Alice and Bob Meet Banach. The Interface of Asymptotic Geometric Analysis and Quantum Information Theory.
To appear in Mathematical Surveys and Monographs,
Amer. Math. Soc., 428 pp. Comic strip and sample chapters (with G. Aubrun)
Bound on Bell Inequalities by Fraction of Determinism and Reverse Triangle Inequality.
Phys. Rev. A. 92, 032329 (2015).
Arxiv.org eprint 11 p.
(with P. Joshi, K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, Ben Li and T. Szarek)
Bound entangled states with extremal properties.
Phys. Rev. A. 90, 012301 (2014).
Arxiv.org eprint 8 p.
(with P. Badziag, K. Horodecki, M. Horodecki and J. Jenkinson)
Entanglement thresholds for random induced states.
Comm. Pure Appl. Math. 67 (2014), 129-171.
Arxiv.org eprint 34 p.
(with G. Aubrun and D. Ye)
Phase transitions for random states and a semicircle law for the partial transpose.
Phys. Rev. A. 85, 030302(R) (2012).
Arxiv.org eprint 5 p.
(with G. Aubrun and D. Ye)
Concentration for noncommutative polynomials in random matrices.
Proc. Amer. Math. Soc. 140, No. 5 (2012), 1803-1813.
Arxiv.org eprint 11 p.
(with M. Meckes)
On some convexity properties of the Least Squares Method for
pairwise comparisons matrices without the reciprocity condition.
J. Global Optim. 54 (2012), no. 4, 689-706.
Journal Website
(with J. Fülöp and W. W. Koczkodaj)
How often is a random quantum state k-entangled?
J. Phys. A: Math. Theor. 44, 045303 (2011).
Arxiv.org eprint 19 p.
(with E. Werner and K. Zyczkowski)
Hastings's additivity counterexample via Dvoretzky's theorem.
Comm. Math. Physics 305, 85-97 (2011).
Arxiv.org eprint 12 p.
(with G. Aubrun and E. Werner)
Almost-Euclidean subspaces of $\ell_1^N$ via tensor products: a simple approach to randomness reduction.
In "APPROX and RANDOM 2010," M. Serna et al (Eds.) LNCS 6302, Springer-Verlag Berlin Heidelberg 2010, 632-641.
Arxiv.org eprint 11 p.
(with P. Indyk)
Non-additivity of Rényi entropy and Dvoretzky's theorem.
J. Math. Phys. 51, 022102 (2010). Arxiv.org eprint 8 p.
(with G. Aubrun and E. Werner)
On the nontrivial projection problem.
Advances in Math. 221 (2009), 331-342.
Arxiv.org eprint 17 p.
(with N. Tomczak-Jaegermann)
On norms of completely positive maps.
In "Proceedings of IWOTA 2008", Oper. Theory Adv. Appl.
Vol. 202, Birkhauser, Basel 2010, 535-538.
Arxiv.org eprint 2 p.
Geometry of sets of quantum maps: a generic positive map acting
on a high-dimensional system is not completely positive.
J. Math. Phys. 49, 032113 (2008)
Arxiv.org eprint 34 p.
(with E. Werner and K. Zyczkowski)
Convexity, Complexity, and High Dimensions.
In "Proceedings of the International Congress of Mathematicians (Madrid, 2006),"
Vol. II. European Math. Soc. 2006, 1599-1621. Available online from
icm2006.org
Decoupling weakly dependent events.
In "Geometric Aspects of Functional Analysis, Israel Seminar 2004-2005",
V.D. Milman and G. Schechtman eds.,
Lecture Notes in Math. 1910, Springer Verlag, Berlin 2007, 297-303.
Preprint in PDF format, 8 p. (with N. Tomczak-Jaegermann)
On the structure of the body of states with positive partial transpose.
