arXiv:2609.10034

One Relative Phase Orders Operational Thresholds of Noisy Bell Pairs

Xuan Du Trinh

quant-phSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations

Abstract

Unlike the two-qubit entangled and CHSH-nonlocal sets, the steerable and Bell-nonlocal ones have no known general closed-form criterion. For any such operational ability whose failing states form a convex set closed under local unitaries, and for a Bell mixture with arbitrary noise, the $X$ part of the noise gives an upper bound on the definitive threshold, the Bell weight above which the mixture attains the ability. That bound is the tightest that the seven of the fifteen Pauli expectation values fixing the $X$ part can support. The same theorem shows that a state has the ability whenever its $X$ part does, so partial tomography of a state can certify its ability. When noise is of $X$ form, the mixture carries a relative phase between the noise and Bell coherences that cannot be gauged away by local unitaries, and it controls the threshold through interference. We prove a monotonicity law: the threshold is nondecreasing in this phase, whether or not a closed-form criterion for the ability exists.

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arXiv:2609.10034

One Relative Phase Orders Operational Thresholds of Noisy Bell Pairs

Xuan Du Trinh

quant-phSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations

arXiv HTML PDF

Abstract

X

part of the noise gives an upper bound on the definitive threshold, the Bell weight above which the mixture attains the ability. That bound is the tightest that the seven of the fifteen Pauli expectation values fixing the

X

part can support. The same theorem shows that a state has the ability whenever its

X

part does, so partial tomography of a state can certify its ability. When noise is of

X

form, the mixture carries a relative phase between the noise and Bell coherences that cannot be gauged away by local unitaries, and it controls the threshold through interference. We prove a monotonicity law: the threshold is nondecreasing in this phase, whether or not a closed-form criterion for the ability exists.

Related papers

Operational thresholds of Bell mixtures with complex X noise

Xuan Du Trinh

Noisy Bell mixtures may carry entanglement, standard teleportation usefulness, projective-measurement steerability, and optimized Cavalcanti-Jones-Wiseman-Reid (CJWR) and Clauser-Horne-Shimony-Holt (CHSH) violations. If the noise already has an ability, the mixture may lose and later recover it at distinct boundary crossings as the Bell weight decreases. We characterize ability-absence intervals and definitive thresholds for mixtures of $|Φ^+\rangle$ with arbitrary complex two-qubit $X$ noise. A closed-form singular-value flow of the Pauli correlation tensor gives the exact teleportation-useless, CJWR-satisfying, and CHSH-local intervals, while the positive partial transpose criterion gives the exact separable interval. At fixed populations and coherence magnitudes, these intervals widen as the relative phase between the $Φ$ -block coherence of the noise and the Bell coherence increases from $0$ to $π$ . The definitive thresholds form a universal chain from entanglement through teleportation usefulness and the three-setting CJWR witness to the common two-setting CJWR witness and CHSH-nonlocal threshold. Replacing teleportation usefulness by steerability gives a second chain, although their thresholds are not mutually ordered. For arbitrary pure two-qubit noise, the steerability thresholds in both directions for projective measurements and arbitrary positive operator-valued measures equal the entanglement threshold. For arbitrary product noise, an effective $X$ -state singular-value flow gives the teleportation usefulness, CJWR witness, and CHSH-nonlocal thresholds. If a local noise factor is pure, the entanglement threshold and all four steerability thresholds are zero. For generic full-rank mixed $X$ noise, finite-setting semidefinite programs give upper bounds on the unknown directional projective-measurement steerability thresholds.Noisy Bell mixtures may carry entanglement, standard teleportation usefulness, projective-measurement steerability, and optimized Cavalcanti-Jones-Wiseman-Reid (CJWR) and Clauser-Horne-Shimony-Holt (CHSH) violations. If the noise already has an ability, the mixture may lose and later recover it at distinct boundary crossings as the Bell weight decreases. We characterize ability-absence intervals and definitive thresholds for mixtures of $|Φ^+\rangle$ with arbitrary complex two-qubit $X$ noise. A closed-form singular-value flow of the Pauli correlation tensor gives the exact teleportation-useless, CJWR-satisfying, and CHSH-local intervals, while the positive partial transpose criterion gives the exact separable interval. At fixed populations and coherence magnitudes, these intervals widen as the relative phase between the $Φ$-block coherence of the noise and the Bell coherence increases from $0$ to $π$. The definitive thresholds form a universal chain from entanglement through teleportation usefulness and the three-setting CJWR witness to the common two-setting CJWR witness and CHSH-nonlocal threshold. Replacing teleportation usefulness by steerability gives a second chain, although their thresholds are not mutually ordered. For arbitrary pure two-qubit noise, the steerability thresholds in both directions for projective measurements and arbitrary positive operator-valued measures equal the entanglement threshold. For arbitrary product noise, an effective $X$-state singular-value flow gives the teleportation usefulness, CJWR witness, and CHSH-nonlocal thresholds. If a local noise factor is pure, the entanglement threshold and all four steerability thresholds are zero. For generic full-rank mixed $X$ noise, finite-setting semidefinite programs give upper bounds on the unknown directional projective-measurement steerability thresholds.

