Most boson quantum states are almost maximally entangled
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- by Shmuel Friedland and Todd Kemp PDF
- Proc. Amer. Math. Soc. 146 (2018), 5035-5049 Request permission
Abstract:
The geometric measure of entanglement $E$ of an $m$ qubit quantum state has maximum value bounded above by $m$. In previous work of Gross, Flammia, and Eisert, it was shown that $E \ge m-O(\log m)$ with high probability as $m\to \infty$. They showed, as a consequence, that the vast majority of states are too entangled to be computationally useful. In this paper, we show that for $m$ qubit Boson quantum states, the maximal possible geometric measure of entanglement is bounded above by $\log _2\! m$, opening the door to many computationally universal states. We further show the corresponding concentration result that $E \ge \log _2\! m - O(\log \log m)$ with high probability as $m\to \infty$. We extend these results also to $m$-mode $n$-bit Boson quantum states.References
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Additional Information
- Shmuel Friedland
- Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045
- MR Author ID: 69405
- Email: friedlan@uic.edu
- Todd Kemp
- Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112
- MR Author ID: 771033
- Email: tkemp@math.ucsd.edu
- Received by editor(s): March 23, 2017
- Received by editor(s) in revised form: August 22, 2017
- Published electronically: September 4, 2018
- Additional Notes: The second author was supported in part by NSF CAREER Award DMS-1254807
- Communicated by: Adrian Ioana
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5035-5049
- MSC (2010): Primary 15A69, 81P40; Secondary 20C35, 60B15
- DOI: https://doi.org/10.1090/proc/13933
- MathSciNet review: 3866844