Pith. sign in

REVIEW 2 major objections 4 minor 43 references

The bare statistics of a quantum measurement determine the least disturbance any implementation of that measurement must cause.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 22:07 UTC pith:ZQ7SACDE

load-bearing objection Core Theorem 1 is new and correct; the experimental estimation technique has an unaddressed finite-statistics gap that should be fixed before publication. the 2 major comments →

arxiv 2607.15874 v1 pith:ZQ7SACDE submitted 2026-07-17 quant-ph

The statistical disturbance bound of quantum measurements

Ties-A. Ohst , Sebastian Schl\"osser , Roope Uola This is my paper
classification quant-ph MSC 81P1581P45 PACS 03.65.Ta03.67.-a
keywords statistical disturbance boundPOVMaverage fidelityLüders instrumentweighted state exclusioninformation-disturbance relationquantum randomness generationsemidefinite programming
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Quantum measurements inevitably disturb the systems they probe, but how much has seemed to depend on how the measurement is implemented. This paper defines a quantity—the statistical disturbance bound—that depends only on the outcome probabilities encoded in a positive operator-valued measure (POVM), together with the chosen ensemble of input states. It shows this bound is the largest average input–output fidelity any measurement channel compatible with those probabilities can achieve, computable as a semidefinite program; for uniformly random (Haar) input states it has the closed form (1 + (1/d)Σ Tr(√A_a)^2)/(d+1), attained by the Lüders instrument. A new technique, weighted state exclusion, lets an experimenter estimate this bound directly from observed conditional probabilities without knowing the POVM effects, and can certify measurement-induced disturbance using only partial state preparations. The result matters because it turns the bare statistics of a measurement into a quantitative guarantee about a dynamical property—state disturbance—that previously required detailed knowledge of the measurement channel.

Core claim

Central claim: for any POVM A on a d-dimensional system and any pure-state ensemble E, the largest average input–output fidelity over all compatible instruments is a value FE(A), computable as a semidefinite program. For uniformly random (Haar) inputs, FE(A) = (1 + (1/d)Σ_a Tr(√A_a)^2)/(d+1), attained by the Lüders instrument. Consequences claimed: Haar bound is 2/(d+1) iff all effects are rank one; weighted state exclusion estimates the bound from probe statistics without knowing the POVM (robust to preparation errors); loss and depolarisation with equal informativeness are distinguished; Lüders is suboptimal for non-Haar ensembles; fidelity bounds limit an eavesdropper's guessing probabili

What carries the argument

Key machinery: the ensemble operator R(E)=d∫dµ(ψ)(|ψ⟩⟨ψ|)^T⊗|ψ⟩⟨ψ|, which turns average input–output fidelity into an inner product Tr(R(E)I) with the Choi operator of an instrument. The central object is the semidefinite program of Definition 1: maximise Σ_a Tr(R(E)I_a) subject to complete positivity and the POVM constraint (Tr₂I_a)^T=A_a/d; strong duality holds. For Haar inputs, Lemma 2 characterises Y⊗1≽|φ⁺⟩⟨φ⁺| as Y≻0 with Tr(Y⁻¹)≤d, giving explicit dual solutions that attain the closed form and prove Lüders optimality. Weighted state exclusion restricts dual variables to Y_a=Σ_i q_ai ρ_i built from trusted probe states, making Tr(Y_a A_a) a linear function of measured conditional probab

Load-bearing premise

The load-bearing premise is that the conditional probabilities p(a|i)=Tr(Aaρi) used in the weighted state exclusion optimisation are known exactly; with only finite measurement samples, the calculated value need not be a valid upper bound on FE(A).

