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Author  |
Arnault, P.; Di Molfetta, G.; Brachet, M.; Debbasch, F. |
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Title |
Quantum walks and non-Abelian discrete gauge theory |
Type |
Journal Article |
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Year |
2016 |
Publication |
Physical Review A |
Abbreviated Journal |
Phys. Rev. A |
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Volume |
94 |
Issue |
1 |
Pages |
012335 - 6pp |
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Abstract |
A family of discrete-time quantum walks (DTQWs) on the line with an exact discrete U(N) gauge invariance is introduced. It is shown that the continuous limit of these DTQWs, when it exists, coincides with the dynamics of a Dirac fermion coupled to usual U(N) gauge fields in two-dimensional spacetime. A discrete generalization of the usual U(N) curvature is also constructed. An alternate interpretation of these results in terms of superimposed U(1) Maxwell fields and SU(N) gauge fields is discussed in the Appendix. Numerical simulations are also presented, which explore the convergence of the DTQWs towards their continuous limit and which also compare the DTQWs with classical (i.e., nonquantum) motions in classical SU(2) fields. The results presented in this paper constitute a first step towards quantum simulations of generic Yang-Mills gauge theories through DTQWs. |
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Address |
[Arnault, Pablo; Debbasch, Fabrice] Univ Paris 06, Univ Paris 04, PSL Res Univ, LERMA,Observ Paris,CNRS,UMR 8112, F-75014 Paris, France, Email: pablo.arnault@upmc.fr; |
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Amer Physical Soc |
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English |
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ISSN |
2469-9926 |
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Conference |
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Notes |
WOS:000380095000005 |
Approved |
no |
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Is ISI |
yes |
International Collaboration |
yes |
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Call Number |
IFIC @ pastor @ |
Serial |
2761 |
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Author |
Arnault, P.; Macquet, A.; Angles-Castillo, A.; Marquez-Martin, I.; Pina-Canelles, V.; Perez, A.; Di Molfetta, G.; Arrighi, P.; Debbasch, F. |
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Title |
Quantum simulation of quantum relativistic diffusion via quantum walks |
Type |
Journal Article |
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Year |
2020 |
Publication |
Journal of Physics A |
Abbreviated Journal |
J. Phys. A |
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Volume |
53 |
Issue |
20 |
Pages |
205303 - 39pp |
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Keywords |
noisy quantum walks; noisy quantum systems; decoherence; Lindblad equation; quantum simulation; relativistic diffusions; telegraph equation |
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Abstract |
Two models are first presented, of a one-dimensional discrete-time quantum walk (DTQW) with temporal noise on the internal degree of freedom (i.e., the coin): (i) a model with both a coin-flip and a phase-flip channel, and (ii) a model with random coin unitaries. It is then shown that both these models admit a common limit in the spacetime continuum, namely, a Lindblad equation with Dirac-fermion Hamiltonian part and, as Lindblad jumps, a chirality flip and a chirality-dependent phase flip, which are two of the three standard error channels for a two-level quantum system. This, as one may call it, Dirac Lindblad equation, provides a model of quantum relativistic spatial diffusion, which is evidenced both analytically and numerically. This model of spatial diffusion has the intriguing specificity of making sense only with original unitary models which are relativistic in the sense that they have chirality, on which the noise is introduced: the diffusion arises via the by-construction (quantum) coupling of chirality to the position. For a particle with vanishing mass, the model of quantum relativistic diffusion introduced in the present work, reduces to the well-known telegraph equation, which yields propagation at short times, diffusion at long times, and exhibits no quantumness. Finally, the results are extended to temporal noises which depend smoothly on position. |
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Address |
[Arnault, Pablo; Angles-Castillo, Andreu; Marquez-Martin, Ivan; Pina-Canelles, Vicente; Perez, Armando; Di Molfetta, Giuseppe] Univ Valencia, Dept Fis Teor, Dr Moliner 50, Burjassot 46100, Spain, Email: pablo.arnault@ic.uv.es |
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Iop Publishing Ltd |
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English |
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ISSN |
1751-8113 |
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Conference |
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Notes |
WOS:000531359000001 |
Approved |
no |
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Is ISI |
yes |
International Collaboration |
yes |
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Call Number |
IFIC @ pastor @ |
