Causal Fisher-Information Inequalities: Classical Causal Model Falsification and Metrological Advantage

Jeongho Bang; Su-Yong Lee

arxiv: 2605.19198 · v1 · pith:YTNZSZO2new · submitted 2026-05-18 · 🪐 quant-ph

Causal Fisher-Information Inequalities: Classical Causal Model Falsification and Metrological Advantage

Jeongho Bang , Su-Yong Lee This is my paper

Pith reviewed 2026-05-20 10:01 UTC · model grok-4.3

classification 🪐 quant-ph

keywords causal Fisher-information inequalitiesclassical causal model falsificationquantum metrologyFisher information synergydirected acyclic graphsmetrological advantagescore correlations

0 comments

The pith

Violating causal Fisher-information inequalities rules out every classical causal model for an experiment and certifies a metrological precision no such model can reach.

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

The paper shows that assuming an experiment follows a classical causal model with a directed acyclic graph, conditional independences, and modular parameter dependence forces the Fisher informations to obey causal Fisher-information inequalities. These inequalities arise from a causal-path series law in which inverse Fisher information adds like series resistance along any classical path. A violation therefore falsifies the entire class of classical causal models. At the same time the violation automatically certifies a metrological resource because it implies a precision gain that no classical causal model can achieve, with the gain coming from off-diagonal score correlations forbidden by classical modularity.

Core claim

Under the assumption of a classical causal model given by a directed acyclic graph, conditional independences, and modular parameter dependence, the Fisher informations must satisfy causal Fisher-information inequalities. The backbone is the causal-path series law: for an additive causal parameter propagating through a classical path A to C to B, the inverse Fisher information behaves as an information resistance and adds in series. Any violation of these inequalities falsifies the entire classical causal model class and certifies a metrological advantage arising from Fisher-information synergy via off-diagonal score correlations that classical modularity forbids.

What carries the argument

The causal-path series law, which requires inverse Fisher information to add in series along any classical causal path.

If this is right

Any CFII violation rules out all possible classical causal models for the observed statistics.
The violation implies a precision bound that no classical causal model can attain.
The precision gain arises specifically from off-diagonal score correlations forbidden by classical modularity.
In long causal decompositions the advantage is chain-amplified.
The witness remains certifiable under realistic visibility loss and readout error.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same series-law logic could be applied to other information measures such as quantum Fisher information or Renyi divergences to obtain broader causal witnesses.
The approach suggests a way to certify quantum resources in experiments where full process tomography is impractical.
It may connect to existing causal inference techniques in quantum networks by providing an operational metrological test for classicality.
Practical sensor design could deliberately engineer causal paths to maximize the synergy gain when classical bounds are exceeded.

Load-bearing premise

The experiment is assumed to admit a classical causal model specified by a directed acyclic graph, conditional independences, and modular parameter dependence.

What would settle it

In a single-qubit coherent-rotation experiment, observe a Fisher information value that exceeds the series sum of inverse informations predicted by any classical causal decomposition of the rotation angle.

Figures

Figures reproduced from arXiv: 2605.19198 by Jeongho Bang, Su-Yong Lee.

**Figure 1.** Figure 1: FIG. 1. AI-assisted adversarial finite-data stress test. (a) Noisy coherent-qubit samples are processed by a classifier score estimator to certify [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Representative single-qubit landscapes for the generic setting [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Achievability of the advantage for the deterministic single-qubit point. Monte-Carlo RMSE of a simple MLE (markers) versus sample [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Adversarial (split-optimized) classical benchmark for the representative single-qubit setting used in Fig. 2. Top: the ratio [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Details of the AI-assisted finite-data stress test. (a) Actual noisy endpoint FI [PITH_FULL_IMAGE:figures/full_fig_p040_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Schematic summary of the logical flow of the paper. Left: under a classical causal-path description, the total parameter [PITH_FULL_IMAGE:figures/full_fig_p041_6.png] view at source ↗