What would settle it

For a fully characterised POVM and input ensemble, solve the SDP for FE(A) and then numerically search over Choi operators of compatible instruments aiming at a higher average fidelity; the theorem says no such instrument exists. For the experimental method, apply weighted state exclusion to a characterised measurement using finite counts and test whether the reported bound ever falls below the directly computed FE(A).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Any instrument that implements a POVM must have input–output fidelity no larger than FE(A) on the chosen ensemble, so bare outcome statistics lower-bound the state disturbance every implementation causes.
  • For Haar inputs, the closed form ties disturbance to the operator square roots of the effects: rank-one effects saturate the minimum possible disturbance 2/(d+1), and effects of rank at most k give bound at most (k+1)/(d+1).
  • The weighted state exclusion bound can certify nonzero disturbance using state preparations that are informationally incomplete for detector tomography, so disturbance detection requires less characterisation than full measurement tomography.
  • Because loss and depolarisation have equal informativeness but different statistical disturbance bounds, informativeness alone is insufficient to predict disturbance; the full POVM statistics carry disturbance information.
  • In the proposed randomness protocol, a certified lower bound on average fidelity directly yields an upper bound on an eavesdropper's guessing probability, linking disturbance certification to randomness generation.

Where Pith is reading between the lines

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

  • An experimentalist applying weighted state exclusion with finite counts should treat the reported value as an estimate, not a guaranteed bound, because the paper's robustness result covers preparation errors only and does not provide finite-sample confidence intervals.
  • Because the bound separates loss from depolarisation, it could provide an operational test for distinguishing detector inefficiency from decoherence in a single device, without tomographic reconstruction.
  • The non-optimality of Lüders for latitude ensembles suggests one can tailor the post-measurement update to the prior ensemble, potentially leading to disturbance-optimal state discrimination or state-preserving measurements in applications with known input distributions.
  • The state-exclusion formulation may carry over to continuous-variable systems, where disturbance bounds could be expressed through analogous dual semidefinite programs, though the paper only gestures at this direction.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces the statistical disturbance bound F_E(A), defined as the largest E-average input-output fidelity achievable by any instrument compatible with a POVM A. The bound is formulated as an SDP (Definition 1, Eq. (4) and Eq. (8)), and a closed-form expression is claimed for Haar-distributed input states (Theorem 1). The paper also proposes the weighted state exclusion technique (Theorem 2) to estimate the bound experimentally without detector tomography, including a robustness extension to state-preparation errors (Section 4.3). Examples demonstrate that the bound distinguishes noise models with identical informativeness, that disturbance can be certified with non-tomographically complete probe sets, and that the Lüders instrument is not optimal for non-Haar ensembles. A simple quantum randomness generation protocol links the certified fidelity to an eavesdropper's guessing probability.

Significance. If fully correct, the framework provides a tight, efficiently computable characterization of how the statistical description of a POVM constrains the disturbance of any compatible instrument, going beyond informativeness-based information-disturbance relations. Strengths include the explicit dual-feasible construction in Appendix A, the rank-based Proposition 1, the clear examples, and the public code repository. However, the experimental claim of determining the bound without detector tomography is not yet supported at finite sample size, and the closed-form theorem statement contains a factor error. The theoretical core is largely sound, but the paper needs revision before publication.

major comments (2)
  1. [Eq. (11), (12), (41)] The closed-form expression is missing a factor 1/(d+1) on the summed term. From Eq. (11), \sum_a Tr(R(H)I_a^L) = (d+S)/(d(d+1)) with S = \sum_a [Tr(\sqrt{A_a})]^2, so the correct value is 1/(d+1) + S/[d(d+1)], not S/d + 1/(d+1). The same error appears in Theorem 1 (Eq. (12)) and its restatement in Appendix A (Eq. (41)). The proof in Appendix A and the examples in Section 5.1 actually use the correct expression. The theorem statement and all derived formulas must be corrected.
  2. [Sec. 4.2 / Theorem 2] The weighted state exclusion bound (Eq. (23)) is defined using exact probabilities p(a|i)=Tr(A_a\rho_i). In any real experiment only finite-sample estimates \hat p(a|i) are available. Since the objective in (23) is linear in these probabilities, substituting empirical frequencies can produce a value below F_E(A,\rho), and consequently below F_E(A). The certification test F_E(A,\rho,\epsilon)<1 is therefore not justified at finite sample size. Section 4.3 only addresses preparation errors (trace distance \epsilon), not counting statistics. To support the claimed experimental determination without detector tomography, the authors should provide finite-sample confidence intervals or a conservative estimator that remains an upper bound with high confidence.
minor comments (4)
  1. [Sec. 5.3] The statement that 'throughout the northern hemisphere, the optimal value is attained by the bit-flip instrument' is not proven analytically. The inequality F_Eθ(B_flip)>F_Eθ(B_L) demonstrates suboptimality of the Lüders instrument but not optimality of the flip instrument. If the 'optimal' curves in Fig. 5 come from numerical SDP, please state this explicitly; otherwise provide a proof.
  2. [Sec. 6, Eq. (36)] The SDP for the guessing probability would benefit from more explanation. In particular, the constraint \sum_a I_a^\dagger(1_d)=1_d/d and the relation between the instrument elements I_a, the channel C, and the certified fidelity f should be spelled out; as written, the instrument appears not to be trace-preserving in the usual sense.
  3. [Fig. 3 caption vs Sec. 5.2] The preparation-uncertainty parameter is given as \epsilon=0.005 in the Figure 3 caption but as \epsilon=0.05 in the main text. Please make these consistent.
  4. [General] Minor typographical issues: 'statsitical' in Definition 1, 'orthornormal' in Section 4.2, and the phrase in Theorem 2 'the upper bound given by F_E(A,\rho)\ge F_E(A)' is awkward—F_E(A,\rho) is an upper bound on F_E(A), so the inequality should be presented as such.

