicity to
prove minimum-error distinguishability bounds.

Paper reference: J. Math. Phys 50, 023106 (2009); Phys Rev A 79,
032343 (2009); to appear

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