Optimal quantum locally differentially private mechanisms in the high-privacy regime

Yuuya Yoshida

arxiv: 2605.27278 · v2 · pith:N5HGDNGTnew · submitted 2026-05-26 · 🪐 quant-ph · cs.IT· math.IT

Optimal quantum locally differentially private mechanisms in the high-privacy regime

Yuuya Yoshida This is my paper

Pith reviewed 2026-06-29 17:33 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT

keywords local differential privacyquantum local differential privacyhigh-privacy regimeHolevo informationhypothesis testingquantum advantageasymptotic ratio

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The pith

Optimal LDP and QLDP mechanisms achieve classical and quantum values with the same asymptotic ratio Q/C independent of utility in the high-privacy regime.

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

The paper establishes optimal mechanisms for local differential privacy and its quantum version that reach the best possible utility values when privacy is taken to the extreme limit. It shows that the ratio of quantum to classical optimal utility settles to one fixed number for any of several different utility measures, including Holevo information and error exponents from hypothesis testing. For data with three or more possible values this fixed ratio is at least 3/2, which means quantum mechanisms can deliver strictly higher utility than classical ones under the same privacy constraint.

Core claim

In the high-privacy regime the optimal QLDP mechanisms attain the quantum value Q and the optimal LDP mechanisms attain the classical value C; the ratio Q/C is independent of the choice among the listed utility functions and satisfies Q/C >= 3/2 whenever the protected data are drawn from an alphabet of size n >= 3.

What carries the argument

The high-privacy regime obtained by sending the privacy parameter epsilon to zero while holding the utility (Holevo information or error exponent) fixed.

If this is right

Explicit optimal LDP and QLDP mechanisms exist that saturate the classical and quantum bounds.
The same limiting ratio Q/C governs every utility function examined.
Quantum advantage appears automatically for any n-ary private data with n at least 3.
The advantage is expressed solely through the ratio and does not depend on further details of the utility.

Where Pith is reading between the lines

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

The independence of the ratio from utility choice may allow designers to pick whichever utility is easiest to compute without changing the predicted quantum gain.
The same limiting construction could be tested on small quantum devices that implement the optimal QLDP map for ternary inputs.
If the ratio remains constant for still other utilities, the result would strengthen the case that quantum local privacy offers a universal multiplicative improvement in the high-privacy limit.

Load-bearing premise

The privacy parameter must be driven to zero while the chosen utility measure is held constant.

What would settle it

An explicit calculation showing that the limiting ratio Q/C differs for two different utility functions, or that no mechanism reaches the stated classical or quantum optimum.

Figures

Figures reproduced from arXiv: 2605.27278 by Yuuya Yoshida.

**Figure 1.** Figure 1: Graphs of S 1 (~σ∗ )/S1 C(n, ε) and A1 (~σ∗ )/A1 C(n, ε). The horizontal axes represent the values of ε, and the vertical axes represent the values of the two ratios. The black dotted lines, the blue dashed lines, and the red solid lines are the cases n = 3, n = 6, and n = 10, respectively. Since the above two upper bounds for ε increase in n, the ranges of the parameter ε are the most restricted in the … view at source ↗

**Figure 2.** Figure 2: Graphs of S 1 (~σ)/S1 C(10, ε) and A1 (~σ)/A1 C(10, ε). The horizontal axes represent the values of ε, and the vertical axes represent the values of the two ratios. The red solid lines and the red dotted lines are the cases ~σ = ~σ∗ and ~σ = ~σ∗∗, respectively. κn(d, r) = r/d. To the best of our knowledge, it remains open whether (∗) κn(Tn) is dense in the interval [1/n, 1/2]. Throughout this section, we … view at source ↗

read the original abstract

We optimize the trade-off between privacy and utility in the high-privacy regime. We adopt local differential privacy (LDP) and its quantum extension, quantum local differential privacy (QLDP), for privacy protection, and investigate utility functions including the Holevo information (which reduces to the mutual information in the classical case) and the error exponents in symmetric and asymmetric hypothesis testing. These utility functions have classical and quantum optimal values, which are denoted by $C$ and $Q$, respectively, in this abstract for simplicity. In this paper, we provide optimal LDP and QLDP mechanisms achieving the classical and quantum optimal values in the high-privacy regime, and prove that the asymptotic ratio $Q/C$ in this regime takes the same value regardless of the utility function. Our results reveal quantum advantages (more precisely, $Q/C\ge3/2$) for the above utility functions when the protected private data are $n$-ary with $n\ge3$.

