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1.
Symmetry Principles in Quantum Systems Theory
Open Access
Title:
Symmetry Principles in Quantum Systems Theory
Author:
Zeier, Robert
;
SchulteHerbrueggen, T.
Zeier, Robert
;
SchulteHerbrueggen, T.
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Description:
General dynamic properties like controllability and simulability of spin systems, fermionic and bosonic systems are investigated in terms of symmetry. Symmetries may be due to the interaction topology or due to the structure and representation of the system and control Hamiltonians. In either case, they obviously entail constants of motion. Conv...
General dynamic properties like controllability and simulability of spin systems, fermionic and bosonic systems are investigated in terms of symmetry. Symmetries may be due to the interaction topology or due to the structure and representation of the system and control Hamiltonians. In either case, they obviously entail constants of motion. Conversely, the absence of symmetry implies irreducibility and provides a convenient necessary condition for full controllability much easier to assess than the wellestablished Liealgebra rank condition. We give a complete lattice of irreducible simple subalgebras of su(2^n) for up to n=15 qubits. It complements the symmetry condition by allowing for easy tests solving homogeneous linear equations to filter irreducible unitary representations of other candidate algebras of classical type as well as of exceptional types.  The lattice of irreducible simple subalgebras given also determines mutual simulability of dynamic systems of spin or fermionic or bosonic nature. We illustrate how controlled quadratic fermionic (and bosonic) systems can be simulated by spin systems and in certain cases also vice versa. ; Comment: For complete table with all irreducible simple subalgebras of su(N) [N up to 2^15] download ancillary file; v2: further results on spin chains with limited local control, minor corrections, and updated references; v3: minor amendments in Sec. 10.6, Appendix C more readerfriendly (see C.3, C.4), electronic supplement list corrected in 2 cases (dim=560, 4928) out of 32768 instances
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Year of Publication:
20101223
Document Type:
text
Subjects:
Quantum Physics ; Mathematical Physics
Quantum Physics ; Mathematical Physics
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DDC:
510 Mathematics
(computed)
URL:
http://arxiv.org/abs/1012.5256
http://arxiv.org/abs/1012.5256
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ArXiv.org (Cornell University Library)
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2.
Controllability and Observability of MultiSpin Systems: Constraints by Symmetry and by Relaxation
Open Access
Title:
Controllability and Observability of MultiSpin Systems: Constraints by Symmetry and by Relaxation
Author:
Sander, U.
;
SchulteHerbrueggen, T.
Sander, U.
;
SchulteHerbrueggen, T.
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Description:
We investigate the universality of multispin systems in architectures of various symmetries of coupling type and topology. Explicit reachability sets under symmetry constraints are provided. Thus for a given (possibly symmetric) experimental coupling architecture several decision problems can be solved in a unified way: (i) can a target Hamilto...
We investigate the universality of multispin systems in architectures of various symmetries of coupling type and topology. Explicit reachability sets under symmetry constraints are provided. Thus for a given (possibly symmetric) experimental coupling architecture several decision problems can be solved in a unified way: (i) can a target Hamiltonian be simulated? (ii) can a target gate be synthesised? (iii) to which extent is the system observable by a given set of detection operators? and, as a special case of the latter, (iv) can an underlying system Hamiltonian be identified with a given set of detection operators? Finally, in turn, lack of symmetry provides a convenient necessary condition for full controllability. Though often easier to assess than the wellestablished Liealgebra rank condition, this is not sufficient unless the candidate dynamic simple Lie algebra can be preidentified uniquely, which is fortunately less complicated than expected. ; Comment: 13 pages, 3 figures; extended update, comments welcome
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Year of Publication:
20090429
Document Type:
text
Subjects:
Quantum Physics
Quantum Physics
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DDC:
531 Classical mechanics; solid mechanics
(computed)
URL:
http://arxiv.org/abs/0904.4654
http://arxiv.org/abs/0904.4654
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3.
Which Quantum Evolutions Can Be Reversed by Local Unitary Operations? Algebraic Classification and GradientFlowBased Numerical Checks
Open Access
Title:
Which Quantum Evolutions Can Be Reversed by Local Unitary Operations? Algebraic Classification and GradientFlowBased Numerical Checks
Author:
SchulteHerbrueggen, T.