quant-pharXiv:2608.17609Aug 19, 20260 citations

Bounding the detection efficiency threshold in Bell tests using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles

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Bounding the detection efficiency threshold in Bell tests using multiple copies of the two-qubit maximally entangled state

István Márton, Erika Bene, Tamás Vértesi

In this paper we investigate the critical efficiency of detectors in order to see Bell nonlocality using multiple copies of the two-qubit maximally entangled state and local Pauli measurements which act in the corresponding qubit subspaces. It is known that for the two-qubit maximally entangled state a symmetric detection efficiency of $82.84\%$ can be tolerated using the Clauser-Horne-Shimony-Holt (CHSH) Bell test. We show that this threshold can be lowered by using multiple copies of the two-qubit maximally entangled state. We get the upper bounds $80.86\%$ , $73.99\%$ and $69.29\%$ on the symmetric detection efficiency threshold for two, three and four copies of the state, where the respective number of measurements per party are 4, 8 and 16. However, in the case of four copies the result is partly due to a heuristic method. In order to get the corresponding Bell inequalities we made use of linear programming for two copies of the state and convex optimization based on Gilbert algorithm for three and four copies of the state.In this paper we investigate the critical efficiency of detectors in order to see Bell nonlocality using multiple copies of the two-qubit maximally entangled state and local Pauli measurements which act in the corresponding qubit subspaces. It is known that for the two-qubit maximally entangled state a symmetric detection efficiency of $82.84\%$ can be tolerated using the Clauser-Horne-Shimony-Holt (CHSH) Bell test. We show that this threshold can be lowered by using multiple copies of the two-qubit maximally entangled state. We get the upper bounds $80.86\%$, $73.99\%$ and $69.29\%$ on the symmetric detection efficiency threshold for two, three and four copies of the state, where the respective number of measurements per party are 4, 8 and 16. However, in the case of four copies the result is partly due to a heuristic method. In order to get the corresponding Bell inequalities we made use of linear programming for two copies of the state and convex optimization based on Gilbert algorithm for three and four copies of the state.

Mar 18, 20211 citation

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Absorption capacity of separable noise: Bell-mixing thresholds on entanglement and teleportation

Xuan Du Trinh

We study Bell-mixing lines $ρ_λ=λΦ^+ +(1-λ)σ$ , where $Φ^+$ is a fixed Bell reference and $σ$ is a separable two-qubit noise state. Along this line there are two operational crossings: the state becomes entangled, and it reaches quantum teleportation advantage over classical strategies. We package these crossings as capacities of the noise state. The entanglement absorption capacity $C_{\rm abs}(σ)$ is the largest amount of Bell reference that $σ$ can absorb while the partial transpose remains positive. The fidelity absorption capacity $C_F(σ)$ is the largest amount of Bell reference that $σ$ can absorb while keeping the maximal teleportation fidelity at or below the classical bound $2/3$ . The thresholds corresponding to the two crossing points are obtained from the same Möbius map, $λ_* = C_{\rm abs}/(1+C_{\rm abs})$ and $λ_F = C_F/(1+C_F)$ . We derive closed-form capacities and thresholds for product noise states and separable complex $X$ noise states. For product noise, $C_{\rm abs}$ depends only on local marginal purities, while $C_F$ also depends on orientation relative to the maximally entangled reference. For $X$ noise states, both capacities are explicit in all four Bell frames. We also study three extensions: arbitrary pure-state references, the evolution of $X$ noise states and their capacities under local amplitude-damping and dephasing channels, and decomposition certificates that give lower bounds on the capacities, hence on the thresholds, for general separable noise.We study Bell-mixing lines $ρ_λ=λΦ^+ +(1-λ)σ$, where $Φ^+$ is a fixed Bell reference and $σ$ is a separable two-qubit noise state. Along this line there are two operational crossings: the state becomes entangled, and it reaches quantum teleportation advantage over classical strategies. We package these crossings as capacities of the noise state. The entanglement absorption capacity $C_{\rm abs}(σ)$ is the largest amount of Bell reference that $σ$ can absorb while the partial transpose remains positive. The fidelity absorption capacity $C_F(σ)$ is the largest amount of Bell reference that $σ$ can absorb while keeping the maximal teleportation fidelity at or below the classical bound $2/3$. The thresholds corresponding to the two crossing points are obtained from the same Möbius map, $λ_* = C_{\rm abs}/(1+C_{\rm abs})$ and $λ_F = C_F/(1+C_F)$. We derive closed-form capacities and thresholds for product noise states and separable complex $X$ noise states. For product noise, $C_{\rm abs}$ depends only on local marginal purities, while $C_F$ also depends on orientation relative to the maximally entangled reference. For $X$ noise states, both capacities are explicit in all four Bell frames. We also study three extensions: arbitrary pure-state references, the evolution of $X$ noise states and their capacities under local amplitude-damping and dephasing channels, and decomposition certificates that give lower bounds on the capacities, hence on the thresholds, for general separable noise.