read the original abstract

Fisher-information inequalities have recently been used as operational witnesses of nonclassical metrological behavior, but their physical meaning is often tied to a particular narrative, such as, segmented dynamics or discrete trajectories. We show that a broader interpretation is available and, in fact, more natural: once an experiment is assumed to admit a classical causal model specified by a directed acyclic graph, conditional independences, and modular parameter dependence, the corresponding Fisher informations are forced to obey causal Fisher-information inequalities (CFIIs). The backbone result is a causal-path series law: for an additive causal parameter that propagates through a classical path $A \to C \to B$, the inverse Fisher information behaves as an information resistance and must add in series. Consequently, any CFII violation is a rigorous falsification of the entire classical causal model class. We then show that the violation is automatically a metrological resource certificate, because it implies a precision that no member of the classical causal class can attain. The gain mechanism is identified as Fisher-information synergy, i.e. off-diagonal score correlations that classical modularity forbids. A single-qubit coherent-rotation example demonstrates the deterministic CFII violation, estimator-level achievability of the resulting gain, robustness against split-optimized classical benchmarks, and a chain-amplified advantage in long causal decompositions. Finally, an AI-assisted adversarial finite-data stress test shows that the witness remains certifiable under the realistic visibility loss and readout error, while optimized modular classical causal models saturate but do not cross the CFII frontier.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives a series law for inverse Fisher information along classical causal paths and shows that violating the resulting inequalities both falsifies the model class and certifies a metrological gain from forbidden score correlations.

read the letter

The main point to take away is that this paper derives causal Fisher-information inequalities from classical model assumptions, and shows that breaking them both falsifies the models and certifies a metrological advantage via forbidden information synergies. What is new is the causal-path series law for inverse Fisher information. For an additive parameter propagating through a path in a directed acyclic graph with conditional independences and modular dependence, the inverses add like series resistances. This follows from the joint distribution factorization leading to block-diagonal score covariances. Any violation then rules out the entire classical causal class, and because it implies unattainable precision, it serves as a metrological resource certificate. The gain is pinned to off-diagonal score correlations that modularity forbids classically. They back this with a deterministic single-qubit coherent-rotation example that achieves the gain at the estimator level, shows robustness to split-optimized classical benchmarks, and demonstrates amplification along longer causal chains. An AI-assisted adversarial test with visibility loss and readout error confirms the witness stays certifiable in realistic conditions while classical models hit but do not exceed the bound. The derivation holds up under the stated premises, with no hidden inconsistencies in the provided text. The example and test are consistent and practical. One soft spot is that the falsification applies only if the setup truly admits a classical causal model of that form; deviations from the assumed DAG or modularity could change things. The metrological claim is tied to this specific mechanism, so it might not cover all quantum advantages. These are not central flaws but limit the scope. This paper is for researchers in quantum metrology and causal inference who want operational witnesses that do double duty. A reader focused on designing sensors or testing causal structures in quantum systems would get concrete tools from it. The combination of derivation, example, and stress test makes it worth a serious referee's time. I recommend putting it through peer review.

Referee Report

2 major / 3 minor

Summary. The manuscript claims that, under the assumptions of a classical causal model (directed acyclic graph, conditional independences, and modular parameter dependence), the Fisher information matrix obeys causal Fisher-information inequalities (CFIIs). The backbone is a causal-path series law stating that the inverse Fisher information for an additive parameter propagating along a path A → C → B adds in series, analogous to information resistance. Any violation rigorously falsifies the entire classical causal model class and automatically certifies a metrological advantage, because the observed precision exceeds what any model in the class can attain; the mechanism is identified as Fisher-information synergy arising from off-diagonal score correlations forbidden by classical modularity. The claims are supported by a deterministic single-qubit coherent-rotation example and an AI-assisted adversarial finite-data stress test demonstrating robustness under visibility loss and readout error.

