Linear analysis and its applications  Page description

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Details of project

  
Identifier
46599
Type K
Principal investigator Petz, Dénes
Title in Hungarian Lineáris analízis és alkalmazásai
Title in English Linear analysis and its applications
Panel Mathematics and Computing Science
Department or equivalent Department of Mathematical Analysis (Budapest University of Technology and Economics)
Participants Andai, Attila
Mosonyi, Milán
Réffy, Júlia
Starting date 2004-01-01
Closing date 2007-12-31
Funding (in million HUF) 4.246
FTE (full time equivalent) 0.00
state closed project





 

Final report

  
Results in Hungarian
A kutatások három témakörre csoportosíthatók: 1. Véletlen mátrixok A nagy eltérés tételek tipikusan egy mátrix sajátértéksűrűségére vonatkoztak. Két véletlen és független projekció esete teljesen újfajta eredményhez vezetett. Vojculescu szabad valószínűség elméleti szállítási költség egyenlőtlenségét több irányba is tovább fejlesztettük, a körvonalon és a számegyenesen adott mértékekre. A bizonyításban véletlen mátrixok sajátértéksűrűségével való megközelítést és nagy eltérés tételeket alkalmaztunk. 2. Mátrixok nyomegyenlőtlenségeinek alkalmazásai Két mátrixra a közepeknek jól kidolgozott elmélete van, de három mátrixra ilyen nem volt Több mátrixra is értelmeztük a közepeket egy algoritmus segítségével. Mátrix egyenlőtlenségek kerültek alkalmazásra a Neumann-entrópia erős szubadditivitásának bizonyításában. 3. A kvantummechanikai rendszerek állapottere Az állapotbecslés az ismeretlen állapot paramétereit mérésekkel kívánja megbecsülni. Bizonyos becslési eljárások statisztikus összehasonlítása és a négyzetes hibamátrix alapján való optimálizálás a kutatott téma. Például, a hibmátrix determinánsa minimális, ha a mérések komplementárisak. Bevezettük a komplementáris részrendszerek fogalmát és számos több problémát meoldottunk a két kvantumbitből álló rendszer esetére. Statisztikailag elégséges részalgebrák több új jellemzése került bizonyításra, közöttük egy faktorizációs tulajdonság.
Results in English
The research can be grouped in three fields: 1. Random matrices A new kind of large deviation result was obtained about the eigenvalue densities of two independent random projections. Voiculescu's free transportation cost inequality was extended to different directions, measures on the circle and on the real line. In the proof approximation by eigenvalue densities of random matrices and large deviation results were used. 2. Matrix inequalities The matrix means for two matrices have a well-known theory. We defined the mean for several matrices by an algorithm. Matrix inequalities were used in a new proof of the strong subadditivity of the von Neumann entropy. 3. State space of quantum systems The state determination means giving an estimate of the unknown system on the basis of measurement. Statistical arguments are used here. Estimation schemes can be compared and optimized. It is proven that the determinant of the quadratic error matrix is minimal, if the complementary measurements are used. The concept of complementarity has been extended to subalgebras and a few relevant questions are answered for a system of two quantum bits.
Full text http://real.mtak.hu/1500/
Decision
Yes





 

