Dr. Achim Kempf Professor and Canada Research Chair  in the Physics of Information Department of Applied Mathematics University of Waterloo Ontario, Canada.

 

Hello! I am a mathematical physicist, working on basic questions related to information theory, quantum theory and general relativity. I am happy to say that we have here in Waterloo quite a number of very good people who work in this general field, at various departments and institutes.

I am based in the Department of Applied Mathematics, and I am also cross-appointed to the Department of Physics at Waterloo, a member of the Guelph-Waterloo Physics Institute, an associate member of the Institute for Quantum Computing and an affiliated member of the independent Perimeter Institute for Theoretical Physics.  No wonder that my workload is high.

 

Current Projects:

 

I. Quantum gravity

• Big question:  How can general relativity and quantum theory be unified in one theory?
   

• Basic idea: I showed that spacetime could be simultaneously continuous and discrete, in the same way that information can. This yields a new connection in between the fields of information theory and quantum gravity, and the emerging methods have the potential to be useful in both fields. For the latest, see here.
   

• Big question: How is the flow of information impacted by special and general relativity, and by quantum effects?
   

• Basic idea: These phenomena can be studied by modeling the interaction of small quantum systems, such as atoms, with another, mediated through their interactions with quantum fields. Recent work with my student Mathieu Cliche clarifies the issues of causality, classical and quantum capacity and entanglement in the quantum channel that arises in this way. The plan is to extend this work to cosmology.
   

• Most recent publications:

  • A. Kempf, Information-theoretic natural ultraviolet cutoff for spacetime, arXiv:0908.3061, in press in Phys. Rev. Lett..

  • M. Cliche, A. Kempf, The relativistic quantum channel of communication through field quanta, arXiv:0908.3144

  • A. Kempf, R. T. Martin, Information Theory, Spectral geometry and Quantum Gravity, Phys. Rev. Lett.,  100, 021304 (2008)    

  • R. T. Martin, A. Kempf, Quantum Uncertainty and the Spectra of Symmetric Operators, to appear in Acta Appl. Math.

II. Cosmology

• Big question:  Air- and space-borne telescopes have started to provide high precision data on the earliest stages of the universe. What information can be extracted about the laws of quantum gravity that governed the big bang?
   

• Basic idea: I implemented key features (the generalized uncertainty relations) of my information-theoretic approach to quantum gravity in the present standard theory for the early universe, cosmic inflation. The work of several groups indicates that it will be very difficult but not impossible to use data from space-based telescopes (such as the recently-launched Planck satellite) to test some theory-specific predictions about the big bang, e.g., concerning the tensor mode spectrum in the CMB. Work with my former student Robert Martin is in progress on a fully relativistic generalization. 
   

• Most recent publications:

  • S. Bachmann, A. Kempf, On the Casimir Effect with General Dispersion Relations, J. Phys. A41, 164021 (2008)

  • A. Kempf, L. Lorenz, Exact solution of inflationary model with minimum length, Phys. Rev. D74, 103517 (2006)

III. Quantum computing

• Big question:  Quantum computers have the potential to speed up certain computations far beyond what can be achieved with computing technology based solely on classical physics. This has been proven mathematically but the physical reasons for the speed-up are not yet clear.
   

• Basic idea: One type of calculation that can be sped up by quantum computers is data search. For example, a classical random search walk proceeds with a speed that relates the distance to the square root of the number of steps taken. Quantum computers can do a random walk that proceeds with a speed that is proportional to the number of steps and therefore much faster. In work with Renato Portugal I showed that quantum walks can often be shown to be equivalent to the propagation of a wave in a medium, which explains why the propagation and therefore the search is with a constant speed.
   

• Big question: How to build quantum computers, given that their reliance on quantum effects makes them vulnerable to even the tiniest disturbances? How to make them resilient against errors?
   