Circularity Check

0 steps flagged

No load-bearing circularity: the SDP derivation is self-contained; the only self-citations are non-central and the finite-statistics caveat is a robustness gap, not circularity.

full rationale

The paper's derivation chain is self-contained. Definition 1 defines F_E(A) as an optimization over instruments compatible with a POVM; Theorem 1 is proved by constructing dual-feasible variables in Appendix A (Lemma 2 and Eq. (44)) and by Eq. (11) for the Lüders instrument, so the Haar formula is an analytic SDP solution rather than a fitted or assumed input. Proposition 1 follows from elementary inequalities. Theorem 2 and Eq. (22) arise by restricting the dual SDP (9) to Y_a = Σ_i q_ai ρ_i; by weak duality this restricted optimum is an upper bound on the dual optimum, so it is an algebraic consequence of the same SDP, not a renamed target. The weighted state exclusion objective (23) uses Born-rule probabilities p(a|i)=Tr(A_a ρ_i), which are inputs from the POVM, not the disturbance bound. The paper's self-citations ([40], [43]) concern an analogy for continuous-variable resources and code availability; neither is load-bearing for the statistical disturbance bound. A skeptical concern about finite-sample statistics is legitimate but is a correctness/robustness gap in the experimental certification claim (Sections 4.2-4.3), not a circularity: the claimed mathematical reduction never assumes the result it derives. Hence no specific circular step is identified; the score reflects only the minor, non-load-bearing self-citation and the unaddressed finite-statistics caveat.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claims rely on standard finite-dimensional quantum information theory: POVMs and instruments, the Choi isomorphism (with 1/d normalisation), SDP duality with Slater's condition, the Haar ensemble operator identity, and the generalised Lüders decomposition. No physical entities are introduced and no constants are fitted; the μ, θ, and ε in the examples are variables or user-chosen bounds. The only numerical inputs are the POVM and ensemble under study.

axioms (6)
  • domain assumption Every instrument compatible with a POVM A can be decomposed via the generalised Lüders rule: I_a(ρ)=C_a(√A_a ρ √A_a) for some channel C_a.
    Stated in Eq. (3) with citation to Hayashi [23]; used to interpret the optimisation and to define the Lüders instrument. The SDP formulation does not strictly need the decomposition, but the paper's narrative relies on it.
  • standard math The Haar ensemble operator is R(H)=(1_{d²}+d|φ+⟩⟨φ+|)/(d+1).
    Eq. (6), from Horodecki et al. [30]; forms the basis for the closed-form Haar bound in Theorem 1 and Proposition 1.
  • standard math The dual SDP (9) is strongly dual to the primal (8) under Slater's condition.
    Invoked after Eq. (9) to ensure the dual upper bound equals the primal value; the paper notes strict feasibility by choosing Y_a with y_a>d.
  • standard math For trace-norm-close states, ρ̃_i ≽ ρ_i − ε 1 when ∥ρ_i−ρ̃_i∥₁≤ε.
    Lemma 1 in Section 4.3; derived from eigenvalue bounds on Hermitian operators.
  • domain assumption The input ensemble E is a probability measure over pure states of a finite-dimensional Hilbert space, and instruments are dimension-preserving.
    Definition 1 and Section 2; the entire framework is restricted to pure inputs and d'=d instruments.
  • standard math With the 1/d-normalised Choi operator, the average fidelity satisfies F_E(I)=Tr(R(E)I).
    Eq. (7); derived from the Choi isomorphism and the definition of R(E).

