


Mathematical Aspects of Quantum Information
Org: Daniel Gottesman (Perimeter Inst.), Achim Kempf (Waterloo), David Kribs (Guelph) and Michele Mosca (Waterloo; Perimeter Inst.) [PDF]
 ANDRIS AMBAINIS, University of Waterloo, 200 University Avenue West, Waterloo,
ON N2L 2T2
A new proof of quantum adiabatic theorem
[PDF] 
Adiabatic quantum algorithms are a new approach to quantum
computation. While being equivalent to standard quantum circuit
model, they present a completely different way of thinking about
quantum algorithms.
Adiabatic algorithms are based on the adiabatic theorem of quantum
mechanics. Informally, this theorem says that, when a Hamiltonian of
a physical system is slowly transformed to a different Hamiltonian,
the lowest energy state of the first Hamiltonian is transformed to the
lowest energy state of the second Hamiltonian.
We present a new proof of quantum adiabatic theorem, in terms of
discrete mathematics.
 ARVID BESSEN, Department of Computer Science, Columbia University,
New York, NY 10027, USA
A Lower Bound for the SturmLiouville Eigenvalue Problem on a
Quantum Computer
[PDF] 
We study the complexity of approximating the lowest eigenvalue of a
SturmLiouville differential equation on a quantum computer. Our
main focus is on the special case of computing the ground state energy
of a quantum system for a given potential.
Recently Papageorgiou and Wo\'zniakowski proved that quantum
computers could achieve exponential speedups compared to classical
computers for certain potentials. Papageorgiou's and
Wo\'zniakowski's method uses the (discretized) unitary propagator
exp(i L_{q}) as a query; here L_{q} is the
differential operator for the SturmLiouville problem. If the
operator exp(i p L_{q}) (a "power query") is computable in
cost comparable to exp(i L_{q}) for any integer p, one can
solve the SturmLiouville problem with O(loge^{1}) power queries.
In this paper we will prove a matching lower bound of W(loge^{1}) power queries, therefore showing that Q(loge^{1}) power queries are sufficient and necessary. Our proof
is based on a frequency analysis technique, which examines the
probability distribution of the final state of a quantum algorithm and
the dependence of its Fourier transform on the input. This method was
first used to give a lower bound for the number of power queries in
the phase estimation problem on quantum computers.
 HILARY CARTERET, Université de Montréal, C.P. 6128 succ. Centreville,
Montréal, Québec H3C 3J7
Noiseless quantum circuits for measuring entanglement
[PDF] 
The nonlocal properties of a density matrix are often defined in
terms of the effects of unphysical maps, such as the Partial Transpose
on the spectrum of the density matrix (PTspectrum). Is it possible
to measure these functions efficiently, or must we use full state
tomography?
Previously proposed methods for measuring these quantities directly
relied on the Structural Physical Approximation, which typically
produce output states with visibilities that scale poorly with the
system size. The moments of the resulting modified density operator
must then be measured in a separate procedure, which can be done using
a set of generalized MachZehnder interferometers. The spectrum can
then be obtained using a little algebra.
I will show how to construct a family of simple quantum circuits that
can determine the PTspectrum for any bipartite state, without
incurring any loss of visibility. These circuits measure the minimum
amount of information required to determine the PTspectrum
completely. They depend only on the dimension of the state and they
will be exact up to the statistical uncertainties inherent in any
experimental data. The analysis of the output of these circuits for
general bipartite states also raises an interesting eigenspectrum
reconstruction problem.
 ANDREW CHILDS, California Institute of Technology, Pasadena, CA, USA
The limitations of nice mutually unbiased bases
[PDF] 
Mutually unbiased bases of a Hilbert space can be constructed by
partitioning a unitary error basis. We consider this construction
when the unitary error basis is a nice error basis. We show that the
number of resulting mutually unbiased bases can be at most one plus
the smallest prime power contained in the dimension, and therefore
that this construction cannot improve upon previous approaches. We
prove this by establishing a correspondence between nice mutually
unbiased bases and abelian subgroups of the index group of a nice
error basis and then bounding the number of such subgroups. This
bound also has implications for the construction of certain
combinatorial objects called nets.
Joint work with Michael Aschbacher and Pawel Wocjan.