Serial |
4390 |
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Author |
Arnault, P.; Pepper, B.; Perez, A. |
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Title |
Quantum walks in weak electric fields and Bloch oscillations |
Type |
Journal Article |
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Year |
2020 |
Publication |
Physical Review A |
Abbreviated Journal |
Phys. Rev. A |
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Volume |
101 |
Issue |
6 |
Pages |
062324 - 12pp |
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Abstract |
Bloch oscillations appear when an electric field is superimposed on a quantum particle that evolves on a lattice with a tight-binding Hamiltonian (TBH), i.e., evolves via what we call an electric TBH; this phenomenon will be referred to as TBH Bloch oscillations. A similar phenomenon is known to show up in so-called electric discrete-time quantum walks (DQWs) [C. Cedzich et al., Phys. Rev. Lett. 111, 160601 (2013);] this phenomenon will be referred to as DQW Bloch oscillations. This similarity is particularly salient when the electric field of the DQW is weak. For a wide, i.e., spatially extended, initial condition, one numerically observes semiclassical oscillations, i.e., oscillations of a localized particle, for both the electric TBH and the electric DQW. More precisely, the numerical simulations strongly suggest that the semiclassical DQW Bloch oscillations correspond to two counterpropagating semiclassical TBH Bloch oscillations. In this work it is shown that, under certain assumptions, the solution of the electric DQW for a weak electric field and a wide initial condition is well approximated by the superposition of two continuous-time expressions, which are counterpropagating solutions of an electric TBH whose hopping amplitude is the cosine of the arbitrary coin-operator mixing angle. In contrast, if one wishes the continuous-time approximation to hold for spatially localized initial conditions, one needs at least the DQW to be lazy, as suggested by numerical simulations and by the fact that this has been proven in the case of a vanishing electric field [F. W. Strauch, Phys. Rev. A 74, 030301(R) (2006)]. |
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Address |
[Arnault, Pablo; Pepper, Benjamin; Perez, A.] Univ Valencia, CSIC, Dept Fis Teor, Cerrer Dr Moliner 50, Burjassot 46100, Spain, Email: pablo.arnault@ific.uv.es; |
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Publisher |
Amer Physical Soc |
Place of Publication |
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English |
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ISSN |
1050-2947 |
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Notes |
WOS:000541400900002 |
Approved |
no |
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Is ISI |
yes |
International Collaboration |
yes |
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Call Number |
IFIC @ pastor @ |
Serial |
4431 |
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Author |
Arnault, P.; Perez, A.; Arrighi, P.; Farrelly, T. |
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Title |
Discrete-time quantum walks as fermions of lattice gauge theory |
Type |
Journal Article |
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Year |
2019 |
Publication |
Physical Review A |
Abbreviated Journal |
Phys. Rev. A |
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Volume |
99 |
Issue |
3 |
Pages |
032110 - 16pp |
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Keywords |
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Abstract |
It is shown that discrete-time quantum walks can be used to digitize, i.e., to time discretize fermionic models of continuous-time lattice gauge theory. The resulting discrete-time dynamics is thus not only manifestly unitary, but also ultralocal, i.e., the particle's speed is upper bounded, as in standard relativistic quantum field theories. The lattice chiral symmetry of staggered fermions, which corresponds to a translational invariance, is lost after the requirement of ultralocality of the evolution; this fact is an instance of Meyer's 1996 no-go results stating that no nontrivial scalar quantum cellular automaton can be translationally invariant [D. A. Meyer, J. Stat. Phys. 85, 551 (1996); Phys. Lett. A 223, 337 (1996)]. All results are presented in a single-particle framework and for a (1+1)-dimensional space-time. |
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Address |
[Arnault, Pablo; Perez, Armando] Univ Valencia, Dept Fis Teor, Dr Moliner 50, E-46100 Burjassot, Spain, Email: pablo.arnault@ific.uv.es |
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Corporate Author |
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Publisher |
Amer Physical Soc |
Place of Publication |
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English |
Summary Language |
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Abbreviated Series Title |
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Series Volume |
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Series Issue |
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ISSN |
2469-9926 |
ISBN |
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Expedition |
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Conference |
