Pith. sign in

REVIEW 2 major objections 6 minor 80 references

Predicting symmetries of quantum dynamics with optimal samples

T0 review · 2 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read Parallel strategies achieve the optimal error for testing quantum dynamics symmetries.

desk verdict A solid general theorem for unitary subgroup testing with real new formulas, but the T-symmetry no-ancilla claim lacks a proof. read the letter →

arxiv 2502.01464 v1 pith:DKTWGYYI submitted 2025-02-03 quant-ph cs.ITmath.IT

Masahito Hayashi , Yu-Ao Chen , Chenghong Zhu , Xin Wang This is my paper · ORCID
classification quant-phcs.ITmath.IT MSC 81P4581P6894A17
keywords quantumhypothesistestingunitarysubgroupmax-relativeentropygrouprepresentationtheoryindefinitecausalordersamplecomplexitysymmetryqubitunitaries
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that deciding whether an unknown unitary operation has a particular symmetry—identity, diagonal (Z), or real (T)—has a fundamental limit set by a single quantity: the quantum max-relative entropy between two average Choi states built from the symmetry subgroup and from the full unitary group. The main theorem states that the smallest possible probability of accepting the symmetry when the operation is actually generic is exactly $(1-\varepsilon)e^{-D_{\max}(\rho^f_{\mu_0}\|\rho^f_\mu)}$, and that simple parallel protocols using several copies and one measurement attain this value, with adaptive and indefinite-causal-order protocols offering no advantage. This yields explicit qubit sample complexities: $O(\delta^{-1/3})$ copies to certify identity and $O(\delta^{-1/2})$ copies to certify Z- or T-symmetry, with zero false rejection and failure probability at most $\delta$. If correct, the result removes the need for complex control sequences and gives a universal, computable formula for symmetry-property testing of quantum dynamics.

What carries the argument

The argument turns on the performance operator $\rho^f_\mu=\mathbb{E}_{U\sim\mu}|f(U)\rangle\!\rangle\langle\!\langle f(U)|$, the average Choi state of the representation $f$ under the group's Haar measure, and the quantum max-relative entropy $D_{\max}(P\|Q)=\min\{t:e^t Q\ge P\}$. The proof shows that the optimization over parallel tests, over adaptive tests, and over indefinite-causal-order tests all reduce to the same unconstrained optimization over positive operators $\min\{{\rm Tr}(T\rho^f_\mu):{\rm Tr}(T\rho^f_{\mu_0})\ge 1-\varepsilon\}$, because twirling any feasible strategy by the group action maps it into an invariant, parallel-achievable form without changing the errors. Theorem 2 then computes $e^{-D_{\max}}$ through the irreducible-representation decompositions of $f$ as a representation of $G$ and of its restriction to $G_0$, using the multiplicities $n_{\eta,\lambda}$.

What would settle it

A concrete check: for a fixed group pair and small $n$, numerically optimize a general adaptive or indefinite-causal-order protocol using the linear conditions of Ref. [46] and compare its worst-case type-II error against $(1-\varepsilon)e^{-D_{\max}(\rho^f_{\mu_0}\|\rho^f_\mu)}$; finding a protocol with strictly smaller error would refute Theorem 1. A simpler experiment would replace the Haar prior by a point mass at a fixed non-identity unitary and show that a sequential strategy beats the claimed parallel bound.

Watch

Extended reading notes

Core claim

The paper's central claim is an exact equality of the optimal type-II error across all three protocol families in unitary subgroup hypothesis testing. For a compact group $G$, a subgroup $G_0$, a unitary representation $f$, and a type-I tolerance $\varepsilon$, the optimal type-II error satisfies $\beta^f_{\mathrm{PAR}}(\varepsilon)=\beta^f_{\mathrm{ICO}}(\varepsilon)=\bar\beta^f_{\mathrm{ICO}}(\varepsilon)=(1-\varepsilon)e^{-D_{\max}(\rho^f_{\mu_0}\|\rho^f_\mu)}$, where $\rho^f_\mu$ and $\rho^f_{\mu_0}$ are the Choi averages (performance operators) of $f$ over the Haar measures of the full group and the subgroup. The paper also proves this value is computable from representation-theoretic data alone, as $\min_{\eta\in\hat G_0^f}\frac{d_{\eta,G_0}}{\sum_{\lambda\in\hat G_f}d_\lambda n_{\eta,\lambda}}$, and works out the qubit cases in closed form: $\frac{6}{(n+1)(n+2)(n+3)}$ for identity testing, $\frac{4}{(n+2)^2}$ (even $n$) and $\frac{4}{(n+1)(n+3)}$ (odd $n$) for Z-symmetry, and $\frac{8}{(n+2)(n+4)}$ (even $n$) and $\frac{8}{(n+1)(n+3)}$ (odd $n$) for T-symmetry, where $n$ is the number of queries.