;
Spoerl, A.
SchulteHerbrueggen, T.
;
Spoerl, A.
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Description:
Generalising in the sense of Hahn's spin echo, we completely characterise those unitary propagators of effective multiqubit interactions that can be inverted solely by {\em local} unitary operations on $n$ qubits (spins$\tfrac{1}{2}$). The subset of $U\in \mathbf{SU}(2^n)$ satisfying $U^{1}=K_1 U K_2$ with pairs of local unitaries $K_1, K_2\i...
Generalising in the sense of Hahn's spin echo, we completely characterise those unitary propagators of effective multiqubit interactions that can be inverted solely by {\em local} unitary operations on $n$ qubits (spins$\tfrac{1}{2}$). The subset of $U\in \mathbf{SU}(2^n)$ satisfying $U^{1}=K_1 U K_2$ with pairs of local unitaries $K_1, K_2\in\mathbf{SU}(2)^{\otimes n}$ comprises two classes: in typeI, $K_1$ and $K_2$ are inverse to one another, while in typeII they are not. {TypeI} consists of oneparameter groups that can jointly be inverted for all times $t\in\R{}$ because their Hamiltonian generators satisfy $K H K^{1} = \Ad K (H) = H$. As all the Hamiltonians generating locally invertible unitaries of typeI are spanned by the eigenspace associated to the eigenvalue 1 of the {\em local} conjugation map $\Ad K$, this eigenspace can be given in closed algebraic form. The relation to the root space decomposition of $\mathfrak{sl}(N,\C{})$ is pointed out. Special cases of typeI invertible Hamiltonians are of $p$quantum order and are analysed by the transformation properties of spherical tensors of order $p$. Effective multiqubit interaction Hamiltonians are characterised via the graphs of their coupling topology. {TypeII} consists of pointwise locally invertible propagators, part of which can be classified according to the symmetries of their matrix representations. Moreover, we show gradient flows for numerically solving the decision problem whether a propagator is typeI or typeII invertible or not by driving the leastsquares distance $\norm{K_1 e^{itH} K_2  e^{+itH}}^2_2$ to zero. ; Comment: 19 pages, 7 figures; comments welcome
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Year of Publication:
20061009
Document Type:
text
Subjects:
Quantum Physics
Quantum Physics
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DDC:
516 Geometry
(computed)
URL:
http://arxiv.org/abs/quantph/0610061
http://arxiv.org/abs/quantph/0610061
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4.
How to Transfer between Arbitrary $n$Qubit Quantum States by Coherent Control and Simplest Switchable Noise on a Single Qubit
Open Access
Title:
How to Transfer between Arbitrary $n$Qubit Quantum States by Coherent Control and Simplest Switchable Noise on a Single Qubit
Author:
Bergholm, V.
;
SchulteHerbrueggen, T.
Bergholm, V.
;
SchulteHerbrueggen, T.
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Description:
We explore reachable sets of open $n$qubit quantum systems, the coherent parts of which are under full unitary control and that have just one qubit whose Markovian noise amplitude can be modulated in time such as to provide an additional degree of incoherent control. In particular, adding bangbang control of amplitude damping noise (nonunital...
We explore reachable sets of open $n$qubit quantum systems, the coherent parts of which are under full unitary control and that have just one qubit whose Markovian noise amplitude can be modulated in time such as to provide an additional degree of incoherent control. In particular, adding bangbang control of amplitude damping noise (nonunital) allows the dynamic system to act transitively on the entire set of density operators. This means one can transform any initial quantum state into any desired target state. Adding switchable bitflip noise (unital), on the other hand, suffices to explore all states majorised by the initial state. We have extended our openloop optimal control algorithm (DYNAMO package) by such degrees of incoherent control so that these unprecedented reachable sets can systematically be exploited in experiments. As illustrated for an ion trap experimental setting, openloop control with noise switching can accomplish all state transfers one can get by the more complicated measurementbased closedloop feedback schemes. ; Comment: v2 streamlined to 5 pages, 3 figures; extended technical details relegated to Supplement (8 pp 2 figs)
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Year of Publication:
20120621
Document Type:
text
Subjects:
Quantum Physics
Quantum Physics
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URL:
http://arxiv.org/abs/1206.4945
http://arxiv.org/abs/1206.4945
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5.
