Achieving the Multi-parameter Quantum Cram\'er-Rao Bound with Antiunitary Symmetry

Aonan Zhang; Ben Wang; Kaimin Zheng; Liang Xu; Lijian Zhang; Qian Xie

REVIEW 3 major objections 5 minor 65 references

Achieving the Multi-parameter Quantum Cram\'er-Rao Bound with Antiunitary Symmetry

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Two-copy conjugate states can saturate the multi-parameter quantum Cramér-Rao bound for both parameters at once, with half the weighted mean squared error of two identical copies.

desk verdict The analytical core is sound and the experiment is plausible, but without raw data and error bars the experimental claim is not yet independently verifiable; worth a serious referee. read the letter →

arxiv 2411.14929 v1 pith:3ANIPNB2 submitted 2024-11-22 quant-ph

Ben Wang , Kaimin Zheng , Qian Xie , Aonan Zhang , Liang Xu , Lijian Zhang This is my paper · ORCID

classification quant-ph MSC 81P4581P5081P15 PACS 03.65.Ta03.65.Wj42.50.Ex

keywords multi-parameterquantummetrologyCramér-Raoboundantiunitarysymmetryweakcommutativityconditionmutuallyconjugatemodelancilla-assistedphotonicexperimentwalk

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

The paper shows that requiring a family of quantum states to be invariant under an antiunitary operation removes the usual incompatibility between estimating different parameters at the same time. For the qubit state $|\psi_\lambda\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle$, the two-copy mutually conjugate model $|\psi_\lambda\rangle \otimes |\psi^*_\lambda\rangle$ obeys the weak commutativity condition everywhere, so the multi-parameter quantum Cramér-Rao bound is saturable for $\theta$ and $\phi$ simultaneously. The paper derives the chain $C^Q_\lambda(|\psi\rangle\otimes|\psi^*\rangle, W) = \tfrac{1}{2} C^Q_\lambda(|\psi\rangle,W) = \tfrac{1}{4} C^H_\lambda(|\psi\rangle,W) = \tfrac{1}{2} C^H_\lambda(|\psi\rangle^{\otimes 2},W)$ with $W=\mathrm{diag}(1,\sin^2\theta)$, and verifies it experimentally on single photons: the weighted mean squared error of the mutually conjugate model reaches the scalar QCRB and is half that of two identical copies. A second model with an ancilla, the AAMCM, reaches the same precision as the maximally entangled model while using only separable measurements.

What carries the argument

The load-bearing object is the conjugation operator $\vartheta$ and the global antiunitary symmetry condition $\Theta\rho_\lambda\Theta^\dagger=\rho_\lambda$ for every $\lambda$. A model with this symmetry has vanishing mean Uhlmann curvature, $U_{ij}=0$, which is precisely the weak commutativity condition needed to saturate the multi-parameter quantum Cramér-Rao bound. The symmetry also yields explicit optimal measurements: when the parameterized states decompose into parameter-independent eigenvectors with real coefficients, those eigenvectors form the optimal POVM; for the MCM this is a four-outcome Bell measurement, and for the AAMCM a separable measurement. The quantum-walk implementation realizes these POVMs deterministically in the polarization and path degrees of freedom of single photons.

What would settle it

Insert a controlled random relative phase between the two paths before the collective Bell measurement: the model then loses global antiunitary symmetry, and the paper's claim implies that no measurement can reach the scalar quantum Cramér-Rao bound value $\mathrm{Tr}(WQ^{-1})/m$ for the MCM, while the unbroken experiment does; observing the predicted degradation would confirm the symmetry as the load-bearing resource.

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Extended reading notes

Core claim

The central claim is that global antiunitary symmetry is a sufficient and practically available resource for simultaneous optimal estimation of multiple parameters. Concretely, the mutually conjugate model $\{|\psi_\lambda\rangle\otimes|\psi^*_\lambda\rangle\}$ for estimating $(\theta,\phi)$ in a qubit satisfies the weak commutativity condition at every point of the parameter space, so an optimal measurement exists that attains the quantum Cramér-Rao bound for both parameters without trade-off. The paper also proves the quantitative improvement over the conventional parallel strategy: the scalar bound for the conjugate pair is exactly half the Holevo bound for two identical copies, a factor-of-two saving in weighted mean squared error. The photonic experiment reproduces the predicted weighted mean squared errors for the MCM and shows that the ancilla-assisted conjugate model matches the maximally entangled model's precision with a separable measurement.