Load-bearing premise

The load-bearing premise is that the 'wrong' case is a unitary drawn uniformly at random from the full group according to its Haar measure (the white-noise model); if the unknown operations come from any other distribution, the claimed optimal errors and sample counts are not guaranteed.

Editorial extensions

If this is right

  • No adaptive feedforward or indefinite causal order is ever needed: for every compact group and subgroup, the optimal symmetry test is a parallel test with a pure input state and a single measurement.
  • The optimal error factorizes linearly in the tolerated false-rejection rate, $\beta(\varepsilon)=(1-\varepsilon)\beta(0)$, so tests designed at zero tolerance remain optimal at any positive tolerance.
  • Qubit identity certification can be done with $\Theta(\delta^{-1/3})$ uses of the unknown operation, while Z-symmetry and T-symmetry certification require $\Theta(\delta^{-1/2})$ uses, at failure probability at most $\delta$ and no false acceptances.
  • The optimal Z-symmetry and T-symmetry tests require no ancilla, while the identity test does; for T-symmetry the error does not improve when the number of queries increases from an even number to the next odd number.
  • If the subgroup $G_0$ forms a unitary $n$-design, no $n$-query test can distinguish it from the full unitary group at all.

Reading between the lines

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

  • Because the collapse relies only on the group structure and the linear feasibility conditions, the same equality is likely to hold for group-covariant channel estimation, complementing the estimation result cited in the paper.
  • The closed-form formulas give an immediate experimental recipe: certifying a qubit gate as identity should consume only about $\delta^{-1/3}$ copies, which a cloud-quantum experiment could verify directly against the claimed bound.
  • Replacing the Haar white-noise prior with a $t$-design or a fixed-gate noise model would likely turn the $D_{\max}$ formula into a valid but looser upper bound; testing that numerically would quantify exactly how much the Haar assumption contributes.
  • Theorem 2's representation-theoretic formula already applies to the Weyl–Heisenberg and Clifford subgroups the paper lists as future directions, so the same machinery is in place for those tests once the branching multiplicities are computed.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper studies hypothesis testing of unitary symmetries: given an unknown unitary drawn either Haar-uniformly from a compact group G or from a subgroup G0, the task is to distinguish the two cases using multiple queries under parallel, adaptive, or indefinite-causal-order (ICO) strategies. The main theorem claims that all three strategy classes have the same optimal type-II error, (1−ε) exp(−Dmax(ρ^f_μ0 ∥ ρ^f_μ)), and that worst-case and average-case type-I constraints are equivalent in this setting. The supplement proves this via a single-shot hypothesis-testing bound together with explicit group-twirling constructions, and derives a representation-theoretic formula for Dmax. Applications to qubit identity testing, Z-symmetry testing, and T-symmetry testing yield explicit formulas for the optimal type-II error and sample complexities O(δ^{−1/3}) and O(δ^{−1/2}). The paper also claims that the optimal parallel tests for Z- and T-symmetry do not require an ancilla system, whereas the identity test does.

Significance. If the main theorem and the application formulas are correct, this is a significant contribution to unitary property testing. It gives an exact, parameter-free characterization of the optimal type-II error for subgroup hypothesis testing, proves that parallel strategies match adaptive and indefinite-causal-order strategies for this group-structured problem, and provides a new operational interpretation of the max-relative entropy. The explicit qubit formulas and sample-complexity scalings are concrete and falsifiable, and the supplement's proof is carefully assembled from standard pieces. The apparent sign discrepancy between main-text Eq. (9) and Supplement Eq. (S30) is resolved by taking reciprocals, and the arithmetic of the three applications checks out. The main weakness is that the ancilla-free claim for T-symmetry testing is not supported by the supplement's own sufficient condition; this is a resource claim that needs either a proof or a qualification. The central Theorem 1 and the sample-complexity formulas are not affected by this gap.