Significance. If the central derivation holds, the work supplies a causally grounded operational witness for nonclassical metrological resources that does not rely on segmented dynamics or discrete trajectories. The explicit link between CFII violation and unattainable classical precision, together with the identification of synergy via score correlations and the demonstration of chain-amplified advantage in long decompositions, could usefully connect causal inference techniques to quantum metrology. The deterministic qubit example and the finite-data adversarial test provide concrete, reproducible illustrations that strengthen the falsification and certification claims.

major comments (2)

[§3] §3 (causal-path series law derivation): the step from the factorization of the joint distribution and the resulting block-diagonal structure of the score covariance to the explicit addition of inverse Fisher informations along the path is load-bearing for both the falsification and metrological claims; the manuscript should supply the intermediate matrix algebra showing how modular parameter dependence eliminates cross-path terms.
[§4.2] §4.2 (single-qubit example): the deterministic CFII violation and the reported estimator-level achievability of the metrological gain are central, yet the explicit numerical values of the Fisher information matrix elements (including off-diagonal terms) and the comparison against the series bound are not tabulated; without these the quantitative advantage cannot be independently verified.

minor comments (3)

[§5] The finite-data adversarial test would be strengthened by reporting the number of trials, the precise visibility and readout-error parameters used, and any data-exclusion criteria.
[§2] Notation for the score vector and the off-diagonal correlations that define Fisher-information synergy should be introduced once in a dedicated subsection rather than inline.
[Figure 1] Figure captions for the qubit example should explicitly state whether the plotted curves are theoretical predictions or Monte-Carlo estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments on our manuscript. We address each major comment below and have revised the manuscript accordingly to improve clarity and reproducibility.

read point-by-point responses

Referee: §3 (causal-path series law derivation): the step from the factorization of the joint distribution and the resulting block-diagonal structure of the score covariance to the explicit addition of inverse Fisher informations along the path is load-bearing for both the falsification and metrological claims; the manuscript should supply the intermediate matrix algebra showing how modular parameter dependence eliminates cross-path terms.

Authors: We agree that the intermediate matrix algebra is important for rigor. In the revised manuscript we have expanded §3 with an explicit derivation: starting from the joint factorization p(x) = ∏ p(v|pa(v)) under the DAG and conditional independences, the score covariance matrix is block-diagonal. Modular parameter dependence further implies that the parameter θ enters only through local conditional distributions, so that cross-derivatives between non-adjacent variables vanish. This eliminates off-block terms and yields the series law I_{AB}^{-1}(θ) = I_{AC}^{-1}(θ) + I_{CB}^{-1}(θ) for additive parameters along the path. The added steps are now shown with the relevant matrix expressions. revision: yes
Referee: §4.2 (single-qubit example): the deterministic CFII violation and the reported estimator-level achievability of the metrological gain are central, yet the explicit numerical values of the Fisher information matrix elements (including off-diagonal terms) and the comparison against the series bound are not tabulated; without these the quantitative advantage cannot be independently verified.

Authors: We accept that tabulated values will aid independent verification. We have added a new table in §4.2 that reports the full Fisher information matrix (including all off-diagonal elements arising from score correlations), the corresponding inverse matrix, the classical series bound, and the observed violation margin for the coherent-rotation example. The table also lists the estimator variance achieved relative to the bound. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from stated causal-model premises

full rationale

The central result—the causal-path series law for inverse Fisher information along additive parameters in a DAG—is obtained directly from the factorization of the joint distribution under conditional independences and modular parameter dependence, which produces a block-diagonal score covariance. This forces the series addition of information resistances without reference to any fitted quantities, data-dependent parameters, or prior results by the same authors. The metrological certificate follows immediately as the logical negation: any observed Fisher information exceeding the derived bound lies outside the entire classical causal model class. No load-bearing step reduces to a self-definition, a renamed empirical pattern, or a self-citation chain; the argument is fully independent of the paper’s own examples or benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption of classical causal models with DAG structure and modular parameters; no free parameters are introduced, and Fisher-information synergy is postulated as an explanatory mechanism without independent falsifiable evidence outside the derivation.

axioms (1)

domain assumption An experiment admits a classical causal model specified by a directed acyclic graph, conditional independences, and modular parameter dependence.
Invoked as the starting point from which the causal-path series law and CFIIs are forced.

invented entities (1)

Fisher-information synergy no independent evidence
purpose: Accounts for the metrological gain arising from off-diagonal score correlations forbidden by classical modularity.
Introduced to explain why CFII violation yields precision unattainable classically; no independent evidence such as a predicted observable outside the model is supplied.