Load-bearing premise

The claimed factor-of-two advantage assumes that producing the conjugate state $|\psi^*_\lambda\rangle$ costs no more than producing $|\psi_\lambda\rangle$; if conjugation is expensive or unavailable, the comparison against the parallel-copy strategy is not resource-fair.

Editorial extensions

If this is right

The mutually conjugate strategy saturates the multi-parameter quantum Cramér-Rao bound for $(\theta,\phi)$ with no trade-off, a limit that two identical copies cannot reach with separable measurements.
The weighted mean squared error of the MCM is exactly half that of the parallel two-copy strategy, so the resource saving is a factor of two whenever preparing the conjugate state costs the same as preparing the original.
The ancilla-assisted mutually conjugate model reaches the same precision as the maximally entangled model while requiring only a separable measurement, replacing entanglement with antiunitary symmetry.
Any quantum statistical model with global antiunitary symmetry inherits compatible optimal measurements, giving a general recipe for parameter-independent optimal POVMs.
For real Hamiltonians, conjugation reduces to inverse evolution; a no-go theorem rules out exact inversion with a single use of $U$, though qubit systems admit exact deterministic time-reversal in special constructions, so the improvement is tied to how the conjugate operation is supplied.

Reading between the lines

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

Editorial inference: the factor-of-two statement is a resource-counting claim; counted in total channel uses rather than state copies, the advantage hinges on the price of implementing $U^*$, and the paper's own discussion shows that price is not always zero.
Editorial inference: the same symmetry argument should extend to tensor powers of conjugate pairs and to mixed states obeying $\Theta\rho_\lambda\Theta^\dagger=\rho_\lambda$, potentially yielding parameter-independent optimal measurements in other estimation problems wherever a conjugation map exists.
Editorial inference: because the AAMCM needs only separable measurements, it is a plausible template for distributed or networked quantum sensing where collective measurements across nodes are costly; an adaptive multi-round version of the experiment would test that directly.
Editorial inference: the paper's link to 'nonlocality without entanglement' suggests the metrological gain is a geometric property of classically correlated quantum states, so analogous gains might appear in classical statistical models whose likelihood functions share the same symmetry.

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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

Bottom line: this is a solid theory result with an under-documented experiment. The analytical core holds up. For the MCM state |ψ>|ψ*>, the Bell POVM gives a classical Fisher information matrix equal to twice the QFIM, so the multiparameter QCRB is saturated without trade-off. The AAMCM QFIM and its separable POVM also check out, and the factor-of-two comparison with the parallel HCRB follows from Eq. (6). The weak-commutativity-to-GAS theorem is imported from Miyazaki and Matsumoto [37], so the conceptual novelty is modest, but the two concrete models, their optimal POVMs, and the photonic implementation are new. No fitted parameters enter the derivation; the benchmarks are independent analytical models. The experiment is well designed: single photons, polarization and path degrees of freedom, and deterministic collective measurements via quantum walks. The plotted mean-square errors are consistent with the theoretical curves. That is all one can say. There are no raw counts, no error bars on the weighted MSE points, no state-preparation fidelity numbers, and no analysis code. With 500 qubits per trial, the statistical uncertainty on each point is non-negligible, so the agreement with the QCRB lines is not independently assessable. The parameter range is restricted to θ,ϕ ∈ [0,π/2] to avoid estimator ambiguity from periodic probabilities; that is disclosed and reasonable, but it means the 'no trade-off' claim is demonstrated only on a subset. The Bell POVM also has singular Fisher information at some parameter values, and those non-regular points are not analyzed. The cost of preparing the conjugate state is a real concern for general systems, and the paper honestly flags the no-go theorem for U^{-1} in the Summary. For the qubit models actually demonstrated, U* = σ_y U σ_y, so conjugate encoding costs the same as U plus two fixed Pauli gates; the resource-fairness worry is therefore minor here. The citation pattern is appropriate, giving clear credit to the antecedent theory. Who this is for: people working on multiparameter quantum estimation, especially the QCRB-versus-HCRB gap, and experimental quantum optics groups. It deserves a serious referee. The referee should request raw data, error bars or credible uncertainty quantification, and ideally the analysis code before accepting the experimental claim. My verdict: the theory is likely correct, the experiment is likely fine, but the evidence as presented is not yet complete.

