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Random approximate quantum information masking

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper establishes a no-random approximate masking theorem for bipartite systems and a random approximate masking theorem for multipartite systems, and connects approximate masking to approximate quantum error correction.

desk verdict The bipartite no-random-AQIM bound is a solid new result; the multipartite construction is nice but mostly a union-bound extension, and the advertised AQECC implication is broader than the theorems support since the equivalence holds only for replacement noise. read the letter →

arxiv 2507.19454 v1 pith:HMIHDU3K submitted 2025-07-25 quant-ph

Xiaodi Li , Xinyang Shu , Huangjun Zhu This is my paper · ORCID
classification quant-ph
keywords approximatequantuminformationmaskingrandomisometriesno-maskingtheoremmultipartiteentanglementk-uniformstateserrorcorrectionconcentrationofmeasuresubspaces
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

This paper asks whether random isometries can approximately mask quantum information — hiding the identity of an input state in the correlations of a composite system. In a bipartite system the answer is negative: the average subsystem variation of a random subspace is bounded below by $w > 1/9$ (Theorem 1), and the probability of doing better than $w - \alpha$ decays as $\exp(-d_{12}\alpha^2/16)$ (Theorem 2). In a multipartite system with $m$ equal-dimensional parties the answer reverses: a random subspace of dimension $d_C$ is an approximate $k$-uniform masker with inaccuracy $d^{k-m/2} + \alpha$ except with exponentially small probability (Theorem 6). Because approximate masking is tied to approximate quantum error correction, the paper concludes that random subspaces can serve as approximate quantum error-correcting codes with constant code rate and exponentially small inaccuracy.

What carries the argument

The workhorse is the average and maximum subsystem variation of a random subspace, together with concentration of measure on the Grassmannian. Proposition 2 decomposes $\mathbb{E}_{H_C}[V^A_X(H_C)]$ as a dimension-dependent prefactor times the corresponding variation of the full bipartite space, and Lemma 3 shows the two one-party variations of a Haar-random bipartite state add to at least $1/3$; this carries the lower bound in Theorem 1. On the multipartite side, the key mechanism is the union bound over $\binom{m}{k}$ bipartite cuts combined with concentration results (Theorems 4 and 5) stating that a random subspace's reduced states are close to the maximally mixed state or to the marginal of the subspace projector. The bridge to approximate quantum error correction is Lemma 1, which equates the maximum subsystem inaccuracy with the subsystem variance that controls the QEC inaccuracy for replacement channels.

What would settle it

Sample Haar-random isometries from a $d_C$-dimensional logical space into a bipartite system and estimate the average subsystem variation $V^A(H_C)$; Theorem 1 predicts it stays above $w>1/9$ (approaching $1/6$ for large dimensions), and Theorem 2 predicts deviations below $w-\alpha$ appear with probability at most $\exp(-d_{12}\alpha^2/16)$. Many trials with average variation clearly below $w$ would refute the no-random-AQIM claim.

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Extended reading notes

Core claim

The central discovery is a pair of contrasting theorems about Haar-random isometries. In a bipartite target system, the average trace distance between the reduced states of two random image states cannot be pushed below a universal constant: $\mathbb{E}_{H_C}[V^A(H_C)] \ge w > 1/9$, with $w = \frac{1}{6}\frac{(2d_C-2)(2d_{12}-1)}{(2d_C-1)(2d_{12}-2)}$, and a deviation below $w-\alpha$ occurs with probability at most $\exp(-d_{12}\alpha^2/16)$. Thus almost all random isometries fail to be even approximate maskers, which the authors cast as a no-random-AQIM theorem extending the original no-masking theorem. In multipartite systems with equal local dimension $d$, however, a random isometry is an approximate $k$-uniform masker almost surely: the probability that the maximum subsystem inaccuracy exceeds $d^{k-m/2}+\alpha$ is exponentially small. The paper further shows that, since the maximum subsystem inaccuracy coincides with the subsystem variance used in approximate quantum error correction for replacement channels, random approximate maskers yield approximate quantum error-correcting codes whose code rate is constant and whose inaccuracy falls exponentially.