pith-pipeline@v0.9.0 · 5806 in / 1516 out tokens · 41891 ms · 2026-05-20T10:01:28.733212+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2 (Causal-path CFII): ... inverse Fisher information behaves as an information resistance and must add in series.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Fisher-information synergy ... off-diagonal score correlations that classical modularity forbids.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

contexts

Preliminaries: two contexts and the NSIT constraint We consider a parameter estimation problem with a single real parameterθ. A probe stateˆρ(θ)is prepared, and the experi- menter can choose between two “contexts” as below: S= 0 :perform measurement ˆBonly, S= 1 :perform measurement ˆAand then ˆB.(A1) Letb∈ Bdenote the outcome of ˆB, and leta∈ Adenote the...

work page
[2]

classical state

NSIT as a classical causal-model constraint We now formulate NSIT as a CI constraint, and thereby, as a causal-model class in the sense of our work. Proposition S2(NSIT is a conditional-independence constraint).Fixθand consider the joint distribution of(S, B)defined by the operational conditional distributionsp(b|S, θ)and an exogenous distributionp(s)of t...

work page
[3]

The OQ is constructed as w(a, b|θ) =p(a, b|S= 1, θ) + 1 2 p(b|S= 0, θ)−p(b|S= 1, θ) .(A10) Note that OQ is constructed purely from observable probabilities

Operational quasiprobability and the reduction under NSIT The central operational move in coQM is to integrate the two contexts into a single statistical model, the operational quasiprob- ability (OQ) [18, 20, 21]. The OQ is constructed as w(a, b|θ) =p(a, b|S= 1, θ) + 1 2 p(b|S= 0, θ)−p(b|S= 1, θ) .(A10) Note that OQ is constructed purely from observable ...

work page
[4]

if the causal model does not collapse, there is no resource to harvest

From witness to resource: Fisher-information consequences Proposition S3already exposes the metrological logic. If NSIT holds, OQ does not introduce a new statistical structure; it merely reproduces the probability of the consecutive measurement. Thus, in the context-free causal regime, coQM cannot create an advantage. To make this statement quantitative,...

work page
[5]

optional

NSIT does not subsume the CFII program The NSIT condition is an extremely sharp and operationally meaningful constraint, but it constrains only one specific kind of conditional independence, namely the absence of a causal arrow from a context choice into a marginal prediction. In the coQM/NSIT setting, one introduces a context variableS(e.g., whether a pr...

work page
[6]

B. R. Frieden,Science from Fisher Information: A Unification(Cambridge University Press, 2004)

work page 2004
[7]

Cram ´er,Mathematical Methods of Statistics(Princeton University Press, Princeton, 1946)

H. Cram ´er,Mathematical Methods of Statistics(Princeton University Press, Princeton, 1946)

work page 1946
[8]

C. W. Helstrom,Quantum Detection and Estimation Theory(Academic Press, New York, 1976)

work page 1976
[9]

S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett.72, 3439 (1994)

work page 1994
[10]

M. G. A. Paris, Quantum estimation for quantum technology, International Journal of Quantum Information7, 125 (2009)

work page 2009
[11]

Zamir, A proof of the fisher information inequality via a data processing argument, IEEE Transactions on Information Theory44, 1246 (1998)

R. Zamir, A proof of the fisher information inequality via a data processing argument, IEEE Transactions on Information Theory44, 1246 (1998)

work page 1998
[12]

K. C. Tan, V . Narasimhachar, and B. Regula, Fisher information from quantum resources, Phys. Rev. Lett.127, 200402 (2021)

work page 2021
[13]

D. R. M. Arvidsson-Shukur, N. Yunger Halpern, and S. Lloyd, The value of temporal correlations in quantum metrology, Nature Commu- nications11, 3775 (2020)

work page 2020
[14]

N. B. Lupu-Gladstein, B. Y . Yilmaz, D. R. M. Arvidsson-Shukur, A. Brodutch, A. O. T. Pang, A. M. Steinberg, and N. Yunger Halpern, Negative quasiprobabilities enhance phase estimation in quantum metrology, Phys. Rev. Lett.128, 220504 (2022)

work page 2022
[15]