J. Phys. A: Math. Gen. 39 (2006) L119-L126.
Arxiv.org eprint 10 p. (with I. Bengtsson and K. Zyczkowski)
Tensor products of convex sets and the volume of separable states on N qudits.
Phys. Rev. A. 73, 022109 (2006).
Arxiv.org eprint 14 p. (with G. Aubrun)
The volume of separable states is super-doubly-exponentially
small in the number of qubits. Phys. Rev. A 72, 032304 (2005).
Arxiv.org eprint 20 p.
Saturating Constructions for Normed Spaces II. J. Funct. Anal. 221 (2005), no. 2, 407-438.
Preprint in PDF format, 35 p.
(with N. Tomczak-Jaegermann)
On convexified packing and entropy duality.
Geom. Funct. Anal. 14 (2004), no. 5, 1134-1141.
Preprint in PDF format, 6p. (with S. Artstein, V. Milman and N. Tomczak-Jaegermann)
Duality of Metric Entropy. Annals of Math. 159 (2004), no. 3, 1313-1328.
Preprint in PDF format, 17 p. (with S. Artstein and V. Milman)
Saturating Constructions for Normed Spaces. Geom. Funct. Anal. 14 (2004), no. 6, 1352-1375.
Preprint in PDF format, 27 p.
(with N. Tomczak-Jaegermann)
Duality of metric entropy in euclidean space.
C. R. Acad. Sci. Paris. 337 (2003), no. 11, 711-714.
Preprint in English in PDF format, 5 p.
(with S. Artstein and V. Milman)
On Gram matrices of uniformly bounded systems of functions.
Trudy MIAN 243 (2003), 237-243 (English transl. Proc. Steklov Inst.
243 (2003), 227--233.)
Preprint in Russian in PDF format, 10 p.English transl.
(with B.S. Kashin)
The Knaster problem and the geometry of high-dimensional cubes.
C. R. Acad. Sci. Paris Sér. I Math. 336 (2003), no. 11, pp. 931--936.
Preprint in English in PDF format, 6 p. (with B. S. Kashin)
More on the Duality Conjecture for Entropy Numbers.
C. R. Acad. Sci. Paris Sér. I Math. 336 (2003), no. 6, 479--482.
(with S. Artstein and V. Milman)
On the geometry of proportional quotients of ${l^m_1}$}.
Studia Math. 155 (2003), 51--66.
Preprint in PDF format, 19 p.
(with P. Mankiewicz)
An Analysis of Completely-Positive Trace-Preserving
Maps on $M_2$. Linear Algebra Appl. 347 (2002), 159--187.
Arxiv.org eprint
(with M.B. Ruskai and E. Werner)
A Geometric Approach to Duality of Metric Entropy.
C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 2, 157-162.
Preprint in PDF format
(with V. Milman)
Shannon's Entropy Power Inequality via Restricted Minkowski Sums.
In "Geometric Aspects of Functional Analysis", V.D. Milman and G. Schechtman eds.,
Lecture Notes in Math. 1745, Springer Verlag, Berlin 2000, 257-262.
Preprint in PDF format
(with D. Voiculescu)
Local operator theory, random matrices and Banach spaces.
In "Handbook on the Geometry of Banach spaces," Volume 1,
W.B. Johnson, J. Lindenstrauss eds., Elsevier Science 2001, pp. 317-366.
Addenda and Corrigenda, in Volume 2, 2003, pp. 1819-1820.
Preprint in PDF format, 59 p.
Addenda and Corrigenda, 2 p.
(with K.R. Davidson)
A Geometric Lemma and Duality of Entropy Numbers.
In "Geometric Aspects of Functional Analysis, Israel Seminar 1996-2000",
V.D. Milman and G. Schechtman eds.,
Lecture Notes in Math. 1745, Springer Verlag, Berlin 2000, 191-222.
