Talks & Papers

What works and mostly, what doesn’t.

Deep learning as optimization, and then as human-in-the-loop meta-optimization.

Joint work with Mosè Giordano, revisiting PackageAnalyzer.jl and what it finds across the Julia ecosystem.

Joint work with Mosè Giordano, presenting results from PackageAnalyzer.jl, which inspects the packages in Julia’s General registry.

VisualStringDistances.jl, which measures how close two strings look when printed, and an application of it: an automated check in the General registry’s auto-merge process to help prevent malicious lookalike registrations.

New features in Convex.jl, a few worked example problems, and where the package was headed next.

A talk for the Quantum Information Processing group meeting at the Cavendish Laboratory discussing the results of arXiv:1902.08173 placed in the physical context of repeated interaction systems. In particular, I introduce the characterization of faithful eventually-entanglement breaking channels through a series of counterexamples.

A talk at Beyond IID 2019 discussing the results of arXiv:1902.08173, which is work done in collaboration with Cambyse Rouzé and Daniel Stilck França.

A brief presentation introducing partial Markovian evolution (classically), entanglement, and some of the results from arXiv:1902.08173, which was done in collaboration with Cambyse Rouzé and Daniel Stilck França. This talk was given at the March 2019 CCIMI retreat.

Winning entry for the 2017 CCIMI video contest.

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

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.

Slides for a talk I gave to the Physics of Information lab at Waterloo, February 2016, about Landauer’s Principle in Repeated Interaction Systems. Based on the preprint arxiv/1510.00533.

“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."

With Vishal Katariya, Nilanjana Datta and Mark M. Wilde. Published in IEEE Transactions on Information Theory 68(1):322-338 (2022).

Abstract: “What is the minimum number of guesses needed on average to correctly guess a realization of a random variable? The answer to this question led to the introduction of the notion of a quantity called guesswork by Massey in 1994, which can be viewed as an alternate security criterion to entropy. In this paper, we consider the guesswork in the presence of quantum side information, and show that a general sequential guessing strategy is equivalent to performing a single measurement and choosing a guessing strategy from the outcome. We use this result to deduce entropic one-shot and asymptotic bounds on the guesswork in the presence of quantum side information, and to formulate a semi-definite program (SDP) to calculate the quantity. We evaluate the guesswork for a simple example involving the BB84 states, both numerically and analytically, and prove a continuity result that certifies the security of slightly imperfect key states when the guesswork is used as the security criterion.”

With Koenraad Audenaert and Nilanjana Datta.

Abstract: “We consider a stochastic network model for epidemics, based on a random graph proposed by Ross [Journal of Applied Probability, 18, 309-315 (1981)]. Members of a population occupy nodes of the graph, with each member being in contact with those who occupy nodes which are connected to his or her node via edges. We prove that the expected number of people who need to be infected initially in order for the epidemic to spread to the entire population, which is given by the expected number of connected components of the random graph, is Lipschitz continuous in the underlying probability distribution of the random graph. We also obtain explicit bounds on the associated Lipschitz constant. We prove this continuity bound via a technique called majorization flow, which provides a general way to obtain tight continuity bounds for Schur concave functions. To establish bounds on the optimal Lipschitz constant we employ properties of the Mills ratio.”

With Nilanjana Datta.

Abstract: “We introduce a notion of majorization flow, and demonstrate it to be a powerful tool for deriving simple and universal proofs of continuity bounds for entropic functions relevant in information theory. In particular, for the case of the alpha-Rényi entropy, whose connections to thermodynamics are discussed in this article, majorization flow yields a Lipschitz continuity bound for the case alpha > 1, thus resolving an open problem and providing a substantial improvement over previously known bounds.”

With Cambyse Rouzé and Daniel Stilck França. Accepted to Annales Henri Poincaré.

Abstract: “We investigate entanglement breaking times of Markovian evolutions in discrete and continuous time. In continuous time, we characterize which Markovian evolutions are eventually entanglement breaking, that is, evolutions for which there is a finite time after which any entanglement initially present has been destroyed by the noisy evolution. In the discrete time framework, we consider the entanglement breaking index, that is, the number of times a quantum channel has to be composed with itself before it becomes entanglement breaking. The PPT-square conjecture is that every PPT quantum channel has an entanglement breaking index of at most 2; we prove that every faithful PPT quantum channel has a finite entanglement breaking index, and more generally, any faithful PPT CP map whose Hilbert-Schmidt adjoint is also faithful is eventually entanglement breaking. We also provide a method to obtain concrete bounds on this index for any faithful quantum channel. To obtain these estimates, we use a notion of robustness of separability to obtain bounds on the radius of the largest separable ball around faithful product states. We also extend the framework of Poincaré inequalities for nonprimitive semigroups to the discrete setting to quantify the convergence of quantum semigroups in discrete time, which is of independent interest.”

With Hao-Chung Cheng, Nilanjana Datta and Min-Hsiu Hsieh.

