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Recoupling Coefficients and Quantum Entropies

Christandl, Matthias and Şahinoglu, M. Burak and Walter, Michael (2018) Recoupling Coefficients and Quantum Entropies. Annales Henri Poincaré, 19 (2). pp. 385-410. ISSN 1424-0637.

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We prove that the asymptotic behavior of the recoupling coefficients of the symmetric group S_k is characterized by a quantum marginal problem: they decay polynomially in k if there exists a quantum state of three particles with given eigenvalues for their reduced density operators and exponentially otherwise. As an application, we deduce solely from symmetry considerations of the coefficients the strong subadditivity property of the von Neumann entropy, first proved by Lieb and Ruskai (J Math Phys 14:1938–1941, 1973). Our work may be seen as a non-commutative generalization of the representation-theoretic aspect of the recently found connection between the quantum marginal problem and the Kronecker coefficient of the symmetric group, which has applications in quantum information theory and algebraic complexity theory. This connection is known to generalize the correspondence between Weyl’s problem on the addition of Hermitian matrices and the Littlewood–Richardson coefficients of SU(d). In this sense, our work may also be regarded as a generalization of Wigner’s famous observation of the semiclassical behavior of the recoupling coefficients (here also known as 6j or Racah coefficients), which decay polynomially whenever a tetrahedron with given edge lengths exists. More precisely, we show that our main theorem contains a characterization of the possible eigenvalues of partial sums of Hermitian matrices thus presenting a representation-theoretic characterization of a generalization of Weyl’s problem. The appropriate geometric objects to SU(d) recoupling coefficients are thus tuples of Hermitian matrices and to S_k recoupling coefficients they are three-particle quantum states.

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Additional Information:© 2017 Springer International Publishing AG, part of Springer Nature. Received: 06 April 2017; Accepted: 07 November 2017; First Online: 15 December 2017. We thank M. Backens, M. Gromov, D. Gross, H. Haggard, F. Hellmann, W. Kamínski, A. Knutson, G. Mitchison, M.B. Ruskai, L. Vinet and R.Werner for valuable discussions. We acknowledge financial support by the German Science Foundation (Grant CH 843/2-1), the Swiss National Science Foundation [Grants PP00P2-128455, 20CH21-138799 (CHIST-ERA project CQC)], the Swiss National Center of Competence in Research ‘Quantum Science and Technology (QSIT)’, the Swiss State Secretariat for Education and Research supporting COST action MP1006, the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 337603, the Simons Foundation, FQXi, the Danish Council for Independent Research (Sapere Aude), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059), and the NWO (Grant No. 680-47-459). M.B. Sahinoğluu acknowledges support of the Excellence Scholarship and Opportunity Programme of ETH Zurich. Part of this work has been carried out while all three authors were with ETH Zurich.
Group:Institute for Quantum Information and Matter, IQIM
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)CH 843/2-1
Swiss National Science Foundation (SNSF)PP00P2-128455
Swiss National Science Foundation (SNSF)20CH21-138799
Swiss National Center of Competence in Research ‘Quantum Science and Technology (QSIT)’UNSPECIFIED
Swiss State Secretariat for Education and ResearchMP1006
European Research Council (ERC)337603
Simons FoundationUNSPECIFIED
Foundational Questions Institute (FQXI)UNSPECIFIED
Danish Council for Independent ResearchUNSPECIFIED
Villum Foundation10059
Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)680-47-459
Record Number:CaltechAUTHORS:20171215-104925515
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Official Citation:Christandl, M., Şahinoğlu, M.B. & Walter, M. Ann. Henri Poincaré (2018) 19: 385.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:83941
Deposited By: George Porter
Deposited On:15 Dec 2017 21:50
Last Modified:23 Jan 2019 22:17

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