Preliminary version:
Erwin Schroedinger Institute, Vienna, preprint # 687, April 1999, 40 p., available via
http://www.esi.ac.at/preprints/ESI-Preprints.html
(with V. Milman)
The Flatness Theorem for Nonsymmetric Convex Bodies
via the Local Theory of Banach Spaces,
Math. of Operation Research 24 (1999), no. 3, 728-750
preprint
(with W. Banaszczyk, A.E. Litvak and A. Pajor)
Non-symmetric Correlation Inequality for Gaussian Measure,
J. Multivariate Analysis 68 (1999), 193-211
Arxiv.org eprint
(with E. Werner)
Metric entropy of homogeneous spaces,
Quantum Probability (Gdansk, 1997), 395-410,
Banach Center Publ. 43, Polish Acad. Sci., Warsaw, 1998
Arxiv.org eprint
Lattice coverings and Gaussian measures of n-dimensional convex bodies,
J. of Discrete and Comp. Geom. 17 (1997), 283-286
abstract
(with W. Banaszczyk)
On the convexified Sauer-Shelah theorem, J. Combinatorial Th. B, 69 No.2 (1997), 183-192
preprint in AMSTeXabstract (with M. Talagrand)
Restricted Minkowski sums and the noncommutative entropy power inequality,
Comm. Math. Physics 178 (1996), 563-570
Arxiv.org eprint (with D.Voiculescu)
Random Banach Spaces.The limitations of the method,
Mathematica 41 (1994), pp.239-250
Arxiv.org eprint
Corrigenda 42 (1995), pp.220-221 (with P.Mankiewicz)
Condition number of random matrices,
J. Complexity 7 (1991), 131-149
An "isomorphic" version of the Sauer-Shelah lemma and
the Banach-Mazur distance to the cube,
GAFA Israel Funct. Anal. Sem., J. Lindenstrauss, V.D. Milman eds., Lecture Notes in Math. vol.1376
Springer Verlag 1989, 105-112 (with M. Talagrand)
On the duality problem for entropy numbers of operators,
as above, 50-63 (with J. Bourgain, A. Pajor and N. Tomczak-Jaegermann)
Spaces with large distance to $l^\infty$ and random matrices,
Amer.J.Math. 112 (1990), 899-942
The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization,
Israel J.Math 62 (1988), 169-180 (with J. Bourgain)
An exotic quasidiagonal operator,
J. Funct. Anal. 89 (1990),274-290
On almost commuting Hermitian matrices,
Rocky Mts.J.Math. 20 (1989), 581-589
A Banach space without a basis which has the bounded approximation property,
Acta Math. 159 (1987), 81-98
On the existence and uniqueness of complex structure and spaces with "few" operators,
Trans. A.M.S. 29 (1986), 339-353
How well can an nxn matrix be approximated by reducible ones?
Duke Math.J. 53 (1986), 233-248 (with D. Herrero)
Nets of Grassmann manifold and orthogonal groups,
Proceedings of Banach Space Workshop, University of Iowa Press 1982, 169-185
reprint
The finite dimensional basis problem with an appendix on nets
of Grassman manifolds Acta Math. 151 (1983), 153-179
Remarks on fixed point problem for nonexpansive mappings,
(in Fixed points and Nonexpansive Mapings), A.M.S. 1983, 87-120
(with J. Elton, P.K. Lin and E. Odell)
On nearly Euclidean decomposition for some classes of Banach spaces,
Compositio Math. 40 (1980), no. 3, 367--385.
Bases and biorthogonal systems in the spaces C and L_1,
Arkiv f. Math 17 (1979), 255-271
On Kashin's almost Euclidean orthogonal decomposition of l_1,
Bull. Acad. Polon. Sci. 26 (1978), 691-694
On the best constant in the Khinchine inequality,
Studia Math. 58 (1976), 197-208
Some of the above materials are based upon work supported by grants from the National
Science Foundation. Any opinions, findings and conclusions or recomendations expressed
therein are those of the author(s) and do not necessarily reflect
the views of the NSF.
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