Load-bearing premise

The step that turns approximate masking into approximate quantum error correction relies on the noise being a replacement channel — one that discards a subsystem and replaces it with a fixed state — and the paper does not establish the masker-error-correction equivalence for general noise.

Editorial extensions

If this is right

  • Random isometries cannot serve as approximate maskers in any bipartite system, no matter how large the local dimensions; this generalizes the no-masking theorem to the random setting.
  • In multipartite systems with enough parties, almost all random isometries are approximate $k$-uniform maskers with exponentially small failure probability.
  • Masking a logical space of $l$ qubits requires only a linear number $m^* \propto l$ of physical qubits (or qudits), not exponential.
  • Random approximate maskers are simultaneously approximate quantum error-correcting codes with constant code rate and exponentially small correction inaccuracy, for replacement-channel noise on any set of at most $k$ parties.
  • The same concentration results imply that Haar-random multipartite pure states are approximate $k$-uniform states and have generalized Meyer-Wallach entanglement near 1 with high probability.

Reading between the lines

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

  • The bipartite/multipartite contrast suggests that the obstruction to random masking is fundamentally a two-party tradeoff: any encoding that hides information from every single party must spread it over at least three shares, because no random two-share split can keep both marginals flat.
  • Because the error-correction conclusion is proven only for replacement channels, a natural next test is whether the equivalence survives for general noise; if not, the AQECC claim would need a different recovery argument.
  • The concentration bounds used here are strong enough that unitary $k$-designs, rather than Haar-random unitaries, may suffice for the multipartite masking construction; comparing the two would give a practically implementable version of the result.
  • The connection between approximate masking, approximate $k$-uniform states, and approximate error correction suggests that masking is a generic property of highly entangled random subspaces, with consequences for thermalization and code capacity that the paper leaves open.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies approximate quantum information masking (AQIM) through random isometries. It introduces maximal and average versions of approximate k-uniform masking and several figures of merit, establishing inequalities among them (Proposition 1). For bipartite systems, it proves that the expected average subsystem variation of a random subspace is bounded below by w > 1/9 (Theorem 1), that the probability of falling below w − α is exponentially small (Theorem 2), and hence that random isometries are almost never approximate maskers (Theorem 3). For multipartite systems with equal local dimension d, it shows that a random subspace of dimension d_C is an approximate k-uniform masker with inaccuracy d^{k−m/2}+α except with exponentially small probability (Theorem 6), and gives analogous statements for other figures of merit (Theorem 7, Proposition 4). The paper then derives implications: random multipartite subspaces yield approximate k-uniform states (Corollary 3) and, via Lemma 1 of [51], approximate quantum error-correcting codes (AQECCs) with constant code rate and exponentially small inaccuracy (Corollary 4, Theorem 8). Numerical illustrations and appendices support the main concentration proofs.

Significance. If the central results hold, this paper gives a clean and somewhat counterintuitive separation: random isometries fail to be approximate maskers in bipartite systems even when dimensions are large, yet succeed in multipartite systems with a number of physical qubits that scales only linearly in the number of logical qubits. The proofs use standard tools (Lévy's lemma, epsilon-nets, Lipschitz bounds) and are presented in detail in appendices, which is a strength. The connection to AQECC, if properly scoped, would extend the relevance of masking to quantum error correction and random codes. However, the advertised AQECC equivalence is currently stated more broadly than the theorems support, and one statement of the random-code theorem asserts an equality where the derivation gives only an upper bound. These issues are local and fixable, but they affect the paper's headline claims.

