Selective and noise-resilient wave estimation with quantum sensor networks

Arne Hamann; Paul Aigner; Pavel Sekatski; Wolfgang D\"ur

REVIEW 3 major objections 4 minor 55 references

Selective and noise-resilient wave estimation with quantum sensor networks

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Quantum sensor networks can be programmed to ignore any chosen plane-wave noise while retaining sensitivity to a target wave, and in strong correlated noise this entangled strategy beats every product-state strategy exponentially when the…

desk verdict A credible constructive method for wave-specific DFS engineering with a real proof of a product-state exponential advantage bound, but the exponential claim rests on an unquantified strong-noise limit that needs to be stated honestly. read the letter →

arxiv 2412.12291 v1 pith:YTCGTAMG submitted 2024-12-16 quant-ph

Arne Hamann , Paul Aigner , Pavel Sekatski , Wolfgang Dür This is my paper · ORCID

classification quant-ph

keywords quantumsensornetworkplane-wavesensingdecoherence-freesubspaceHeisenbergscalingexponentialadvantagelock-inamplificationFisherinformationcorrelatednoise

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quantum sensor networks made of qubits spread out in space are sensitive to every plane wave that passes through them, so estimating one target wave is spoiled by other waves acting as noise. This paper argues that the problem can be solved by engineering: for a chosen set of sensor positions, a local bit-flip control sequence built by Gram–Schmidt orthogonalization of the signal field against the noise fields places the states $|z\rangle$ and $|-z\rangle$ in a decoherence-free subspace, where all noise waves cancel exactly while the target wave still accumulates phase. The same construction works when the noise phases are unknown, at the cost of treating cosine and sine components as independent noise generators. The paper then compares entangled strategies, which prepare an equal superposition of $|z\rangle$ and $|-z\rangle$ inside such a subspace, with all strategies that start from an unentangled product state. Its central quantitative claim is that, in the strong-noise regime, every product-state contribution to the quantum Fisher information (the quantity whose inverse bounds the variance of unbiased estimators) is exponentially suppressed by the minimal number of sensors needed to form a decoherence-free subspace, so an exponential precision advantage appears whenever at most polynomially many useful subspaces exist, for instance when the sensor count equals that minimum.

What carries the argument

The argument runs on three mechanisms. First, lock-in amplification: flipping each sensor qubit in resonance with the signal frequency makes that frequency's phase accumulate while off-resonant waves average out, filtering the signal in time. Second, a decoherence-free subspace for waves: for noise generators $\hat G_j$, the affine subspace ${\rm DFS}_\kappa = \{z : g_z(\omega_j,\vec k_j,\phi_j)=\kappa_j\ \forall j\}$ collects the bitstrings whose couplings to every noise wave are equal, so a superposition of such bitstrings loses no coherence to the noise; here the Hamming distance between two bitstrings is the number of sensor positions at which they differ. The construction itself is Gram–Schmidt orthogonalization of the field matrix against the $z$-weighted inner product, chosen so that the overlap $\langle C,F_j\rangle$ reproduces the noise eigenvalue of Eq. (7); the normalized orthogonal signal component is the control sequence, and a rectangular-wave version implements the same protection with slow control at a $4/\pi$ coupling improvement. Third, the product-state bound: a combinatorial lemma shows that any product distribution can place at most $1/\sum_{\ell=0}^{\lfloor (m-1)/2\rfloor}\binom{m}{\ell} \leq 2^{-m}$ total probability on the less likely bitstrings of a subspace with minimum Hamming distance $m$, and converting this into a variance estimate yields the quantum Fisher information bound $\|G_\kappa\|_{\rm Spec}^2\,2^{-m+1}$. Comparing the entangled value with this bound, Eqs. (45) and (46), is the formal heart of the exponential-advantage claim.

