Connecting unextendible maximally entangled base with partial Hadamard matrices

Connecting unextendible maximally entangled base with partial Hadamard matrices We study the unextendible maximally entangled bases (UMEB) in $$\mathbb {C}^{d}\bigotimes \mathbb {C}^{d}$$ C d ⨂ C d and connect the problem to the partial Hadamard matrices. We show that for a given special UMEB in $$\mathbb {C}^{d}\bigotimes \mathbb {C}^{d}$$ C d ⨂ C d , there is a partial Hadamard matrix which cannot be extended to a Hadamard matrix in $$\mathbb {C}^{d}$$ C d . As a corollary, any $$(d-1)\times d$$ ( d - 1 ) × d partial Hadamard matrix can be extended to a Hadamard matrix, which answers a conjecture about $$d=5$$ d = 5 . We obtain that for any d there is a UMEB except for $$d=p\ \text {or}\ 2p$$ d = p or 2 p , where $$p\equiv 3\mod 4$$ p ≡ 3 mod 4 and p is a prime. The existence of different kinds of constructions of UMEBs in $$\mathbb {C}^{nd}\bigotimes \mathbb {C}^{nd}$$ C n d ⨂ C n d for any $$n\in \mathbb {N}$$ n ∈ N and $$d=3\times 5 \times 7$$ d = 3 × 5 × 7 is also discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

Connecting unextendible maximally entangled base with partial Hadamard matrices

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Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media New York
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-017-1537-7
Publisher site
See Article on Publisher Site

Abstract

We study the unextendible maximally entangled bases (UMEB) in $$\mathbb {C}^{d}\bigotimes \mathbb {C}^{d}$$ C d ⨂ C d and connect the problem to the partial Hadamard matrices. We show that for a given special UMEB in $$\mathbb {C}^{d}\bigotimes \mathbb {C}^{d}$$ C d ⨂ C d , there is a partial Hadamard matrix which cannot be extended to a Hadamard matrix in $$\mathbb {C}^{d}$$ C d . As a corollary, any $$(d-1)\times d$$ ( d - 1 ) × d partial Hadamard matrix can be extended to a Hadamard matrix, which answers a conjecture about $$d=5$$ d = 5 . We obtain that for any d there is a UMEB except for $$d=p\ \text {or}\ 2p$$ d = p or 2 p , where $$p\equiv 3\mod 4$$ p ≡ 3 mod 4 and p is a prime. The existence of different kinds of constructions of UMEBs in $$\mathbb {C}^{nd}\bigotimes \mathbb {C}^{nd}$$ C n d ⨂ C n d for any $$n\in \mathbb {N}$$ n ∈ N and $$d=3\times 5 \times 7$$ d = 3 × 5 × 7 is also discussed.

Journal

Quantum Information ProcessingSpringer Journals

Published: Feb 13, 2017

References

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