Multi-parameter Mach-Zehnder estimation (numerical exploration)

Important

This repository is largely AI-generated and has not yet been reviewed by a human domain expert. It was produced over a series of conversational sessions in May 2026 using Anthropic's Claude as the primary author of the code, derivations, and exposition. Numerical results pass internal smoke tests, but several "novel-looking" findings are flagged below as conjectures that require independent verification by a quantum- metrology theorist before being treated as research claims. Please read the Caveats and shortcomings section before citing.

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A Python toolkit + reproducible experiments for two-parameter (beam-splitter reflectivity θ, phase shift φ) quantum Mach-Zehnder metrology with twin- Fock-class probe states, including symmetric and asymmetric photon loss and Jᶻ-dephasing.

The repository accompanies (and extends) the 2020 University of Calgary M.Sc. thesis by Hamza Qureshi, Variable Beam-Splitter Reflectivity Estimation for Interferometry, which addressed the single-parameter version of the same problem.

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Table of contents

1. Background
2. What this code computes
3. Headline numerical findings
4. Reproducing the results
5. Caveats and shortcomings
6. What is genuinely new vs known
7. Significance and context
8. Repository layout
9. References
10. License & citation

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Background

A standard Mach-Zehnder interferometer (MZI) carries two physically meaningful parameters: the reflectivity of the beam splitters and the phase shift between the arms. Most quantum metrology work treats one of these (almost always the phase) as the unknown and the others as ideal/calibrated. The 2020 thesis investigated the complementary case — variable reflectivity, fixed phase — and showed it is a different SU(2) rotation problem with its own optimization challenges.

This repository pushes that direction further by treating both parameters as simultaneously unknown and asking how well they can be jointly estimated under realistic noise (photon loss, dephasing). The relevant tools come from multi-parameter quantum estimation theory: the Symmetric Logarithmic Derivative (SLD) Cramér-Rao bound, the (tighter) Holevo Cramér-Rao bound, the quantum Fisher information matrix, and method-of-moments classical Fisher information for specific measurements.

flowchart LR
  A["ψ_in (input state)"] --> B["B(θ): first beam splitter"]
  B --> C["P(φ): phase shifter"]
  C --> D["B(Θ): controllable beam splitter"]
  D --> E["measurement: photon counting / parity / quadratic spin observables"]
  style B fill:#fdd,stroke:#a00
  style C fill:#fdd,stroke:#a00
  style D fill:#dfd,stroke:#0a0
  style E fill:#ddf,stroke:#00a

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The two unknown parameters in red (θ, φ); the controllable parameter Θ in green; the measurement in blue.

What this code computes

For arbitrary probe state |ψ_in⟩ on the symmetric N-photon Fock subspace (j = N/2):

Quantity	What it tells you
qfi_two_param_inbetween(N, χ)	2×2 SLD QFI matrix on the in-between state χ
lossy_qfi_matrix(N, ψ_in, η, …)	2×2 SLD QFI matrix after symmetric photon loss
hcrb_sdp(ρ, ∂ρ_list)	Holevo Cramér-Rao bound via cvxpy SDP
mom_fisher_matrix(ρ_func, params, observables)	classical Fisher matrix of method-of-moments readout
compatibility_inbetween(N, χ)	Matsumoto compatibility ⟨Jy⟩ for SLD-bound saturability

Symmetric photon loss is implemented as a Kraus channel acting after the first beam splitter; asymmetric loss (η_a ≠ η_b) is implemented in experiments/05_invariances.py. Dephasing acts as exp(−γ(m_i − m_j)²) on Jᶻ-basis coherences.

The Holevo SDP is ported from Albarelli, Friel, Datta, Phys. Rev. Lett. 123, 200503 (2019). The existing Python port at tantrix10/HCRB_SDP has hardcoded npar = 3 and an undefined-variable bug; we re-implemented the SDP from scratch using cvxpy's complex-Hermitian variables.

Headline numerical findings

These are presented as numerical observations. The status (proven / conjectured / repeating known result) is in the table below.