 J. IGNACIO CIRAC, MaxPlanck Institute for Quantum Optics,
HansKopferemannstr. 1, D85748 Garching, Germany
Simulating quantum systems
[PDF] 
Manybody quantum systems are very hard to simulate since the
dimension of the corresponding Hilbert space scales exponentially with
the number of particles N. In practice, however, the quantum states
that typically appear in nature may be described with fewer
parameters. In this talk I will review a novel description of quantum
states which was introduced by F. Verstraete and myself, and it is
based on projecting twoparticle entangled states of a given dimension
D onto subspaces of dimension d, which is the one of the Hilbert
spaces corresponding to the original particles. The complexity of
these states scales polynomially in d, D and N. Moreover, most
of the states that appear in Quantum Information Theory, like cluster,
GHZ, W, graphs, etc., have D=2. We have also developed
(classical) numerical algorithms based on this description which allow
us to simulate quantum manybody systems in 1 and higher spatial
dimensions.
 RICHARD CLEVE, Universiy of Waterloo
Nonlocality and limits of faulttolerant computation
[PDF] 
We present various results concerning nonlocal behavior in an abstract
setting, and then show how to use them to establish upper bounds on
the errorthreshold below which computations can be made
faulttolerant.
This is joint work with Harry Buhrman, Noah Linden, and Falk Unger.
 PATRICK HAYDEN, McGill University, Montreal
On private communication using a shared reference frame
[PDF] 
A private shared Cartesian frame is a novel form of private shared
correlation that allows for both private classical and quantum
communication. Cryptography using a private shared Cartesian frame
has the remarkable property that, if perfect security is demanded, the
private classical capacity is roughly three times the private quantum
capacity. I'll present work done with Stephen Bartlett and Rob
Spekkens demonstrating that if the requirement for perfect security is
relaxed, then it is possible to use the properties of random subspaces
to nearly triple the private quantum capacity.
 MARK HILLERY, Hunter College of the City University of New York, Department
of Physics, 695 Park Avenue, New York, NY 10020, USA
Programmable quantum circuits
[PDF] 
Many quantum circuits are designed to accomplish a single task, such
as approximate cloning or teleportation. It is often useful to have
more versatile circuits that can perform many tasks. Such a circuit
has two inputs, data and a program, both of which are quantum states.
The data has an operation performed on it, and the program controls
what the operation is. If the circuit is to be able to perform an
infinite number of operations with a finitedimensional program space,
it must be either probabilistic or approximate. That is, it either
succeeds only part of the time, or the operations are performed to
some level of approximation, and not exactly. The operation of
probabilistic and approximate programmable circuits will be
discussed.
 PETER HØ YER, Calgary

 LOUIS KAUFFMAN, University of Illinois at Chicago
Topological Quantum Computation
[PDF] 
This talk will survey joint work with Sam Lomonaco on the use of
topology in relation to quantum computation. We begin by a discussion
of universal gates based on unitary solutions to the YangBaxter
equation. We then discuss models based on topological quantum field
theory and show, in particular, how to construct the Fibonacci model
of Freedman, Kitaev and Wang by using knot theoretic methods based on
qdeformations of Penrose spin networks (Temperley Lieb recoupling
theory). These methods provide a very direct way to construct
representations of the Artin Braid groups that are dense in the
unitary groups. Many questions will be discussed in the light of
these constructions.
 CHRISTOPHER KING, Northeastern University, Boston
Matrix inequalities and multiplicativity results
[PDF] 
The multiplicativity of the q ® p norm of a completely
positive qubit channel for q < 2 < p follows from some Hannertype
inequalities involving pnorms of positive semidefinite matrices.
These inequalities are reviewed, together with their application to
the multiplicativity question. Some extensions of the inequalities
are described, and some open problems for qubit and higher dimensional
channels are also discussed.
 GREG KUPERBERG, University of California, Davis
Hybrid quantum memory and its capacity
[PDF] 
What is the most general possible kind of memory consistent with
quantum mechanics? The only commonly considered kinds are qudits and
classical digits, but a hybrid modelled by an arbitrary C^{*}algebra
is more generally possible. The important ChoiEffros theorem
implies that it is the most general possible quantum memory model
modulo certain (debatable) assumptions. In particular it generalizes
the theory of "decoherencefree subspaces".