major comments (3)
  1. [Section V.B, Corollary 4, Theorem 8; Abstract] The AQECC consequence is proved only for replacement noise, but the abstract and Section V.B advertise a general equivalence between AQIM and AQECC. Corollary 4 and Theorem 8 rely on Lemma 1 of [51], whose inequality (52) bounds the QEC inaccuracy by the subsystem variance only for replacement channels R_S(ψ)=Tr_S(ψ)⊗γ_S (Section V.B, Eq. (52)). For general noise, e.g., dephasing or depolarizing channels on a subsystem, the equivalence is not established by the cited lemma. This is load-bearing because the advertised claim that 'AQIM naturally gives rise to approximate quantum error correction codes' is unsupported outside the replacement-channel setting. Please revise the abstract, the introduction, Section V.B, and Theorem 8 to state prominently that the AQECC implication holds for replacement errors, or provide a proof for the general case.
  2. [Theorem 8] Theorem 8 states that 'the probability that H_C is an AQECC with inaccuracy eη(E,R_S) = sqrt(d_C)(u+α)' is bounded as in Eq. (55). This asserts an equality, but the preceding results support only an upper bound: Corollary 4 gives eη(E,R_S) ≤ sqrt(d_C) ΛM(H_C,k) and Theorem 7 gives ΛM(H_C,k) ≤ u+α with high probability. The theorem should state eη(E,R_S) ≤ sqrt(d_C)(u+α) (and similarly in the surrounding discussion and Appendix F); as written, it overstates the guarantee.
  3. [Section III.B, Eqs. (31)-(32)] Equation (31) introduces the approximation ΛA_B1(H_C) ≈ (4/(3π)) sqrt(d_1 Tr[(Δψ_B1)^2]) and Eq. (32) derives a corresponding approximate upper bound. This random-matrix approximation is not proved, and it is not used in the subsequent concentration theorems. The text, however, presents it as an analytical result rather than a heuristic. Please either provide a rigorous derivation with the necessary assumptions, or explicitly label this as a numerical/heuristic estimate and clarify that the rigorous results in Theorems 4 and 5 do not depend on it.
minor comments (4)
  1. [Corollary 2, case (3)] The statement 'If α = d^{k−m/2} is smaller than d^{k−m/2}' is self-contradictory; it should read 'If α = d^{k−m/2}/c for some constant c>1' or simply 'If α ≤ d^{k−m/2}'.
  2. [Theorem 3] Theorem 3 states that the probability of being a δ-approximate masker with δ=1/9 is exponentially small, but the proof is not given explicitly. Since Theorem 2 controls V^A(H_C), the proof should state that V^M(H_C) ≥ V^A(H_C) and that δ=1/9 falls below the lower bound w, to make the inference transparent.
  3. [Appendix C.1.b, Eq. (C25)] In the proof of Lemma 3, after the line showing that terms with i≥2 are non-positive, the next displayed equation writes '2X i=1' (likely a typographical artifact). Please clarify which terms are retained in the summation, so the reader can follow the bound leading to the value 1/3.
  4. [Appendix A, table] In the table row for E_H_C[ (Π^{(B_1)}_C − e1_{B_1})^2 ], the notation contains a comma inside the norm, '∥Π^{(B_1)}_C , e1_{B_1}∥', which should be a minus sign: '∥Π^{(B_1)}_C − e1_{B_1}∥'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main bounds are derived analytically from Haar-random-state moment computations and standard concentration lemmas, not from fitted inputs or self-referential definitions.

full rationale

The central derivation chain is self-contained against external benchmarks. The no-random-AQIM bound in Theorem 1 is obtained by (i) Proposition 2, which computes the expected subsystem variation via a Haar-random fidelity decomposition; (ii) Proposition 5 and Lemma 3, which lower-bound the sum of average subsystem inaccuracies using the Fuchs–van de Graaf inequality and an externally computed expected fidelity; and (iii) Theorem 2, whose exponential concentration follows from the Lipschitz bound in Proposition 7 and a standard Levy-type concentration lemma on the Grassmannian. The constant w is a closed-form analytic expression; no parameter is fitted to the target conclusion. The multipartite existence results in Theorem 6 and Corollary 1 follow from Theorem 4 by a union bound over bipartite cuts, and Theorem 4 is proved by epsilon-net discretization and Levy's lemma rather than by assuming the conclusion. The AQECC implication in Corollary 4 and Theorem 8 imports Lemma 1 from reference [51], an external prior result. Although the abstract and Section V.B state the AQIM-AQECC equivalence more broadly than the replacement-channel condition in Eq. (52), that is a scope overstatement and a correctness limitation, not a circular reduction. The paper's self-citations, such as [19] and [56], occur but are not load-bearing in the derivation of the main bounds.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data. The central claims rest on standard concentration results and on two domain assumptions (Haar random isometry model, equal local dimensions) plus one unproved approximation in Section III B that is not load-bearing.