What would settle it

Take a minimal network with $n=m=4$ sensors, three noise waves, and one signal wave, and compute the exact quantum Fisher information under Gaussian noise amplitudes of increasing finite variance $\sigma^2$ for the optimal balanced-superposition strategy and the optimal product-state strategy with optimized local control. If the ratio of the two quantum Fisher informations grows only polynomially in $n$ as $\sigma^2$ increases instead of exponentially, then the exponential-advantage claim holds only in the idealized twirling limit; a direct check of how well Eq. (20) approximates Eq. (9) for finite $\sigma^2$ would locate where the separation begins.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a constructive recipe that turns wave estimation into a decoherence-free subspace problem, plus a sharp bound showing how badly product states fare in the worst case. Given $d$ noise plane waves, $n$ sensor locations, and a signal wave of the same frequency, the authors build the field matrix $F$ whose rows are the noise fields (both cosine and sine components when the phases are unknown) and whose last row is the signal field. Orthogonalizing these rows with the $z$-weighted inner product $\langle x,y\rangle = \sum_i z_i \int x_i(t)y_i(t)\,dt$ yields an orthogonal signal component $s^\perp$; using a normalized version of $s^\perp$ as the local control sequence makes the noise generators act trivially on $|z\rangle$ and $|-z\rangle$ while the signal generator keeps a nonzero coupling. Under the strong-noise assumption, where the noise channel is exactly the twirling projection onto affine decoherence-free subspaces, the quantum Fisher information of a balanced superposition state in the best subspace equals $\max_\kappa \|G_\kappa\|_{\rm Spec}^2$, while for any product initial state the contribution of a subspace whose bitstrings are pairwise at Hamming distance at least $m$ is at most $\|G_\kappa\|_{\rm Spec}^2\,2^{-m+1}$. The claimed exponential separation follows because the entangled strategy needs only one useful subspace, whereas the product state must distribute probability over many well-separated bitstrings, and each bitstring carries exponentially little weight.

Load-bearing premise

The exponential-advantage proof assumes the ideal strong-noise limit, where the noise channel is exactly the twirling projection onto decoherence-free subspaces, meaning the noise amplitudes have effectively infinite variance; the paper does not say how large the variance must be for this projection to be a good approximation, so finite-noise corrections could shrink or eliminate the exponential gap.

Editorial extensions

If this is right

A network with $n>d$ sensors (known noise phases) or $n>2d$ sensors (unknown phases) can be made simultaneously insensitive to $d$ noise waves and sensitive to a target wave of the same frequency using only local bit-flip control, with no global operations during the sensing interval.
Preparing an equal superposition of $|z\rangle$ and $|-z\rangle$ inside the engineered decoherence-free subspace restores Heisenberg scaling, a quadratic improvement in precision with sensor number, because the quantum Fisher information is set by the squared spectral range of the signal generator rather than by the sum of single-sensor couplings.
In the strong-noise regime any product-state strategy is bounded by Eq. (46), so the entangled strategy wins exponentially whenever the number of decoherence-free subspaces with nonzero signal coupling grows at most polynomially in the sensor count; a minimal network with $n=m$ is one such case.
Point-symmetric sensor layouts reduce the requirement for unknown phases from $n>2d$ to $n>d$, because the sine components of the noise fields cancel by symmetry.
The same orthogonalization recipe generalizes to other monochromatic wave families, to synchronized time-dependent fields whose time translates span a finite-dimensional space, and to higher-dimensional decoherence-free subspaces built with auxiliary control qubits.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Although the paper states the exponential bound for waves, the proof only uses the Hamming-distance structure of the decoherence-free subspaces, so the same $2^{-m+1}$ suppression should apply to any commuting dephasing noise whose decoherence-free subspace consists of well-separated bitstrings; the wave model is best read as a concrete instance of that more general mechanism.
A natural next step the authors leave implicit is a finite-noise cross-over analysis: replacing the infinite-variance twirling map by Gaussian noise with variance $\sigma^2$ would presumably turn the exponential advantage into a continuous gain that weakens as $\sigma$ drops, and quantifying that threshold would make the result directly usable for experiments.
The numerical comparison shows that the quantum Fisher information of product states can require entangling measurements to extract, so a formal statement about the measurement entanglement needed to saturate Eq. (46) would make the practical gap between entangled and unentangled strategies even larger than the state-preparation gap alone.
The Fourier-transform viewpoint in the appendix points to an uncertainty trade-off between frequency and direction selectivity on one side and the spacetime extent of the sensor network on the other; spelling this out as a quantitative bound could guide how many sensors are needed to separate a given set of waves.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The genuinely new thing here is the constructive three-step procedure (field matrix, Gram-Schmidt, control sequence) that turns any generic sensor placement into a protocol protecting |z> and |-z> from d noise waves while keeping signal sensitivity. That part is clean, worked out for fast and slow control, and the paper earns credit for spelling out the n>d and n>2d conditions, even if the genericity caveat is buried in a footnote. The slow-control factor of 4/pi is nice and the examples actually illustrate the method.