Which Quantum Evolutions Can Be Reversed by Local Unitary Operations? Algebraic Classification and GradientFlowBased Numerical Checks
Title:
Which Quantum Evolutions Can Be Reversed by Local Unitary Operations? Algebraic Classification and GradientFlowBased Numerical Checks
Author:
SchulteHerbrueggen, T
;
Spoerl, A
SchulteHerbrueggen, T
;
Spoerl, A
Minimize authors
Description:
Generalising in the sense of Hahn's spin echo, we completely characterise those unitary propagators of effective multiqubit interactions that can be inverted solely by {\em local} unitary operations on $n$ qubits (spins$\tfrac{1}{2}$). The subset of $U\in \mathbf{SU}(2^n)$ satisfying $U^{1}=K_1 U K_2$ with pairs of local unitaries $K_1, K_2\i...
Generalising in the sense of Hahn's spin echo, we completely characterise those unitary propagators of effective multiqubit interactions that can be inverted solely by {\em local} unitary operations on $n$ qubits (spins$\tfrac{1}{2}$). The subset of $U\in \mathbf{SU}(2^n)$ satisfying $U^{1}=K_1 U K_2$ with pairs of local unitaries $K_1, K_2\in\mathbf{SU}(2)^{\otimes n}$ comprises two classes: in typeI, $K_1$ and $K_2$ are inverse to one another, while in typeII they are not. {TypeI} consists of oneparameter groups that can jointly be inverted for all times $t\in\R{}$ because their Hamiltonian generators satisfy $K H K^{1} = \Ad K (H) = H$. As all the Hamiltonians generating locally invertible unitaries of typeI are spanned by the eigenspace associated to the eigenvalue 1 of the {\em local} conjugation map $\Ad K$, this eigenspace can be given in closed algebraic form. The relation to the root space decomposition of $\mathfrak{sl}(N,\C{})$ is pointed out. Special cases of typeI invertible Hamiltonians are of $p$quantum order and are analysed by the transformation properties of spherical tensors of order $p$. Effective multiqubit interaction Hamiltonians are characterised via the graphs of their coupling topology. {TypeII} consists of pointwise locally invertible propagators, part of which can be classified according to the symmetries of their matrix representations. Moreover, we show gradient flows for numerically solving the decision problem whether a propagator is typeI or typeII invertible or not by driving the leastsquares distance $\norm{K_1 e^{itH} K_2  e^{+itH}}^2_2$ to zero.
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Year of Publication:
20061009
Language:
eng
Subjects:
General Theoretical Physics
General Theoretical Physics
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DDC:
516 Geometry
(computed)
URL:
http://cds.cern.ch/record/988586
http://cds.cern.ch/record/988586
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CERN Document Server (CDS)
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6.
Exploiting Matrix Symmetries and Physical Symmetries in Matrix Product States and Tensor Trains
Open Access
Title:
Exploiting Matrix Symmetries and Physical Symmetries in Matrix Product States and Tensor Trains
Author:
Huckle, T.
;
Waldherr, K.
;
SchulteHerbrueggen, T.
Huckle, T.
;
Waldherr, K.
;
SchulteHerbrueggen, T.
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Description:
We focus on symmetries related to matrices and vectors appearing in the simulation of quantum manybody systems. Spin Hamiltonians have special matrixsymmetry properties such as persymmetry. Furthermore, the systems may exhibit physical symmetries translating into symmetry properties of the eigenvectors of interest. Both types of symmetry can b...