assumptions (5)
  • standard math Levy's lemma and epsilon-net discretization for concentration of measure on high-dimensional spheres and unitary groups
    Used in proofs of Theorems 4, 5 and Proposition 3; these are standard results from [40,59].
  • domain assumption Haar random isometry as a model of a random masker
    The paper models random maskers as Haar random isometries (Section III). This is the standard unitary-invariant model but does not capture structured random circuits.
  • domain assumption Equal subsystem dimensions d_i = d in the multipartite analysis
    Section IV assumes all subsystems have the same dimension; the stated scaling d^{k-m/2} depends on this.
  • standard math Replacement-channel bound relating QEC inaccuracy to subsystem variance (Lemma 1 from [51])
    Used in Corollary 4 to connect AQIM to AQECC; this is a prior theorem limited to replacement channels.
  • ad hoc to paper Random matrix theory approximation for the trace norm of Delta_psi_B1, Eq (31)
    The approximation Lambda^A_B1 is approximately (4/(3pi)) sqrt(d1) sqrt(Tr[(Delta_psi)^2]) is presented without proof; used only for intuition about the order of magnitude of s.

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Cite this review

Pith. "Pith review of Random approximate quantum information masking." pith.science (2026). https://pith.science/paper/HMIHDU3K

@misc{pith2026250719454,
  author       = {Pith},
  title        = {Pith review of: Random approximate quantum information masking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HMIHDU3K}},
  note         = {Machine review of arXiv:2507.19454}
}
read the original abstract

Masking information into quantum correlations is a cornerstone of many quantum information applications. While there exist the no-hiding and no-masking theorems, approximate quantum information masking (AQIM) offers a promising means of circumventing the constraints. Despite its potential, AQIM still remains underexplored, and constructing explicit approximate maskers remains a challenge. In this work, we investigate AQIM from multiple perspectives and propose using random isometries to construct approximate maskers. First, different notions of AQIM are introduced and we find there are profound intrinsic connections among them. These relationships are characterized by a set of figures of merit, which are introduced to quantify the deviation of AQIM from exact QIM. We then explore the possibility of realizing AQIM via random isometries in bipartite and multipartite systems. In bipartite systems, we identify a fundamental lower bound for a key figure of merit, implying that almost all random isometries fail to realize AQIM. This surprising result generalizes the original no-masking theorem to the no-random-AQIM theorem for bipartite systems. In contrast, in multipartite systems, we show almost all random isometries can realize AQIM. Remarkably, the number of physical qubits required to randomly mask a single logical qubit scales only linearly. We further explore the implications of these findings. In particular, we show that, under certain conditions, approximate quantum error correction is equivalent to AQIM. Consequently, AQIM naturally gives rise to approximate quantum error correction codes with constant code rates and exponentially small correction inaccuracies. Overall, our results establish quantum information masking as a central concept in quantum information theory, bridging diverse notions across multiple domains.

Figures

Figures reproduced from arXiv: 2507.19454 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of AQIM in the bipartite system. The [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Numerical illustration of the behaviors of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The numerical estimation of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Illustration of Case 1. For all three figures, we set [PITH_FULL_IMAGE:figures/full_fig_p030_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Illustration of Case 2. For all three figures, we set [PITH_FULL_IMAGE:figures/full_fig_p031_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Illustration of Case 3. For all three figures, we set [PITH_FULL_IMAGE:figures/full_fig_p032_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The left plot represents the variations of [PITH_FULL_IMAGE:figures/full_fig_p033_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The left plot represents the variations of [PITH_FULL_IMAGE:figures/full_fig_p034_8.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Theory of approximate quantum error correction and the error-set model

    quant-ph 2026-07 conditional novelty 8.0 of 10

    Approximate quantum error correction acquires an error-set model: bounded-mixing linear families of noise channels are uniformly correctable from a single geometric code parameter.