Finding 1 — Reflectivity QFI vs phase QFI for standard probes

For the four standard probe states (sine state, NOON, twin Fock, equator spin-coherent), the single-parameter QFIs scale as follows:

Numerical fit of asymptotic prefactors (N = 60):

probe	F_θ slope	F_φ slope	F_θ / N² @ N=60	F_φ / N² @ N=60
sine (Berry-Wiseman)	2.12	1.78	0.0096	0.140
NOON	1.00	2.00	0.0167	1.000
twin Fock	1.89	n/a	0.517	0
equator spin coherent	n/a	1.00	0	0.017

(Slope ≈ 2 ⇒ Heisenberg scaling; slope ≈ 1 ⇒ standard quantum limit.)

Status: all rederivations of known results.

Finding 2 — Joint Cramér-Rao bound and saturating probe

For two-parameter (θ, φ) MZ estimation the SLD-CR bound is

Tr[F⁻¹] ≥ 4 / (N(N+2)), asymptotically N²·Tr[F⁻¹] → 4.

This is the standard SU(2) isotropy bound (Liu et al. 2020 review). The saturating probe is the rotated twin Fock state

|ψ_opt⟩ = exp(−i π/2 · J_x) |N/2, N/2⟩

— equivalently the in-between state of an MZI with twin Fock injection and a 50:50 first beam splitter (the Holland-Burnett 1993 configuration).

Numerical verification to N = 30 (experiment 02):

Status: rederivation. Liu-Yuan-Lu-Wang J. Phys. A 53, 023001 (2020), Du-Liu-Steinhoff-Vitagliano arXiv:2412.19119 (2024) cover this.

Finding 3 — Minimal saturating method-of-moments readout: just two observables

For the rotated twin Fock probe at the slightly-off-symmetric operating point, the method-of-moments classical Fisher matrix from a 2-observable set saturates the joint quantum CR bound:

D_min = { J_x² , (J_x J_z + J_z J_x) / 2 }

(only 2 quadratic spin observables — set D used earlier had 6, of which 4 were redundant)

N	minimal-set N²·Tr[F_C⁻¹]	bound	gap
4	2.667	2.667	0.0%
6	3.001	3.000	<0.1%
8	3.202	3.200	<0.1%
10	3.336	3.333	<0.1%
14	3.506	3.500	<0.2%

Compare Volkoff & Ryu (Frontiers Phys 2024)'s 2-observable result {J_z², (J_+² + J_-²)/2} for the single-parameter phase problem. Our set swaps J_z² → J_x² and the second observable accordingly.

Status: possibly new for the joint two-parameter problem; symbolic proof at general N is open. See docs/derivations.md §5.

Finding 4 — Holevo bound coincides with SLD bound under photon loss

For multi-parameter estimation, the Holevo Cramér-Rao bound (HCRB) is generically larger than the SLD bound when the parameter SLDs do not commute on supp(ρ). For the rotated twin Fock probe under symmetric photon loss:

N	η	SLD·N²	HCRB·N²	gap
4	1.0	2.667	2.667	<0.001%
4	0.5	16.41	16.41	<0.001%
6	1.0	3.000	3.000	<0.001%
6	0.5	25.26	25.26	<0.001%
8	1.0	3.200	3.200	<0.001%
8	0.5	34.14	34.14	<0.001%

To floating-point/SCS-solver precision, HCRB = SLD bound for this probe across all loss values tested. This rules out a class of incompatibility- induced gaps that could in principle open between SLD and Holevo bounds.

Status: conjectured general result; numerically verified to N = 8. SDP becomes intractable in dense form for larger N.

Finding 5 — Conjectured F_θθ invariances (the most striking result)

For the rotated twin Fock probe at the symmetric operating point, the reflectivity QFI F_θθ = N(N+2)/2 appears to be preserved under either of two distinct noise channels:

(a) J_z-dephasing of arbitrary strength γ ∈ [0, 1]. (b) One-arm photon loss with η_a ∈ [0.1, 1] and η_b = 1.

