My research has been concerned with mathematical problems which arise in the study of quantum theory. My PhD thesis in quantum chemistry was on something called the N-representability Problem, a mathematical problem of interest to quantum chemists who hoped it could be used to simplify complicated many-electron computations. Like many others, I concluded that there was "no free lunch" and moved on to a postdoc in Geneva on the foundation of quantum theory.
This was followed by a postdoc with E. Lieb at MIT, which led to the proof of the Strong Subadditivity of Quantum Entropy (SSA). At that time, this problem was of interest to a small cadre of mathematical physicists, but I was asked by the session chair at one physics conference, "It's a beautiful theorem, but what use is it?"
So I moved on to other things, primarily Schrödinger operators, particularly the binding of electrons in atoms and molecules. In 1982 I gave the first proof that an atom with a fixed nuclear charge could bind only finitely many electrons. This was dubbed the Ruskai-Sigal Theorem by Barry Simon.
In the mid 1990's, after Shor's proof that a quantum computer could factor large numbers, which could threaten the security of current encryption schemes, the field of quantum information theory developed rapidly. The old SSA theorem on quantum entropy played an important role and I began to work in this area. It was even stated that SSA is "the key result on which virtually every nontrivial coding theorem relies".
In 2005, Klyachko announced the solution of the pure state N-representability for the one-body reduced density matrix, and in 2006 it was shown that the aspect of the N-representability problem that I had worked on was QMA-complete, which means that its solution would require exponential time, even on a quantum computer. Although this was a mathematical vindication of the "no free lunch" principle, it generated new interest in the topic along with closely related questions in quantum information theory (in which reduced density matrices were known as quantum marginals). I worked with a young group of people at MIT who showed that quantum error correcting codes could be used to solve a question about N-representability that had been open for almost 40 years.
In 2012, on the 40th anniversary of the proof of SSA, Isaac Kim (then a graduate student at Caltech) proved a stronger version of SSA, which I did not believe was possible. I discuss this work in a recent article, as well as in my talk for Dyson's 90th birthday.
As I near the end of my research career, I feel that in many ways I have come full circle. I enjoy meeting and working with young people who have new insights which enable them to move forward and prove stronger theorems.
Preprints of most of my publications in the past 10-15 years are available on arxiv.org. Search on Ruskai in the quant-ph section.
Download a pdf with a complete list of my research publications.
- Evolution of a Fundamental Theorem on Quantum Entropy
- Reduced Density Matrices and Quantum Marginals
- Concavity Bounds for Quantum Entropy
- Contraction Coefficients for Noisy Quantum Channels
- Entropy of Stabilizer States
- The Gour-Friedland Proof of Local Additivity
- A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, with Conditions for Equality
- Properties of Quantum Entropy and Related Convex Trace Functions
Short Course at the International Center for Theoretical Physics, Trieste, Italy
NSF-CBMS Conference on Wavelets (University of Massachusetts, Lowell, June 1990)
Download pdf of conference program
This was the first US conference on wavelets on which Ingrid Daubechies gave the talks which led to the now legendary Ten Lectures on Wavelets. Papers based on most of the other talks appeared in Wavelets and Their Applications, edited by Ruskai et al. Download pdf of the introduction
The US National Science Foundation provides funding for 30-40 participants at each Conference Board of the Mathematical Sciences conference. This meeting attracted the exceptional number of nearly 200 registered participants, most of whom came at their own expense, including many major figures in signal processing. Daubechies' work literally changed the face of imaged processing. Among the younger participants was Chris Brislawn, who subsequently developed the FBI's fingerprint compression algorithm using wavelet techniques.
It led to a featured story in the Science Section of the Boston Globe (by Richard Saltus), as well as coverage in the Lowell Sun (pdf), A New Wave in Applied Mathematics [Barry A. Cipra, "Science Vol 249 pp. 858-859 (24 August 1990)], and SIAM News (pdf).
- Ruskai's reviews of other books on Wavelets:
- Wavelets: Applications and Algorithms by Y. Meyer, for Math. Intelligencer 17, 70-73 (1995).
The World According to Wavelets by B.B. Hubbard for American Scientist 85(1), 85-86 (1997). Note: This review is of the first edition. Most criticisms were addressed in the second edition. Download the pdf
Co-organizer of Special Sessions
- AMERICAN MATHEMATICAL SOCIETY Meetings
- Many-Body Quantum Theory, Lexington, Kentucky (March, 1994)
Mathematical Physics of Many-Body Systems, Melbourne, Australia (July, 1999)
Quantum Information Theory, Lowell, Massachusetts (April, 2000)
Quantum Error Correction, Hoboken, New Jersey (April, 2001)
Quantum Information Theory, Boston, Massachusetts (October, 2002)
Quantum Information Theory, Warsaw, Poland (August, 2007)
(co-organized with Robert Alicki)
Mathematical Sciences Research Institute (by invitation)
- Quantum Information and Cryptography, Berkeley, California (November, 2002)
- 28th International Conference on Quantum Probability (by invitation)
- Quantum Information Theory, Guanajuato, Mexico (September, 2007)
- International Congress of Mathematical Physics (by invitation)
- Quantum Information Theory, Rio de Janeiro, Brazil (July, 2006)
- Quantum Information Theory, Santiago, Chile (July, 2014)
- American Association for Advancement of Science: Symposia at annual meetings
- Women in Physics: Why so Few? (Joint with APS/AAPT) (January, 1989)
- Mathematics in the Science Policy Arena (February, 1991)
- Mathematical Advances in Signal Processing (February, 1995)
- Quantum Information Theory, Boston Massachusetts (February, 2008)
- Participant in workshops on Quantum Information Theory at Centro de Ciencias de Benasque Pedro Pascual
- Photos from 2003
Photos from 2009
Photos from 2011
Photos from 2013