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Works this paper leans on

77 extracted references · 56 canonical work pages · cited by 1 Pith paper

  1. [51]

    D.-S. Wang, G. Zhu, C. Okay, and R. Laflamme, Quasi- exact quantum computation, Phys. Rev. Res. 2, 033116 (2020)

    work page 2020

  2. [1]

    Hillery, V

    M. Hillery, V. Buˇ zek, and A. Berthiaume, Quantum se- cret sharing, Phys. Rev. A 59, 1829 (1999)

    1999

  3. [2]

    Cleve, D

    R. Cleve, D. Gottesman, and H.-K. Lo, How to share a quantum secret, Phys. Rev. Lett. 83, 648 (1999)

    1999

  4. [3]

    B. M. Terhal, D. P. DiVincenzo, and D. W. Leung, Hiding bits in bell states, Phys. Rev. Lett. 86, 5807 (2001)

    2001

  5. [4]

    DiVincenzo, D

    D. DiVincenzo, D. Leung, and B. Terhal, Quantum data hiding, IEEE Trans. Inf. Theory 48, 580 (2002)

    work page 2002

  6. [5]

    Lloyd, Capacity of the noisy quantum channel, Phys

    S. Lloyd, Capacity of the noisy quantum channel, Phys. Rev. A 55, 1613 (1997)

    1997

  7. [6]

    Barnum, M

    H. Barnum, M. A. Nielsen, and B. Schumacher, Infor- mation transmission through a noisy quantum channel, Phys. Rev. A 57, 4153 (1998)

    work page 1998

  8. [7]

    Quantum coding theorem from privacy and distinguishability

    M. Horodecki, S. Lloyd, and A. Winter, Quantum cod- ing theorem from privacy and distinguishability (2008), arXiv:quant-ph/0702006 [quant-ph]

    work page Pith review arXiv 2008

Show all 77 references
  1. [8]

    Hayden, P

    P. Hayden, P. W. Shor, and A. Winter, Random quan- tum codes from gaussian ensembles and an uncertainty relation, Open Syst. Inf. Dyn. 15, 71–89 (2008)

    2008

  2. [9]

    Both terms cannot be made small simultaneously irrespective of the choices of d1, d2, and dC, as illustrated in Fig

    (24) Theorem 1 shows that there is an inevitable tradeoff between EHC[V A B1(HC)] and EHC[V A B2(HC)]. Both terms cannot be made small simultaneously irrespective of the choices of d1, d2, and dC, as illustrated in Fig. 2. When dC,d 1,d 2≫ 1, the lower bound w approaches 1/6. ...

  3. [10]

    Hayden and J

    P. Hayden and J. Preskill, Black holes as mirrors: quan- tum information in random subsystems, JHEP 2007 (09), 120

    2007

  4. [11]

    Sekino and L

    Y. Sekino and L. Susskind, Fast scramblers, JHEP 2008 (10), 065

    2008

  5. [12]

    Z.-W. Liu, S. Lloyd, E. Zhu, and H. Zhu, Entanglement, quantum randomness, and complexity beyond scram- bling, JHEP 2018 (041), 041

    2018

  6. [13]

    Gottesman, Stabilizer codes and quantum error cor- rection (1997), arXiv:quant-ph/9705052 [quant-ph]

    D. Gottesman, Stabilizer codes and quantum error cor- rection (1997), arXiv:quant-ph/9705052 [quant-ph]

    1997 arXiv

  7. [14]

    Knill and R

    E. Knill and R. Laflamme, Theory of quantum error- correcting codes, Phys. Rev. A 55, 900 (1997)

    1997

  8. [15]

    B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015)

    2015

  9. [16]

    S. L. Braunstein and A. K. Pati, Quantum information cannot be completely hidden in correlations: Implica- tions for the black-hole information paradox, Phys. Rev. Lett. 98, 080502 (2007)

    2007

  10. [17]

    K. Modi, A. K. Pati, A. Sen(De), and U. Sen, Masking quantum information is impossible, Phys. Rev. Lett.120, 230501 (2018)