Result 2 is the load-bearing piece for the exponential advantage, and it is a real theorem: for product states and DFSs with Hamming distance at least m, the QFI contribution is bounded by 2^{-m+1} ||G_k||^2. The proof in Appendix E is detailed and self-contained. That is a solid contribution on its own and generalizes the specific examples in [17] and [29].

The soft spot is the one the stress-test flags, and it is real. Result 2 and equations (45)-(46) are derived in the strong-noise limit where the map in Eq. (20) is exactly the twirling projection. The paper never quantifies how large the noise variance must be for that approximation to hold, and finite variance leaves inter-DFS coherence that can bridge the exponential gap. The noiseless QFI of |+>^n is O(n), so a small residual coherence can restore polynomial scaling. The authors should either prove a stability bound in terms of the variance or state plainly that the exponential advantage is established in the infinite-variance limit, not in the n~d regime per se. This is a fixable gap but it should be fixed before the claim is taken at face value.

The secondary issue is the abstract overstating the scope: the exponential advantage statement should carry the strong-noise qualifier. The genericity of sensor positions is also under-emphasized; n>d alone is not sufficient when the signal is linearly dependent on the noise fields at the sensor locations. These are tightening issues, not fatal ones. The derivations are self-contained and I do not see circularity or hidden fitting.

The paper deserves a serious referee. It is a theory paper with a real constructive method, a genuine product-state bound, and honest generalizations. I would send it to review with a request that the authors quantify the strong-noise regime and qualify the n>d claim in the abstract and Result 1. I would cite it for Result 2 and the construction once those qualifications are in place.

Referee Report

3 major / 4 minor

Summary. This paper studies estimation of plane-wave amplitudes by networks of qubit sensors in the presence of other plane waves treated as noise. It proposes a Gram-Schmidt-based method (Result 1) to construct control sequences and decoherence-free subspaces (DFSs) that contain |z> and |-z> and are insensitive to d noise waves, for n>d (or n>2d for unknown phases) sensor locations. It then derives general bounds (Result 2, Eqs. (41)-(46)) showing that, under a strong-noise twirling approximation, product-state strategies have exponentially suppressed QFI contributions from any DFS whose bitstrings are pairwise Hamming-separated by m, implying an exponential entanglement advantage when the number of non-trivial DFSs is polynomial. Numerical examples illustrate the construction and the advantage, and extensions to spherical harmonics/time-dependent fields and higher-dimensional DFSs are outlined.

Significance. The constructive protocol is clear and potentially useful; the QFI-based comparison between entangled and product strategies is a valuable general framework, and the numerical section suggests the effect is real. The paper is self-contained, provides detailed analytical proofs and concrete examples, and extends prior work [17-20] in a natural way. However, the advertised exponential advantage rests on two currently unsecured pillars: a proof of the combinatorial QFI bound that contains invalid algebraic steps, and an unquantified strong-noise limit. If these are repaired, the paper would be a solid contribution to quantum sensing.

major comments (3)