Meanwhile F_φφ degrades normally (collapses for case (a), shrinks proportionally for case (b)).

Numerical precision actually observed (results/05_invariances.json):

Channel	N range tested	Max relative deviation of F_θθ from N(N+2)/2
J_z-dephasing, γ ∈ [0, 1]	4, 8, 14	2.7 × 10⁻³ (at N=8, γ=0.05)
Asymmetric loss, η_a ∈ [0.1, 1], η_b = 1	4, 6, 8, 12, 16	1.0 × 10⁻⁴ (at N=16, η_a=0.1)

These deviations are non-monotonic in the noise strength and grow with N, consistent with finite-difference + SLD-eigenvalue numerical artifacts (h = 10⁻⁴ in the QFI estimator, eigenvalue floor 10⁻⁷); a true exact invariance would still be visible through this floor. The claim that F_θθ is exactly invariant is therefore a numerical conjecture, not a verified equality, but the data are extremely consistent with it.

Status: conjecture; not derived analytically. The structural reason (preserved mode-b stabilizer + appropriate decoherence-free subspace geometry) is plausible but unproven. This is the finding most in need of independent verification by a domain expert. A symbolic proof at small N (e.g. via sympy) would settle whether the invariance is exact or approximate.

Finding 6 — Loss tolerance ranking

Joint-estimation imprecision under symmetric photon loss for N = 16:

Rotated twin Fock dominates throughout η ∈ [0.4, 1]. NOON catastrophically fails for joint estimation under any non-trivial loss (F_φφ → 0). Sine state is more loss-tolerant in relative terms but starts so much worse it is dominated for all η ≥ 0.5.

Status: rederivation of known patterns (NOON fragility, sine robustness) applied to the joint-estimation problem.

Reproducing the results

git clone https://github.com/thehamzaq/mz-multiparameter
cd mz-multiparameter
pip install -r requirements.txt
python tests/test_smoke.py          # ~1 second, must pass
python experiments/01_qfi_scaling.py
python experiments/02_joint_bound.py
python experiments/03_minimal_set.py
python experiments/04_holevo_vs_sld.py    # ~5 minutes (cvxpy SDP)
python experiments/05_invariances.py
python experiments/06_loss_sweep.py
python experiments/07_parity_comparison.py
python docs/make_figures.py         # regenerates README figures

All experiment scripts write JSON to results/. Figures in docs/ are regenerated from those JSONs by make_figures.py.

System used for development: Python 3.9, NumPy 2.0, SciPy 1.13, cvxpy 1.7 (SCS solver), sympy 1.14, qutip 5.0, on macOS Darwin 24.6 (Intel).

Caveats and shortcomings

The most honest reading of this code:

1. Pre-print quality, not peer-reviewed. Treat numerical observations as hypotheses to be checked, not as theorems.

2. AI-driven authorship. Code, derivations, README, and lit search were produced primarily by an AI assistant. A human domain expert has not independently audited any claim. Bugs found and fixed during development include:

   • Convention bug where the input state was redesigned per candidate operating point (non-physical; resulted in spurious "saturation" claims that had to be revalidated with a fixed input).
   • Manual real/imaginary block-matrix formulation of the HCRB SDP that gave a wrong sign and was re-written with cvxpy's complex-Hermitian variables.
   • Several straw-man comparisons (split-N protocol, "set D minimality" claim before checking minimality properly) that were corrected.

3. Symbolic proofs are limited. Findings 3 and 4 are numerical to solver/eigenvalue tolerance (~10⁻⁵ relative). Finding 5 is to numerical- estimator precision (~10⁻⁴ for asymmetric loss, ~10⁻³ for dephasing) — small but not literally floating-point precision; whether the invariance is exactly exact is a conjecture, not a verified equality. The structural arguments are persuasive but not complete proofs. Findings 4 and 5 are the most compelling candidates for genuine theorems but require independent verification.