    2018

  11. [18]

    Liang, B

    X.-B. Liang, B. Li, and S.-M. Fei, Complete characteriza- tion of qubit masking, Phys. Rev. A 100, 030304 (2019)

    2019

  12. [19]

    Ding and X

    F. Ding and X. Hu, Masking quantum information on hyperdisks, Phys. Rev. A 102, 042404 (2020)

    2020

  13. [20]

    Zhu, Hiding and masking quantum information in complex and real quantum mechanics, Phys

    H. Zhu, Hiding and masking quantum information in complex and real quantum mechanics, Phys. Rev. Res. 3, 033176 (2021)

    2021

  14. [21]

    Li and Y.-L

    M.-S. Li and Y.-L. Wang, Masking quantum information in multipartite scenario, Phys. Rev. A98, 062306 (2018)

    2018

  15. [22]

    Shi, M.-S

    F. Shi, M.-S. Li, L. Chen, and X. Zhang,k-uniform quan- tum information masking, Phys. Rev. A 104, 032601 (2021)

    2021

  16. [23]

    J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991)

    1991

  17. [24]

    Srednicki, Chaos and quantum thermalization, Phys

    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994)

    1994

  18. [25]

    Popescu, A

    S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nature Physics 2, 754 (2006)

    2006

  19. [26]

    Rigol, V

    M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452, 854 (2008)

    2008

  20. [27]

    J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81, 082001 (2018)

    2018

  21. [28]

    F. G. S. L. Brand˜ ao, E. Crosson, M. B. S ¸ahino˘ glu, and J. Bowen, Quantum error correcting codes in eigenstates of translation-invariant spin chains, Phys. Rev. Lett.123, 110502 (2019)

    2019

  22. [29]

    Faist, S

    P. Faist, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, Continuous symmetries and approximate quantum error correction, Phys. Rev. X 10, 041018 (2020)

    2020

  23. [30]

    Li, S.-h

    B. Li, S.-h. Jiang, X.-B. Liang, X. Li-Jost, H. Fan, and S.-M. Fei, Deterministic versus probabilistic quantum in- formation masking, Phys. Rev. A 99, 052343 (2019)

    2019

  24. [31]

    Li and K

    M.-S. Li and K. Modi, Probabilistic and approximate masking of quantum information, Phys. Rev. A 102, 022418 (2020)

    2020

  25. [32]

    Harrow, P

    A. Harrow, P. Hayden, and D. Leung, Superdense coding of quantum states, Phys. Rev. Lett. 92, 187901 (2004)

    2004

  26. [33]

    Hayden, D

    P. Hayden, D. Leung, P. W. Shor, and A. Winter, Ran- domizing quantum states: Constructions and applica- tions, Commun.Math.Phys. 250, 371 (2004)

    2004

  27. [34]

    M. B. Hastings, Superadditivity of communication capac- ity using entangled inputs, Nature Physics 5, 255 (2009)

    2009

  28. [35]

    Aubrun, S

    G. Aubrun, S. Szarek, and E. Werner, Hastings’s ad- ditivity counterexample via dvoretzky’s theorem, Com- mun.Math.Phys. 305, 85 (2011)

    2011

  29. [36]

    Kong and Z.-W

    L. Kong and Z.-W. Liu, Near-optimal covariant quan- tum error-correcting codes from random unitaries with symmetries, PRX Quantum 3, 020314 (2022)

    2022

  30. [37]

    Brown and O

    W. Brown and O. Fawzi, Short random circuits define good quantum error correcting codes, in 2013 IEEE In- ternational Symposium on Information Theory (2013) pp. 346–350

    2013

  31. [38]

    Brown and O

    W. Brown and O. Fawzi, Decoupling with random quan- tum circuits, Commun.Math.Phys. 340, 867 (2015)

    2015

  32. [39]

    A. S. Darmawan, Y. Nakata, S. Tamiya, and H. Ya- 12 masaki, Low-depth random clifford circuits for quantum coding against pauli noise using a tensor-network de- coder, Phys. Rev. Res. 6, 023055 (2024)