[Eqs. (41)-(44) and Appendix E4b/c] The proof of Result 2 is not valid as written. In Appendix E4b the claimed identities, e.g. $\binom{m}{(m-1)/2}^{-m}=2^{-m}((m-1)/m)^m$, are algebraically false, and the intermediate bound $1/\sum_{\ell=0}^{\lfloor(m-1)/2\rfloor}\binom{m}{\ell} \le \binom{m}{\lfloor(m-1)/2\rfloor}^{-m}$ fails already at m=4 (the left side is 1/5 and the right side is 4^{-4}). Moreover, Eq. (43) does not follow from the maximization in Appendix E4c: when $p_1>p_\kappa/2$, the derivation gives $4p_\kappa\mathrm{Var}\le 4p_1p_\perp/p_\kappa \|G_\kappa\|^2_{\rm Spec}$, which can exceed $2p_\perp\|G_\kappa\|^2_{\rm Spec}$. A concrete product-state example with n=m=3 and iid per-sensor weights $q_i=0.7368$ gives $p_1=0.4$, $p_2=0.0182$, $p_\kappa=0.4182$, and $4p_\kappa\mathrm{Var}=0.277g^2$ against $2p_2\|G_\kappa\|^2_{\rm Spec}=0.146g^2$, contradicting Eq. (43). Since Eqs. (42) and (46) are derived from (43), the stated $2^{-m+1}$ constants are not established. The exponential-in-m separation may survive with a corrected constant, but the present derivation needs to be reworked.
[Sec. IV C 1, Eqs. (20), (35)-(36), (46)] The exponential-advantage bound assumes that the noise map is exactly the twirling projection Eq. (20), which holds only in the limit of infinite noise variance (or, as stated in Sec. III C, 'very large' variance). For finite variance, coherences between different affine DFSs survive and can contribute to the QFI of product states. The noiseless product state $|+\rangle^{\otimes n}$ has QFI $4\sum_i g_i^2=O(n)$, whereas the twirled bound Eq. (46) can be exponentially small in m; hence a residual coherence of size $O(2^{-m}/n)$ is enough in principle to erase the claimed separation. The manuscript does not quantify how large the noise variance must be, nor how finite-variance corrections scale with n and m. The abstract states the exponential advantage without this qualification; at minimum the statement should be restricted to the strong-noise limit and, ideally, supplemented by a perturbation bound.
[Result 1 and Sec. IV A2 / footnote 2 in Sec. IV C2] Result 1's guarantee that n>d (or n>2d) sensors suffice is too strong without a genericity assumption. As acknowledged in Sec. IV A2, the orthogonal signal component $s^\perp$ is nonzero only if the signal restricted to the sensor locations is linearly independent of the noise fields; this is necessary but not sufficient, and footnote 2 in Sec. IV C2 concedes that for some sensor sizes the signal can be linearly dependent with the noise waves on the sensor locations. For point-symmetric networks the same caveat applies to the symmetric components. Result 1 should state 'for generic sensor positions such that the signal is not in the span of the noise fields at the sensors' (with the analogous condition for point-symmetric networks), otherwise the claimed guarantee is false.

minor comments (4)

[Eq. (41)] The notation $P_{\lfloor(m-1)/2\rfloor}\binom{m}{l}$ is unclear; it should be written as $\sum_{\ell=0}^{\lfloor(m-1)/2\rfloor}\binom{m}{\ell}$.
[Throughout] There are several typographical errors: 'tansform' (Sec. III B), 'applyed' (Sec. IV A), 'risidual' (Sec. IV A3b), 'follwos' (Sec. VI C1), and 'grantees' (Sec. VI C1).
[Fig. 4(bottom)] The quantity 'max F.I.' for separable measurements is not defined in the caption; the optimization over projective qubit measurements should be described explicitly in the text or caption.
[Appendix E4c] The stated maximizer $x_{\max}=p_j-p_\kappa/2$ has the wrong sign; it should be $p_\kappa/2-p_j$.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivations are self-contained constructions and packing bounds.