We focus on symmetries related to matrices and vectors appearing in the simulation of quantum manybody systems. Spin Hamiltonians have special matrixsymmetry properties such as persymmetry. Furthermore, the systems may exhibit physical symmetries translating into symmetry properties of the eigenvectors of interest. Both types of symmetry can be exploited in sparse representation formats such as Matrix Product States (MPS) for the desired eigenvectors. This paper summarizes symmetries of Hamiltonians for typical physical systems such as the Ising model and lists resulting properties of the related eigenvectors. Based on an overview of Matrix Product States (Tensor Trains or Tensor Chains) and their canonical normal forms we show how symmetry properties of the vector translate into relations between the MPS matrices and, in turn, which symmetry properties result from relations within the MPS matrices. In this context we analyze different kinds of symmetries and derive appropriate normal forms for MPS representing these symmetries. Exploiting such symmetries by using these normal forms will lead to a reduction in the number of degrees of freedom in the MPS matrices. This paper provides a uniform platform for both wellknown and new results which are presented from the (multi)linear algebra point of view.
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Year of Publication:
20130104
Document Type:
text
Subjects:
Mathematical Physics ; Quantum Physics ; 15A69 ; 15B57 ; 8108 ; 15A18
Mathematical Physics ; Quantum Physics ; 15A69 ; 15B57 ; 8108 ; 15A18
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DDC:
531 Classical mechanics; solid mechanics
(computed)
URL:
http://arxiv.org/abs/1301.0746
http://arxiv.org/abs/1301.0746
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7.
Computations in Quantum Tensor Networks
Open Access
Title:
Computations in Quantum Tensor Networks
Author:
Huckle, T.
;
Waldherr, K.
;
SchulteHerbrueggen, T.
Huckle, T.
;
Waldherr, K.
;
SchulteHerbrueggen, T.
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Description:
The computation of the ground state (i.e. the eigenvector related to the smallest eigenvalue) is an important task in the simulation of quantum manybody systems. As the dimension of the underlying vector space grows exponentially in the number of particles, one has to consider appropriate subsets promising both convenient approximation properti...
The computation of the ground state (i.e. the eigenvector related to the smallest eigenvalue) is an important task in the simulation of quantum manybody systems. As the dimension of the underlying vector space grows exponentially in the number of particles, one has to consider appropriate subsets promising both convenient approximation properties and efficient computations. The variational ansatz for this numerical approach leads to the minimization of the Rayleigh quotient. The Alternating Least Squares technique is then applied to break down the eigenvector computation to problems of appropriate size, which can be solved by classical methods. Efficient computations require fast computation of the matrixvector product and of the inner product of two decomposed vectors. To this end, both appropriate representations of vectors and efficient contraction schemes are needed. Here approaches from manybody quantum physics for onedimensional and twodimensional systems (Matrix Product States and Projected Entangled Pair States) are treated mathematically in terms of tensors. We give the definition of these concepts, bring some results concerning uniqueness and numerical stability and show how computations can be executed efficiently within these concepts. Based on this overview we present some modifications and generalizations of these concepts and show that they still allow efficient computations such as applicable contraction schemes. In this context we consider the minimization of the Rayleigh quotient in terms of the {\sc parafac} (CP) formalism, where we also allow different tensor partitions. This approach makes use of efficient contraction schemes for the calculation of inner products in a way that can easily be extended to the mps format but also to higher dimensional problems. ; Comment: Presented in part at the 26th GAMM Seminar on Tensor Approximations at the MaxPlanckInstitute for Mathematics in the Sciences in Leipzig, February 2010
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Year of Publication:
20121220
Document Type:
text
Subjects:
Quantum Physics ; Mathematical Physics
Quantum Physics ; Mathematical Physics
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DDC:
518 Numerical analysis
(computed)
URL:
http://arxiv.org/abs/1212.5005
http://arxiv.org/abs/1212.5005
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8.
Controlling Several Atoms in a Cavity
Open Access
Title:
Controlling Several Atoms in a Cavity
Author:
Keyl, Michael
;
Zeier, Robert
;
SchulteHerbrueggen, T.
Keyl, Michael
;
Zeier, Robert
;
SchulteHerbrueggen, T.
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Description:
We treat control of several twolevel atoms interacting with one mode of the electromagnetic field in a cavity. This provides a useful model to study pertinent aspects of quantum control in infinite dimensions via the emergence of infinitedimensional system algebras. Hence we address problems arising with infinitedimensional Lie algebras and t...
