bibkey: mehboudi2025lossincompatibility
authors: Mohammad Mehboudi, Fatemeh Rezaeinia, Saleh Rahimi-Keshari
year: 2025
title: "Measurement incompatibility under loss"
doi: 10.1103/s9sp-2fd7
url: https://arxiv.org/abs/2411.05920v3
claim: "A pure-loss channel of transmissivity τ ≤ 1/n makes every set of n measurements jointly measurable (Result 2). Result 3 asserts that some n + 1 measurements, displaced on-off photodetections, stay incompatible at τ = 1/n; the paper proves this numerically for 2 ≤ n ≤ 13 and conjectures it for arbitrary n."
strata_touched:
  - D5/S3/Quantum/Measurement/PureLossIncompatibility
license: citation-only
triage: anchor

Mehboudi–Rezaeinia–Rahimi-Keshari, measurement incompatibility under loss

M. Mehboudi, F. Rezaeinia and S. Rahimi-Keshari, Measurement incompatibility under loss, arXiv:2411.05920v3 [quant-ph] (17 January 2025); Phys. Rev. Lett. 135, 100202 (2025), DOI 10.1103/s9sp-2fd7.

Verified locator

DOI: 10.1103/s9sp-2fd7 (Phys. Rev. Lett. 135, 100202, published 4 September 2025; the journal full text was not read). Primary version read: https://arxiv.org/abs/2411.05920v3 (the latest arXiv version). Source: https://arxiv.org/pdf/2411.05920v3 (text read from the PDF):

• the paragraph “The loss channel”, Eq. (2) and the beam-splitter model of E_τ;
• Result 2, the dual E*_τ and joint measurability;
• Result 3, Eq. (4) and Fig. 2, the displaced on-off measurements;
• the introduction and the conclusions, where the general case is stated as a conjecture.

Source statements

The loss channel: “E_τ(|α⟩⟨α|) := |√τ α⟩⟨√τ α|, (2) where 0 ≤ τ ≤ 1 is the transmissivity parameter … we can describe the action of a loss channel on state ρ as E_τ(ρ) = tr₂(U_BS ρ ⊗ |0⟩⟨0| U_BS†). The unitary operator U_BS describes the beam splitter, and its action on coherent and vacuum states U_BS|α⟩|0⟩ = |√τ α⟩|√(1−τ) α⟩ gives the definition (2).”

Result 2: “A loss channel with transmissivity τ ≤ 1/n breaks the incompatibility of any arbitrary set of n measurements … denoting E*τ as the dual of the loss channel defined through tr(M^j{a_j} E_τ(ρ)) =: tr(ρ E*τ(M^j{a_j})), the set of measurements {{E*τ(M^j{a_j})}{a_j}}{j=1}^n with n ≤ ⌊1/τ⌋ is jointly measurable.”

Result 3: “There exist a set of n + 1 measurements that remain incompatible under τ = 1/n transmissivity. We prove this result numerically for 2 ≤ n ≤ 13, and conjuncture it to hold for arbitrary n.”

Eq. (4): the two-outcome measurements M^k_1 = |μ_k⟩⟨μ_k| and M^k_2 = I − M^k_1 with coherent states |μ_k⟩ of equal modulus r, separated by the phase 2π/(n + 1) (Fig. 2).

Conclusions: “We explicitly show this for 2 ≤ n ≤ 13. However, we conjecture that the same is true for higher n.”