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Cited by 44Quantum Information and CryptographyQuantum Mechanics and ApplicationsQuantum Computing Algorithms and Architecture

Better bound on the exponent of the radius of the multipartite separable ball

Open access
Gurvits, Leonid, Barnum, Howard·2005·Physical Review A
DOI 10.1103/physreva.72.032322
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Abstract

We show that for an $m$-qubit quantum system, there is a ball of radius asymptotically approaching $\ensuremath{\kappa}{2}^{\ensuremath{-}\ensuremath{\gamma}m}$ in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent $\ensuremath{\gamma}=0.5(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}3∕\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2\ensuremath{-}1)\ensuremath{\approx}0.292\phantom{\rule{0.2em}{0ex}}481\phantom{\rule{0.2em}{0ex}}25$ much smaller in magnitude than the best previously known exponent, from our earlier work, of $1∕2$. For normalized $m$-qubit states, we get a separable ball of radius $\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}(1+\ensuremath{\gamma})m}\ensuremath{\equiv}\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{6}^{\ensuremath{-}m∕2}$ (note that $\ensuremath{\kappa}=\sqrt{3}$), compared to the previous $2\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}3m∕2}$. This implies that with parameters realistic for current experiments, nuclear magnetic resonance (NMR) with standard pseudopure-state preparation techniques can access only unentangled states if 36 qubits or fewer are used (compared to 23 qubits via our earlier results). We also obtain an improved exponent for $m$-partite systems of fixed local dimension ${d}_{0}$, although approaching our earlier exponent as ${d}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$.

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  • For N quantum systems of d-dimensions, there exists a ball of radius R(N,d), centred on the maximally mixed state, containing only separable states.

    “For N quantum systems of d-dimensions, there exists a ball of radius R ( N,d ) , centred on the maximally mixed state, containing only separable states [35].”

    From Enhancing the Charging Power of Quantum Batteries (2017) · cites this paper as [35]

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  1. [1]K. ˙ Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Physical Review A 58 , 883 (1998).
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  4. [4]P. Rungta, W. J. Munro, K. Nemoto, P. Deuar, G. J. Milburn, and C. M. Caves, in Directions in Quantum Optics: A Collection of Papers Dedicated to the Memory of Dan Walls , edited by D. Walls, R. Glauber, M. Scully, and H. Carmichael (Springer, New York and Berlin, 2001), also arXiv.org e-print quant-ph/0001075. · arXiv
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  7. [7]S. Szarek (2004), arXiv.org e-print quant-ph/0310061 v.2 . · arXiv
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  15. [20]L. J. Schulman and U. Vazirani, Proceedings of the 31st Annual ACM Symposium on the Theory of Computing (STOC) pp. 322-329 (1999), earlier version is quant-ph/9804060. · arXiv

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Authors
Gurvits, L., & Barnum, H.
Affiliations
Los Alamos National Laboratory (US)
ORCID
Barnum, Howard
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2005-09-16
Volume
72
Issue
3
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.72.032322
OpenAlex
W2154497602
Cited by
44
References
14
Citations by year
2026: 2 · 2025: 4 · 2024: 1 · 2021: 3 · 2019: 1 · 2018: 1 · 2017: 2 · 2016: 2 · 2015: 2 · 2014: 2 · 2013: 4 · 2012: 2
FWCI
3.52
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
Keywords
multipartite entanglement, qubits, Frobenius norm, positive semidefinite matrices, nuclear magnetic resonance, entanglement bounds, local dimension

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