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 optimal QLDP mechanisms in the epsilon to zero limit and proves the Q/C utility ratio is invariant across functions and at least 3/2 for n-ary inputs with n at least 3.

read the letter

The main takeaway is that this work constructs optimal mechanisms for both LDP and QLDP when privacy goes to the high-privacy limit, then shows the resulting quantum-to-classical utility ratio stays fixed no matter whether you measure utility by Holevo information or by symmetric or asymmetric hypothesis-testing exponents, and that the ratio is at least 3/2 once the input alphabet size reaches three or more.

The paper does two things cleanly. It supplies explicit optimal mechanisms in that limit, and it proves the ratio does not depend on the choice of utility. The invariance result is new relative to the classical LDP papers cited and to earlier quantum privacy work. A concrete lower bound on the advantage is also useful for anyone thinking about when quantum resources might matter for privacy.

The limitations are mostly about scope rather than internal problems. All claims are asymptotic as epsilon approaches zero while the utility stays fixed, so they give no direct guidance for any fixed positive epsilon that would be used in practice. The advantage appears only for n greater than or equal to 3, which is consistent but narrows the immediate applicability. The derivations themselves cannot be checked from the abstract alone, but the listed assumptions are standard definitions and there is no sign of circularity or free parameters.

This is a paper for people working on quantum differential privacy and on asymptotic trade-offs between privacy and information quantities. A reader interested in quantum advantages in privacy mechanisms would find the invariance result worth seeing. It is worth sending to peer review because the claims are concrete enough to be tested and the ratio result, if it holds, is a clear addition to the literature.

Referee Report

0 major / 2 minor

Summary. The paper optimizes the privacy-utility trade-off under local differential privacy (LDP) and its quantum extension (QLDP) in the high-privacy regime (ε → 0 with utility fixed). It constructs explicit optimal mechanisms achieving the classical optimum C and quantum optimum Q for utilities including Holevo information (reducing to mutual information classically) and symmetric/asymmetric hypothesis-testing error exponents. The central results are that the asymptotic ratio Q/C is independent of the utility function and that Q/C ≥ 3/2 for n-ary inputs with n ≥ 3, establishing quantum advantages.

Significance. If the derivations hold, the utility-independence of the limiting Q/C ratio is a strong structural result that unifies several privacy-utility settings. The explicit optimal mechanisms and the concrete lower bound of 3/2 for n ≥ 3 provide both constructive and information-theoretic value for quantum private data processing. The work strengthens the case for quantum advantages in local privacy beyond specific utilities.

minor comments (2)

[Introduction / Section 2] The high-privacy regime is defined via ε → 0 while holding utility fixed; a brief remark on the order of limits (e.g., whether n is fixed or grows) would clarify applicability to finite-alphabet settings.
[Abstract] Notation for the classical and quantum optimal values (C and Q) is introduced in the abstract but the precise definitions appear only after the utility functions are stated; moving the definitions forward would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No major comments were listed in the report, so there are no specific points requiring point-by-point response or manuscript changes at this stage.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs optimal LDP/QLDP mechanisms for the ε→0 high-privacy limit while holding utility (Holevo information or hypothesis-testing error exponents) fixed, then proves the resulting asymptotic Q/C ratio is utility-independent and ≥3/2 for n-ary inputs with n≥3. These steps rest on the standard definitions of LDP, QLDP, and the listed utility functions; the limiting procedure and explicit mechanism constructions do not reduce the claimed ratio to a fitted parameter, self-citation, or input-by-construction. No load-bearing step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on the standard mathematical definitions of local differential privacy, quantum local differential privacy, Holevo information, and symmetric/asymmetric hypothesis testing error exponents; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (3)

standard math Standard definition of epsilon-local differential privacy for classical mechanisms
Invoked when stating optimal LDP mechanisms in the high-privacy regime
standard math Standard definition of quantum local differential privacy for quantum channels
Invoked when stating optimal QLDP mechanisms
domain assumption Holevo information and hypothesis-testing error exponents are the utility functions under optimization
Used to define the classical and quantum optimal values C and Q

pith-pipeline@v0.9.1-grok · 5688 in / 1512 out tokens · 33523 ms · 2026-06-29T17:33:56.821812+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 7 canonical work pages · 3 internal anchors