Abstract: “In this paper, we establish an interesting duality between two different quantum information-processing tasks, namely, classical source coding with quantum side information, and channel coding over c-q channels. The duality relates the optimal error exponents of these two tasks, generalizing the classical results of Ahlswede and Dueck. We establish duality both at the operational level and at the level of the entropic quantities characterizing these exponents. For the latter, the duality is given by an exact relation, whereas for the former, duality manifests itself in the following sense: an optimal coding strategy for one task can be used to construct an optimal coding strategy for the other task. Along the way, we derive a bound on the error exponent for c-q channel coding with constant composition codes which might be of independent interest.”

With Hao-Chung Cheng, Nilanjana Datta and Min-Hsiu Hsieh.

Abstract: “In this paper, we analyze classical data compression with quantum side information (also known as the classical-quantum Slepian-Wolf protocol) in the so-called large and moderate deviation regimes. In the non-asymptotic setting, the protocol involves compressing classical sequences of finite length n and decoding them with the assistance of quantum side information. In the large deviation regime, the compression rate is fixed, and we obtain bounds on the error exponent function, which characterizes the minimal probability of error as a function of the rate. Devetak and Winter showed that the asymptotic data compression limit for this protocol is given by a conditional entropy. For any protocol with a rate below this quantity, the probability of error converges to one asymptotically and its speed of convergence is given by the strong converse exponent function. We obtain finite blocklength bounds on this function, and determine exactly its asymptotic value. In the moderate deviation regime for the compression rate, the latter is no longer considered to be fixed. It is allowed to depend on the blocklength n, but assumed to decay slowly to the asymptotic data compression limit. Starting from a rate above this limit, we determine the speed of convergence of the error probability to zero and show that it is given in terms of the conditional information variance. Our results complement earlier results obtained by Tomamichel and Hayashi, in which they analyzed the so-called small deviation regime of this protocol.”

With Nilanjana Datta.

Abstract: “We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the \(\alpha\)-Rényi entropy, \(S_\alpha\), for \(\alpha\in (0,1)\). We establish necessary and sufficient conditions for equality in the continuity bound and prove that these conditions are the same for every member of the family. Our result builds on recent work in which we constructed a state which was majorized by every state in a neighbourhood (\(\varepsilon\)-ball) of a given state, and thus was the minimal state in majorization order in the \(\varepsilon\)-ball. This minimal state satisfies a particular semigroup property, which we exploit to prove our bound.”

We consider the geometry of the trace-ball of quantum states, find maximal and minimal states in a particular partial order called majorization, and use these states to construct local continuity bounds for quantum entropies. We also apply the theory of convex optimization to motivate the construction of the maximal state, and to find general optimality conditions for a particular class of functions subject to a trace-ball constraint. Abstract: “Given an arbitrary quantum state (\(\sigma\)), we obtain an explicit construction of a state \(\rho^*_\varepsilon(\sigma)\) (resp. \(\rho_{*,\varepsilon}(\sigma)\)) which has the maximum (resp. minimum) entropy among all states which lie in a specified neighbourhood (\(\varepsilon\)-ball) of \(\sigma\). Computing the entropy of these states leads to a local strengthening of the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannes inequality. Our bound is local in the sense that it depends on the spectrum of \(\sigma\). The states \(\rho^*_\varepsilon(\sigma)\) and \(\rho_{*,\varepsilon}(\sigma)\) depend only on the geometry of the \(\varepsilon\)-ball and are in fact optimizers for a larger class of entropies. These include the Rényi entropy and the min- and max- entropies. This allows us to obtain local continuity bounds for these quantities as well. In obtaining this bound, we first derive a more general result which may be of independent interest, namely a necessary and sufficient condition under which a state maximizes a concave and Gâteaux-differentiable function in an \(\varepsilon\)-ball around a given state \(\sigma\). Examples of such a function include the von Neumann entropy, and the conditional entropy of bipartite states. Our proofs employ tools from the theory of convex optimization under non-differentiable constraints, in particular Fermat’s Rule, and majorization theory."

A refinement and generalization of our previous work on Landauer’s Principle in repeated interaction systems. We consider a two-time measurement protocol of the energy of the quantum probes, and recover a large deviations principle and a central limit theorem in the adiabatic limit of a repeated interaction system. Abstract: “We analyze Landauer’s principle for repeated interaction systems consisting of a reference quantum system \(\mathcal{S}\) in contact with a environment \(\mathcal{E}\) consisting of a chain of independent quantum probes. The system \(\mathcal{S}\) interacts with each probe sequentially, for a given duration, and the Landauer principle relates the energy variation of \(\mathcal{E}\) and the decrease of entropy of \(\mathcal{S}\) by the entropy production of the dynamical process. We consider refinements of the Landauer bound at the level of the full statistics (FS) associated to a two-time measurement protocol of, essentially, the energy of \(\mathcal{E}\) . The emphasis is put on the adiabatic regime where the environment, consisting of \(T\gg 1\) probes, displays variations of order \(T^{-1}\) between the successive probes, and the measurements take place initially and after \(T\) interactions. We prove a large deviations principle and a central limit theorem as \(T\to\infty\) for the classical random variable describing the entropy production of the process, with respect to the FS measure. In a special case, related to a detailed balance condition, we obtain an explicit limiting distribution of this random variable without rescaling. At the technical level, we generalize the discrete non-unitary adiabatic theorem by the present authors in [Commun. Math. Phys. (2017) 349: 285] and analyze the spectrum of complex deformations of families of irreducible completely positive trace-preserving maps."