Assuming this model, when is one hybrid memory worth more than
another? I will give a characterization of when many copies of a
memory A embed (or blindly encode with perfect fidelity) into
slightly more copies of another memory B. In particular, either
there is such an embedding, or A admits a state that does not
visibly encode into B with high fidelity. The second half of this
alternative depends on a Holder inequality for hybrid memories that
generalizes the classical pigeonhole principle.
Reference: quantph/0203105.
 MAIA LESOSKY, University of Guelph
On Generalized Noiseless Subsystems
[PDF] 
A generalized notion of noiseless subsystems was recently introduced
by Kribs, Laflamme and Poulin as part of a unified and generalized
approach to quantum error correction called operator quantum
error correction. One advantage to generalized noiseless
subsystems is that they are not restricted to unital channels. In
this talk I will present some simple examples and outline necessary
and sufficient conditions that describe the existence of generalized
noiseless subsystems.
 DEBBIE LEUNG, University of Waterloo and Caltech
Composition of randomization maps
[PDF] 
Recently, various quantum communication tasks have been related to the
ability to randomize the state of a quantum system. In this talk, we
will see how to obtain communication protocols for multiple receivers
by understanding the composition of randomization maps.
 ROBERT MARTIN, University of Waterloo, Waterloo, ON N2L 3G1
On the relationship between discrete and continuous
representations of quantum information
[PDF] 
In classical information theory, sampling theory provides the
connection between discrete and continuous representations of
information. Here we present a generalization to quantum field theory
in which the bandwidth is provided by an ultraviolet cutoff. We
investigate, in particular, the role of quantum fluctations as noise
and implications for the channel capacity.
 ASHWIN NAYAK, University of Waterloo, 200 University Ave. W., Waterloo, ON
N2L 3G1
A quantum test for group commutativity
[PDF] 
We consider the computational problem of testing whether an implicitly
specified group is commutative. The group is defined by its k
generators, and a procedure that implements group operations. The
computational complexity (in terms of k) of this problem was first
considered by Pak (2000). We construct a quite optimal quantum
algorithm for this problem whose complexity is in
[(O)\tilde](k^{2/3}). The algorithm uses and highlights the power of
the quantization method of Szegedy (2004). For the lower bound
W(k^{2/3}), we introduce a new technique of reduction for
quantum query complexity. We also prove an W(k) lower bound
for classical algorithms, which shows that the algorithm of Pak is
optimal.
This is joint work with Frédéric Magniez (CNRSLRI, France).
 JONATHAN OPPENHEIM, University of Cambridge
Quantum information can be negative
[PDF] 
Even the most ignorant among us cannot know less than nothing. What
could negative knowledge mean? In the everyday world we are
accustomed to, negative knowledge makes no sense. But in the world
where the laws of quantum mechanics hold sway, knowledge can be
negative. In essence, one can have situations where someone knows
more than everything, and it is in these situations where one finds
negative knowledge. This negative knowledge turns out to be precisely
the right amount to cancel the fact that we can know too much.
Negative knowledge is due to exotic features of quantum information
theory and by understanding that quantum knowledge can be negative, we
gain deeper insights into such phenomena as quantum networks, quantum
teleportation, quantum computation, and the very structure of the
quantum world.
In more detail, given part of an unknown quantum state, we determine
how much quantum communication is needed to obtain the full state.
This is the partial information we need conditional on our previous
information. It turns out to be given by an extremely simple formula,
the conditional entropy. In the classical case, partial information
must always be positive, but we find that in the quantum world this
physical quantity can be negative. If the partial information is
positive, the sender of the partial information needs to communicate
this number of quantum bits to the receiver; if it is negative, they
instead gain the corresponding potential for quantum communication in
the future. The primitive that is introducedquantum state
mergingenables a systematic understanding of quantum network
theory, and several such applications will be discussed.
 CARLOS PEREZ, University of Waterloo, 200 University Ave. West, Waterloo,
ON N2L 3G1
Models of Quantum Cellular Automata
[PDF] 
In this talk we present a systematic view of Quantum Cellular Automata
(QCA), a mathematical formalism of quantum computation. We present
four QCA models, and compare them. One model we discuss is the
traditional QCA, similar to those introduced by Shumacher and Werner,
Watrous, and Van Dam. We discuss also Margolus QCA, also discussed by
Schumacher and Werner. We introduce two new models, Coloured QCA, and
Continuous QCA. We also compare our models with the established
models. We give proofs of computational equivalence for several of
these models. We show the strengths of each model, and provide
examples of how our models can be useful to come up with algorithms,
and implement them in realworld physical devices.