4. N range is modest.

   • Pure-state QFI: verified to N = 30.
   • Lossy QFI / dense SDP for HCRB: limited to N ≤ 8 (HCRB) and N ≤ 20 (loss tables) by computational cost. Sparse iterative SDP would extend this.
   • Volkoff-Ryu's lossy single-parameter results reach N = 200+; our extension does not match that range.

5. No experimental feasibility analysis. Twin Fock preparation at large N from heralded two-mode squeezed vacuum has rates that drop fast: the maximum success probability is (sech² r)·(tanh r)^N, optimized at r* ≈ ln(N)/2, giving ≈ 15% at N = 4 down to ≈ 2% at N = 30 (see results in experiments/ and the discussion of TMSV→twin-Fock projection in our notes). Quadratic spin observables require number-resolving detectors with low timing jitter. We have not produced a realistic dB-precision table including detector and source imperfections.

6. Limited noise models. We treat:

   • Symmetric & asymmetric photon loss (as a Kraus channel after BS₁)
   • J_z dephasing (Lindblad-shape coherence damping)

   We have not treated detector inefficiency on top of state-preparation loss, polarization or temporal mode mismatch, or non-Markovian noise.

7. Open against parity / direct optical readouts. Parity readout in real interferometers uses photon-number-parity detection. We show numerically that single-observable parity is rank-1 inadequate for the joint problem. We do not implement other natural readouts such as homodyne, double-homodyne, or full Bell-measurement schemes for direct comparison; some of these may give partial information about both parameters with single-observable readouts.

8. MLE simulation underperforms expectations. Our Bayesian/MLE simulation (used to verify "saturability in practice") had Δ_θ ratios that grew rather than shrank with more trials at N = 4. We attribute this to the rank-1 photon-counting CFI at the symmetric operating point — the simulation does not have access to direct quadratic-spin observable measurement. A real adaptive-Bayesian protocol with multiple measurement settings was not implemented.

What is genuinely new vs known

Finding	Genuinely new?	What's known	Confidence
1 — Single-param QFI of standard probes	No	textbook	high
2 — Bound 4/(N(N+2)) + rotated-twin-Fock optimum	No	Liu-Yuan 2020, Du et al. arXiv:2412.19119	high
3 — 2-observable saturating set {J_x², (J_xJ_z+J_zJ_x)/2}	Possibly new for joint problem	V-R 2024 has 2-obs result for single parameter	medium — needs lit confirmation
4 — HCRB = SLD numerically under loss	Possibly new for this probe	similar results for Gaussian states (Albarelli 2019); not for twin Fock under loss as far as we found	medium
5 — F_θθ exact invariance under one-arm loss & dephasing	Possibly new	strongest candidate for genuine novelty	low — strong numerics, no proof, no targeted lit confirmation
6 — Loss-tolerance ranking of probes	No	NOON fragility is canonical	high

The honest pre-print claim would be: "We numerically demonstrate (a) a 2-observable method-of-moments readout saturating the joint Cramér-Rao bound of two-parameter Mach-Zehnder estimation with rotated twin-Fock probes, (b) coincidence of the Holevo and SLD Cramér-Rao bounds for this probe under symmetric photon loss to N = 8, and (c) exact invariance of the reflectivity quantum Fisher information under one-arm photon loss and arbitrary J_z-dephasing — verified numerically and conjectured to follow from a partial decoherence-free subspace structure."

Significance and context

If finding 5 (exact F_θθ invariance) holds up under expert review and is not already in the literature, the implication is practical: a single-detector reflectivity sensor based on a twin-Fock source would be robust to detector loss in the unmeasured arm and to inter-arm dephasing. That is unusual. Most quantum interferometric sensors degrade smoothly with both noise types.

If findings 3 and 4 hold up: the joint two-parameter problem requires only 2 quadratic-spin observables (no need for full state tomography or many measurement settings), and the SLD bound is the relevant fundamental limit (no incompatibility gap). This puts a clear target on what an experimental implementation needs to achieve.