List of publications

  
M. Matolcsi, J. Réffy and F. Szöllösi: Constructions of complex Hadamard matrices via tiling Abelian groups, benyújtva az Opens Systems folyóirathoz, 2007
D. Petz: A course on Algebraic methods for Quantum Mechanics, School on Theory and Technology in Quantum Information, Communication, Computation and Cryptography,Trieste, June, 2006
D. Petz: Large Deviations, The Centre for Mathematical Physics, The University of Queensland, 23 March, 2006
D. Petz: From the Monotonicity of Quantum Relative Entropy to SSA and Sufficient Coarse-grainings, Entanglement, Information & Noise, Krzyzowa, Poland, 2004
D. Petz: Free analogue of transportation cost inequalities, ''Free probability theory'', Banff, 2004
P. Hayden, R. Jozsa, D. Petz and A. Winter: Structure of states which satisfy strong subadditivity, Commun. Math. Phys.246(2004), 359-374, 2004
D. Petz and J. Réffy: On asymptotics of large Haar distributed unitary matrices, Period. Math. Hungar. 49(2004), 103-117, 2004
M. Mosonyi and D. Petz: Structure of sufficient quantum coarse-grainings, Lett. Math. Phys. 68(2004), 19-30, 2004
F. Hiai, D. Petz and Y. Ueda: Free transportation cost inequalities via random matrix approximation, Prob. Theory Rel. Fields. 130(2004), 199-221, 2004
M. Ohya and D. Petz: Quantum Entropy and Its Use (Second Edition), Springer-Verlag, Heidelberg, 2004
J. Réffy: Asymptotics of large truncated Haar unitary matrices, in Quantum, From Foundations to Applications vol. XVIII, eds. M. Schürmann and U. Franz, World Scientific, 2005, 448-456, 2005
A. Andai: Information geometry in quantum mechanics, BME Budapest, 2004
A. Andai: On the curvature of the space of qubits, to appear in Bedlewo proceedings, 2004
Petz Dénes: From the Monotonicity of Quantum Relative Entropy to SSA and Sufficiency, Quantum Information Geometry, McMaster University, Hamilton, Canada, 2004
D. Petz: Free transportation cost inequalities via random matrix approximation, Random Matrices and Probability, Warwick, UK, 2004
J. Réffy: Asymptotics of random unitary matrices, Probability Theory on Groups and Related Structures, Budapest, 2004
D. Petz: Truncated random unitary matrices, 3rd Sendai Workshop ''Free Probability and Random Matrices'', Japan, 2004
D. Petz: Perturbed Wigner matrices and a conjecture, 3rd Sendai Workshop ''Free Probability and Random Matrices'', Japan, 2004
J. Réffy: Asymptotics of Haar unitaries and their truncation, Workshop on Random Matrices and Probability, Warwick, 2004
D. Petz: Sufficient statistical inference in quantum mechanics, Quantum Statistics - Quantum Measurements, Estimation and Related Topics, Cambridge, 2004
A. Andai: On the curvature of the space of density matrices, 25th QP Conference on Quantum Probability and Related Topics, Bedlevo, 2004
J. Réffy: Asymptotics of random unitary matrices, Probability Theory on Groups and Related Structures, Budapest, 2004
J. Réffy: Asymptotics of Haar unitaries and their truncation, Workshop on Random Matrices and Probability, Warwick, 2004
D. Petz and J. Réffy,: Large deviation theorem for empirical eigenvalue density of truncated Haar unitary matrices, Prob. Theory Rel. Fields. 133(2005), 175--189., 2005
M. A. Nielsen and D. Petz: A simple proof of the strong subadditivity inequality, Quantum Information & Computation, 5(2005), 507--513., 2005
N. Akiho, F. Hiai and D. Petz,: Equilibrium states and entropy densities in gauge-invariant C*-systems, Rev. Math. Phys. 17(2005), 365--389., 2005
J. Réffy: Asymptotics of random unitaries, doktori disszertáció, BME, 2005
J. Réffy: Orthogonal polynomials and random matrices, Potenciálelméleti Nyáriiskola, Debrecen, 2005
J. Réffy: Közepek és egyenlőtlenségek, Polygon XIV (2005) no. 1, 61--70, 2005
A. Andai: On scalar curvatures of state space, Lecture at the University of Wroclaw, 2005
Á. Császár and D. Petz,: A panorama of Hungarian real and functional analysis in the 20th century, A panorama of Hungarian Mathematics in the Twentieth Century I, ed. J. Horváth, pp. 211--244, Springer, 2005., 2005
Petz Dénes: Lax Péter Abel-dijas, Népszabadság, 2005. március 19., 2005
D. Petz: An application of random matrix methods: The free analogue of the transportation cost inequality, Lecture in Mexiko, Iztapalapa, January, 2005
D. Petz: Large deviations for functions of two random projections, Lecture at the Workshop on non-commutative harmonic analysis, Bedlevo, August, 2005
D. Petz: Large deviations, Lecture at the Mathematical Physics Days, Leuven, September, 2005
D. Petz: Information geometry and quantum statistical inference, Lecture at the IGAIA2 conference, Tokyo,December, 2005
D. Petz: Entropy, information geometry and quantum statistical inference, Lecture in the Schrödinger Institute, Vienna, December, 2005
A. Jencova and D. Petz: Sufficiency in quantum statistical inference:, Commun. Math. Phys. 203(2006), 259--276, 2006
D. Petz and R. Temesi: Means of positive numbers and matrices, SIAM Journal on Matrix Analysis and Applications, 27(2006), 712-720., 2006
A. Jencova and D. Petz: Sufficiency in quantum statistical inference: A survey with examples, J. Infinite Dimensional Analysis and Quantum Probability, 9(2006), 331--352., 2006
D. Petz: Saturation of entropy bounds and quantum Markov states, Quantum Information and Computing, eds. L. Accardi, M. Ohya and N. Watanabe, World Scientific, 2006., 2006
D. Perez-Garcia, M.M. Wolf, D. Petz and M.B. Ruskai: Contractivity of positive and trace preserving maps under $L^p$ norms, J. Math. Phys. 47(2006), 083506., 2006
F. Hiai and D. Petz: Large deviations for functions of two random projections, Acta Sci. Math. (Szeged) 72(2006), 581--609, 2006
D. Petz: Bregman divergence as relative operator entropy, Acta Math. Hungar. megjelenés alatt, 2007
D. Petz and J. Kahn: Complementary reductions for two qubits, J. Math. Phys. 48(2007) 012107., 2007
D. Petz: On Sufficient Coarse-Grainings, XXXVIII Symposium on Mathematical Physics, Quantum Entanglement & Geometry, Torun, June 4-7., 2006
D. Petz: Relative Entropy in Quantum Information Theory, Quantum Information Workshop, Pécs, 12-15 July, 2006
D. Petz: State Tomography for Qubits, Quantum Probability, Information and Control, Nottingham 15-22 July, 2006
A. Andai: Volume of the quantum mechanical state space, J. Phys. A: Math. Gen. 39 (2006) 13641-13657, 2006
A. Andai: On the curvature of the quantum state space with pull-back metrics, benyújtva a Lin. Alg. Appl. folyóirathoz, 2007



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