We treat control of several twolevel atoms interacting with one mode of the electromagnetic field in a cavity. This provides a useful model to study pertinent aspects of quantum control in infinite dimensions via the emergence of infinitedimensional system algebras. Hence we address problems arising with infinitedimensional Lie algebras and those of unbounded operators. For the models considered, these problems can be solved by splitting the set of control Hamiltonians into two subsets: The first obeys an abelian symmetry and can be treated in terms of infinitedimensional Lie algebras and strongly closed subgroups of the unitary group of the system Hilbert space. The second breaks this symmetry, and its discussion introduces new arguments. Yet, full controllability can be achieved in a strong sense: e.g., in a time dependent JaynesCummings model we show that, by tuning coupling constants appropriately, every unitary of the coupled system (atoms and cavity) can be approximated with arbitrarily small error.
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Year of Publication:
20140122
Document Type:
text
Subjects:
Quantum Physics
Quantum Physics
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URL:
http://arxiv.org/abs/1401.5722
http://arxiv.org/abs/1401.5722
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9.
Illustrating the Geometry of Coherently Controlled Quantum Channels
Open Access
Title:
Illustrating the Geometry of Coherently Controlled Quantum Channels
Author:
O'Meara, C.
;
Dirr, G.
;
SchulteHerbrueggen, T.
O'Meara, C.
;
Dirr, G.
;
SchulteHerbrueggen, T.
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Description:
We extend standard Markovian open quantum systems (quantum channels) by allowing for Hamiltonian controls and elucidate their geometry in terms of Lie semigroups. For standard dissipative interactions with the environment and different coherent controls, we particularly specify the tangent cones (Lie wedges) of the respective Lie semigroups of q...
We extend standard Markovian open quantum systems (quantum channels) by allowing for Hamiltonian controls and elucidate their geometry in terms of Lie semigroups. For standard dissipative interactions with the environment and different coherent controls, we particularly specify the tangent cones (Lie wedges) of the respective Lie semigroups of quantum channels. These cones are the counterpart of the infinitesimal generator of a single oneparameter semigroup. They comprise all directions the underlying open quantum system can be steered to and thus give insight into the geometry of controlled open quantum dynamics. Such a differential characterisation is highly valuable for approximating reachable sets of given initial quantum states in a plethora of experimental implementations. ; Comment: condensed and updated version; 14 pages; comments welcome
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Year of Publication:
20110314
Document Type:
text
Subjects:
Quantum Physics
Quantum Physics
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URL:
http://arxiv.org/abs/1103.2703
http://arxiv.org/abs/1103.2703
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10.
LeastSquares Approximation by Elements from Matrix Orbits Achieved by Gradient Flows on Compact Lie Groups
Open Access
Title:
LeastSquares Approximation by Elements from Matrix Orbits Achieved by Gradient Flows on Compact Lie Groups
Author:
Li, C. K.
;
Poon, Y. T.
;
SchulteHerbrueggen, T.
Li, C. K.
;
Poon, Y. T.
;
SchulteHerbrueggen, T.
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Description:
Let $S(A)$ denote the orbit of a complex or real matrix $A$ under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradientflow algorithms are constructed to determine the best approximation of a given matrix $A_0$ by the sum of matrices in $S(A_1), ., S(A_N)$ in the sense of fin...
Let $S(A)$ denote the orbit of a complex or real matrix $A$ under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradientflow algorithms are constructed to determine the best approximation of a given matrix $A_0$ by the sum of matrices in $S(A_1), ., S(A_N)$ in the sense of finding the Euclidean leastsquares distance $$\min \{\X_1+ . + X_N  A_0\: X_j \in S(A_j), j = 1, >., N\}.$$ Connections of the results to different pure and applied areas are discussed.
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Year of Publication:
20081209
Document Type:
text
Subjects:
Mathematics  Numerical Analysis ; Mathematics  Dynamical Systems ; Mathematics  Optimization and Control ; Quantum Physics ; 15A18 ; 15A60 ; 15A90 ; 37N30
Mathematics  Numerical Analysis ; Mathematics  Dynamical Systems ; Mathematics  Optimization and Control ; Quantum Physics ; 15A18 ; 15A60 ; 15A90 ; 37N30
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URL:
http://arxiv.org/abs/0812.1817
http://arxiv.org/abs/0812.1817
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Author
(22) SchulteHerbrueggen, T.
