trureturing — Discovering truth

bibkey: zhanguptahillery2024detectorsensor
authors: C. Zhan, H. Gupta, M. Hillery
year: 2024
title: Optimizing Initial State of Detector Sensors in Quantum Sensor Networks
doi: 10.1145/3655028
url: https://arxiv.org/abs/2306.17401v6
claim: Section 5, Conjecture 1 proposes a uniform coefficient-square state on any weight partition minimizing the common final-state overlap as an optimal initial detector-sensor state; Conjecture 2 claims that the coefficient-square average of two probes with equal error and unequal coefficient-squares has strictly smaller error.
strata_touched:
  - D5/S3/Quantum/Measurement/DetectorSensorDickeOptimality
  - D5/S3/Quantum/Measurements/DetectorSensorStrictAveragingRefutation
license: citation-only
triage: anchor

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DOI: 10.1145/3655028

Source: https://arxiv.org/abs/2306.17401v6

ACM Transactions on Quantum Computing 5(2), Article 10 (2024).

Statement

Section 5, Conjecture 1, page 10:

Consider the ISO problem, with the unitary operator 𝑈, initial state |𝜓⟩, and final states {|𝜙ᵢ⟩} as defined therein. Recall that the eigenvalues of 𝑈 are {𝑒^{+𝑖𝜃}, 𝑒^{−𝑖𝜃}}. Let 𝑛 ≥ 3 be the number of sensors. For a given 𝜃 ∈ (0,𝑇] ∪ [180 − 𝑇, 180), degrees, where 𝑇 is from Theorem 2, the optimal initial state |𝜓⟩ for the ISO problem is as follows. Let 𝑆ₗ be the partition that minimizes (𝑅ₗ + cos(2𝜃)𝐿ₗ)/(𝑅ₗ + 𝐿ₗ), where 𝑅ₗ and 𝐿ₗ are as defined in the previous section. The coefficients of the optimal solution are such that their coefficient-squares are given by: |𝜓ⱼ|² = 1/|𝑆ₗ| ∀ |𝜓ⱼ|² ∈ 𝑆ₗ |𝜓ⱼ|² = 0 ∀ |𝜓ⱼ|² ∉ 𝑆ₗ

Section 5, Conjecture 2 (arXiv v6, optimal.tex, label conj:avg):

For a given $U$, consider two initial states $|\psi ⟩={\sum }_j{\psi }_j|j⟩$ and $|{\psi }^{\prime }⟩={\sum }_j{\psi }_j^{\prime }|j⟩$ such that (i) they are two unequal coefficient-squares, i.e., for some $j$, $|{\psi }_j{|}^2\ne |{\psi }_j^{\prime }{|}^2$, and (ii) they have the same objective function value, i.e., $P(|\psi ⟩,U)=P(|{\psi }^{\prime }⟩,U)$. We claim that the “average” state given by $|{\psi }_{avg}⟩={\sum }_j\sqrt{\frac{|{\psi }_j{|}^2+|{\psi }_j^{\prime }{|}^2}{2}}|j⟩$ has a lower objective function value, i.e., $P(|{\psi }_{avg}⟩,U)<P(|{\psi }^{\prime }⟩,U)$.

Encoding and boundary

The tensor eigenbasis is Fin n → Bool, with true denoting u₊ and false denoting u₋. The number of true entries gives the weight. Angles are measured in radians. The threshold is Real.arccos (-((q - 1) / q)) / 2, where q = (n + 1) / 2 uses natural-number division. The real-equation counts are the cardinalities of weight-k words with equal and unequal selected bits. A weight-l class contains Nat.choose n l words.

The repository conclusion compares every normalized competitor and every global equal-prior POVM with an attaining POVM for every proposed state, allowing arbitrary coefficient phases. Permutation averaging of Gram decompositions, a positive semidefinite trace certificate, monotonicity of the square-root diagonal, and square-root-measurement attainment supply the proof. The source’s stronger strict-averaging Conjecture 2 is not required.

The full-angle combination, ancillary and mixed probes, unequal priors, multiple simultaneous firing events, and noisy channels are outside this conclusion.

For Conjecture 2 the objective $P(\psi ,U)$ is one minus the supremum of isoSuccess over all equal-prior POVMs with outcomes Fin n, and the average state takes the nonnegative square root of the mean coefficient-square at every word. The ratio minimized in Conjecture 1 is symmetric under exchanging weights $k$ and $n-k$, so for odd $n$ two distinct layers are optimal; the repository refutes Conjecture 2 with the two optimal layers at $n=3$, $\theta =\pi /4$.