full rationale

The paper's main claims are derived from explicit definitions rather than imported conclusions. Result 1 is a constructive Gram-Schmidt procedure: the scalar product (25) is chosen so that the orthogonality condition (28) reproduces the generator eigenvalues (7), and the control sequence is then engineered to place |z> and |-z> in a DFS; this is a construction, not a fit. Result 2 follows from the Hamming-distance packing argument of Lemma 1 and the product-distribution lemmas proved in Appendix E, leading to Eqs. (41)-(44); these inequalities do not assume the claimed exponential advantage. The exponential-advantage comparison (45)-(46) combines the achievable GHZ-state QFI with the product-state upper bound, both derived in the paper. The self-citations [17]-[20] are used for DFS optimality and for the strong-noise twirling form (20), but Eq. (20) is stated as the infinite-variance limit of the noise map (9) and is not fitted, while the optimality results are independent prior work. The assumption that the noise variance is large enough to make Eq. (20) exact is a robustness concern about finite-noise corrections, not a circular step, so it does not affect the circularity score.

Assumptions & free parameters 3 free parameters · 7 assumptions · 2 invented entities

The central claim rests on the commuting dephasing sensor model, instantaneous control, the strong-noise twirling limit, and generic sensor positions. No free parameters are fitted to data; the example-specific choices (delta, virtual wave count, lambda) are construction knobs that do not determine the qualitative results. No new physical entities are postulated.

free parameters (3)

offset angle delta = pi/20
Hand-chosen in the numerical comparison (Sec. IV C 2) to avoid accidental linear dependence between the signal and noise fields on the sensor circle; it affects the numerical QFI values but not the qualitative exponential trend.
number of virtual noise waves = n/2
Hand-chosen in the approximate-DFS example (Sec. V B); determines the degree of protection across the noisy half-plane but is a construction choice, not fitted to data.
slowdown parameter lambda = 0 < lambda < 1 (arbitrary)
Introduced in Sec. VI C 2 to build higher-dimensional DFSs by controlling the relative phase accumulation; it is a free design knob, not fitted.

assumptions (7)

domain assumption All wave generators commute, so the total evolution is a product of commuting unitaries diagonal in the sigma_z basis (Eq. (5)).
Fundamental to the model in Sec. II; if the waves coupled via non-commuting operators, the DFS construction would break.
domain assumption Bit-flip control is instantaneous; fast control can realize any integrable C_i(t) in [-1,1], while slow control is restricted to +/-1 (Eq. (3) and Sec. II).
Used throughout; finite-duration pulses would modify the Fourier response in Sec. III B.
domain assumption In the strong-noise limit, the noise map N in Eq. (9) becomes the twirling projector onto affine DFSs (Eq. (20)).
Load-bearing for the QFI decomposition (Eq. (36)) and the product-state bound (Result 2); finite-noise corrections are not analyzed.
domain assumption Unknown wave phases are uniformly distributed over [0,2*pi], allowing cosine/sine decomposition (Eq. (24)).
Used in Sec. IV A 1 to double the field matrix rows for unknown phases.
domain assumption Sensor positions are generic, i.e., the signal field restricted to the sensor locations is linearly independent of the noise fields; otherwise the n > d guarantee fails (see footnote in Sec. IV C 2).
Needed for the orthogonalization to have a nonvanishing signal component in Result 1.
domain assumption A point-symmetric sensor array with equal z_i in the initial state is sensitive only to the symmetric (cosine) wave components (Sec. IV A 4).
Used to halve the number of rows for unknown phases in the point-symmetric case.
standard math Fourier transform of control sequences, Gram-Schmidt orthogonalization, harmonic addition theorem, and quantum Fisher information formulas are used as standard tools.
Standard background in quantum metrology and Fourier analysis; relied on without proof in Secs. III and IV.

invented entities (2)

virtual noise waves
purpose: Construct an approximate DFS in Sec. V B by treating n/2 fictitious noise waves in the uncertain direction sector as exact noises to be decoupled.
These are auxiliary mathematical constructs, not physical sources; they do not represent new physics and have no falsifiable handle.
auxiliary control qubits per sensor
purpose: In Sec. VI C 2, auxiliary qubits controlled by a logical state allow realization of a higher-dimensional DFS with signal generator proportional to (|z><z| - |-z><-z|) tensored with (|0><0| + lambda |1><1|).
A device-level addition, not a new physical entity; no independent evidence beyond the construction itself.