major comments (2)
  1. [Supplement III C (T-symmetry), Theorem 6, Eq. (S63); main text Introduction] The claim that the optimal parallel T-symmetry test does not require an ancilla is not supported by the supplied proof. The only ancilla-free criterion in the supplement is Theorem 6, whose condition Eq. (S63) requires d_{η,G0} n_{η,λ} ≤ n_λ for every λ in the optimal irreducible representation η0. In the T-symmetry application with d=2 and odd n=2k−1, the optimal η0 is the two-dimensional irrep with l=k−1 (Supp. Eqs. (S78)–(S79)). For n=3 (k=2, l=1), this η0 appears in the λ with d_λ=4, whose multiplicity is n_λ=1, while d_{η,G0} n_{η,λ}=2·1=2; hence Eq. (S63) fails. The same violation occurs for every odd n≥3 in the highest-weight λ, whose multiplicity is 1. Since no alternative ancilla-free construction is provided, the headline assertion in the Introduction that optimal parallel T-symmetry tests are ancilla-free is unproven. This does not invalidate Theorem 1 or the explicit type-II error formulas, but the claim must be either proved with a different construction or removed/qualified.
  2. [Supplement III C and main text around Eq. (13)] For T-symmetry testing, the supplement computes the optimal type-II error but never verifies the ancilla-free condition Eq. (S63) for the optimal η in either the even-n or odd-n case. Since the paper's only general ancilla-free theorem is Theorem 6, the statement in the Introduction that optimal parallel T-symmetry tests do not require ancilla needs an explicit check of Eq. (S63) or a separate construction. As written, Section III C establishes the value of the type-II error but not the ancilla-free achievability claimed in the main text.
minor comments (6)
  1. [Theorem 2 and Supplement Eq. (S30)] The main text's Eq. (9) states e^{−Dmax} = min_η d_{η,G0} / (Σ_λ d_λ n_{η,λ}), while the supplement derives e^{Dmax} = max_η d_{η,G0}^{−1} Σ_λ d_λ n_{η,λ}. These are reciprocals of each other and are consistent, but the main text should say this explicitly to avoid the apparent contradiction.
  2. [Throughout] The subgroup is denoted K in Theorem 1 and G0 elsewhere; please unify the notation.
  3. [Supplement II C] The phrase 'maxin text' should be 'main text'.
  4. [Figure 1 caption and concluding remarks] 'Indefinite Casual Order' should be 'Indefinite Causal Order'.
  5. [Section 'Main Results', implication d] The sentence 'This result also implies that when G0 is a unitary n-design, it is impossible to distinguish between U∼μ_G0 and U∼μ_U(d) within n uses' is stated without proof or citation; a one-line justification would help because it is not immediate from Eq. (9).
  6. [Applications, Eqs. (12) and (13)] The application formulas are stated for G=U(d) while the supplement's derivations use G=SU(d); the global phase drops out of the Choi operator, so the results are consistent, but the equivalence should be noted.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem and formulas are derived from first principles, with self-citations only contextual.

full rationale

The central derivation is self-contained. The optimal type-II error is obtained by solving the hypothesis-testing optimization directly: the lower bound (S24) follows from the definition of Dmax, and achievability is shown by a vector that saturates the generalized eigenvalue in (S25). The parallel strategy constructed in Section II D supplies the matching upper bound, so Theorem 1 is not assumed or fitted. Theorem 2 is proven in Supplement Section II C by decomposing the performance operators and computing Dmax from the block structure; the branching formula (S46) is derived, not imported. The application formulas for identity, Z-symmetry, and T-symmetry testing are obtained by explicit dimension and multiplicity computations in the supplement. While the paper cites earlier works by the same authors (e.g., Refs. [26], [60], [73]), these citations are used for context, comparison, or standard combinatorial identities; they are not load-bearing for the proof of the main theorem. The skeptic's concern about the no-ancilla T-symmetry claim is a potential correctness gap in a resource statement, not a circularity, since it does not reduce any derived result to an input by construction. No fitted parameter is renamed as a prediction, and no uniqueness or ansatz is smuggled in via self-citation. Hence there is no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The 'performance operator' is a defined mathematical object, not an invented entity. The central claim rests on standard representation theory and the Haar-prior model.

assumptions (4)
  • domain assumption The unknown unitary, when it does not belong to the subgroup G0, is drawn from the Haar measure mu on G.
    This is the white-noise model stated in the introduction and formalized by rho^f_mu in Eq. (2). All type-II error and sample-complexity claims depend on this prior.
  • domain assumption Any admissible strategy can be represented by a measurement operator T satisfying the linear ICO conditions of Ref. [46].
    Used in Theorem 1 and Section II A to define beta_ICO. The proof also assumes twirling preserves this feasible set.
  • standard math Standard results of compact group representation theory, including Peter-Weyl orthogonality and the decomposition (S5) into irreps, are correct.
    Used throughout Theorem 2, Theorem 4, and all application calculations.
  • standard math The relevant multiplicities n_eta,lambda and dimensions d_lambda can be computed for the tensor-product representations; the SU(2) and O(2) decompositions quoted from Refs. [55,60] are accurate.
    Used in the qubit identity, Z, and T formulas.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Predicting symmetries of quantum dynamics with optimal samples." pith.science (2026). https://pith.science/paper/DKTWGYYI

