arXiv:1504.05934 · Physical Review A

Closing the detection loophole in tripartite Bell tests using the W state

Károly F. Pál, Tamás Vértesi

Published Aug 4, 2015• 14 citations

Abstract

We study the problem of closing the detection loophole in three-qubit Bell tests, the experimentally most relevant case beyond the usual bipartite scenario, and show that the minimal detection efficiencies required can be considerably lowered compared to the two-qubit case. The lowest reported detection efficiency thresholds for two and three qubits so far are $\ensuremath{\sim}66.7$ and $60%$, respectively. Using the three-qubit $W$ state and a three-setting Bell inequality, we beat these thresholds and with an eight-setting Bell inequality we reach $50.13%$. We also investigate generic three-qubit states which allow us to attain a detection efficiency of $50%$ in a four-setting Bell test. We conjecture that the limit of $50%$ is unbeatable using three-qubit states and any number of measurements.

DOI: 10.1103/physreva.92.022103

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arXiv:1504.05934 · Physical Review A

Closing the detection loophole in tripartite Bell tests using the W state

Károly F. Pál, Tamás Vértesi

Published Aug 4, 2015• 14 citations

arXiv HTML PDF

Abstract

\ensuremath{\sim}66.7

and

60%

, respectively. Using the three-qubit

W

state and a three-setting Bell inequality, we beat these thresholds and with an eight-setting Bell inequality we reach

50.13%

. We also investigate generic three-qubit states which allow us to attain a detection efficiency of

50%

in a four-setting Bell test. We conjecture that the limit of

50%

is unbeatable using three-qubit states and any number of measurements.

DOI: 10.1103/physreva.92.022103

Related papers

Closing the detection loophole in multipartite Bell tests using Greenberger-Horne-Zeilinger states

Károly F. Pál, Tamás Vértesi, Nicolas Brunner

We investigate the problem of closing the detection loophole in multipartite Bell tests, and show that the required detection efficiencies can be significantly lowered compared to the bipartite case. In particular, we present Bell tests based on $n$ -qubit Greenberger-Horne-Zeilinger states, which can tolerate efficiencies as low as $38%$ for a reasonable number of parties and measurements. Even in the presence of a significant amount of noise, efficiencies below $50%$ can be tolerated, which is encouraging given recent experimental progress. Finally, we give strong evidence that, for a sufficiently large number of parties and measurements, arbitrarily small efficiencies can be tolerated, even in the presence of an arbitrary large amount of noise.We investigate the problem of closing the detection loophole in multipartite Bell tests, and show that the required detection efficiencies can be significantly lowered compared to the bipartite case. In particular, we present Bell tests based on $n$-qubit Greenberger-Horne-Zeilinger states, which can tolerate efficiencies as low as $38%$ for a reasonable number of parties and measurements. Even in the presence of a significant amount of noise, efficiencies below $50%$ can be tolerated, which is encouraging given recent experimental progress. Finally, we give strong evidence that, for a sufficiently large number of parties and measurements, arbitrarily small efficiencies can be tolerated, even in the presence of an arbitrary large amount of noise.

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Minimum detection efficiencies for a loophole-free observable-asymmetric Bell-type test

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Loophole-free quantum nonlocality often demands experiments with high complexity (defined by all parties' settings and outcomes) and multiple efficient detectors. Here, we identify the fundamental efficiency and complexity thresholds for quantum steering using two-qubit entangled states. Remarkably, it requires only one photon detector on the untrusted side, with efficiency $ε> 1/X$ , where $X \geq 2$ is the number of settings on that side. This threshold applies to all pure entangled states, in contrast to analogous Bell-nonlocality tests, which require almost unentangled states to be loss-tolerant. We confirm these predictions in a minimal-complexity ( $X = 2$ for the untrusted party and a single three-outcome measurement for the trusted party), detection-loophole-free photonic experiment with $ε= (51.6 \pm 0.4)\% $.Loophole-free quantum nonlocality often demands experiments with high complexity (defined by all parties' settings and outcomes) and multiple efficient detectors. Here, we identify the fundamental efficiency and complexity thresholds for quantum steering using two-qubit entangled states. Remarkably, it requires only one photon detector on the untrusted side, with efficiency $ε> 1/X$, where $X \geq 2$ is the number of settings on that side. This threshold applies to all pure entangled states, in contrast to analogous Bell-nonlocality tests, which require almost unentangled states to be loss-tolerant. We confirm these predictions in a minimal-complexity ($X = 2$ for the untrusted party and a single three-outcome measurement for the trusted party), detection-loophole-free photonic experiment with $ε= (51.6 \pm 0.4)\% $.

quant-pharXiv:2601.03817Jan 8, 20260 citations

\ensuremath{\sim}66.7

and

60%

, respectively. Using the three-qubit

W

state and a three-setting Bell inequality, we beat these thresholds and with an eight-setting Bell inequality we reach

50.13%

. We also investigate generic three-qubit states which allow us to attain a detection efficiency of

50%

in a four-setting Bell test. We conjecture that the limit of

50%

is unbeatable using three-qubit states and any number of measurements.

n

-qubit Greenberger-Horne-Zeilinger states, which can tolerate efficiencies as low as

38%

for a reasonable number of parties and measurements. Even in the presence of a significant amount of noise, efficiencies below

50%

can be tolerated, which is encouraging given recent experimental progress. Finally, we give strong evidence that, for a sufficiently large number of parties and measurements, arbitrarily small efficiencies can be tolerated, even in the presence of an arbitrary large amount of noise.

82.84%

can be tolerated for the two-qubit maximally entangled state. We beat this enigmatic threshold by entangling two particles with multiple degrees of freedom. The obtained upper bounds of the symmetric detection efficiency thresholds are

80.86%

73.99%

, and

69.29%

for two, three, and four copies of the two-qubit maximally entangled state, respectively. The number of measurements and outcomes in the respective cases are 4, 8, and 16. To find the improved thresholds, we use large-scale convex optimization tools, which allows us to significantly go beyond state-of-the-art results. The proof is exact up to three copies, while for four copies it is due to reliable numerical computations. Specifically, we used linear programming to obtain the two-copy threshold and the corresponding Bell inequality, and convex optimization based on Gilbert's algorithm for three and four copies of the two-qubit state. We show analytically that the symmetric detection efficiency threshold decays exponentially with the number of copies of the two-qubit state. Our techniques can also be applied to more general Bell nonlocality scenarios with more than two parties.

ε> 1/X

, where

X \geq 2

is the number of settings on that side. This threshold applies to all pure entangled states, in contrast to analogous Bell-nonlocality tests, which require almost unentangled states to be loss-tolerant. We confirm these predictions in a minimal-complexity (

X = 2

for the untrusted party and a single three-outcome measurement for the trusted party), detection-loophole-free photonic experiment with $ε= (51.6 \pm 0.4)\% $.