pith-pipeline@v1.3.0-alltime-deepseek · 17334 in / 35691 out tokens · 298196 ms · 2026-08-01T22:07:24.138625+00:00 · methodology

0 comments
read the original abstract

Quantifying the disturbance caused by a quantum measurement typically requires detailed knowledge of the underlying measurement channel. In this work, we introduce a statistical disturbance bound, which connects the statistical properties of a quantum measurement to the state disturbance induced by any compatible measurement channel. Specifically, we show that the average fidelity between input and output with respect to an arbitrary ensemble of pure input states is fundamentally bounded in terms of the measurement, described as a positive operator-valued measure (POVM). We further develop the weighted state exclusion technique, which enables an experimental determination of the statistical disturbance bound without requiring explicit knowledge of the measurement effects. To see the advantages of our approach over existing information-disturbance relations, we show that our bound distinguishes between measurements with equivalent informativeness. Furthermore, we demonstrate that the weighted state exclusion technique can detect and quantify measurement-induced disturbance using state preparations that are insufficient for tomographic reconstruction of the measurement operators. Finally, we illustrate how disturbance bounds defined with respect to specific input ensembles can be used to bound an eavesdropper's guessing probability in a simple protocol for quantum randomness generation.

Figures

Figures reproduced from arXiv: 2607.15874 by Roope Uola, Sebastian Schl\"osser, Ties-A. Ohst.

Figure 1
Figure 1. Figure 1: Sketch of the statistical disturbance estimation [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison between depolarisation noise and [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Weighted state exclusion with incomplete state [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Equivalently, these are the pure states satisfy￾ing ⟨ψ| σz |ψ⟩ = cos(θ). Unlike the Haar ensemble, latitude ensembles encode prior information about the likely input states. This additional structure allows the post-measurement state update to be tailored to the ensemble, potentially increasing the average fidelity. To compute the statistical disturbance bound with respect to the ensemble Eθ, we first noti… view at source ↗
Figure 5
Figure 5. Figure 5: Statistical disturbance bounds for latitude [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Guessing probability versus average fidelity. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

43 extracted references · 2 linked inside Pith

  1. [1]

    Über den anschaulichen In- halt der quantentheoretischen Kinematik und Mechanik

    W. Heisenberg, “Über den anschaulichen In- halt der quantentheoretischen Kinematik und Mechanik”, Zeitschrift für Physik43, 172–198 (1927)

    1927

  2. [2]

    Heisenberg,Physical principles of the quantum theory(Dover Publications, 1949)

    W. Heisenberg,Physical principles of the quantum theory(Dover Publications, 1949)

    1949

  3. [3]

    The se- curity of practical quantum key distribution

    V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The se- curity of practical quantum key distribution”, Re- views of Modern Physics81, 1301–1350 (2009)

    2009

  4. [4]

    Quantum cryptog- raphy: public key distribution and coin tossing

    C. H. Bennett and G. Brassard, “Quantum cryptog- raphy: public key distribution and coin tossing”, Theoretical Computer Science560, 7–11 (2014)

    2014

  5. [5]

    Quantum-state distur- bance versus information gain: uncertainty rela- tions for quantum information

    C. A. Fuchs and A. Peres, “Quantum-state distur- bance versus information gain: uncertainty rela- tions for quantum information”, Physical Review A53, 2038–2045 (1996)

    2038

  6. [6]

    Information-tradeoff relations for finite-strength quantum measure- ments

    C. A. Fuchs and K. Jacobs, “Information-tradeoff relations for finite-strength quantum measure- ments”, Physical Review A63, 062305 (2001)

    2001

  7. [7]

    Information-disturbance tradeoff in quantum measurements

    L. Maccone, “Information-disturbance tradeoff in quantum measurements”, Physical Review A73, 042307 (2006)

    2006

  8. [8]