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Notes |
WOS:000461896700002 |
Approved |
no |
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Is ISI |
yes |
International Collaboration |
yes |
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Call Number |
IFIC @ pastor @ |
Serial |
3950 |
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Author |
Jay, G.; Arnault, P.; Debbasch, F. |
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Title |
Dirac quantum walks with conserved angular momentum |
Type |
Journal Article |
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Year |
2021 |
Publication |
Quantum Studies-Mathematics and Foundations |
Abbreviated Journal |
Quantum Stud. Math. Found. |
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Volume |
8 |
Issue |
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Pages |
419-430 |
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Keywords |
Quantum walks; Quantum simulation; Lattice field theory |
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Abstract |
A quantum walk (QW) simulating the flat (1+2)D Dirac equation on a spatial polar grid is constructed. Because fermions are represented by spinors, which do not constitute a representation of the rotation group SO(3), but rather of its double cover SU(2), the QW can only be defined globally on an extended spacetime where the polar angle extends from 0 to 4 pi. The coupling of the QW with arbitrary electromagnetic fields is also presented. Finally, the cylindrical relativistic Landau levels of the Dirac equation are computed explicitly and simulated by the QW. |
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Address |
[Jay, Gareth] Univ Western Australia, Phys Dept, Perth, WA 6009, Australia, Email: gareth.jay@uwa.edu.au; |
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Publisher |
Springer |
Place of Publication |
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Editor |
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Language |
English |
Summary Language |
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Original Title |
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Series Editor |
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Abbreviated Series Title |
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Series Volume |
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Edition |
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ISSN |
2196-5609 |
ISBN |
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Conference |
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Notes |
WOS:000697709700001 |
Approved |
no |
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Is ISI |
yes |
International Collaboration |
yes |
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Call Number |
IFIC @ pastor @ |
Serial |
4975 |
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Author |
Marquez-Martin, I.; Arnault, P.; Di Molfetta, G.; Perez, A. |
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Title |
Electromagnetic lattice gauge invariance in two-dimensional discrete-time quantum walks |
Type |
Journal Article |
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Year |
2018 |
Publication |
Physical Review A |
Abbreviated Journal |
Phys. Rev. A |
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Volume |
98 |
Issue |
3 |
Pages |
032333 - 8pp |
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Keywords |
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Abstract |
Gauge invariance is one of the more important concepts in physics. We discuss this concept in connection with the unitary evolution of discrete-time quantum walks in one and two spatial dimensions, when they include the interaction with synthetic, external electromagnetic fields. One introduces this interaction as additional phases that play the role of gauge fields. Here, we present a way to incorporate those phases, which differs from previous works. Our proposal allows the discrete derivatives, that appear under a gauge transformation, to treat time and space on the same footing, in a way which is similar to standard lattice gauge theories. By considering two steps of the evolution, we define a density current which is gauge invariant and conserved. In the continuum limit, the dynamics of the particle, under a suitable choice of the parameters, becomes the Dirac equation and the conserved current satisfies the corresponding conservation equation. |
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Address |
[Marquez-Martin, Ivan; Arnault, Pablo; Di Molfetta, Giuseppe; Perez, Armando] Univ Valencia, Dept Fis Teor, Dr Moliner 50, E-46100 Burjassot, Spain, Email: ivan.marquez@uv.es; |
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Publisher |
Amer Physical Soc |
Place of Publication |
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Language |
English |
Summary Language |
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Original Title |
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Series Editor |
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Series Title |
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Abbreviated Series Title |
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Series Volume |
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Series Issue |
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Edition |