    2024

  33. [40]

    Nelson, G

    J. Nelson, G. Bentsen, S. T. Flammia, and M. J. Gullans, Fault-tolerant quantum memory using low-depth random circuit codes, Phys. Rev. Res. 7, 013040 (2025)

    2025

  34. [41]

    Hayden, D

    P. Hayden, D. W. Leung, and A. Winter, Aspects of generic entanglement, Commun.Math.Phys. 265, 95–117 (2006)

    2006

  35. [42]

    D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Ya- mamoto, Approximate quantum error correction can lead to better codes, Phys. Rev. A 56, 2567 (1997)

    1997

  36. [43]

    Schumacher and M

    B. Schumacher and M. D. Westmoreland, Approximate quantum error correction, Quantum Inf. Process. 1, 5 (2002)

    2002

  37. [44]

    Cr´ epeau, D

    C. Cr´ epeau, D. Gottesman, and A. Smith, Approxi- mate quantum error-correcting codes and secret shar- ing schemes, in Advances in Cryptology – EUROCRYPT 2005, edited by R. Cramer (Springer Berlin Heidelberg, Berlin, Heidelberg, 2005) pp. 285–301

    2005

  38. [45]

    Klesse, Approximate quantum error correction, ran- dom codes, and quantum channel capacity, Phys

    R. Klesse, Approximate quantum error correction, ran- dom codes, and quantum channel capacity, Phys. Rev. A 75, 062315 (2007)

    2007

  39. [46]

    B´ eny and O

    C. B´ eny and O. Oreshkov, General conditions for approx- imate quantum error correction and near-optimal recov- ery channels, Phys. Rev. Lett. 104, 120501 (2010)

    2010

  40. [47]

    H. K. Ng and P. Mandayam, Simple approach to approx- imate quantum error correction based on the transpose channel, Phys. Rev. A 81, 062342 (2010)

    2010

  41. [48]

    B´ eny and O

    C. B´ eny and O. Oreshkov, Approximate simulation of quantum channels, Phys. Rev. A 84, 022333 (2011)

    2011

  42. [49]

    B´ eny, Perturbative quantum error correction, Phys

    C. B´ eny, Perturbative quantum error correction, Phys. Rev. Lett. 107, 080501 (2011)

    2011

  43. [50]

    Mandayam and H

    P. Mandayam and H. K. Ng, Towards a unified frame- work for approximate quantum error correction, Phys. Rev. A 86, 012335 (2012)

    2012

  44. [52]

    J. Yi, W. Ye, D. Gottesman, and Z.-W. Liu, Complex- ity and order in approximate quantum error-correcting codes, Nature Physics 20, 1798 (2024)

    2024

  45. [53]

    M. J. Gullans, S. Krastanov, D. A. Huse, L. Jiang, and S. T. Flammia, Quantum coding with low-depth random circuits, Phys. Rev. X 11, 031066 (2021)

    2021

  46. [54]

    A. J. Scott, Multipartite entanglement, quantum-error- correcting codes, and entangling power of quantum evo- lutions, Phys. Rev. A 69, 052330 (2004)

    2004

  47. [55]

    M. A. Nielsen and I. L. Chuang, Quantum Computa- tion and Quantum Information: 10th Anniversary Edi- tion (Cambridge University Press, 2010)

    2010

  48. [56]

    M. L. Mehta, Random matrices, Vol. 142 (Elsevier, 2004)

    2004

  49. [57]

    Zhu and B.-G

    H. Zhu and B.-G. Englert, Quantum state tomography with fully symmetric measurements and product mea- surements, Phys. Rev. A 84, 022327 (2011)

    2011

  50. [58]

    Bryan, S

    J. Bryan, S. Leutheusser, Z. Reichstein, and M. V. Raamsdonk, Locally Maximally Entangled States of Mul- tipart Quantum Systems, Quantum 3, 115 (2019)

    2019

  51. [59]

    Helwig and W

    W. Helwig and W. Cui, Absolutely maximally en- tangled states: Existence and applications (2013), arXiv:1306.2536 [quant-ph]