Referee Report

3 major / 5 minor

Summary. The paper theoretically proposes and experimentally demonstrates two photonic quantum statistical models with global antiunitary symmetry (GAS): the mutually conjugate model (MCM), consisting of |ψλ⟩⊗|ψλ*⟩, and the ancilla-assisted mutually conjugate model (AAMCM). For estimating (θ,ϕ) of a qubit with weight matrix W=diag(1,sin²θ), the authors show that these models saturate the multi-parameter scalar quantum Cramér-Rao bound (QCRB) with explicit parameter-independent POVMs, while the MCM achieves half the weighted mean square error of the parallel-copy strategy. A single-photon experiment using polarization and path degrees of freedom reports WMSE values for the MCM, parallel model, AAMCM, and maximally entangled model, claiming agreement with the theoretical QCRB/HCRB curves.

Significance. The theoretical core is sound and valuable: the closed-form derivations of the Fisher information matrices and the explicit Bell-type POVMs provide a clear, parameter-free demonstration of how GAS resolves the incompatibility of multi-parameter estimation. The factor-of-two improvement of the MCM over the parallel model is a clean and falsifiable prediction, and for qubit systems the conjugate unitary U* can be implemented with a single use of U plus fixed gates, which mitigates the resource-cost concern that naturally arises for general U. However, the experimental evidence is not reported to the standard needed to verify the central experimental claim: there are no error bars, raw counts, fidelity calibrations, or statistical analysis. The significance of the paper therefore rests on whether the experimental data can be provided and quantified; as written, the experimental demonstration is unverified.

major comments (3)

[Experimental setup and results, Figs. 3 and 4] The manuscript reports no error bars, no raw detection counts, no number of repeated trials, no state-preparation fidelity calibration, and no analysis code for any of the four models. With m=500 qubits per trial, the statistical uncertainty of each WMSE point is substantial, and the plotted points alone cannot establish that the experiment "matches" the theoretical curves. Please provide uncertainty quantification (e.g., bootstrap over repeated experimental runs or Bayesian credible intervals), raw data for a representative set of parameters, and a precise description of how the Bayesian estimator was implemented (including the update rule and prior). This is load-bearing because the central claim is an experimental demonstration of QCRB saturation.
[Experimental setup and results, parameter restriction] The authors restrict θ and ϕ to the interval [0, π/2] "to ensure a one-to-one correspondence between estimators and measurement results." However, the theoretical saturation claim in Eq. (6) and the claimed optimality of the POVMs are stated in the text for the full parameter range θ∈[0,π], ϕ∈[0,2π]. Moreover, the QFIMs of both the MCM and the AAMCM are singular at θ=0 (and the MCM also at θ=π), so the notion of saturating the QCRB at those points is not well defined. The manuscript should state the exact parameter domain on which the experimental claim holds and clarify whether the one-to-one restriction reflects a limitation of the specific estimator, the POVM, or the model itself.
[Experimental setup and results, Fig. 3 comparison to asymptotic bounds] The experimental WMSE points are compared directly to asymptotic QCRB and HCRB lines computed from Fisher information. With m=500 and a Bayesian estimator using a uniform prior on a restricted interval, the expected mean square error contains finite-sample, prior, and boundary corrections that are not quantified. A valid comparison requires either deriving the finite-sample expected WMSE of the implemented estimator (analytically or by simulation) or demonstrating numerically that all finite-size corrections are below the resolution of the plot. As written, the statement that the experiment "match[es] the theoretical predictions" is not yet a falsifiable claim.

minor comments (5)

[References] The reference list is disordered: Ref. [64] is inserted between [37] and [39], and the same reference appears twice (as [64] and again at the end). Please renumber and deduplicate the bibliography.
[Abstract and Eq. (6)] The abstract states the precision "is improved at least twice compared to conventional encoding strategies," but the paper establishes exactly a factor of two relative to the parallel model, not a lower bound against all conventional strategies. Please either prove a general lower bound or qualify the statement to name the specific comparison.
[Fig. 3 legend] The legend contains "MUB Theory" but the text never defines what this line represents; please specify whether it is the asymptotic HCRB of the parallel model under the MUB-like POVM and how it was computed.
[Fig. 4 labels] The axis labels and legend in Fig. 4 are garbled ("2 2 QCRB of ..."), making it impossible to read which curve corresponds to which model. Please regenerate the figure with clear labels.
[Summary and discussions] For qubit systems, the conjugation property U* = σ_y U σ_y means the MCM encoding can be implemented with a single use of U plus fixed Pauli gates; the paper should state this explicitly in the main text, since the discussion of the no-go theorem for general U may otherwise leave readers with the impression that the MCM advantage is not resource-fair even for the qubit platform demonstrated here.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theoretical derivation is self-contained and the external antiunitary-symmetry theorem is independent support.

