bibkey: shchukinvanloock2016higherorderepr
authors: E. Shchukin and P. van Loock
year: 2016
title: "Higher-order Einstein-Podolsky-Rosen correlations and inseparability conditions for continuous variables"
doi: 10.1103/PhysRevA.93.032114
url: "https://arxiv.org/abs/1511.08444v1"
claim: "For two bosonic modes and every positive integer n, ⟨(a†ⁿ ± bⁿ)(aⁿ ± b†ⁿ)⟩ ≥ n! for separable states and ≥ 0 for all states; for n = 2 the bound 0 is proved tight, and the paper conjectures that the bound 0 is tight for every n and that the separable bound cannot be violated by Gaussian states."
strata_touched:
  - D5/S3/Quantum/FockSpace/HigherOrderEPRTightness
license: citation-only
triage: anchor

Shchukin and van Loock, Higher-order EPR correlations and inseparability conditions for continuous variables

E. Shchukin and P. van Loock, Phys. Rev. A 93(3), 032114 (2016); arXiv:1511.08444v1. Quotations are from the v1 TeX source.

For two bosonic modes with annihilation operators $\hat{a},\hat{b}$ the paper proves, for every positive integer $n$, $$⟨({\hat{a}}^{†n}\pm {\hat{b}}^n)({\hat{a}}^n\pm {\hat{b}}^{†n})⟩\ge \{\begin{array}{ll}n!& \text{for separable states (nS:1)}\\ 0& \text{for all states (nS:2)},\end{array}$$ proves for $n=2$ that (bS:2) “is always strict and tight, i.e., the left-hand side can be arbitrarily close to zero (but never equal to zero)”, and for $n\ge 3$ exhibits a family whose limiting value $N_n^{2}n/4$ violates (nS:1) but stays positive. It concludes:

Our conjecture is that the inequality (nS:1) cannot be violated by Gaussian states (including all inseparable ones) and the inequality (nS:2) is tight. We verified this by numerical computations, but we could not find rigorous proofs.

Verified locator

Source: https://arxiv.org/abs/1511.08444v1

• DOI: https://doi.org/10.1103/PhysRevA.93.032114 (Physical Review A 93(3), 032114, 2016; title, authors, journal, volume, issue and article number from Crossref).
• URL: https://arxiv.org/abs/1511.08444v1. The e-print member bip-ent2.tex has 1772 lines and md5 97ea0260ffd180db07b62fdb44c91bc5. Locations:
  • the theorem for $n=2$ and its tightness, l. 804–812;
  • the theorem (nS) for general $n$, l. 908–922;
  • the family $|{\psi }_{\xi }^{(n)}⟩$ and its limit, l. 924–958;
  • the conjecture, l. 961–963.