• Basic idea: I have worked on the generalization of so-called quantum error correction algorithms that improve the resilience of quantum computers to noise. This work also yields new insight into the transfer of information from one system to another during a measurement process. This is work with David Kribs and my student Cedric Beny. Cedric pioneered this study.
   

• Most recent publications: 

  • A. Kempf, R. Portugal, Group velocity of discrete-time quantum walks, Physical Review A79, 052317 (2009)

  • C. Beny, A. Kempf, D. Kribs, Qantum Error Correction on Infinite-Dimensional Hilbert Spaces, quant-ph/0811.0421, J. Math. Phys. 50, 062108 (2009)

  •  C. Bény, A. Kempf, and D.W. Kribs,  Generalization of Quantum Error Correction via the Heisenberg Picture, Phys. Rev. Lett. 98, 100502 (2007)

  • C. Bény, A. Kempf, D.W. Kribs, Quantum error correction of observables, Phys. Rev. A76, 042303 (2007)

IV. Communication engineering

• Big question:  It is commonly believed that a signal cannot vary on time scales that are smaller than its bandwidth would indicate. As was shown by Sir Michael Berry, Yakir Aharonov and others, this is not so: there are signals that locally oscillate faster than expected. Why is that so, what is the significance and how can we make super-oscillatory signals or wave functions?
   

• Basic idea: I showed that superoscillations can exist (and not violate Shannon's channel capacity theorem) because they trade signal-to-noise ratio for bandwidth. With Paulo Ferreira I also calculated the most efficient method for designing such signals. We can now easily make acoustic superoscillations. I am currently mostly interested in the behaviours of superoscillations in quantum wave functions.
   

• Most recent publications:

  • P.J.S.G. Ferreira, A. Kempf, M.J.C.S. Reis, Construction of Aharonov–Berry's superoscillations, J. Phys. A40, 5141 (2007)

  • P.J.S.G. Ferreira and A. Kempf, Superoscillations: Faster than the Nyquist Rate, IEEE Trans. Signal Processing, 54, 3732 (2006)
     

V. Shannon sampling theory / Data compression

• Big question: How to compress continuous data, such as music recordings, most efficiently?
   

• Basic idea: The crucial link between continuous representations of information, such as music, and discrete representations of information, such as sequences of numbers, is provided by a mathematical discipline called sampling theory. The basic Shannon sampling theorem, for example, shows that if a music signal is bandlimited to 20KHz, then it suffices to record its amplitudes 40000 times per second (those numbers are stored in a computer file) and from those numbers, it then, in principle, possible to reconstruct the music perfectly (within the precision to which the amplitudes were measured. I develop and use sampling theory for two purposes: A) I generalized Shannon sampling theory to time-varying bandlimits by using functional analytic tools that generalize Fourier theory. This improves sampling efficiency because it allows one to sample and reconstruct signals at time-varying rates that are adjusted to the time-varying properties of individual signals.  I hold US patent #6531971 on a resulting new data compression method. A recent result with my student Yufang Hao adds a very valuable new tool to that method, namely a completely adapted frequency filtering method. My student James "Chuck" Bronson works on the implementation. B) I am generalizing sampling theory to the sampling and reconstruction of fields on spacetime and also spacetime itself. This provides mathematical tools for my project I) above.  
   

• Most recent publications:

• Y. Hao, A. Kempf, Generalized Shannon Sampling Method reduces the Gibbs Overshoot in the Approximation of a Step Function, accepted for publication in J. Conc. & Appl. Math.

• Y. Hao, A. Kempf, Filtering, sampling and reconstruction with time-varying bandwidth, submitted July 2009.

• R. Martin, A. Kempf, Approximation of Bandlimited Functions on a Non-Compact Manifold by Bandlimited Functions on Compact Submanifolds, Sampl. Theor. in Sign. and Image Processing, 7, 281 (2008)

• Y. Hao, A. Kempf, On a Non-Fourier Generalization of Shannon Sampling Theory, IEEE Information Theory CWIT 2007, 193 (2007)

VI. Mathematical biology

• Big question: What was the environment like in which life first arose? What was the temperature, the acidity and the mineral content of the primordial soup in which life first evolved? New information about the primordial environment could help us narrow down the possible mechanisms by which life first arose on earth.
   

