
Local continuity bounds for entropies of finite probability distributions (May 2017, 7 slides)

A presentation on my work with my supervisor Nilanjana Datta on local continuity bounds for the entropies of finite distributions, presented at the oneday event High Dimensional Mathematics. These are the slides I gave accompanying my short talk, the video of which is here.

Landauer’s Principle in Repeated Interaction Systems (July 2016, 34 slides)

The slides for the talk I gave at the Autrans summer school Stochastic Methods in Quantum Mechanics on my work with Alain Joye, Yan Pautrat, and Renaud Raquepas on Landauer’s Principle in repeated interaction systems.

Landauer's Principle in RIS (Winter 2016; 31 slides)

Landauer's Principle (Summer 2014; 53 slides)

“Landauer’s principle states that the energy cost to erase one bit of information by the action of a thermal reservoir at equilibrium temperature is always at least , and provides an interesting link between information theory and physics. My goal is to discuss Landauer’s principle generally, and for the case of a 2 level system coupled to a thermal reservoir modelled by an XY spin chain (or sequence of spin1/2 particles), using the tools of quantum statistical mechanics.”

QSEPState is in QMIPne (Winter 2014; 17 slides)

“The promise problem QSEPSTATE asks if a quantum state described by a circuit is close to a separable (not entangled) state across a given cut, or not. A quantum multiprover protocol was found to give an upper bound for the computational complexity of QMIPne, which is the class of problems that can be solved by a computationally bounded quantum verifier exchanging messages with unentangled, computationally bounded, quantum provers.”

Is QSEPCIRCUIT QMAhard? (Summer 2013; 60 slides)

“QSEPCIRCUIT, as defined in a recent paper by Patrick Hayden, Kevin Milner, and Mark Wilde (http://arxiv.org/abs/1211.6120) in the computer science department here at McGill, is the problem of determining whether or not a quantum state given as a circuit is separable or not. The problem has been shown to be in QIP(2), and to be NPhard and QSZKhard, which lends credence to the idea that the problem is QMAhard. The goal of this talk is to provide background for and explain in detail what QSEPCIRCUIT and QMA are, explain why we care whether or not QSEPCIRCUIT is QMAhard, and hopefully showcase quantum information theory as an exciting and dynamic field. No background knowledge will be assumed.”

Local continuity bounds for entropies of finite probability distributions (May 2017)

A poster on my work with my supervisor Nilanjana Datta on local continuity bounds for the entropies of finite distributions, presented at the oneday event High Dimensional Mathematics. I gave a short “elevator pitch” to advertise the poster, the video recording of which is posted here.

Landauer's Principle in Repeated Interation Systems (November 2016)

QSEPSTATE in QMIPne (September 2013)

A poster on my work with Chris Bahr, under the supervision of Patrick Hayden, on a quantum complexity theory problem about entanglement, presented at McGill University for an undergraduate computer science poster session. Abstract: The promise problem QSEPSTATE asks if a quantum state described by a circuit is close to a separable (not entangled) state across a given cut, or not. A quantum multiprover protocol was found to give an upper bound for the computational complexity problem of QMIPne, which is the class of problems that can be solved by a computationally bounded quantum verifier exchanging messages with unentangled, computationally unbounded, quantum provers.

The existence and uniqueness of the Haar Measure (Fall 2014; 11 pages)

“The Haar measure allows integration over topological groups, which has many applications; for example, the use of the Haar measure in proving the Fully Quantum Slepian Wolf (or state transfer) protocol fundamental to quantum information theory is briefly discussed in section VI. But first we build the definitions necessary to define the Haar measure, and state some fundamental theorems. Next, we show a left Haar measure exists on locally compact Hausdorff topological groups, and is unique (up to scale) on such groups which are also compact. While of course the Haar measure is unique up to scale on all locally compact Hausdorff groups, the simpler case admits an elegant proof. Lastly, the relationship between the right and left Haar measure is investigated.”

The black hole information paradox (Winter 2014; 9 pages)

“The black hole information paradox concerns the intersection of quantum mechanics and general relativity, so its resolution could point the way towards a quantum theory of gravity. The paradox is a contradiction between our understanding of the limits of our physical theories and the consequences of them. With curvature on the scales of that in our solar system, current physical theories have been very successful experimentally; thus, there seems to exist a “solar system limit” in which the laws of physics reduce to those of quantum field theory. However, Hawking’s argument [Haw75], as formalized into a theorem by Mathur [Mat11] (theorem 3 in this report), shows that a natural interpretation of this limit, along with the existence of a “traditional” black hole(such as a Schwarzschild black hole), leads to a violation of the unitarity of quantum mechanics or physicallyobjectionable “remnants”. Two resolutions to this paradox are presented. The “firewalls” resolution formulated by [AMPS13] considers the observations of an external observer and an infalling observer, and concludes that highenergy quanta near the horizon of the black hole would prevent the paradox (and evade the theorem, as the black hole would no longer be “traditional”). A subsequent argument by Hayden and Harlow examines the computational complexity of the task presented to the infalling observer, and suggests that perhaps quantum field theory does not hold between “computationallyinaccessible” observables, allowing a resolution of the paradox without firewalls [HH13] (thus evading the theorem by interpreting the solar system limit in a different way).”

Nonlocal games and communication complexity (Winter 2013; 16 pages)

“Synthesizing some of the literature on nonlocal games and communication complexity scenarios provides a deeper understanding of the fundamental differences between classical mechanics and quantum mechanics, as exemplified by the quantum violation of Bell inequalities, which hold in classical mechanics. Quantum mechanical protocols are shown to provide significant advantages in certain tasks over classical protocols, and could create the first loopholefree demonstration of quantum nonlocality. Nonlocal games are a scenario where two players attempt to perform a task without communication; communication complexity scenarios generalize nonlocal games by allowing but attempting to minimize communication. Completing the circle, communication complexity tasks naturally draw forth new nonlocality scenarios.”

Life on Phorcys (Fall 2013; 20 pages)

“In this article, we investigate the possible forms of life that may exist on planet Phorcys. Our approach is to determine key parameters describing the planet: temperature variation, ice cover, surface pressure and ocean presence and depth. We find that Phorcys is covered in a global ocean, with large ice caps and the possibility of an ice free equatorial region. The presence and size of this region is determined by a combination of the salinity of the ocean water, which affects the freezing point of water, and the day length of the planet, which affects how temperature varies with latitude. earth levels of salinity and day length lead to an ice covered planet, but salinity levels tolerable by many known species lead to an ice free region. Alternatively, an ice free region is predicted if the day on Phorcys is threequarters that of earth. After gaining this understanding of the climate on Phorcys, we investigate life. Our approach is to investigate life through the various layers of the deep oceans of Phorcys, which we find extend to 33 km, three times deeper than earths oceans. At 7.5 km on Phorcys we identify an unusual behaviour in the viscosity of water with implications for life at this depth. Additionally, we identify a depth bound of 22 km for organisms with a cell membrane similar to earth life by estimating the magnitude of pressure fluctuations at depth.”

Combinatorics Notes (Winter 2016; 90 pages)

Extremal combinatorics: Sperner systems, the LittlewoodOfford problem, intersecting hypergraphics, compression, Turan type problems, Ramsey theory, convexity, incidence problems, and algebraic methods. Contribute typo fixes here.

Math 354 Honours Analysis 3 Notes (Fall 2013; 128 pages)

Introduction to metric spaces, topological spaces.