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J. Calsamiglia, R. Mu˜ noz Tapia, Ll. Masanes, A. Acin, and E. Bagan . Quantum Chernoﬀ bound as a measure of distinguishability between density matrices: Applicat ion to qubit and gaussian states. Phys. Rev. A , 77(3):032311, 2008

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Q. Geng and P. Viswanath. The optimal noise-adding mechanism in diﬀerential privacy. IEEE Trans. Inform. Theory , 62(2):925–951, 2016

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M. Hayashi. Quantum Information Theory: Mathematical Foundation, Sec ond Edition . Springer, Berlin, Heidelberg, 2017. OPTIMAL QLDP MECHANISMS IN THE HIGH-PRIV ACY REGIME 45

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P. Kairouz, S. Oh, and P. Viswanath. Extremal mechanisms for local diﬀerential privacy. J. Mach. Learn. Res., 17:Paper No. 17, 51, 2016

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E. J. King. New constructions and characterizations of ﬂat an d almost ﬂat Grassmannian fusion frames. Preprint, available at https://arxiv.org/abs/1612.05784, 2016

work page arXiv 2016
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D. W. Kribs, D. Mammarella, and R. Pereira. Isoclinic subspaces a nd quantum error correction. Oper. Matrices, 15(2):571–580, 2021

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A. Lesniewski and M. B. Ruskai. Monotone Riemannian metrics an d relative entropy on noncom- mutative probability spaces. J. Math. Phys. , 40(11):5702–5724, 1999

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S.-H. Nam, H.-Y. Park, S.-H. Lee, and J. Bae. Quantum advanta ge in locally diﬀerentially private hypothesis testing. Preprint, available at https://arxiv.org/abs/2501.10152v3, 2025

work page internal anchor Pith review arXiv 2025
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H. M. Nguyen, H. A. Nguyen, and C. T. Le. Optimization of maxima l quantum f -divergences between unitary orbits. Preprint, available at https://arxiv.org/abs/2601.08268, 2026

work page arXiv 2026
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Nuradha, S

T. Nuradha, S. Bhalerao, and F. Leditzky. Privacy-utility trad eoﬀs in quantum information pro- cessing. Preprint, available at https://arxiv.org/abs/2602.10510v1, 2026

work page arXiv 2026
[30]

Park, S.-H

H.-Y. Park, S.-H. Nam, and S.-H. Lee. Exactly optimal and commu nication-eﬃcient private esti- mation via block designs. IEEE J. Sel. Areas Inf. Theory , 5:123–134, 2024

2024
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D. Petz. Monotone metrics on matrix spaces. Linear Algebra Appl. , 244:81–96, 1996

1996
[32]

D. Petz. Information-geometry on quantum state. Quantum Probab. Commun. , X:135–157, 1998

1998
[33]

Petz and C

D. Petz and C. Sud´ ar. Geometries of quantum states. J. Math. Phys. , 37(6):2662–2673, 1996

1996
[34]

S. Wang, L. Huang, P. Wang, Y. Nie, H. Xu, W. Yang, X.-Y. Li, and C. Qiao. Mu- tual information optimally local private discrete distribution estimat ion. Preprint, available at https://arxiv.org/abs/1607.08025v1, 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016
[35]

Ye and A

M. Ye and A. Barg. Optimal schemes for discrete distribution es timation under locally diﬀerential privacy. IEEE Trans. Inform. Theory , 64(8):5662–5676, 2018

2018
[36]

Y. Yoshida. Mathematical comparison of classical and quantum mechanisms in optimization under local diﬀerential privacy. J. Phys. A , 58(3):035301, 2025

2025
[37]

Yoshida and M

Y. Yoshida and M. Hayashi. Classical mechanism is optimal in classic al-quantum diﬀerentially private mechanisms. In 2020 IEEE International Symposium on Information Theory (I SIT), pages 1973–1977. 2020. Yuuya Yoshida, Independent Researcher, Komaki, Aichi, 485 -0003, Japan Email address : yyoshida9130@gmail.com

2020

[1] [1]

K. M. R. Audenaert, J. Calsamiglia, R. Mu˜ noz Tapia, E. Bagan, Ll. Masanes, A. Acin, and F. Ver- straete. Discriminating states: the quantum Chernoﬀ bound. Phys. Rev. Lett. , 98(16):160501, 2007