@misc{pith2026250201464,
  author       = {Pith},
  title        = {Pith review of: Predicting symmetries of quantum dynamics with optimal samples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DKTWGYYI}},
  note         = {Machine review of arXiv:2502.01464}
}
abstract

Identifying symmetries in quantum dynamics, such as identity or time-reversal invariance, is a crucial challenge with profound implications for quantum technologies. We introduce a unified framework combining group representation theory and subgroup hypothesis testing to predict these symmetries with optimal efficiency. By exploiting the inherent symmetry of compact groups and their irreducible representations, we derive an exact characterization of the optimal type-II error (failure probability to detect a symmetry), offering an operational interpretation for the quantum max-relative entropy. In particular, we prove that parallel strategies achieve the same performance as adaptive or indefinite-causal-order protocols, resolving debates about the necessity of complex control sequences. Applications to the singleton group, maximal commutative group, and orthogonal group yield explicit results: for predicting the identity property, Z-symmetry, and T-symmetry of unknown qubit unitaries, with zero type-I error and type-II error bounded by $\delta$, we establish the explicit optimal sample complexity which scales as $\mathcal{O}(\delta^{-1/3})$ for identity testing and $\mathcal{O}(\delta^{-1/2})$ for T/Z-symmetry testing. These findings offer theoretical insights and practical guidelines for efficient unitary property testing and symmetry-driven protocols in quantum information processing.

Figures

Figures reproduced from arXiv: 2502.01464 by the authors.

Figure 1
Figure 1. FIG. 1: An illustration of hypothesis testing for quantum [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The optimal type-II error scaling with respect to [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

80 extracted references · 73 canonical work pages

  1. [26]

    Y. A. Chen, C. Zhu, K. He, Y. Liu, and X, Wang, Hy- pothesis testing of symmetry in quantum dynamics , arXiv:2411.14292, (2024)

    arXiv 2024

  2. [60]

    Heisenberg scaling based on population coding

    M. Hayashi, “Heisenberg scaling based on population cod- ing,” arXiv:2410.13399

    work page Pith review arXiv

  3. [1]

    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information , (Cambridge University Press, Cam- bridge, England, 2000), (2009)

    work page 2009

  4. [2]

    Chuang and M

    I. Chuang and M. Nielsen, Prescription for experimental de- termination of the dynamics of a quantum black box, J. Mod. Opt. 44, 2455 (1997)

    work page 1997

  5. [3]

    J. F . Poyatos, J. I. Cirac, and P . Zoller,Complete Characteriza- tion of a Quantum Process: The Two-Bit Quantum Gate, Phys. Rev. Lett. 78, 390 (1997)

    work page 1997

  6. [4]

    D. F . James, P . G. Kwiat, W . J. Munro, and A. G. White,Mea- surement of qubits, Physical Review A, 64, 052312 (2001)

    work page 2001

  7. [5]

    Bouchard, F

    F . Bouchard, F . Hufnagel, D. Koutný , A. Abbas, A. Sit, K. He- shami, R. Fickler, and E. Karimi, Quantum process tomogra- phy of a high-dimensional quantum communication channel , Quantum, 3, 138 (2019)

    work page 2019

  8. [6]

    A. I. Lvovsky ,Iterative maximum-likelihood reconstruction in quantum homodyne tomography, Journal of Optics B: Quan- tum and Semiclassical Optics, 6, S556-S559 (2004). 6

    work page 2004

Show all 80 references
  1. [7]

    Blume-Kohout, Hedged maximum likelihood quantum state estimation, Phys

    R. Blume-Kohout, Hedged maximum likelihood quantum state estimation, Phys. Rev. Lett. 105, 200504 (2010)

    2010

  2. [8]

    J. A. Smolin, J. M. Gambetta, and G. Smith, Efficient method for computing the maximum-likelihood quantum state from measurements with additive gaussian noise , Phys. Rev. Lett. 108, 070502 (2012)

    2012

  3. [9]

    Granade, C

    C. Granade, C. Ferrie, and S. T . Flammia, Practical adaptive quantum tomography, New Journal of Physics 19, 113017 (2017)

    2017

  4. [10]