full rationale

The paper's central claims are derived analytically from closed-form expressions and an external theorem, not from fitted parameters or self-citation. The MCM and AAMCM states are defined explicitly; their QFIMs are computed directly (e.g., Eq. (6) and Eq. (8)). The Bell POVM of Eq. (10) and the separable POVM of Eq. (11) are constructed explicitly, and the saturation of the QCRB follows from the classical Fisher information matrix equalling the QFIM. The key theoretical input, that global antiunitary symmetry implies the weak commutativity condition Uij=0, is cited to Ref. [37] by Miyazaki and Matsumoto, which are not authors of the present paper; this is independent, external support rather than self-citation. The comparison benchmarks (parallel model, MEM) are also independent analytical models with their own HCRB and QFI expressions. The weight matrix W=diag(1,sin^2 theta) is chosen as the Fubini-Study metric, not fitted to data. The experimental section lacks raw counts and uncertainty bars, which is a verification or reporting weakness, but it is not a circularity: the reported agreement with theory does not reduce to a fitted parameter or a self-referential definition. Therefore no circular step satisfying the evidentiary standard is present.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on existing theoretical results (weak commutativity, GAS theorem) and on two concrete states whose QFIs and POVMs are computed analytically. No new physical entities are postulated. The only hand-chosen object relevant to the stated precision numbers is the weight matrix W, and the experimental comparison relies on an unquantified fidelity assumption.

free parameters (1)

Weight matrix W = diag(1, sin^2 theta)
Chosen to equal the Fubini-Study metric. All scalar QCRB and HCRB values, including the factor-of-two improvement, are defined with respect to this weighting; a different W changes the ratios.

assumptions (5)

domain assumption The weak commutativity condition Uij=0 is necessary and sufficient for saturating the multi-parameter QCRB.
Invoked in Theoretical framework to connect QCRB attainability to vanishing mean Uhlmann curvature; attributed to Refs. [24,32-34].
domain assumption Parameterized states with global antiunitary symmetry always satisfy the weak commutativity condition and admit optimal POVMs.
Core theoretical input imported from Ref. [37] (Miyazaki and Matsumoto); the paper builds its two models on this theorem rather than proving it.
domain assumption For pure states the Holevo bound is asymptotically attained by separable measurements on single copies and scales as 1/n with the number of copies.
Used to compute the benchmark HCRB for the parallel model in Eq. (6); attributed to Refs. [40,41,22].
domain assumption The experimental wave-plate and beam-displacer network realizes the intended unitary operations and POVMs with negligible or unquantified error.
No fidelity calibration, count rates, or error analysis is reported; the plotted MSE is assumed to come from the ideal POVMs.
ad hoc to paper True parameters are restricted to [0, pi/2] to ensure a one-to-one map between estimators and outcomes.
Introduced in Experimental setup and results; the demonstrated parameter range excludes the full sphere used in the theoretical model.

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Pith. "Pith review of Achieving the Multi-parameter Quantum Cram\'er-Rao Bound with Antiunitary Symmetry." pith.science (2026). https://pith.science/paper/3ANIPNB2

@misc{pith2026241114929,
  author       = {Pith},
  title        = {Pith review of: Achieving the Multi-parameter Quantum Cram\'er-Rao Bound with Antiunitary Symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3ANIPNB2}},
  note         = {Machine review of arXiv:2411.14929}
}

read the original abstract

The estimation of multiple parameters is a ubiquitous requirement in many quantum metrology applications. However, achieving the ultimate precision limit, i.e. the quantum Cram\'er-Rao bound, becomes challenging in these scenarios compared to single parameter estimation. To address this issue, optimizing the parameters encoding strategies with the aid of antiunitary symmetry is a novel and comprehensive approach. For demonstration, we propose two types of quantum statistical models exhibiting antiunitary symmetry in experiments. The results showcase the simultaneous achievement of ultimate precision for multiple parameters without any trade-off and the precision is improved at least twice compared to conventional encoding strategies. Our work emphasizes the significant potential of antiunitary symmetry in addressing multi-parameter estimation problems.