    Entropic information-disturbance tradeoff

    L. Maccone, “Entropic information-disturbance tradeoff”, Europhysics Letters (EPL)77, 40002 (2007)

    2007

  9. [9]

    Global information balance in quantum mea- surements

    F. Buscemi, M. Hayashi, and M. Horodecki, “Global information balance in quantum mea- surements”, Physical Review Letters100, 210504 (2008)

    2008

  10. [10]

    The information-disturbance tradeoff and the continuity of Stinespring’s representa- tion

    D. Kretschmann, D. Schlingemann, and R. F. Werner, “The information-disturbance tradeoff and the continuity of Stinespring’s representa- tion”, IEEE Transactions on Information Theory 54, 1708–1717 (2008)

    2008

  11. [11]

    Universally valid reformulation of the heisenberg uncertainty principle on noise and disturbance in measurement

    M. Ozawa, “Universally valid reformulation of the heisenberg uncertainty principle on noise and disturbance in measurement”, Physical Review A 67, 042105 (2003)

    2003

  12. [12]

    Universal uncertainty principle in the measurement operator formalism

    M. Ozawa, “Universal uncertainty principle in the measurement operator formalism”, Journal of Optics B: Quantum and Semiclassical Optics7, S672–S681 (2005). 14

    2005

  13. [13]

    Proof of Heisenberg’s error-disturbance relation

    P. Busch, P. Lahti, and R. F. Werner, “Proof of Heisenberg’s error-disturbance relation”, Physi- cal Review Letters111, 160405 (2013)

    2013

  14. [14]

    Qualitative noise- disturbance relation for quantum measurements

    T. Heinosaari and T. Miyadera, “Qualitative noise- disturbance relation for quantum measurements”, Physical Review A88, 042117 (2013)

    2013

  15. [15]

    Sending entanglement through noisy quantum channels

    B. Schumacher, “Sending entanglement through noisy quantum channels”, Physical Review A54, 2614–2628 (1996)

    1996

  16. [16]

    Fidelity balance in quantum oper- ations

    K. Banaszek, “Fidelity balance in quantum oper- ations”, Physical Review Letters86, 1366–1369 (2001)

    2001

  17. [17]

    Balance between information gain and reversibility in weak mea- surement

    Y. W. Cheong and S.-W. Lee, “Balance between information gain and reversibility in weak mea- surement”, Physical Review Letters109, 150402 (2012)

    2012

  18. [18]

    Trade-off relation between information and disturbance in quantum measurement

    T. Shitara, Y. Kuramochi, and M. Ueda, “Trade-off relation between information and disturbance in quantum measurement”, Physical Review A93, 032134 (2016)

    2016

  19. [19]

    Complete informa- tion balance in quantum measurement

    S.-W. Lee, J. Kim, and H. Nha, “Complete informa- tion balance in quantum measurement”, Quantum 5, 414 (2021)

    2021

  20. [20]

    Optimal quantum measurements for additive information and disturbance mea- sures

    H. Terashima, “Optimal quantum measurements for additive information and disturbance mea- sures”, arXiv:2507.03346 (2025)

    Pith/arXiv arXiv 2025

  21. [21]

    Introduction to Haar measure tools in quantum information: a beginner’s tutorial

    A. A. Mele, “Introduction to Haar measure tools in quantum information: a beginner’s tutorial”, Quantum8, 1340 (2024)

    2024

  22. [22]

    M. A. Nielsen and I. L. Chuang,Quantum com- putation and quantum information(Cambridge University Press, 2000)

    2000

  23. [23]

    Hayashi,Quantum information(Springer, 2006)

    M. Hayashi,Quantum information(Springer, 2006)

    2006

  24. [24]

    Lüders theorem for un- sharp quantum measurements

    P. Busch and J. Singh, “Lüders theorem for un- sharp quantum measurements”, Physics Letters A249, 10–12 (1998)

    1998

  25. [25]

    Boyd and L

    S. Boyd and L. Vandenberghe,Convex optimiza- tion(Cambridge University Press, 2004)

    2004

  26. [26]

    Watrous,The theory of quantum information (Cambridge University Press, 2018)

