Parametrizations of density matrices

This article gives a brief overview of some recent progress in the characterization and parametrization of density matrices of finite dimensional systems. We discuss in some detail the Bloch-vector and Jarlskog parametrizations and mention briefly the coset parametrization. As applications of the Bloch parametrization we discuss the trace invariants for the case of time dependent Hamiltonians and in some detail the dynamics of three-level systems. Furthermore, the Bloch vector of two-qubit systems as well as the use of the polarization operator basis is indicated. As the main application of the Jarlskog parametrization we construct density matrices for composite systems. In addition, some recent related articles are mentioned without further discussion.

[1]  The classification of three-parameter density matrices for a qutrit , 2006 .

[2]  M. Teich,et al.  Quantum entanglement and the two-photon Stokes parameters , 2001, quant-ph/0110172.

[3]  E. Sudarshan,et al.  Generalized Euler angle parametrization for SU(N) , 2002, math-ph/0205016.

[4]  János A. Bergou,et al.  Complementarity and entanglement in bipartite qudit systems , 2007 .

[5]  D. Pegg.  Two-photon resonance with fields amplitude modulated in quadrature , 1985 .

[6]  Solving the von Neumann equation with time-dependent Hamiltonian. Part I: Method , 2008, 0805.4487.

[7]  John E. Harriman,et al.  Geometry of density matrices. I. Definitions,Nmatrices and 1 matrices , 1978 .

[8]  Giuseppe Dattoli,et al.  An algebraic view to the operatorial ordering and its applications to optics , 1988 .

[9]  Pérès.  Separability Criterion for Density Matrices. , 1996, Physical review letters.

[10]  D. Chruściński,et al.  Circulant states with positive partial transpose , 2007, 0705.3534.

[11]  Todd Tilma,et al.  Generalized Euler angle parameterization for U(N) with applications to SU(N) coset volume measures , 2004 .

[12]  J. Elgin.  Semiclassical formalism for the treatment of three-level systems , 1980 .

[13]  J. Eberly,et al.  Nonlinear constants of motion for three-level quantum systems , 1982 .

[14]  Lendi.  Entropy production in coherence-vector formulation for N-level systems. , 1986, Physical review. A, General physics.

[15]  J. E. Harriman.  Geometry of density matrices. IV. The relationship between density matrices and densities , 1983 .

[16]  L. Jakóbczyk,et al.  Geometry of Bloch vectors in two-qubit system , 2001 .

[17]  Francesco Petruccione,et al.  Density Matrices and Their Time Evolution , 2008, Open Syst. Inf. Dyn..

[18]  R. Gilmore,et al.  Lie Groups, Lie Algebras, and Some of Their Applications , 1974 .

[19]  Harriman.  Densities, operators, and basis sets. , 1986, Physical review. A, General physics.

[20]  Gregg Jaeger,et al.  Quantum Lorentz-group invariants of n-qubit systems , 2003 .

[21]  H. Gottlieb.  Linear constants of motion for a three-level atom excited by two modulated electromagnetic waves , 1982 .

[22]  J. V. Corbett,et al.  The geometry of state space , 1993 .

[23]  C. Jarlskog.  Recursive parametrization and invariant phases of unitary matrices , 2006 .

[24]  Philipp Krammer.  Characterizing entanglement with geometric entanglement witnesses , 2008, 0807.4830.

[25]  Schlienz,et al.  Description of entanglement. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[26]  S. Kryszewski,et al.  Positivity of the N × N density matrix expressed in terms of polarization operators , 2006 .

[27]  J. E. Harriman.  Geometry of density matrices. II. Reduced density matrices and N representability , 1978 .

[28]  Claudio Altafini,et al.  Tensor of coherences parametrization of multiqubit density operators for entanglement characterization , 2003, quant-ph/0308019.

[29]  J. Eberly,et al.  N-Level Coherence Vector and Higher Conservation Laws in Quantum Optics and Quantum Mechanics , 1981 .

[30]  C. Altafini,et al.  QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC) 2357 Controllability properties for finite dimensional quantum Markovian master equations , 2002, quant-ph/0211194.