(8) Glaser, S. J.
(6) SchulteHerbrueggen, T
(5) SchulteHerbrüggen, T.
(4) Dirr, G.
(4) Glaser, S J
(4) Nielsen, Niels Christian
(3) Helmke, U.
(3) Khaneja, N
(3) Spoerl, A
(3) Spoerl, A.
(3) Zeier, Robert
(2) Bergholm, V.
(2) Griesinger, C.
(2) Huckle, T.
(2) Keyl, Michael
(2) Khaneja, N.
(2) Sander, U.
(2) Spoerl, A K
(2) Spoerl, A. K.
(2) Sørensen, Ole Winnesch
(2) Untidt, T.
(2) Waldherr, K.
(2) Wilhelm, F K
(2) Wilhelm, F. K.
(1) Abe, H.
(1) Arbogast, Helmut
(1) Arndt, H.
(1) Benner, S A
(1) Bergholm, V
(1) Bergholm, Ville
(1) Biamonte, Jacob
(1) Daniel, P.
(1) Dolde, Florian
(1) Egli, M
(1) Ferber, J
(1) Ferber, J.
(1) Fisher, R.
(1) Glaser, J. S.
(1) Gruslys, A.
(1) Heitmann, B
(1) Helmer, F.
(1) Hyrup, B
(1) Isoya, J.
(1) Jakobi, Ingmar
(1) Jamali, M.
(1) Kleinsteuber, M.
(1) Kurniawan, I.
(1) Li, C. K.
(1) Luy, B.
(1) Machnes, S.
(1) Marquardt, F.
(1) Meijer, Jan
(1) Neumann, P.
(1) Neumann, Philipp
(1) O'Meara, C.
(1) Ohshima, T.
(1) Pezzagna, Sebastien
(1) Poon, Y. T.
(1) Rebentrost, P
(1) Rebentrost, P.
(1) Richert, C
(1) Schedletzky, O.
(1) Schirmer, S.
(1) SchulteHerbrüggen, T
(1) Serban, I
(1) Serban, I.
(1) Sieveking, M.
(1) Sprengard, M.
(1) Storcz, M J
(1) Storcz, M. J.
(1) Szinicz, L.
(1) Sørensen, O.W.
(1) Sørensen, Ole Winneche
(1) Untidt, T.S.
(1) Waldherr, G.
(1) Wang, Y.
(1) Wang, Ya
(1) Wrachtrup, J.
(1) Wrachtrup, Jörg
(1) Yuan, H
(1) Zaiser, S.
(1) Zimborás, Zoltán
(1) de Fouquieres, P.
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(22) quantum physics
(7) mathematical physics
(6) general theoretical physics
(4) condensed matter mesoscale and nanoscale physics
(4) mathematics optimization and control
(2) 15a18
(1) 15a60
(1) 15a69
(1) 15a90
(1) 15b57
(1) 37n30
(1) 81 08
(1) ddc 610
(1) mathematics dynamical systems
(1) mathematics numerical analysis
(1) medizin
Subject:
Dewey Decimal Classification (DDC)
(6) Mathematics [51*]
(6) Physics [53*]
(1) Science [50*]
Dewey Decimal Classification (DDC):
Year of Publication
(7) 2006
(4) 2005
(3) 2008
(3) 2012
(3) 2013
(2) 1998
(2) 2007
(2) 2009
(2) 2010
(1) 1988
(1) 1995
(1) 1999
(1) 2001
(1) 2011
(1) 2014
Year of Publication:
Content Provider
(22) ArXiv.org
(6) CERN (Switzerland)
(4) Aarhus Univ.: Pure
(1) Munich LMU: Open Access
(1) PubMed Central
Content Provider:
Language
(22) Unknown
(11) English
(1) German
Language:
Document Type
(23) Text
(6) Unknown
(4) Article, Journals
(1) Reports, Papers, Lectures
Document Type:
Access
(23) Open Access
(11) Unknown
Access:
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