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Cite this review

Pith. "Pith review of Selective and noise-resilient wave estimation with quantum sensor networks." pith.science (2026). https://pith.science/paper/YTCGTAMG

@misc{pith2026241212291,
  author       = {Pith},
  title        = {Pith review of: Selective and noise-resilient wave estimation with quantum sensor networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YTCGTAMG}},
  note         = {Machine review of arXiv:2412.12291}
}

read the original abstract

We consider the selective sensing of planar waves in the presence of noise. We present different methods to control the sensitivity of a quantum sensor network, which allow one to decouple it from arbitrarily selected waves while retaining sensitivity to the signal. Comparing these methods with classical (non-entangled) sensor networks we demonstrate two advantages. First, entanglement increases precision by enabling the Heisenberg scaling. Second, entanglement enables the elimination of correlated noise processes corresponding to waves with different propagation directions, by exploiting decoherence-free subspaces. We then provide a theoretical and numerical analysis of the advantage offered by entangled quantum sensor networks, which is not specific to waves and can be of general interest. We demonstrate an exponential advantage in the regime where the number of sensor locations is comparable to the number of noise sources. Finally, we outline a generalization to other waveforms, e.g., spherical harmonics and general time-dependent fields.

Figures

Figures reproduced from arXiv: 2412.12291 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Constructing a sensor protected from 3 waves is done [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 8.** Figure 8: FIG. 8. In this example, the sensor positions are chosen such [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 9.** Figure 9: FIG. 9. The plots show the control sequences [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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It is diagonal in the product basis |z⟩, and denote the corespondent real eigenvalues as gz(ω, ⃗k, ϕ) such that gz(ω, ⃗k, ϕ) |z⟩ = ˆG(ω, ⃗k, ϕ) |z⟩

Coupling to a wave The previous section has shown that the total time evolution factors into the evolution with respect to the individual waves, but what is the relation between the waves parameter, the sensor positions and the applied time control of the network? This section...
[48]

QFI for a GHZ state This section contains a straightforward generalization of [18] The QFI for a signal with ( ω, ⃗k, ϕ) and the GHZ state |z⟩+|−z⟩√ 2 is given by Fω,⃗k,ϕ = 4gz(ω, ⃗k, ϕ)2|d|2 = 4 X i zi cos(ϕ + γi)|F(Ci(⃗ x, t))| !2 |d|2 (A1) with γi = −i log F(Ci(⃗ x,t)) |F(C...
[49]

The DFS is given by {z|gz(ω, ⃗k, ϕ) ∀ω, ⃗k, ϕ}, where ω, ⃗k and ϕ are the parameters descibing the noise waves

QFIM for the initial state within DFS This is a straightforward generalization of [20]. The DFS is given by {z|gz(ω, ⃗k, ϕ) ∀ω, ⃗k, ϕ}, where ω, ⃗k and ϕ are the parameters descibing the noise waves. The QFIM elements for a signal w = (ω, ⃗k, ϕ) and w′ = (ω′, ⃗k′, ϕ′) and for ...
[50]

QFI for classical strategies F + w := 4 X z∈{±1}N 1 2N gz(ω, ⃗k, ϕ)2 (A4) = 4 X z∈{±1}N 1 2N gz(ω, ⃗k, ϕ)2 (A5) = 4 X z∈{±1}N 1 2N Re NX i=1 zieiϕF(Ci(⃗ x, t)) !2 (A6) introducing F = Re(eiϕ(F(C1(⃗ x, t)), F(C2(⃗ x, t)), ...)⊺) = 4 2N X z∈{±1}N (z⊺F )2 (A7) = 4 2N X z∈{±1}N (F...
[51]

General DFS Here, we explicitly provide the 3 steps to construct a DFS for the scenario in Section V A. a. Determine the field matrix The field matrix contains the local field strength and its temporal correlation. Determining it differs depending on whether the phases are kno...
[52]

iid strategy

Preparatory results for separable strategies Let DFS be a set containing some bitstrings of lenght n. We say that it has Hamming distance m DH (z, z′) ≥ m ∀z ̸= z′ ∈ DFS. (E1) 20 FIG. 9. The plots show the control sequences C f ast k , C slow k for the example V A with known p...
[53]