[31]  P. A. Ivanov,et al.  Engineering of arbitrary U ( N ) transformations by quantum Householder reflections , 2006, 0708.2811.

[32]  K. Lendi,et al.  Quantum Dynamical Semigroups and Applications , 1987 .

[33]  M. Reed,et al.  Methods of Modern Mathematical Physics. 2. Fourier Analysis, Self-adjointness , 1975 .

[34]  Howard Barnum,et al.  Separable balls around the maximally mixed multipartite quantum states , 2003 .

[35]  N. Vitanov,et al.  Navigation between quantum states by quantum mirrors , 2007, 0712.1272.

[36]  Francesco Petruccione,et al.  Parametrizing Density Matrices for Composite Quantum Systems , 2008, Open Syst. Inf. Dyn..

[37]  F. T. Hioe.  Analytic solutions of density-matrix evolutions with the use of Racah tensorial decompositions , 1984 .

[38]  Multilevel inversion schemes in and beyond the adiabatic limit. , 1985, Physical review. A, General physics.

[39]  M. Horodecki,et al.  Separability of mixed states: necessary and sufficient conditions , 1996, quant-ph/9605038.

[40]  Lendi.  Extension of quantum dynamical semigroup generators for open systems to time-dependent Hamiltonians. , 1986, Physical review. A, General physics.

[41]  P. Dita.  Factorization of unitary matrices , 2001, math-ph/0103005.

[42]  Hioe.  Gell-Mann dynamic symmetry for N-level quantum systems. , 1985, Physical review. A, General physics.

[43]  U. Fano.  Description of States in Quantum Mechanics by Density Matrix and Operator Techniques , 1957 .

[44]  D. Mattis.  Quantum Theory of Angular Momentum , 1981 .

[45]  Andrew G. White,et al.  Measurement of qubits , 2001, quant-ph/0103121.

[46]  N. Khaneja,et al.  Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation , 2003, quant-ph/0302024.

[47]  Bahaa E. A. Saleh,et al.  Multiphoton Stokes-parameter invariant for entangled states , 2003 .

[48]  Finite-level systems, Hermitian operators, isometries and a novel parametrization of Stiefel and Grassmann manifolds , 2003, quant-ph/0305156.

[49]  Comment on "A Recursive Parametrisation of Unitary Matrices" , 2005, math-ph/0505047.

[50]  Philippe Blanchard,et al.  Mathematical methods in physics : distributions, Hilbert space operators and variational methods , 2003 .

[51]  S. G. Schirmer,et al.  Stabilizing open quantum systems by Markovian reservoir engineering , 2009, 0909.1596.

[52]  S. Schirmer,et al.  Orbits of quantum states and geometry of Bloch vectors for N-level systems , 2003, quant-ph/0308004.

[53]  G. Kimura.  The Bloch Vector for N-Level Systems , 2003, quant-ph/0301152.

[54]  R. Bertlmann,et al.  Bloch vectors for qudits , 2008, 0806.1174.

[55]  C. Jarlskog.  A recursive parametrization of unitary matrices , 2005 .

[56]  N. Vitanov,et al.  Simple implementation of a quantum search with trapped ions , 2008, 0909.5401.

[57]  M. Lewenstein,et al.  Quantum Entanglement , 2020, Quantum Mechanics.

[58]  U. Fano,et al.  Pairs of two-level systems , 1983 .

[59]  B. Esser.  Density Matrix Theory and Applications , 1998 .

[60]  Marcus Huber,et al.  A composite parameterization of unitary groups, density matrices and subspaces , 2010, 1004.5252.

[61]  P. K. Aravind.  Geometry of the Schmidt decomposition and Hardy’s theorem , 1996 .

[62]  G. Tóth,et al.  Entanglement detection , 2008, 0811.2803.

[63]  James D. Louck,et al.  Angular Momentum in Quantum Physics: Theory and Application , 1984 .

[64]  Jarlskog's Parametrization of Unitary Matrices and Qudit Theory , 2005, quant-ph/0508006.

[65]  S. J. Akhtarshenas.  Coset parameterization of density matrices , 2007 .