Figures

Figures reproduced from arXiv: 2411.14929 by the authors.

**Figure 2.** FIG. 2. The experimental setup for proving the metrological advantages of quantum states with GAS. There are five modules [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Experimental results for estimating ( [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Experimental results for estimating ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

65 extracted references · 54 canonical work pages

[37]

Chaichian and R

M. Chaichian and R. Hagedorn, Symmetries in quantum mechanics: from angular momentum to supersymmetry (IoP, 1998)

work page 1998
[1]

Tsang, H

M. Tsang, H. M. Wiseman, and C. M. Caves, Physical review letters 106, 090401 (2011)

work page 2011
[2]

Similar to MCM, two HWPs are used to put |10⟩ and |11⟩ of the quantum state into the same path and they would interfere at an HWP with a ro- tation angle 22.5◦ to realize the measurement in Eq. (11). The MEM is shown in Fig. 2(e). After passing through φ 0 8 π 4 π 3 8 π 2 π 8 π 4 π 3 8 π 2 π θ 0.004 0.006 0.008 0.010 WMSE HCRB QCRB MUB Theory MCM Exp MUB ...

work page
[3]

The POVM in Eq

The polarization qubit will be manipulated by the group of wave plates to generate the quantum state in Eq (5) and the path DOF acts as an ancillary qubit. The POVM in Eq. (12) is realized by a BD and a HWP with a rotation angle 22.5 ◦. The measurement results of all of the models are recorded by single-photon detectors. The experimental results of the MC...

work page
[4]

C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017)

2017
[5]

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, R. Ad- hikari, et al., Classical and quantum gravity 32, 074001 (2015)

work page 2015
[6]

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, R. Ad- hikari, et al., Nature Photonics 7, 613 (2013)

work page 2013
[7]

Z. Hou, Z. Zhang, G.-Y. Xiang, C.-F. Li, G.-C. Guo, H. Chen, L. Liu, and H. Yuan, Phys. Rev. Lett. 125, 020501 (2020)

work page 2020

Show all 65 references

[8]

B. Xia, J. Huang, H. Li, H. Wang, and G. Zeng, Nature Communications 14, 1021 (2023)

2023
[9]

Hou, J.-F

Z. Hou, J.-F. Tang, H. Chen, H. Yuan, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Science Advances7, eabd2986 (2021)

2021
[10]

Z. Hou, Y. Jin, H. Chen, J.-F. Tang, C.-J. Huang, H. Yuan, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 126, 070503 (2021)

2021
[11]

Yuan, Phys

H. Yuan, Phys. Rev. Lett. 117, 160801 (2016)

2016
[12]

Liu and H

J. Liu and H. Yuan, Phys. Rev. A 96, 042114 (2017)

2017
[13]

L. O. Conlon, T. Vogl, C. D. Marciniak, I. Pogorelov, S. K. Yung, F. Eilenberger, D. W. Berry, F. S. Santana, R. Blatt, T. Monz, P. K. Lam, and S. M. Assad, Nature Physics 19, 351 (2023)

2023
[14]

Hou, J.-F

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.- D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Nature Communications 9, 1414 (2018)

2018
[15]

S. L. Braunstein and C. M. Caves, Physical Review Let- ters 72, 3439 (1994)

1994
[16]

J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Journal of Physics A: Mathematical and Theoretical 53, 023001 (2019)

2019
[17]

M. G. A. PARIS, International Journal of Quantum In- formation 07, 125 (2009)

2009
[18]

Lu and X

X.-M. Lu and X. Wang, Phys. Rev. Lett. 126, 120503 (2021)

2021
[19]

H. Chen, Y. Chen, and H. Yuan, Phys. Rev. Lett. 128, 250502 (2022)

2022
[20]

A. S. Holevo, Probabilistic and statistical aspects of quan- tum theory, Vol. 1 (Springer Science & Business Media, 2011)

2011
[21]

Hayashi, Asymptotic theory of quantum statistical in- ference: selected papers(World Scientific, 2005)

M. Hayashi, Asymptotic theory of quantum statistical in- ference: selected papers(World Scientific, 2005)

2005
[22]

