## Profile
Coding Theory
Design Theory
Finite Geometry
Algebraic Graph Theory
Quantum Computing
Cryptography
Communication Sequences ## Outputs
Title | Category | Date | Authors |
The chromatic number and rank of the complements of the Kasami graphs University of Calgary | Publication | 2007-01-01 | A. Roy, G. F. Royle | Bounds for codes and designs in complex subspaces University of Calgary | Publication | 2009-02-01 | A. Roy | Minimal Euclidean representations of graphs University of Calgary | Publication | 2010-02-01 | A. Roy | Equiangular lines, mutually unbiased bases, and spin models University of Calgary | Publication | 2009-01-01 | A. Roy, C. Godsil | Two characterizations of crooked functions University of Calgary | Publication | 2008-01-01 | A. Roy, C. Godsil | The chromatic number and rank of the complements of the Kasami University of Calgary | Publication | 2007-01-01 | A. Roy, G. F. Royle | Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements University of Calgary | Presentation | 2007-06-18 | A. Roy, A. Scott | Quantum state tomography using two-outcome measurementsWe investigate the process of quantum state tomography in which the observer may only access measurements on a single qubit. We assume that all measurements are projective and have only two possible outcomes, so that measurements on a system of dimension n are represented by POVMs in which the elements of each POVM are projections onto orthogonal subspaces of dimension n/2.
We claim that the optimal measurements for non-adaptive quantum state tomography using two-outcome measurements are described by {\em complex Grassmannian 2-designs}. We also give lower bounds for the size of a Grassmannian 2-design: in a system of dimension n, at least n^2-1 two-outcome measurements must occur in order for the union of the POVM elements to form a 2-design. Finally, we show that this lower bound on the size of a 2-design is tight by constructing 2-designs of minimal size in certain dimensions. In particular, for every n such that there exists a Hadamard matrix of order n, we construct n^2-1 two-outcome measurements which form a complex Grassmannian 2-design in an n-dimensional system. University of Calgary | Presentation | 2008-08-23 | A. Roy, M. Roetteler, C. Godsil | Graphs and codes from nonlinear functionsCertain nonlinear codes, such as the binary Preparata codes, have better error-correcting properties than any linear codes of the same length and size. Classically, these codes are described using functions on finite vector spaces which are far from linear; these "crooked" functions can also be used to construct interesting distance-regular graphs. In this talk I will explain some of the interesting connections between these structures, including some new results which chracterize crooked functions in terms of their graphs and codes.
University of Calgary | Presentation | 2004-05-07 | A. Roy | Association schemes in systems of lines and subspaces University of Calgary | Presentation | 2007-05-29 | A. Roy | Quantum state tomography and 2-designsIn 1989, Wootters and Fields showed that mutually unbiased bases (MUBs) are examples of optimal orthogonal measurements for quantum state tomography. As Roetteler and Klappenecker observed, MUBs are examples of complex projective
2-designs. In 2006, Scott showed that a general rank-one measurement is optimal for quantum state tomography if and only if it is a 2-design; this
result is also true for orthogonal rank-one measurements and "two-outcome" measurements. In this talk, we explain the connection between designs and measurements, and we describe some new constructions and important open problems. University of Calgary | Presentation | 2008-06-16 | A. Roy | Unitary t-designsIn this talk, we introduce the concept of unitary t-designs. A unitary design is a finite set of unitary matrices that nicely approximates the en-
tire unitary group, much as spherical design approximates the unit sphere. Unitary t-designs were introduced by Dankert, Cleve, Emerson, and Livine in 2006, and in 2007 Scott showed that unitary t-designs serve as optimal choices for quantum process tomography. Using representations of the unitary group, we give a Delsarte-style lower bound on the size of a unitary t-design and some necessary conditions for the bound to be tight. This is Unijoint work with Andrew Scott. University of Calgary | Presentation | 2008-07-08 | A. Roy, A. Scott | Minimal Euclidean representations of graphsA simple graph G is representable in a real vector space V if there is an embedding of the vertex set in V such that the Euclidean distance between two vertices u and v depends only on whether or not u and v are adjacent. The Euclidean representation number of G is the smallest dimension in which G is representable. Representations of graphs were introduced by Pouzet in 1977 as a means of showing that certain graph invariants of Tutte are reconstructible.