Pearl,Causality: Models, Reasoning, and Inference, 2nd ed

J. Pearl,Causality: Models, Reasoning, and Inference, 2nd ed. (Cambridge University Press, Cambridge, 2009)

work page 2009
[16]

Demkowicz-Dobrza ´nski, W

R. Demkowicz-Dobrza ´nski, W. G´orecki, and M. Gut ¸˘a, Multi-parameter estimation beyond quantum fisher information, Journal of Physics A: Mathematical and Theoretical53, 363001 (2020)

work page 2020
[17]

J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum fisher information matrix and multiparameter estimation, Journal of Physics A: Mathematical and Theoretical53, 023001 (2020)

work page 2020
[18]

J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens, Quantum common causes and quantum causal models, Phys. Rev. X7, 031021 (2017)

work page 2017
[19]

Barrett, R

J. Barrett, R. Lorenz, and O. Oreshkov, Cyclic quantum causal models, Nature Communications12, 885 (2021)

work page 2021
[20]

Cranmer, J

K. Cranmer, J. Brehmer, and G. Louppe, The frontier of simulation-based inference, Proceedings of the National Academy of Sciences 117, 30055 (2020)

work page 2020
[21]

D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, inInternational Conference on Learning Representations(2015)

work page 2015
[22]

L. A. Rozema, T. Str ¨omberg, and P. Walther, Experimental aspects of indefinite causal order in quantum mechanics, Nature Reviews Physics6, 483 (2024)

work page 2024
[23]

J. Jae, J. Lee, M. S. Kim, K.-G. Lee, and J. Lee, Contextual quantum metrology with operational quasiprobabilities, npj Quantum Infor- mation10, 68 (2024)

work page 2024
[24]

J. J. Halliwell, The interpretation of no-signalling in time, Phys. Rev. A96, 012121 (2017)

work page 2017
[25]

J. Ryu, J. Lim, S. Hong, and J. Lee, Operational quasiprobabilities for qudits, Phys. Rev. A88, 052123 (2013)

work page 2013
[26]

J. Jae, J. Ryu, and J. Lee, Operational quasiprobabilities for continuous variables, Phys. Rev. A96, 042121 (2017)

work page 2017

[1] [1]

contexts

Preliminaries: two contexts and the NSIT constraint We consider a parameter estimation problem with a single real parameterθ. A probe stateˆρ(θ)is prepared, and the experi- menter can choose between two “contexts” as below: S= 0 :perform measurement ˆBonly, S= 1 :perform measurement ˆAand then ˆB.(A1) Letb∈ Bdenote the outcome of ˆB, and leta∈ Adenote the...

work page

[2] [2]

classical state

NSIT as a classical causal-model constraint We now formulate NSIT as a CI constraint, and thereby, as a causal-model class in the sense of our work. Proposition S2(NSIT is a conditional-independence constraint).Fixθand consider the joint distribution of(S, B)defined by the operational conditional distributionsp(b|S, θ)and an exogenous distributionp(s)of t...

work page

[3] [3]

The OQ is constructed as w(a, b|θ) =p(a, b|S= 1, θ) + 1 2 p(b|S= 0, θ)−p(b|S= 1, θ) .(A10) Note that OQ is constructed purely from observable probabilities

Operational quasiprobability and the reduction under NSIT The central operational move in coQM is to integrate the two contexts into a single statistical model, the operational quasiprob- ability (OQ) [18, 20, 21]. The OQ is constructed as w(a, b|θ) =p(a, b|S= 1, θ) + 1 2 p(b|S= 0, θ)−p(b|S= 1, θ) .(A10) Note that OQ is constructed purely from observable ...

work page

[4] [4]

if the causal model does not collapse, there is no resource to harvest

From witness to resource: Fisher-information consequences Proposition S3already exposes the metrological logic. If NSIT holds, OQ does not introduce a new statistical structure; it merely reproduces the probability of the consecutive measurement. Thus, in the context-free causal regime, coQM cannot create an advantage. To make this statement quantitative,...

work page

[5] [5]

optional

NSIT does not subsume the CFII program The NSIT condition is an extremely sharp and operationally meaningful constraint, but it constrains only one specific kind of conditional independence, namely the absence of a causal arrow from a context choice into a marginal prediction. In the coQM/NSIT setting, one introduces a context variableS(e.g., whether a pr...