“We study Landauer’s Principle for Repeated Interaction Systems (RIS) consisting of a reference quantum system \(\mathcal{S}\) in contact with a structured environment \(\mathcal{E}\) made of a chain of independent quantum probes; \(\mathcal{S}\) interacts with each probe, for a fixed duration, in sequence. We first adapt Landauer’s lower bound, which relates the energy variation of the environment \(\mathcal{E}\) to a decrease of entropy of the system \(\mathcal{S}\) during the evolution, to the peculiar discrete time dynamics of RIS. Then we consider RIS with a structured environment \(\mathcal{E}\) displaying small variations of order \(T^{-1}\) between the successive probes encountered by \(\mathcal{S}\), after \(n\simeq T\) interactions, in keeping with adiabatic scaling. We establish a discrete time non-unitary adiabatic theorem to approximate the reduced dynamics of \(\mathcal{S}\) in this regime, in order to tackle the adiabatic limit of Landauer’s bound. Our analysis shows that, in general, Landauer’s bound for RIS is not saturated in the adiabatic regime, due to the entropy production inherent to the RIS dynamics. This is to be contrasted with the saturation of Landauer’s bound known to hold for continuous time evolution of open quantum system in the adiabatic regime."

Eric P. Hanson, Hannah Robertson, Kendal Sandridge, Franz Fürbass, Alexander M. Chan, Dave F. Kleinschmidt, Alex Arslan, Phillip M. Alday, Jay Pathmanathan, M. Brandon Westover, Jacob Donoghue — Beacon Biosignals.

From the abstract: “Here, we found no evidence of experimental model bias favoring Caucasian subjects – but had insufficient data to assess for bias against any specific racial group.”

Hannah Robertson, Eric P. Hanson, Nader Bagherzadeh, Kendal Sandridge, Michelle Fogerson, Alex Arslan, Alexander M. Chan, Jay Pathmanathan, M. Brandon Westover, Jacob Donoghue, Franz Fürbass — Beacon Biosignals.

From the abstract: “IED detection is feasible using deep learning networks, can be robust to electrode loss, and a single algorithm may be capable of handling a wide variety of electrode configurations.”

Jay Pathmanathan, Kim Laberinto, Hannah Robertson, Eric Hanson, Jacob Donoghue — Beacon Biosignals.

From the abstract: “Machine learning augmented detection of IEDs allows for rapid, quantitative identification of individuals with subclinical epileptiform activity in patients with AD and MCI.”

H. Robertson, Z. Beckman, E. Hanson, D. Kleinschmidt, A. Arslan, T. Volkamer, J. Revels, J. Donoghue — Beacon Biosignals.

A poster on my work with my supervisor Nilanjana Datta on uniform continuity bounds for the single-partite entropies, presented at the conference Fifth London Symposium on Information Theory in May 2019. This is an updated version of the poster which includes new results giving necessary and sufficient conditions for the Rényi entropy to be Lipschitz continuous.

A poster on my work with Hao-Chung Cheng, my supervisor Nilanjana Datta, and Min-Hsiu Hsieh on operational duality of some quantum information-theoretic protocols, presented at the conference Beyond IID 2018 in Cambridge, UK in July 2018, and at Quantum Information Processing in Boulder, Colorado in January 2019.

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

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.

My PhD thesis, supervised by Nilanjana Datta and Yan Pautrat.

Abstract: “In the first part of this thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions. Our technique uses a particular relationship between majorization and the trace distance between quantum states. Namely, the majorization pre-order attains a minimum over \(\epsilon\)-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance \(\epsilon\), we obtain the path of “majorization flow”. This yields a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy in a universal framework which extends to the other functions, including the \(\alpha\)-Rényi entropy, for which we obtain novel bounds in the case \(\alpha > 1\). We apply this technique to other Schur concave functions, such as the number of connected components of a certain random graph model, and the number of distinct realizations of a random variable. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes one at a time. We characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference after finitely many steps. Additionally, when the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer’s bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction.”

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.

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] (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 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] (thus evading the theorem by interpreting the solar system limit in a different way)."

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."

In Lent 2018 and 2019, I gave the example classes and revision class for Cambridge’s Part III course “Quantum Information Theory”. The current version of this course is here: http://www.qi.damtp.cam.ac.uk/part-iii-quantum-information-theory.

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."