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ISSN |
2469-9926 |
ISBN |
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Area |
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Expedition |
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Conference |
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Notes |
WOS:000446163200006 |
Approved |
no |
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Is ISI |
yes |
International Collaboration |
yes |
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Call Number |
IFIC @ pastor @ |
Serial |
3750 |
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Author |
Nzongani, U.; Zylberman, J.; Doncecchi, C.E.; Perez, A.; Debbasch, F.; Arnault, P. |
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Title |
Quantum circuits for discrete-time quantum walks with position-dependent coin operator |
Type |
Journal Article |
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Year |
2023 |
Publication |
Quantum Information Processing |
Abbreviated Journal |
Quantum Inf. Process. |
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Volume |
22 |
Issue |
7 |
Pages |
270 - 46pp |
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Keywords |
Quantum walks; Quantum circuits; Quantum simulation |
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Abstract |
The aim of this paper is to build quantum circuits that implement discrete-time quantum walks having an arbitrary position-dependent coin operator. The position of the walker is encoded in base 2: with n wires, each corresponding to one qubit, we encode 2(n) position states. The data necessary to define an arbitrary position-dependent coin operator is therefore exponential in n. Hence, the exponentiality will necessarily appear somewhere in our circuits. We first propose a circuit implementing the position-dependent coin operator, that is naive, in the sense that it has exponential depth and implements sequentially all appropriate position-dependent coin operators. We then propose a circuit that “transfers” all the depth into ancillae, yielding a final depth that is linear in n at the cost of an exponential number of ancillae. Themain idea of this linear-depth circuit is to implement in parallel all coin operators at the different positions. Reducing the depth exponentially at the cost of having an exponential number of ancillae is a goal which has already been achieved for the problem of loading classical data on a quantum circuit (Araujo in Sci Rep 11:6329, 2021) (notice that such a circuit can be used to load the initial state of the walker). Here, we achieve this goal for the problem of applying a position-dependent coin operator in a discrete-time quantum walk. Finally, we extend the result of Welch (New J Phys 16:033040, 2014) from position-dependent unitaries which are diagonal in the position basis to position-dependent 2 x 2-block-diagonal unitaries: indeed, we show that for a position dependence of the coin operator (the block-diagonal unitary) which is smooth enough, one can find an efficient quantum-circuit implementation approximating the coin operator up to an error epsilon (in terms of the spectral norm), the depth and size of which scale as O(1/epsilon). A typical application of the efficient implementation would be the quantum simulation of a relativistic spin-1/2 particle on a lattice, coupled to a smooth external gauge field; notice that recently, quantum spatial-search schemes have been developed which use gauge fields as the oracle, to mark the vertex to be found (Zylberman in Entropy 23:1441, 2021), (Fredon arXiv:2210.13920). A typical application of the linear-depth circuit would be when there is spatial noise on the coin operator (and hence a non-smooth dependence in the position). |
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Address |
[Nzongani, Ugo; Doncecchi, Carlo-Elia; Arnault, Pablo] Univ Paris Saclay, CNRS, INRIA, Lab Methodes Formelles,ENS Paris Saclay, F-91190 Gif Sur Yvette, France, Email: ugo.nzongani@universite-paris-saclay.fr; |
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Corporate Author |
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Thesis |
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Publisher |
Springer |
Place of Publication |
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Editor |
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Language |
English |
Summary Language |
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Original Title |
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Series Editor |
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Abbreviated Series Title |
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Series Volume |
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Series Issue |
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Edition |
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ISSN |
1570-0755 |
ISBN |
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Conference |
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Notes |
WOS:001022408900002 |
Approved |
no |
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Is ISI |
yes |
International Collaboration |
yes |
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Call Number |
IFIC @ pastor @ |
Serial |
5587 |
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