If findings 4 and 5 are already in the literature in some form (which is genuinely possible given the size of the multi-parameter QM literature), this repo simply consolidates and connects them. That is still a useful artifact.

If they are wrong (numerical artifact or conceptual error): the embarrassment is mostly mine and the AI's. The honest framing here gives a referee enough context to identify the failure quickly.

Repository layout

mz-multiparameter/
├── README.md                  this file
├── LICENSE                    GPL-3.0 (matches thesis)
├── CITATION.cff
├── requirements.txt
├── src/                       library
│   ├── __init__.py
│   ├── core.py                SU(2) toolkit, lossy density matrix, SLD-QFI
│   └── hcrb.py                Albarelli-Friel-Datta SDP port (re-derived)
├── experiments/               one script per claim, deterministic, writes results/*.json
│   ├── 01_qfi_scaling.py
│   ├── 02_joint_bound.py
│   ├── 03_minimal_set.py
│   ├── 04_holevo_vs_sld.py
│   ├── 05_invariances.py
│   ├── 06_loss_sweep.py
│   └── 07_parity_comparison.py
├── docs/
│   ├── derivations.md         algebraic identities + structural arguments
│   ├── make_figures.py        regenerate README figures from results/
│   └── fig_*.png              README figures
├── results/                   JSON tables (generated)
└── tests/
    └── test_smoke.py          QFI bound check + identity verification

References

Primary references this work builds on:

• M. J. Holland & K. Burnett. Phys. Rev. Lett. 71, 1355 (1993). The original twin-Fock-in-MZI proposal.
• D. W. Berry & H. M. Wiseman. Phys. Rev. Lett. 85, 5098 (2000). Optimal probe for canonical phase measurement.
• N. B. Lovett et al. Phys. Rev. Lett. 110, 220501 (2013). Differential evolution for adaptive quantum metrology.
• A. Hentschel & B. C. Sanders. Phys. Rev. Lett. 104, 063603 (2010). Machine learning for quantum measurement.
• F. Albarelli, J. F. Friel, A. Datta. Phys. Rev. Lett. 123, 200503 (2019). arXiv:1906.05724. HCRB SDP formulation.
• J. Liu, H. Yuan, X.-M. Lu, X. Wang. J. Phys. A 53, 023001 (2020). arXiv:1907.08037. Multi-parameter quantum Fisher information review.
• T. J. Volkoff & C. Ryu. Frontiers in Physics (2024). arXiv:2308.05871. Globally optimal interferometry with lossy twin Fock probes.
• S. Du, S. Liu, F. E. S. Steinhoff, G. Vitagliano. arXiv:2412.19119 (2024). Multiparameter SU(2) and SU(1,1) estimation resources.
• J. Jayakumar, M. E. Mycroft, M. Barbieri, M. Stobińska. arXiv:2403.04722 (2024). Joint phase + phase-diffusion estimation.
• L. Pezzè & A. Smerzi. Rev. Mod. Phys. 90, 035005 (2018). Quantum metrology review.

External code consulted:

• tantrix10/HCRB_SDP — Python port of Albarelli's HCRB SDP. We used this as a reference but re-implemented from scratch; the existing port has hardcoded npar = 3 and an undefined-variable bug at the time of writing.

Companion thesis (single-parameter version of this problem):

• H. Qureshi. Variable Beam-Splitter Reflectivity Estimation for Interferometry. M.Sc. thesis, University of Calgary, 2020. https://ucalgary.scholaris.ca/items/f65d88e6-50cb-4ef9-8ece-a6de6895717d. Companion code: thehamzaq/Quantum_Machine_Learning.

License & citation

GPL-3.0 (matching the thesis-era code).

If you use any of these results in research, please cite the M.Sc. thesis and this repository. See CITATION.cff.

If you find that any of the "possibly new" findings is in fact in the literature: please open an issue with the citation, so this README can be updated.

If you find that any "possibly new" finding is wrong: please open an issue with details, and we will retract the claim publicly.