In this talk, we use Euclidean distance matrices to give an exact formula for the Euclidean representation number of any graph in terms of its spectrum and main values. University of Calgary | Presentation | 2009-06-07 | A. Roy | Weighted complex projective 2-designs from basesWe introduce the problem of constructing weighted complex projective 2-designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases, being equivalent whenever the design is composed of d+1 bases in dimension d. We show that, for the purpose of quantum state determination, these designs specify an optimal collection of orthogonal measurements. Using highly nonlinear functions on abelian groups, we construct explicit examples from d+2 orthonormal bases whenever d+1 is a prime power, covering dimensions d+d=6, 10, and 12, for example, where no complete sets of mutually unbiased bases have thus far been found.
University of Calgary | Presentation | 2007-06-02 | A. Roy, A. Scott | Maximal graphs and graphs with maximal spectral radius University of Calgary | Publication | 2002-05-01 | D. D. Olesky, A. Roy, P. v. Driessche | On the epistemic view of quantum states University of Calgary | Publication | 2008-01-01 | M. Skotiniotis, A. Roy, B. C. Sanders | Entanglement of subspaces in terms of entanglement of superpositions University of Calgary | Publication | 2008-01-01 | G. Gour, A. Roy | Secret sharing using graph states University of Calgary | Presentation | 2008-08-20 | D. Markham, A. Roy, B. C. Sanders | Communication of information in the absence of a shared frame of referenceIn a communication protocol the sender, Alice, encodes classical messages by preparing a quantum system in a particular state and sending it to the receiver, Bob, who decodes the message by an appropriate quantum measurement. Implicit in the protocol is the assumption that whatever the physical encoding employed by Alice, whether it is the spin of particle, or the energy levels of an atom, is known to Bob. This assumption amounts to Alice and Bob sharing a common reference frame relative to which the states of physical systems are described. The lack of a shared frame of reference imposes severe restrictions on many communication and computational tasks. We obtain the optimal protocols for two cases: where invariant subspaces are available and where they are not.
University of Calgary | Presentation | 2011-03-09 | M. Skotiniotis, A. Roy, G. Gour, B. C. Sanders | Fully epistemic toy theoryThe Spekkens toy model is an interesting example of how to modify classical physics in order to perform several quantum information processing tasks. Spekkens\' toy model has four axioms concerning toy states, valid operations, measurements, and composition of single toy systems. Motivated by the empirical indistinguishability of epistemic vs. ontic states in the toy universe, we show that relaxing valid operations to mappings of epistemic rather than ontic states preserves the features of the toy model. Similarly we show that relaxing the axiom regarding the composition of single toy systems also preserves the toy model. Relaxing both axioms simultaneously, however, breaks the correspondence of the toy model with quantum theory because the tensor product composition rule is violated, but these two relaxations together produce a group of operations on epistemic states that is isomorphic to the projected extended Clifford Group. University of Calgary | Presentation | 2008-03-11 | M. Skotiniotis, A. Roy, B. C. Sanders | Entanglement-enhanced classical communication without a shared frame of referenceTwo parties, Alice and Bob, share a communication channel but lack a shared reference frame.
Alice's task is to communicate a message to Bob, and she does so by preparing an object in a state
that represents the message, for example as a rotation, and transmitting this object to Bob who
measures the state of the object to reveal the message. Due to the lack of a shared reference frame,
Bob may not be able to perform the appropriate measurement to learn the message. For example
Bob may be lacking the reference angle against which to measure the rotation. Here we tackle the
problem of how two parties, lacking a shared reference frame, could prepare and measure a message
in order to communicate successfully. We deem a prepare-and-measure procedure to be successful
if it minimizes the average error over all received messages.
In our communication protocol the parties circumvent the lack of a shared reference frame by
preparing and sending two objects such that the message is the relative transformation parameter
from the state of the rst object into the state of the second object. Bob performs joint measurements
on the pair of received objects to infer the message from the measurement outcomes. Our aim is
to devise a prepare-and-measure scheme that ensures the highest average success rate for sending
messages as relative transformation parameters between two objects.
We use Schur's lemmas, group representation theory, and quantum estimation theory to derive
optimal measurements given constraints imposed on Alice's preparations. We can nd closed-form
solutions for prepare-and-measure schemes for some constraints and employ numerical methods to
obtain optimal protocols in the more general cases. In particular we discover that, whereas preparing
objects in an entangled state is sucient for success, entanglement is not always necessary. Our
theory lays the groundwork for circumventing a lack of reference frames between parties by sending
messages through the parameter of a relative transformation between two objects. University of Calgary | Presentation | 2010-08-26 | M. Skotiniotis, A. Roy, G. Gour, B. C. Sanders | A contextual toy model University of Calgary | Presentation | 2008-08-21 | M. Skotiniotis, G. Gour, A. Roy, B. C. Sanders | Local extrema of entropy functions under tensor products University of Calgary | Publication | 2011-09-01 | S. Friedland, G. Gour, A. Roy |
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