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Most boson quantum states are almost maximally entangled

Authors: Shmuel Friedland and Todd Kemp
Journal: Proc. Amer. Math. Soc. 146 (2018), 5035-5049
MSC (2010): Primary 15A69, 81P40; Secondary 20C35, 60B15
Published electronically: September 4, 2018
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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.

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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

Todd Kemp
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112

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
Article copyright: © Copyright 2018 American Mathematical Society

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