Graph-theoretic approach to Bell experiments with low detection
efficiency
- URL: http://arxiv.org/abs/2205.05098v4
- Date: Thu, 9 Feb 2023 11:40:55 GMT
- Title: Graph-theoretic approach to Bell experiments with low detection
efficiency
- Authors: Zhen-Peng Xu, Jonathan Steinberg, Jaskaran Singh, Antonio J.
L\'opez-Tarrida, Jos\'e R. Portillo, Ad\'an Cabello
- Abstract summary: We introduce a method to identify Bell tests requiring low $eta_rmcrit$ and relatively low dimension $d$ of the local quantum systems.
The tools presented here may allow for developing high-dimensional loophole-free Bell tests and loophole-free Bell nonlocality over long distances.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Bell inequality tests where the detection efficiency is below a certain
threshold $\eta_{\rm{crit}}$ can be simulated with local hidden-variable
models. Here, we introduce a method to identify Bell tests requiring low
$\eta_{\rm{crit}}$ and relatively low dimension $d$ of the local quantum
systems. The method has two steps. First, we show a family of bipartite Bell
inequalities for which, for correlations produced by maximally entangled
states, $\eta_{\rm{crit}}$ can be upper bounded by a function of some
invariants of graphs, and use it to identify correlations that require small
$\eta_{\rm{crit}}$. We present examples in which, for maximally entangled
states, $\eta_{\rm{crit}} \le 0.516$ for $d=16$, $\eta_{\rm{crit}} \le 0.407$
for $d=28$, and $\eta_{\rm{crit}} \le 0.326$ for $d=32$. We also show evidence
that the upper bound for $\eta_{\rm{crit}}$ can be lowered down to $0.415$ for
$d=16$ and present a method to make the upper bound of $\eta_{\rm{crit}}$
arbitrarily small by increasing the dimension and the number of settings. All
these upper bounds for $\eta_{\rm{crit}}$ are valid (as it is the case in the
literature) assuming no noise. The second step is based on the observation
that, using the initial state and measurement settings identified in the first
step, we can construct Bell inequalities with smaller $\eta_{\rm{crit}}$ and
better noise robustness. For that, we use a modified version of Gilbert's
algorithm that takes advantage of the automorphisms of the graphs used in the
first step. We illustrate its power by explicitly developing an example in
which $\eta_{\rm{crit}}$ is $12.38\%$ lower and the required visibility is
$14.62\%$ lower than the upper bounds obtained in the first step. The tools
presented here may allow for developing high-dimensional loophole-free Bell
tests and loophole-free Bell nonlocality over long distances.
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