An essay submitted in the competition for a Smith-Knight or Rayleigh-Knight prize. This essay combines my work on uniform continuity bounds (arXiv:1707.04249) with that on local continuity bounds (arXiv:1706.02212), and unifies the notation. Abstract: “Majorization is a pre-order of vectors, giving a sense in which one vector can be said to be more disordered than another. Two given vectors, however, may be incomparable. This concept has been extended to quantum mechanical states, to particular success in the theory of entanglement manipulation. Here, we investigate the majorization pre-order over the ε-ball of quantum states defined by the trace distance. We find, suprisingly, that the ε-ball admits a maximal and minimal state in majorization order, which are comparable to every other state in the set. This property depends in particular on the geometry described by the trace distance. We apply these minimal and maximal states to find explicit formulas for “smooothed” single-partite entropies, briefly consider implications for state estimation via the MaxEnt principle, and establish local continuity bounds for a broad class of functions. Next, we investigate further properties of the minimal state in majorization order, and in particular of the map taking the center of the ε-ball to the minimal state in this ball. We find that this map satisfies a so-called semigroup property, which allows us to derive continuity bounds which are uniform over the set of states from our local results. The construction of the minimal state is motivated by a more general result we derive via Fermat’s rule of convex optimization, providing a first step to extending our results to functions of bipartite states, such as the conditional entropy.
My masters thesis; essentially, an expanded version of most of arXiv:1510.00533, in my own words. The abstract: “Landauer’s Principle states that there is a lower bound on the energy required to change the state of a small system from an initial state to a final state by interacting with a thermodynamic reservoir; of particular interest is when the bound is saturated and the minimal energy cost obtained for a given state transformation. We investigate Landauer’s Principle in the context of repeated interaction systems (RIS), a class of physical systems in which a small system of interest interacts with a sequence of thermal probes. In particular, we show that for RIS, Lan- dauer’s bound is not saturated generically in the adiabatic limit, in which time evolution can be thought of as proceeding infinitely slowly, in contrast to the case of the interaction of a system and a single thermodynamic reservoir. However, for a specific RIS which models the small sys- tem and the probes as 2-level systems interacting via a dipole interaction in the rotating wave approximation, Landauer’s bound is saturated adiabatically. In this work, we also formulate and prove a discrete non-unitary adiabatic theorem to use for RIS.
Winning entry for the 2017 CCIMI video contest. Video:
Many thanks to falcxne for allowing use of his songs “Said You Wouldn’t Change” (from the album Timbits III), “I Hate Homework (Interlude)” (from Timbits II) and “NYC (Interlude)” (from Timbits III); you can find these and more at https://falcxne.bandcamp.com/. And thanks to Phillipe Faist for creating the entropy zoo and allowing me to use it in the video. For more details on the continuity bounds mentioned, you can find a preprint here, which is joint work with my supervisor, Dr. Nilanjana Datta.
Lastly, one clarification: in the video, I talk about the data-compression example in the language of classical information theory, for simplicity. However, the quantum case can be described in a very similar way, with the von Neumann entropy replacing the Shannon entropy.
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 states that the energy cost to erase one bit of information by the action of a thermal reservoir at equilibrium temperature $T$ is always at least $kT$, 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 spin-1⁄2 particles), using the tools of quantum statistical mechanics.”
“The promise problem QSEP-STATE 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.”
“QSEP-CIRCUIT, 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 NP-hard and QSZK-hard, which lends credence to the idea that the problem is QMA-hard. The goal of this talk is to provide background for and explain in detail what QSEP-CIRCUIT and QMA are, explain why we care whether or not QSEP-CIRCUIT is QMA-hard, and hopefully showcase quantum information theory as an exciting and dynamic field. No background knowledge will be assumed.”
A poster on my work with my supervisor Nilanjana Datta on uniform continuity bounds for the single-partite entropies, presented at the conference Beyond IID 2017 in Singapore in August 2017, and the main conference of the thematic semester Analysis in Quantum Information Theory in Paris in December 2017.
A poster on my work with Alain Joye, Yan Pautrat, and Renaud Raquepas on Landauer’s Principle in repeated interaction systems, presented at the one-day event CCIMI New Directions in the Mathematics of Information.
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 QSEP-STATE 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.
Extremal combinatorics: Sperner systems, the Littlewood-Offord problem, intersecting hypergraphics, compression, Turan type problems, Ramsey theory, convexity, incidence problems, and algebraic methods. Contribute typo fixes here.
“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 $\sigma$-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 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, 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 physically-objectionable “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 high-energy 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 “computationally-inaccessible” observables, allowing a resolution of the paradox without firewalls HH13.”
Introduction to metric spaces, topological spaces.
“Synthesizing some of the literature on non-local 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 loophole-free demonstration of quantum non-locality. Non-local games are a scenario where two players attempt to perform a task without communication; communication complexity scenarios generalize non-local games by allowing but attempting to minimize communication. Completing the circle, communication complexity tasks naturally draw forth new non-locality scenarios.”
2013 University Physics Competition submisssion. “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 three-quarters 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.”