    Christandl, R

    M. Christandl, R. König, and R. Renner, Postselection tech- nique for quantum channels with applications to quantum cryptography, Phys. Rev. Lett. 102, 020504 (2009)

    2009

  5. [11]

    Blume-Kohout, Optimal, reliable estimation of quantum states, New Journal of Physics 12, 043034 (2010)

    R. Blume-Kohout, Optimal, reliable estimation of quantum states, New Journal of Physics 12, 043034 (2010)

    2010

  6. [12]

    A. V . Rodionov, A. Veitia, R. Barends, J. Kelly , D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, Compressed sensing quantum process tomography for super- conducting quantum gates , Physical Review B 90, 144504 (2014)

    2014

  7. [13]

    Torlai, C

    G. Torlai, C. J. Wood, A. Acharya, G. Carleo, J. Carrasquilla, and L. Aolita,Quantum process tomography with unsupervised learning and tensor networks , Nature Communications 14, 10.1038/s41467-023-38332-9 (2023)

    2023 doi

  8. [14]

    G. C. Knee, E. Bolduc, J. Leach, and E. M. Gauger, Quan- tum process tomography via completely positive and trace- preserving projection, Physical Review A 98, 062336 (2018)

    2018

  9. [15]

    Surawy-Stepney , J

    T . Surawy-Stepney , J. Kahn, R. Kueng, and M. Guta,Projected least-squares quantum process tomography, Quantum 6, 844 (2022)

    2022

  10. [16]

    Zhang, M

    G. Zhang, M. Huo, and Y. Li,Quantum operation of fermionic systems and process tomography using majorana fermion gates , (2021), arXiv:2102.00620 [quant-ph]

    2021 arXiv

  11. [17]

    G. A. L. White, F . A. Pollock, L. C. L. Hollenberg, K. Modi, and C. D. Hill, Non-markovian quantum process tomography, PRX Quantum, 3, 020344 (2022)

    2022

  12. [18]

    L. C. G. Govia, G. J. Ribeill, D. Ristè, M. Ware, and H. Krovi, Bootstrapping quantum process tomography via a perturbative ansatz, Nature Communications, 11, 1084 (2020)

    2020

  13. [19]

    S. Xue, Y. Liu, Y. Wang, P . Zhu, C. Guo, and J. Wu,Variational quantum process tomography of unitaries, Physical Review A, 105, 032427 (2022)

    2022

  14. [20]

    Property testing of unitary operators,

    G. Wang, “Property testing of unitary operators,”Physical Re- view A, 84, 052328 (2011)

    2011

  15. [21]

    Montanaro and R

    A. Montanaro and R. de Wolf, A Survey of Quantum Property Testing, Theory of Computing 1, 1 (2016)

    2016

  16. [22]

    Non-identity-check

    D. Janzing, P . Wocjan, and T . Beth, “ Non-identity-check” is QMA-complete, Int. J. Quantum Inf. 3, 463 (2005)

    2005

  17. [23]

    Ji and X

    Z. Ji and X. Wu, Non-identity check remains QMA-complete for short circuits, arXiv:0906.5416, (2009)

    2009 arXiv

  18. [24]

    She, and H

    A. She, and H. Yuen, Unitary Property Testing Lower Bounds by Polynomials, , arXiv:2210.05885, (2022)

    2022 arXiv

  19. [25]

    Aharonov, J

    D. Aharonov, J. Cotler, and X.-L. Qi, Quantum algorithmic measurement, Nature Communications 13, 887 (2022)

    2022

  20. [27]

    and Kashefi, E., Quantum certi- fication and benchmarking, Nature Reviews Physics , 2(7), pp.382-390, (2020)

    Eisert, J., Hangleiter, D., Walk, N., Roth, I., Markham, D., Parekh, R., Chabaud, U. and Kashefi, E., Quantum certi- fication and benchmarking, Nature Reviews Physics , 2(7), pp.382-390, (2020)

    2020

  21. [28]

    Renou, D

    M.-O. Renou, D. Trillo, M. Weilenmann, T . P . Le, A. Tavakoli, N. Gisin, A. Acin, and M. Navascu´es, Quantum theory based on real numbers can be experimentally falsified,Nature 600, 625 (2021)

    2021

  22. [29]

    M.-C. Chen, C. Wang, F .-M. Liu, J.-W . Wang, C. Ying, Z.-X. Shang, Y. Wu, M. Gong, H. Deng, F .-T . Liang, et al., Ruling out real-valued standard formalism of quantum theory ,Physical Review Letters 128, 040403 (2022)

    2022

  23. [30]