work page

[6] [6]

B. R. Frieden,Science from Fisher Information: A Unification(Cambridge University Press, 2004)

work page 2004

[7] [7]

Cram ´er,Mathematical Methods of Statistics(Princeton University Press, Princeton, 1946)

H. Cram ´er,Mathematical Methods of Statistics(Princeton University Press, Princeton, 1946)

work page 1946

[8] [8]

C. W. Helstrom,Quantum Detection and Estimation Theory(Academic Press, New York, 1976)

work page 1976

[9] [9]

S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett.72, 3439 (1994)

work page 1994

[10] [10]

M. G. A. Paris, Quantum estimation for quantum technology, International Journal of Quantum Information7, 125 (2009)

work page 2009

[11] [11]

Zamir, A proof of the fisher information inequality via a data processing argument, IEEE Transactions on Information Theory44, 1246 (1998)

R. Zamir, A proof of the fisher information inequality via a data processing argument, IEEE Transactions on Information Theory44, 1246 (1998)

work page 1998

[12] [12]

K. C. Tan, V . Narasimhachar, and B. Regula, Fisher information from quantum resources, Phys. Rev. Lett.127, 200402 (2021)

work page 2021

[13] [13]

D. R. M. Arvidsson-Shukur, N. Yunger Halpern, and S. Lloyd, The value of temporal correlations in quantum metrology, Nature Commu- nications11, 3775 (2020)

work page 2020

[14] [14]

N. B. Lupu-Gladstein, B. Y . Yilmaz, D. R. M. Arvidsson-Shukur, A. Brodutch, A. O. T. Pang, A. M. Steinberg, and N. Yunger Halpern, Negative quasiprobabilities enhance phase estimation in quantum metrology, Phys. Rev. Lett.128, 220504 (2022)

work page 2022

[15] [15]

Pearl,Causality: Models, Reasoning, and Inference, 2nd ed

J. Pearl,Causality: Models, Reasoning, and Inference, 2nd ed. (Cambridge University Press, Cambridge, 2009)

work page 2009

[16] [16]

Demkowicz-Dobrza ´nski, W

R. Demkowicz-Dobrza ´nski, W. G´orecki, and M. Gut ¸˘a, Multi-parameter estimation beyond quantum fisher information, Journal of Physics A: Mathematical and Theoretical53, 363001 (2020)

work page 2020

[17] [17]

J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum fisher information matrix and multiparameter estimation, Journal of Physics A: Mathematical and Theoretical53, 023001 (2020)

work page 2020

[18] [18]

J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens, Quantum common causes and quantum causal models, Phys. Rev. X7, 031021 (2017)

work page 2017

[19] [19]

Barrett, R

J. Barrett, R. Lorenz, and O. Oreshkov, Cyclic quantum causal models, Nature Communications12, 885 (2021)

work page 2021

[20] [20]

Cranmer, J

K. Cranmer, J. Brehmer, and G. Louppe, The frontier of simulation-based inference, Proceedings of the National Academy of Sciences 117, 30055 (2020)

work page 2020

[21] [21]

D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, inInternational Conference on Learning Representations(2015)

work page 2015

[22] [22]

L. A. Rozema, T. Str ¨omberg, and P. Walther, Experimental aspects of indefinite causal order in quantum mechanics, Nature Reviews Physics6, 483 (2024)

work page 2024

[23] [23]

J. Jae, J. Lee, M. S. Kim, K.-G. Lee, and J. Lee, Contextual quantum metrology with operational quasiprobabilities, npj Quantum Infor- mation10, 68 (2024)

work page 2024

[24] [24]

J. J. Halliwell, The interpretation of no-signalling in time, Phys. Rev. A96, 012121 (2017)

work page 2017

[25] [25]

J. Ryu, J. Lim, S. Hong, and J. Lee, Operational quasiprobabilities for qudits, Phys. Rev. A88, 052123 (2013)

work page 2013

[26] [26]

J. Jae, J. Ryu, and J. Lee, Operational quasiprobabilities for continuous variables, Phys. Rev. A96, 042121 (2017)

work page 2017