2007

[2] [2]

R. Bhatia. Matrix Analysis . Springer, 2013

2013

[3] [3]

B¨ ottcher and I

A. B¨ ottcher and I. M. Spitkovsky. A gentle guide to the basics o f two projections theory. Linear Algebra Appl., 432(6):1412–1459, 2010

2010

[4] [4]

Calsamiglia, R

J. Calsamiglia, R. Mu˜ noz Tapia, Ll. Masanes, A. Acin, and E. Bagan . Quantum Chernoﬀ bound as a measure of distinguishability between density matrices: Applicat ion to qubit and gaussian states. Phys. Rev. A , 77(3):032311, 2008

2008

[5] [5]

J. C. Duchi, M. I. Jordan, and M. J. Wainwright. Local privacy an d statistical minimax rates. In 2013 IEEE 54th Annual Symposium on Foundations of Computer S cience—FOCS 2013 , pages 429–438. IEEE Computer Soc., Los Alamitos, CA, 2013

2013

[6] [6]

C. Dwork. Diﬀerential privacy. In Automata, languages and programming. Part II , volume 4052 of Lecture Notes in Comput. Sci. , pages 1–12. Springer, Berlin, 2006

2006

[7] [7]

Dwork, F

C. Dwork, F. McSherry, K. Nissim, and A. Smith. Calibrating noise t o sensitivity in private data analysis. In Theory of cryptography, volume 3876 of Lecture Notes in Comput. Sci. , pages 265–284. Springer, Berlin, 2006

2006

[8] [8]

Fickus, E

M. Fickus, E. Gomez-Leos, and J. W. Iverson. Radon–Hurwitz G rassmannian codes. IEEE Trans. Inform. Theory, 71(4):3203–3213, 2025

2025

[9] [9]

Fickus, J

M. Fickus, J. W. Iverson, J. Jasper, and D. G. Mixon. Harmonic G rassmannian codes. Appl. Comput. Harmon. Anal. , 65:1–39, 2023

2023

[10] [10]

Fickus, J

M. Fickus, J. W. Iverson, J. Jasper, and D. G. Mixon. Totally sy mmetric Grassmannian codes. Preprint, available at https://arxiv.org/abs/2406.19542v2, 2026

work page arXiv 2026

[11] [11]

Fickus, J

M. Fickus, J. Jasper, D. G. Mixon, and C. E. Watson. A brief intr oduction to equi-chordal and equi-isoclinic tight fusion frames. In Wavelets and Sparsity XVII , volume 10394, pages 186–194. SPIE, 2017

2017

[12] [12]

Tables of the existence of equiangular tight frames

M. Fickus and D. G. Mixon. Tables of the existence of equiangular tight frames. Preprint, available at https://arxiv.org/abs/1504.00253v2, 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016

[13] [13]

Q. Geng, P. Kairouz, S. Oh, and P. Viswanath. The staircase me chanism in diﬀerential privacy. IEEE J. Sel. Topics Signal Process. , 9(7):1176–1184, 2015

2015

[14] [14]

Geng and P

Q. Geng and P. Viswanath. The optimal noise-adding mechanism in diﬀerential privacy. IEEE Trans. Inform. Theory , 62(2):925–951, 2016

2016

[15] [15]

Geng and P

Q. Geng and P. Viswanath. Optimal noise adding mechanisms for a pproximate diﬀerential privacy. IEEE Trans. Inform. Theory , 62(2):952–969, 2016

2016

[16] [16]

M. R. Grace and S. Guha. Perturbation theory for quantum inf ormation. In 2022 IEEE Informa- tion Theory Workshop (ITW) , pages 500–505, 2022

2022

[17] [17]

J. Guan. Optimal mechanisms for quantum local diﬀerential priv acy. In Proceedings of the 2025 ACM SIGSAC Conference on Computer and Communications Secur ity, pages 3737–3749. Asso- ciation for Computing Machinery, New York, 2025

2025

[18] [18]

M. Hayashi. Quantum Information Theory: Mathematical Foundation, Sec ond Edition . Springer, Berlin, Heidelberg, 2017. OPTIMAL QLDP MECHANISMS IN THE HIGH-PRIV ACY REGIME 45

2017

[19] [19]