For any product probability density on m-bitstrings (with m ≥ d ≥ 0), and some bitstrings z such that P (z) ≥ P (¬z) one has Pr(Cd(¬z)) ≥ P (¬z) m d P (z) P (¬z) d m ≥ P (¬z) m d

Lemmas required for Result 2 Lemma 5. For any product probability density on m-bitstrings (with m ≥ d ≥ 0), and some bitstrings z such that P (z) ≥ P (¬z) one has Pr(Cd(¬z)) ≥ P (¬z) m d P (z) P (¬z) d m ≥ P (¬z) m d . (E24) Proof. Without loss of generality we can chose z = 0...
[54]

Let us now come back to our set DFS= {z1,

Proof of the Result 2 Proof. Let us now come back to our set DFS= {z1, . . . ,zk}. For a given product probability density, we would like to understand what are the possibility values of P (zi). Without loss of generality we can arrange the vecotors such that these probabiliti...
[55]

F urther derivations within Section IV C 1 a. Explicit derivation of equation (36) FG = F M DFSκ Πκ |Ψ⟩0 ⟨Ψ|0 Πκ ! = X κ pκF (|Ψ⟩κ ⟨Ψ|κ) (QFI linear in direct sums) = 4 X κ pκVarΨκ (G) (QFI formula) = 4 X κ pκ ⟨Ψ|0 ΠκG2Πκ |Ψ⟩0 pκ − ⟨Ψ|0 ΠκGΠκ |Ψ⟩0 pκ 2! (Definition variance) =...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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S. W¨ olk, P. Sekatski, and W. D¨ ur, Quantum Science and Technology 5, 045003 (2020)

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Determine the field matrix F

work page

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Notice that this scalar product is chosen to reproduce the eigenvalues in Eq

Orthogonalize F To construct a DFS that contains the states |z⟩ and |−z⟩, use the Gram–Schmidt algorithms with rows of F as vectors and with the scalar product ⟨x, y⟩ = nX i=1 zi Z T 2 − T 2 xi(t)yi(t)dt (25) to orthogonalize F . Notice that this scalar product is chosen to reproduce the eigenvalues in Eq. (7), for certain input vectors. After orthogonali...

work page

[5] [3]

For point symmetric sensors, the phases can be considered to be known, and n > dis sufficient

Identify the control sequence For known (unknown) wave phases, the protocol is guar- anteed to be successful if n > d (n > 2d). For point symmetric sensors, the phases can be considered to be known, and n > dis sufficient

work page

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(23) The first d rows correspond to the noise fields, and the last row corresponds to the signal field

Determine the field matrix F If the noise phases are known, the field matrix is an d + 1 times n function-valued matrix with the elements Fji = cos ⃗k⊺ j ⃗ xi − ωt + ϕj . (23) The first d rows correspond to the noise fields, and the last row corresponds to the signal field. For unknown noise phases, notice that cos ⃗k⊺ j ⃗ xi − ωt + ϕj = Aj cos ⃗k⊺ j ⃗ xi...

work page

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Identify the control sequence a. For fast control, i.e., when C f ast i (t) can effec- tively be any integrable function between −1 and 1, we can directly use a normalized orthogonal signal compo- nent as a control sequence C f ast i (t) = s⊥ i (t) maxi′ sup− T 2 ≤t≤ T 2 |s⊥ i′ (t)| . (27) The normalization comes from the fact that the absolute value of t...

work page

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almost full

Point symmetric sensors We say that a sensor network with the initial state |ψ⟩ is point symmetric with respect to the point ⃗ pif for each sensor position ⃗ xi there exists another sensor at position ⃗ xi′ such that ⃗ xi − ⃗ p= −⃗ xi′ + ⃗ p and zi = zi′ (34) for all z such that ⟨z|ψ⟩ ̸= 0. For a sensor network symmetric around a point ⃗ pit is natural to...

work page