 MARTIN ROETTELER, NEC Laboratories America, Inc.
On the Power of Random Bases in Fourier Sampling: Hidden
Subgroup Problem in the Heisenberg Group
[PDF] 
The hidden subgroup problem (HSP) provides a unified framework to
study problems of grouptheoretical nature in quantum computing such
as order finding and the discrete logarithm problem. While it is
known that Fourier sampling provides an efficient solution in the
abelian case, not much is known for general nonabelian groups.
Recently, some authors raised the question as to whether
postprocessing the Fourier spectrum by measuring in a random
orthonormal basis helps for solving the HSP. Several negative results
on the shortcomings of this random strong method are known. In this
talk I will show that the random strong method can be quite powerful
under certain conditions on the group G. In particular the HSP for
finite Heisenberg groups can be solved using polynomially many random
strong Fourier samplings followed by a possibly exponential classical
postprocessing without further queries.
Joint work with Jaikumar Radhakrishnan and Pranab Sen.
 MARY BETH RUSKAI, Tufts University, Medford, MA 02155, USA
Completely bounded pnorms in quantum information theory
[PDF] 
Proof of the multiplicativity of maximal pnorms of noisy quantum
channels has been conjectured and is known to imply additivity of
minimal entropy and several equivalent conjectures. The concept of a
completely bounded norm has been defined in the context of operator
spaces. A channel is a completely positive, tracepreserving (CPT)
map F acting on the d ×d matrices, which can be regarded
as forming a Banach space associated with the Schatten pnorm. The
completely bounded norm of F is defined in terms of the action of
I_{m} ÄF on tensor products of matrices, which generate a
noncommutative vectorvalued L_{p} space. The completely bounded
norm of a tensor product of CPT maps is multiplicative. This implies
that a certain type of minimal conditional entropy is additive.
This talk is based on joint work with I. Devetak, M. Junge and
C. King. It is intended to be accessible to both operator algebraists
and quantum information theorists.
 ANDREAS WINTER, University of Bristol, Department of Mathematics, Bristol
BS8 1TW, UK
Random coding for quantum information
[PDF] 
In this talk, the conceptual and mathematical ideas in a LloydShor
type proof of the quantum channel capacity theorem will be presented.
To be precise, we will look at a general quantum channel
N, i.e., a completely positive and trace
preserving linear map on density operators, and study block coding of
quantum information for n instances N^{Än} of the
channel, for large n. After introducing these concepts in
mathematical terms, we will study a specific random coding procedure,
which we call Haarrandom codes. These are akin to P. Shor's
proposal [MSRI talk, Nov. 2002; lecture notes online at
http://www.msri.org/publications/ln/msri/2002/quantumcrypto/shor/1/]:
the code is (essentially) a subspace of the sender's typical space,
chosen according to the unitarily invariant measure.
The "standard" approach to analysing the performance of quantum
codes [see S. Lloyd, PRA 1997; I. Devetak, IEEE IT 2005] proceeds by
showing
(i) that a basis of the code subspace can be distinguished
reliably by the receiver;
(ii) that the channel environment has almost no information
about this basis;
(iii) finally, how these two elements imply that
superpositions of the basis vectors can be errorcorrected with
high fidelity.
After highlighting this strategy and some of its technical
difficulties, we will demonstrate a new strategy which has the
advantage of leading to the result with minimal technical effort, and
which is also conceptually nice. It entirely avoids the difficult
step (ii), the privacy of the code against the environment, which
comes out automatically. Instead, we show
(a) that a basis and its Fourier conjugate basis of the code
subspace each can be distinguished reliably by the receiver;
(b) how a recently discovered information uncertainty relation
[M. Christandl and AW, quantph/0501090] then implies that
the quantum mutual information between sender and receiver
is close to maximumand the quantum mutual information between
sender and environment is close to 0;
(c) finally, a simple algebraic reasoning [B. Schumacher and
M. Westmoreland, Quantum Inf. Proc. 2002] shows the existence of a
decoding procedure.
Time permitting, variations and other applications of the Haarrandom
coding will be shown.