Albarelli, J

F. Albarelli, J. F. Friel, and A. Datta, Phys. Rev. Lett. 123, 200503 (2019)

2019
[23]

J. S. Sidhu, Y. Ouyang, E. T. Campbell, and P. Kok, 6 Phys. Rev. X 11, 011028 (2021)

2021
[24]

Demkowicz-Dobrza´ nski, W

R. Demkowicz-Dobrza´ nski, W. G´ orecki, and M. Gut ¸˘ a, Journal of Physics A: Mathematical and Theoretical 53, 363001 (2020)

2020
[25]

Gut ¸˘ a and J

M. Gut ¸˘ a and J. Kahn, Physical Review A 73, 052108 (2006)

2006
[26]

Hayashi and K

M. Hayashi and K. Matsumoto, Journal of Mathematical Physics 49, 102101 (2008)

2008
[27]

Kahn and M

J. Kahn and M. Gut ¸˘ a, Communications in Mathematical Physics 289, 597 (2009)

2009
[28]

Yamagata, A

K. Yamagata, A. Fujiwara, and R. D. Gill, The Annals of Statistics 41, 2197 (2013)

2013
[29]

Chen and H

H. Chen and H. Yuan, Tradeoff relations for simul- taneous measurement of multiple incompatible observ- ables and multi-parameter quantum estimation (2023), arXiv:2310.11925 [quant-ph]

2023 arXiv
[30]

Zhang and J

J. Zhang and J. Suzuki, Qestoptpovm: An iterative al- gorithm to find optimal measurements for quantum pa- rameter estimation (2024), arXiv:2403.20131 [quant-ph]

2024 arXiv
[31]

Zhang, H.-M

M. Zhang, H.-M. Yu, H. Yuan, X. Wang, R. Demkowicz- Dobrza´ nski, and J. Liu, Phys. Rev. Res.4, 043057 (2022)

2022
[32]

Yu and J

H.-M. Yu and J. Liu, Quanestimation.jl: An open- source julia framework for quantum parameter estima- tion (2024), arXiv:2405.12066 [quant-ph]

2024 arXiv
[33]

Ozawa, Physics Letters A 320, 367 (2004)

M. Ozawa, Physics Letters A 320, 367 (2004)

2004
[34]

S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza´ nski, Physical Review A 94, 052108 (2016)

2016
[35]

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, M. Kim, A. Datta, M. Barbieri, and I. A. Walmsley, Nature communications 5, 1 (2014)

2014
[36]

Carollo, B

A. Carollo, B. Spagnolo, and D. Valenti, Scientific reports 8, 1 (2018)

2018
[38]

Uhlmann, Science China Physics, Mechanics & As- tronomy 59, 630301 (2016)

A. Uhlmann, Science China Physics, Mechanics & As- tronomy 59, 630301 (2016)

2016
[39]

Miyazaki and K

J. Miyazaki and K. Matsumoto, Quantum 6, 665 (2022)

2022
[40]

Matsumoto, Journal of Physics A: Mathematical and General 35, 3111 (2002)

K. Matsumoto, Journal of Physics A: Mathematical and General 35, 3111 (2002)

2002
[41]

S. M. Kay, Fundamentals of statistical signal processing: estimation theory (Prentice-Hall, Inc., 1993)

1993
[42]

[12, 13, 63, 64]

See Supplemental Material at [url] for additional infor- mation about the detailed calculation and experimental methods, which includes Refs. [12, 13, 63, 64]
[43]

L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad, npj Quantum Information 7, 1 (2021)

2021
[44]

Carollo, B

A. Carollo, B. Spagnolo, A. A. Dubkov, and D. Valenti, Journal of Statistical Mechanics: Theory and Experi- ment 2019, 094010 (2019)

2019
[45]

Bengtsson and K

I. Bengtsson and K. ˙Zyczkowski, Geometry of quantum states: an introduction to quantum entanglement(Cam- bridge university press, 2017)

2017
[46]

Ac ´ ın, Phys

A. Ac ´ ın, Phys. Rev. Lett.87, 177901 (2001)

2001
[47]

Tsang, F

M. Tsang, F. Albarelli, and A. Datta, Physical Review X 10, 031023 (2020)

2020
[48]

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, Nature Physics 4, 282 (2008)

2008
[49]