Hiai and D

F. Hiai and D. Petz. Introduction to Matrix Analysis and Applications . Springer, 2014

2014

[20] [20]

Holohan, D

N. Holohan, D. J. Leith, and O. Mason. Optimal diﬀerentially priva te mechanisms for randomised response. IEEE Trans. Inf. Forensics Secur. , 12(11):2726–2735, 2017

2017

[21] [21]

Kairouz, S

P. Kairouz, S. Oh, and P. Viswanath. Extremal mechanisms for local diﬀerential privacy. J. Mach. Learn. Res., 17:Paper No. 17, 51, 2016

2016

[22] [22]

E. J. King. New constructions and characterizations of ﬂat an d almost ﬂat Grassmannian fusion frames. Preprint, available at https://arxiv.org/abs/1612.05784, 2016

work page arXiv 2016

[23] [23]

D. W. Kribs, D. Mammarella, and R. Pereira. Isoclinic subspaces a nd quantum error correction. Oper. Matrices, 15(2):571–580, 2021

2021

[24] [24]

D. W. Kribs, R. Pereira, and M. Taank. Generalized Knill–Laﬂamme theorem for families of isoclinic subspaces. Can. Math. Bull. , 68(3):992–1010, 2025

2025

[25] [25]

Lesniewski and M

A. Lesniewski and M. B. Ruskai. Monotone Riemannian metrics an d relative entropy on noncom- mutative probability spaces. J. Math. Phys. , 40(11):5702–5724, 1999

1999

[26] [26]

Nagaj, P

D. Nagaj, P. Wocjan, and Y. Zhang. Fast ampliﬁcation of QMA. Quantum Inf. Comput. , 9(11&12):1053–1068, 2009

2009

[27] [27]

Nam, H.-Y

S.-H. Nam, H.-Y. Park, S.-H. Lee, and J. Bae. Quantum advanta ge in locally diﬀerentially private hypothesis testing. Preprint, available at https://arxiv.org/abs/2501.10152v3, 2025

work page internal anchor Pith review arXiv 2025

[28] [28]

H. M. Nguyen, H. A. Nguyen, and C. T. Le. Optimization of maxima l quantum f -divergences between unitary orbits. Preprint, available at https://arxiv.org/abs/2601.08268, 2026

work page arXiv 2026

[29] [29]

Nuradha, S

T. Nuradha, S. Bhalerao, and F. Leditzky. Privacy-utility trad eoﬀs in quantum information pro- cessing. Preprint, available at https://arxiv.org/abs/2602.10510v1, 2026

work page arXiv 2026

[30] [30]

Park, S.-H

H.-Y. Park, S.-H. Nam, and S.-H. Lee. Exactly optimal and commu nication-eﬃcient private esti- mation via block designs. IEEE J. Sel. Areas Inf. Theory , 5:123–134, 2024

2024

[31] [31]

D. Petz. Monotone metrics on matrix spaces. Linear Algebra Appl. , 244:81–96, 1996

1996

[32] [32]

D. Petz. Information-geometry on quantum state. Quantum Probab. Commun. , X:135–157, 1998

1998

[33] [33]

Petz and C

D. Petz and C. Sud´ ar. Geometries of quantum states. J. Math. Phys. , 37(6):2662–2673, 1996

1996

[34] [34]

S. Wang, L. Huang, P. Wang, Y. Nie, H. Xu, W. Yang, X.-Y. Li, and C. Qiao. Mu- tual information optimally local private discrete distribution estimat ion. Preprint, available at https://arxiv.org/abs/1607.08025v1, 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016

[35] [35]

Ye and A

M. Ye and A. Barg. Optimal schemes for discrete distribution es timation under locally diﬀerential privacy. IEEE Trans. Inform. Theory , 64(8):5662–5676, 2018

2018

[36] [36]

Y. Yoshida. Mathematical comparison of classical and quantum mechanisms in optimization under local diﬀerential privacy. J. Phys. A , 58(3):035301, 2025

2025

[37] [37]

Yoshida and M

Y. Yoshida and M. Hayashi. Classical mechanism is optimal in classic al-quantum diﬀerentially private mechanisms. In 2020 IEEE International Symposium on Information Theory (I SIT), pages 1973–1977. 2020. Yuuya Yoshida, Independent Researcher, Komaki, Aichi, 485 -0003, Japan Email address : yyoshida9130@gmail.com

2020