    Li, Y.-L

    Z.-D. Li, Y.-L. Mao, M. Weilenmann, A. Tavakoli, H. Chen, L. Feng, S.-J. Yang, M.-O. Renou, D. Trillo, T . P . Le, et al., Testing real quantum theory in an optical quantum network,Physical Review Letters 128, 040402 (2022)

    2022

  24. [31]

    Khanna, A

    K. Khanna, A. Kumar, R. Vasseur, A W . W . Ludwig, Ran- dom Quantum Circuits with Time-Reversal Symmetry , arXiv preprint arXiv:2501.13161

  25. [32]

    Huang, M

    H.-Y. Huang, M. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, and J. R. McClean, Quantum advantage in learning from experi- ments, Science 376, 1182 (2022)

    2022

  26. [33]

    S. Chen, J. Cotler, H.-Y. Huang, and J. Li, Exponential separa- tions between learning with and without quantum memory , in 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2022) pp. 574–585

    2021

  27. [34]

    K.-D. Wu, T . V . Kondra, S. Rana, C. M. Scandolo, G.-Y. Xi- ang, C.-F . Li, G.-C. Guo, and A. Streltsov, Resource theory of imaginarity: Quantification and state conversion, , Physical Review A 103, 032401 (2021)

    2021

  28. [35]

    Hickey and G

    A. Hickey and G. Gour, Quantifying the imaginarity of quan- tum mechanics, , Journal of Physics A: Mathematical and The- oretical 51, 414009 (2018)

    2018

  29. [36]

    Nakata, M

    Y. Nakata, M. Koashi, and M. Murao, Generating a state t- design by diagonal quantum circuits, New Journal of Physics 16, 053043 (2014)

    2014

  30. [37]

    Nakata, C

    Y. Nakata, C. Hirche, M. Koashi, and A. Winter, Efficient quantum pseudorandomness with nearly time-independent hamiltonian dynamics , Physical Review X 7, 021006 (2017)

    2017

  31. [38]

    Nakata and M

    Y. Nakata and M. Murao, Diagonal quantum circuits: their computational power and applications , The European Phys- ical Journal Plus 129, 1 (2014)

    2014

  32. [39]

    Zhang, K

    S. Zhang, K. Huang, and L. Li, Automatic depth-optimized quantum circuit synthesis for diagonal unitary matrices with asymptotically optimal gate count , arXiv:2212.01002, (2022)

    2022 arXiv

  33. [40]

    M. T . Quintino and D. Ebler, Deterministic transforma- tions between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality , Quantum 6, 679 (2022)

    2022

  34. [41]

    Tight asymptotic bounds on lo- cal hypothesis testing between a pure bipartite state and the white noise state

    M. Hayashi and O. Masaki, “Tight asymptotic bounds on lo- cal hypothesis testing between a pure bipartite state and the white noise state” IEEE Transactions on Information Theory, Volume: 63, Issue: 6, 4008–4036 (2017)

    2017

  35. [42]

    Local hypothesis testing between a pure bipartite state and the white noise state

    O. Masaki and M. Hayashi, “Local hypothesis testing between a pure bipartite state and the white noise state” IEEE Trans- actions on Information Theory, Volume: 61, Issue: 12, 6995– 7011 (2015)

    2015

  36. [43]

    Asymptotic local hypothesis test- ing between a pure bipartite state and the completely mixed state

    O. Masaki and M. Hayashi, “Asymptotic local hypothesis test- ing between a pure bipartite state and the completely mixed state” , Physical Review A Vol.90, No.3, 032327 (2014)

    2014

  37. [44]

    Two-way classical communi- cation remarkably improves local distinguishability

    O. Masaki and M. Hayashi, “Two-way classical communi- cation remarkably improves local distinguishability” , New Journal of Physics Vol.10, No.1, 013006 (2008)

    2008

  38. [45]

    T . S. Han, Information-Spectrum Methods in Information Theory , (Springer, Berlin Heidelberg New York, 2002) (orig- inally appeared in Japanese in 1998)

    2002

  39. [46]

    Bavaresco, M

    J. Bavaresco, M. Murao, and M. T . Quintino, Strict hierar- chy between parallel, sequential, and indefinite-causal-order strategies for channel discrimination, Physical Review Letters 127, 200504 (2021). 7

    2021

  40. [47]

    Chiribella, M

    G. Chiribella, M. Banik, S. S. Bhattacharya, T . Guha, M. Al- imuddin, A. Roy , S. Saha, S. Agrawal, and G. Kar, Indefinite causal order enables perfect quantum communication with zero capacity channels , New Journal of Physics 23, 033039 (2021)

