Entanglement of the Antisymmetric State
Abstract
We analyse the entanglement of the antisymmetric state in dimension d×d and present two main results. First, we show that the amount of secrecy that can be extracted from the state is low, more precisely, the distillable key is bounded by O(1/d). Second, we show that the state is highly entangled in the sense that a large number of ebits are needed in order to create the state: entanglement cost is larger than a constant, independent of d. The second result is shown to imply that the regularised relative entropy with respect to separable states is also lower bounded by a constant. Finally, we note that the regularised relative entropy of entanglement is asymptotically continuous in the state. Elementary and advanced facts from the representation theory of the unitary group, including the concept of plethysm, play a central role in the proofs of the main results.
Additional Information
© 2012 Springer-Verlag. Received: 21 January 2011; Accepted: 7 October 2011; Published online: 7 March 2012. Communicated by M. B. Ruskai. After completion of this work, F. Brandão kindly pointed out to us that the states from [27] can be used to construct states with EC(ρ) ≥ 1/2 and KD(ρ) ≤ 2/log2d. MC was supported by the Swiss National Science Foundation (grant PP00P2-128455), the National Centre of Competence in Research 'Quantum Science and Technology' and the German Science Foundation (grants CH 843/1-1 and CH 843/2-1). NS acknowledges support by the EU (QUEVADIS, SCALA), the German cluster of excellence project MAP, the Gordon and Betty Moore Foundation through Caltech's Center for the Physics of Information, and the NSF Grant No. PHY-0803371. AW is supported by the European Commission, the U.K. EPSRC, the Royal Society, and a Philip Leverhulme Prize. The Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Excellence programme.Additional details
- Eprint ID
- 31275
- DOI
- 10.1007/s00220-012-1446-7
- Resolver ID
- CaltechAUTHORS:20120502-131130430
- PP00P2-128455
- Swiss National Science Foundation (SNSF)
- National Centre of Competence in Research 'Quantum Science and Technology'
- CH 843/1-1
- German Science Foundation
- CH 843/2-1
- German Science Foundation
- European Union (EU) QUEVADIS
- European Union (EU) SCALA
- German Cluster of Excellence Project MAP
- Gordon and Betty Moore Foundation
- PHY-0803371
- NSF
- European Commission
- Engineering and Physical Sciences Research Council (EPSRC)
- Royal Society
- Philip Leverhulme Prize
- Singapore Ministry of Education
- National Research Foundation
- Created
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2012-05-02Created from EPrint's datestamp field
- Updated
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2021-11-09Created from EPrint's last_modified field