B. P. Williams, R. J. Sadlier, and T. S. Humble, Phys. Rev. Lett. 118, 050501 (2017)

2017
[50]

C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 59, 1070 (1999)

1999
[51]

Schuck, G

C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, Phys. Rev. Lett. 96, 190501 (2006)

2006
[52]

Chiribella and D

G. Chiribella and D. Ebler, New Journal of Physics 18, 093053 (2016)

2016
[53]

G. J. Pryde, J. L. O’Brien, A. G. White, and S. D. Bartlett, Phys. Rev. Lett. 94, 220406 (2005)

2005
[54]

M. T. Quintino, Q. Dong, A. Shimbo, A. Soeda, and M. Murao, Phys. Rev. A 100, 062339 (2019)

2019
[55]

Sedl´ ak, A

M. Sedl´ ak, A. Bisio, and M. Ziman, Phys. Rev. Lett.122, 170502 (2019)

2019
[56]

Yoshida, A

S. Yoshida, A. Soeda, and M. Murao, Phys. Rev. Lett. 131, 120602 (2023)

2023
[57]

M. T. Quintino, Q. Dong, A. Shimbo, A. Soeda, and M. Murao, Phys. Rev. Lett. 123, 210502 (2019)

2019
[58]

Rubino, L

G. Rubino, L. A. Rozema, A. Feix, M. Ara´ ujo, J. M. Zeuner, L. M. Procopio, ˇCaslav Brukner, and P. Walther, Science Advances 3, e1602589 (2017)

2017
[59]

Schiansky, T

P. Schiansky, T. Str¨ omberg, D. Trillo, V. Saggio, B. Dive, M. Navascu´ es, and P. Walther, Optica10, 200 (2023)

2023
[60]

Chang, N

L. Chang, N. Li, S. Luo, and H. Song, Physical Review A 89, 042110 (2014)

2014
[61]

Str¨ omberg, P

T. Str¨ omberg, P. Schiansky, M. T. Quintino, M. An- tesberger, L. Rozema, I. Agresti, ˇC. Brukner, and P. Walther, arXiv:2211.01283 (2022)

2022 arXiv
[62]

Gisin and S

N. Gisin and S. Popescu, Physical Review Letters 83, 432 (1999)

1999
[63]

J.-F. Tang, Z. Hou, J. Shang, H. Zhu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Physical review letters 124, 060502 (2020)

2020
[65]

Kurzy´ nski and A

P. Kurzy´ nski and A. W´ ojcik, Phys. Rev. Lett. 110, 200404 (2013)

2013
[66]

Z. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, Phys. Rev. Lett. 114, 203602 (2015)

2015

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [37]

Chaichian and R

M. Chaichian and R. Hagedorn, Symmetries in quantum mechanics: from angular momentum to supersymmetry (IoP, 1998)

work page 1998

[2] [1]

Tsang, H

M. Tsang, H. M. Wiseman, and C. M. Caves, Physical review letters 106, 090401 (2011)

work page 2011

[3] [2]

Similar to MCM, two HWPs are used to put |10⟩ and |11⟩ of the quantum state into the same path and they would interfere at an HWP with a ro- tation angle 22.5◦ to realize the measurement in Eq. (11). The MEM is shown in Fig. 2(e). After passing through φ 0 8 π 4 π 3 8 π 2 π 8 π 4 π 3 8 π 2 π θ 0.004 0.006 0.008 0.010 WMSE HCRB QCRB MUB Theory MCM Exp MUB ...

work page

[4] [3]

The POVM in Eq

The polarization qubit will be manipulated by the group of wave plates to generate the quantum state in Eq (5) and the path DOF acts as an ancillary qubit. The POVM in Eq. (12) is realized by a BD and a HWP with a rotation angle 22.5 ◦. The measurement results of all of the models are recorded by single-photon detectors. The experimental results of the MC...

work page

[5] [4]

C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017)

2017

[6] [5]

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, R. Ad- hikari, et al., Classical and quantum gravity 32, 074001 (2015)

work page 2015

[7] [6]

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, R. Ad- hikari, et al., Nature Photonics 7, 613 (2013)

work page 2013

[8] [7]

Z. Hou, Z. Zhang, G.-Y. Xiang, C.-F. Li, G.-C. Guo, H. Chen, L. Liu, and H. Yuan, Phys. Rev. Lett. 125, 020501 (2020)

work page 2020