    J. Watrous,The theory of quantum information (Cambridge University Press, 2018)

    2018

  27. [27]

    Skrzypczyk and D

    P. Skrzypczyk and D. Cavalcanti,Semidefinite pro- gramming in quantum information science, IOP Se- ries in Quantum Technology (Institute of Physics Publishing, 2023)

    2023

  28. [28]

    CSDP, a C library for semidefinite programming

    B. Borchers, “CSDP, a C library for semidefinite programming”, Optimization Methods & Soft- ware11, 613–623 (1999)

    1999

  29. [29]

    Positive linear maps on C*-algebras

    M.-D. Choi, “Positive linear maps on C*-algebras”, Canadian Journal of Mathematics24, 520–529 (1972)

    1972

  30. [30]

    General teleportation channel, singlet fraction, and quasidistillation

    M. Horodecki, P. Horodecki, and R. Horodecki, “General teleportation channel, singlet fraction, and quasidistillation”, Physical Review A60, 1888–1898 (1999)

    1999

  31. [31]

    Lagrange multipliers revisited

    M. Slater, “Lagrange multipliers revisited”, in Traces and emergence of nonlinear programming (Springer Basel, 2013), pp. 293–306

    2013

  32. [32]

    Resource- efficient high-dimensional entanglement detec- tion via symmetric projections

    S. Morelli, M. Huber, and A. Tavakoli, “Resource- efficient high-dimensional entanglement detec- tion via symmetric projections”, Physical Review Letters131, 170201 (2023)

    2023

  33. [33]

    Mea- suring measurement: theory and practice

    A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Mea- suring measurement: theory and practice”, New Journal of Physics11, 093038 (2009)

    2009

  34. [34]

    Robustness of measurement, discrimination games, and acces- sible information

    P. Skrzypczyk and N. Linden, “Robustness of measurement, discrimination games, and acces- sible information”, Physical Review Letters122, 140403 (2019)

    2019

  35. [35]

    Conclusive exclusion of quantum states

    S. Bandyopadhyay, R. Jain, J. Oppenheim, and C. Perry, “Conclusive exclusion of quantum states”, Physical Review A89, 022336 (2014)

    2014

  36. [36]

    Symmetric informationally com- plete quantum measurements

    J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally com- plete quantum measurements”, Journal of Mathe- matical Physics45, 2171–2180 (2004)

    2004

  37. [37]

    A com- prehensive review of quantum random number generators: concepts, classification and the origin of randomness

    V. Mannalath, S. Mishra, and A. Pathak, “A com- prehensive review of quantum random number generators: concepts, classification and the origin of randomness”, Quantum Information Process- ing22(2023)

    2023

  38. [38]

    Quantum to classical randomness extractors

    M. Berta, O. Fawzi, and S. Wehner, “Quantum to classical randomness extractors”, IEEE Trans- actions on Information Theory60, 1168–1192 (2014)

    2014

  39. [39]

    Quantum randomness extraction for various lev- els of characterization of the devices

    Y. Z. Law, L. P. Thinh, J.-D. Bancal, and V. Scarani, “Quantum randomness extraction for various lev- els of characterization of the devices”, Journal of Physics A: Mathematical and Theoretical47, 424028 (2014)

    2014

  40. [40]

    Operational characterization of infinite- dimensional quantum resources

    E. Haapasalo, T. Kraft, J. -P. Pellonpää, and R. Uola, “Operational characterization of infinite- dimensional quantum resources”, Physical Re- view Letters127(2021)

    2021

  41. [41]

    Op- erational quantification of continuous-variable quantum resources

    B. Regula, L. Lami, G. Ferrari, and R. Takagi, “Op- erational quantification of continuous-variable quantum resources”, Physical Review Letters126 (2021). 15

    2021

  42. [42]

    Compact convex structure of measurements and its applications to simu- lability, incompatibility, and convex resource theory of continuous-outcome measurements

    Y. Kuramochi, “Compact convex structure of measurements and its applications to simu- lability, incompatibility, and convex resource theory of continuous-outcome measurements”, arXiv:2002.03504 (2020)

    Pith/arXiv arXiv 2002

  43. [43]

    T. A. Ohst,Statistical disturbance bounds, GitLab repository, (2026). 16

    2026