    2021

  41. [48]

    X. Zhao, Y. Yang, and G. Chiribella, Quantum metrology with indefinite causal order , Physical Review Letters 124, 190503 (2020)

    2020

  42. [49]

    Chapeau-Blondeau, Noisy quantum metrology with the as- sistance of indefinite causal order , Physical Review A 103, 032615 (2021)

    F . Chapeau-Blondeau, Noisy quantum metrology with the as- sistance of indefinite causal order , Physical Review A 103, 032615 (2021)

    2021

  43. [50]

    Datta, Min- and Max-Relative Entropies and a New Entan- glement Monotone , IEEE Trans

    N. Datta, Min- and Max-Relative Entropies and a New Entan- glement Monotone , IEEE Trans. Inf. Theory55, 2816 (2009)

    2009

  44. [51]

    K. Bu, U. Singh, S.-M. Fei, A. K. Pati, and J. Wu, Maxi- mum relative entropy of coherence: An operational coher- ence measure , Physical Review Letters 119, 150405 (2017)

    2017

  45. [52]

    X. Wang, M. M. Wilde, and Y. Su, Efficiently computable bounds for magic state distillation , Physical Review Letters 124, 090505 (2020)

    2020

  46. [53]

    C. Zhu, C. Zhu, and X. Wang, Estimate distillable entangle- ment and quantum capacity by squeezing useless entangle- ment , IEEE Journal on Selected Areas in Communications42, 1850-1860 (2024)

    2024

  47. [54]

    A. S. Holevo, Probabilistic and statistical aspects of quantum theory, Translated from the Russian by the author, North- Holland Series in Statistics and Probability , 1, North-Holland Publishing Co., Amsterdam, 1982 , xii+312 pp

    1982

  48. [55]

    Hayashi, Group Representation for Quantum Theory , Springer (2017)

    M. Hayashi, Group Representation for Quantum Theory , Springer (2017). (Originally published from Kyoritsu Shup- pan in 2014 with Japanese.)

    2017

  49. [56]

    Hayashi, A Group Theoretic Approach to Quantum In- formation, Springer (2017)

    M. Hayashi, A Group Theoretic Approach to Quantum In- formation, Springer (2017). (Originally published from Ky- oritsu Shuppan in 2014 with Japanese.)

    2017

  50. [57]

    Reversing Un- known Qubit-Unitary Operation, Deterministically and Ex- actly ,

    Yoshida, S., Soeda, A., and Murao, M., “Reversing Un- known Qubit-Unitary Operation, Deterministically and Ex- actly ,”Physical Review Letters, 131, 120602 (2023)

    2023

  51. [58]

    Quan- tum advantage in reversing unknown unitary evolutions,

    Chen, Y. A., Mo, Y., Liu, Y., Zhang, L., and Wang, X., “Quan- tum advantage in reversing unknown unitary evolutions,” arXiv preprint arXiv:2403.04704

  52. [59]

    Covariant measurements and uncertainty re- lations,

    A. S. Holevo, “Covariant measurements and uncertainty re- lations,” Rep. Math. Phys., 16, 3 (1979) 385 – 400

    1979

  53. [61]

    Parallel Treatment of Estimation of SU(2) and Phase Estimation,

    M. Hayashi, “Parallel Treatment of Estimation of SU(2) and Phase Estimation,” Physics Letters A, Vol.354, No.3, 183-189 (2006)

    2006

  54. [62]

    Optimal esti- mation of group transformations using entanglement,

    G. Chiribella, G. M. D’Ariano, and M. F . Sacchi, “Optimal esti- mation of group transformations using entanglement,” Phys. Rev. A 72, 042338 (2005)

    2005

  55. [63]

    Classification of topological insulators and superconductors in three spatial dimensions

    A. P . Schnyder, S. Ryu, A. Furusaki, and A. W . W . Ludwig, “Classification of topological insulators and superconductors in three spatial dimensions”, Phys. Rev. B 78, 195125 (2008)

    2008

  56. [64]

    S. Ryu, A. P . Schnyder, A. Furusaki, and A. W . W . Lud- wig, Topological insulators and superconductors: tenfold way and dimensional hierarchy , New Journal of Physics 12, 065010 (2010)

    2010

  57. [65]

    Asymptotically Secure Network Code for Active Attacks

    M. Hayashi and N. Cai, “Asymptotically Secure Network Code for Active Attacks” IEEE Transactions on Communica- tions, Volume: 69, Issue: 5, 3245 – 3259 (2021)

    2021

  58. [66]