quant-pharXiv:2606.31243Jul 1, 20260 citations

X

X

part can support. The same theorem shows that a state has the ability whenever its

X

part does, so partial tomography of a state can certify its ability. When noise is of

X

|Φ^+\rangle

with arbitrary complex two-qubit

X

noise. A closed-form singular-value flow of the Pauli correlation tensor gives the exact teleportation-useless, CJWR-satisfying, and CHSH-local intervals, while the positive partial transpose criterion gives the exact separable interval. At fixed populations and coherence magnitudes, these intervals widen as the relative phase between the

Φ

-block coherence of the noise and the Bell coherence increases from

0

π

. The definitive thresholds form a universal chain from entanglement through teleportation usefulness and the three-setting CJWR witness to the common two-setting CJWR witness and CHSH-nonlocal threshold. Replacing teleportation usefulness by steerability gives a second chain, although their thresholds are not mutually ordered. For arbitrary pure two-qubit noise, the steerability thresholds in both directions for projective measurements and arbitrary positive operator-valued measures equal the entanglement threshold. For arbitrary product noise, an effective

X

-state singular-value flow gives the teleportation usefulness, CJWR witness, and CHSH-nonlocal thresholds. If a local noise factor is pure, the entanglement threshold and all four steerability thresholds are zero. For generic full-rank mixed

X

noise, finite-setting semidefinite programs give upper bounds on the unknown directional projective-measurement steerability thresholds.

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.

82.84\%

can be tolerated using the Clauser-Horne-Shimony-Holt (CHSH) Bell test. We show that this threshold can be lowered by using multiple copies of the two-qubit maximally entangled state. We get the upper bounds

80.86\%

73.99\%

and

69.29\%

on the symmetric detection efficiency threshold for two, three and four copies of the state, where the respective number of measurements per party are 4, 8 and 16. However, in the case of four copies the result is partly due to a heuristic method. In order to get the corresponding Bell inequalities we made use of linear programming for two copies of the state and convex optimization based on Gilbert algorithm for three and four copies of the state.

We study Bell-mixing lines

ρ_λ=λΦ^+ +(1-λ)σ

, where

Φ^+

is a fixed Bell reference and

σ

is a separable two-qubit noise state. Along this line there are two operational crossings: the state becomes entangled, and it reaches quantum teleportation advantage over classical strategies. We package these crossings as capacities of the noise state. The entanglement absorption capacity

C_{\rm abs}(σ)

is the largest amount of Bell reference that

σ

can absorb while the partial transpose remains positive. The fidelity absorption capacity

C_F(σ)

is the largest amount of Bell reference that

σ

can absorb while keeping the maximal teleportation fidelity at or below the classical bound

2/3

. The thresholds corresponding to the two crossing points are obtained from the same Möbius map,

λ_* = C_{\rm abs}/(1+C_{\rm abs})

and

λ_F = C_F/(1+C_F)

. We derive closed-form capacities and thresholds for product noise states and separable complex

X

noise states. For product noise,

C_{\rm abs}

depends only on local marginal purities, while

C_F

also depends on orientation relative to the maximally entangled reference. For

X

noise states, both capacities are explicit in all four Bell frames. We also study three extensions: arbitrary pure-state references, the evolution of

X

noise states and their capacities under local amplitude-damping and dephasing channels, and decomposition certificates that give lower bounds on the capacities, hence on the thresholds, for general separable noise.