• Basic idea: The genetic code is the code that all organisms, from carrots to humans, use to interpret the 64 codons as commands for the production of the 20 amino acids (and the stop).  There is evidence that the genetic code is a passively error correcting code in the information theoretic sense.  So I asked, what can we learn from the structure of the code about the environment that shaped it? Basically, I suggested to interpret the genetic ode as the oldest available fossil (because it has not changed since our last common ancestor) and to analyze it as such. Its structure should be able to tell us about the evolutionary pressures that were at work when the code formed. With my student Sasha Gutfraind, I published work showing that the code structure indicates that it evolved in organisms that were probably not in a hot environment. (We inferred this when we found the code to be ill adapted to protect organisms from the pattern of mutations that is typical for organisms that are in hot environments - those organisms have an elevated frequency of G's and C's because these base pairs are thermodynamically more stable due to their triple bond). In very recent work with my collaborator J. Jestin we used this approach to study the concentrations of Mg and Mn cations in the primordial environment.
   

• Most recent publications:

  • J.-L. Jestin, A. Kempf, Optimization models and the structure of the genetic code, to appear in a special issue of The Journal of Molecular Evolution devoted to the Origin of Life

  • A. Gutfraind, A. Kempf, , Error-reducing Structure of the Genetic Code Indicates Code Origin in Non-thermophile Organisms, Orig. Life Evol. Biosph., 38, 75 (2008)

  • J.-L. Jestin, A. Kempf, Degeneracy in the genetic code: how and why?, Genes, Genomes and Genomics 1, 100-103 (2007)

VII. Radar signal design for maximum information return

• This is a project for which a patent is currently pending. My student Raymond Su is working on optimized implementations. Stay tuned.

VIII. The Casimir effect in layered superconductors

• Big question:  Much is known about high temperature superconductors but little is known about their energetics. Why is it energetically favorable for electrons to team up to form Cooper pairs? This question is important because the answer could tell us how the transition temperature of superconductors can be raised further and if room temperature superconductivity will ever be possible.
   

• Basic idea: At the 10th Marcel Grossmann meeting in Rio de Janeiro in 2003 I introduced the idea that the condensation energy of high temperature superconductors arises from their layering, namely through the Casimir effect, (see Sec.5 of the proceedings paper). While the predictions from my ansatz are in the right order of magnitude, more work needs to be done to make precision comparisons between theory and experiment. A solid prediction of my ansatz is that to increase Tc, one needs to make materials which in the superconducting state consist of very closely spaced superconducting layers separated by low-conductivity layers, and which in the normal state are of very low conductivity in all layers. My students Simon Foreman and Peter Forbes are working in this field. 
   

• Most recent publications:

  • A. Kempf, On the Casimir Effect in the High Tc Cuprates, J. Phys. A41, 164038 (2008) 
     

IX. Combinatorics in quantum field theory

• Big question:  The quantum field theoretic path integral of interacting quantum fields is analytically ill-defined and yet it is a very successful tool for predicting experimental data. What gives?
   

• Basic idea: Much of the perturbative structure of QFT may ultimately be combinatorial in nature and for that reason insensitive to analytic issues. Indeed, with my collaborators D.M. Jackson and A. Morales, I showed that for example a key property of the QFT path integral, whose conventional derivation involves an analytically ill-defined Legendre transform, can be combinatorially proven (namely the fact that the Legendre transform of the effective action yields the sum of connected graphs and vice versa).  
   

• Most recent publications:

  • D. M. Jackson, A. Kempf, A. Morales, On the Structure of QFT in the Particle Picture of the Path Integral Formulation, hep-th/0810.4293
     

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