    Reduction The- orem for Secrecy over Linear Network Code for Active At- tacks,

    M. Hayashi, M. Owari, G. Kato, and N. Cai, “Reduction The- orem for Secrecy over Linear Network Code for Active At- tacks,” Entropy, Information Theory , Probability and Statis- tics Section, Special Issue: Multiuser Information Theory III, 22 (9), 1053 (2020)

    2020

  59. [67]

    Secure Network Code for Adap- tive and Active Attacks with No-Randomness in Intermediate Nodes,

    N. Cai and M. Hayashi, “Secure Network Code for Adap- tive and Active Attacks with No-Randomness in Intermediate Nodes,” IEEE Transactions on Information Theory , Volume: 66, Issue: 3, 1428 – 1448 (2020)

    2020

  60. [68]

    Secure network coding with adaptive and ac- tive attack,

    M. Hayashi, “Secure network coding with adaptive and ac- tive attack,” Festschrift in Memory of Ning Cai Information Theory and Related Fields, Bielefeld, Germany , November 24– 26, 2023; The proceedings is in Press

    2023

  61. [69]

    Single-Shot Secure Quantum Network Coding for General Multiple Unicast Network with Free One-Way Public Communication,

    G. Kato, M. Owari, and M. Hayashi, “Single-Shot Secure Quantum Network Coding for General Multiple Unicast Network with Free One-Way Public Communication,” IEEE Transactions on Information Theory , Volume: 67, Issue: 7, 4564 – 4587 (2021)

    2021

  62. [70]

    Single-Shot Secure Quantum Network Coding on Butterfly Network with Free Public Communication,

    M. Owari, G. Kato, and M. Hayashi, “Single-Shot Secure Quantum Network Coding on Butterfly Network with Free Public Communication,” Quantum Science and Technology , Vol. 3, 014001 (2017)

    2017

  63. [71]

    Quantum state transmission over partially corrupted quantum information network,

    M. Hayashi and S. Song, “Quantum state transmission over partially corrupted quantum information network,” Physical Review Research, 2, 033079 (2020)

    2020

  64. [72]

    Secure non-linear network code over one-hop relay network,

    M. Hayashi and N. Cai, “Secure non-linear network code over one-hop relay network,” IEEE Journal on Selected Areas in Information Theory vol. 2, no. 1, 296 - 305 (2021)

    2021

  65. [73]

    Indefinite causal order strategy nor adaptive strategy does not improve the estimation of group action,

    M. Hayashi, “Indefinite causal order strategy nor adaptive strategy does not improve the estimation of group action,” arXiv preprint arXiv:2501.09312 (2025)

    2025

  66. [74]

    Unitary sub- group testing,

    Z. Brakerski, D. Sharma, and G. Weissenberg, “Unitary sub- group testing,” arXiv preprint arXiv:2104.03591 (2021)

    2021 arXiv

  67. [75]

    Bavaresco, M

    J. Bavaresco, M. Murao, and M. T . Quintino, Unitary chan- nel discrimination beyond group structures: Advantages of sequential and indefinite-causal-order strategies, J. Math. Phys. 63, (2022)

    2022

  68. [76]

    F . G. S. L. Brandao and M. B. Plenio, A reversible theory of entanglement and its relation to the second law, Communi- cations in Mathematical Physics 295, 791–828 (2010)

    2010

  69. [77]

    Hayashi and H

    M. Hayashi and H. Yamasaki, Generalized quantum stein’s lemma and second law of quantum resource, arXiv preprint arXiv:2408.02722 (2024)

    2024 arXiv

  70. [78]

    Lami, A solution of the generalised quantum Stein’s lemma, arxiv:2408.06410 (2024)

    L. Lami, A solution of the generalised quantum Stein’s lemma, arxiv:2408.06410 (2024)

    2024 arXiv

  71. [79]

    Predicting symmetries of quantum dynamics with optimal samples

    See Supplemental Material for technical details. 8 Supplemental Material for “Predicting symmetries of quantum dynamics with optimal samples” In the supplemental material, we first introduce some preliminaries and elaborate on the basic setup of this work. Next, we provide the...

  72. [80]

    self-entanglement

    introduced the word “self-entanglement” to characterize this phenomena. When n = 2m− 1, we have X λ∈ ˆGf2m−1 d2 λ = mX j=1 (2j)2 = 2m(m + 1)(2m + 1) 3 . (S69) In this case, the multiplicity nj is 2m−1 m−j − 2m−1 m−j−1 for j = 0, . . . ,m− 1, and nm is 1. Since dj = 2j≤ nj = 2...

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.