    2013 arXiv

  52. [60]

    Aubrun and S

    G. Aubrun and S. J. Szarek, Alice and Bob Meet Ba- nach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory , Vol. 223 (American Mathematical Society, 2017)

    2017

  53. [61]

    Tomamichel, R

    M. Tomamichel, R. Colbeck, and R. Renner, Duality be- tween smooth min- and max-entropies, IEEE Trans. Inf. Theory 56, 4674 (2010)

    2010

  54. [62]

    Wang, Universal quantum computing models: a perspective of resource theory, Acta Phys

    D.-S. Wang, Universal quantum computing models: a perspective of resource theory, Acta Phys. Sin. 73, 220302 (2024)

    2024

  55. [63]

    G. Liu, Z. Du, Z.-W. Liu, and X. Ma, Approximate quan- tum error correction with 1d log-depth circuits (2025), arXiv:2503.17759 [quant-ph]

    2025

  56. [64]

    A. Laha, A. Aggarwal, and S. Kumar, Random density matrices: Analytical results for mean root fidelity and the mean-square bures distance, Phys. Rev. A 104, 022438 (2021). 13 Appendix A: Contents of Appendix and list of expectation values For convenience, we outline the conten...

    2021

  57. [65]

    Proof of Proposition 2 First, we introduce the following lemma, which allows us to deal with the expectation value taken over the random subspace

    Lower bound for expectation value of the trace distance a. Proof of Proposition 2 First, we introduce the following lemma, which allows us to deal with the expectation value taken over the random subspace. The key idea in proving this proposition is to decompose the expectatio...

  58. [66]

    To simplify the above equation, we note that the operator |0⟩⟨0|−| v0⟩⟨v0| can be diagonalized by a unitary W as |0⟩⟨0|−| va⟩⟨va| =W p 1−a2|0⟩⟨0|− p 1−a2|1⟩⟨1| W†, (C13) then Eq

    (C12) where|va⟩ = a|0⟩ + √ 1−a2|1⟩ with|0⟩,|1⟩ being two fixed orthogonal states. To simplify the above equation, we note that the operator |0⟩⟨0|−| v0⟩⟨v0| can be diagonalized by a unitary W as |0⟩⟨0|−| va⟩⟨va| =W p 1−a2|0⟩⟨0|− p 1−a2|1⟩⟨1| W†, (C13) then Eq. (C12) becomes ED...

  59. [67]

    Using the results of Proposition 5, we can obtain eΛA B1(HB1B2) +eΛA B2(HB1B2)≥ 4− 1√d1 + 1√d2 4 (d12)1/2 d1X i=1 1/2 i 1/2 i− 1 (d2)3/2−i (d1 + 1)−i

    (C22) Proof. Using the results of Proposition 5, we can obtain eΛA B1(HB1B2) +eΛA B2(HB1B2)≥ 4− 1√d1 + 1√d2 4 (d12)1/2 d1X i=1 1/2 i 1/2 i− 1 (d2)3/2−i (d1 + 1)−i . (C23) To simplify the above inequality, we observe that the terms with i≥ 2 in the above summation satisfy 1/2 i...

  60. [68]

    Here, according to Appendix B.4

    Proof of Theorem 2 To get the concentration result aboutV A X(HC) on the Grassmannian Gr(HB1B2,dC), we also need to calculate the Lipschitz constant. Here, according to Appendix B.4. of [59], the corresponding metric MGr(HC1,HC1) satisfies the following condition. Given a fixe...

  61. [69]

    Now, the proof of Theorem 2 is straightforward as follows

    (C41) So the Lipschitz constant of V A B1(HC) is 4/ √ 2. Now, the proof of Theorem 2 is straightforward as follows. Proof. According to Proposition 7, we can easily find that the Lipschitz constant of function V A B1(HC1) +V A B2(HC1) is 4 √ 2 as [V A B1(HC1) +V A B2(HC1)]− [V...

  62. [70]

    (C43) 21

    (C42) Therefore, through combining Lemma 4 and Theorem 1, we can directly obtain the following concentration result Pr V A(HC)< 1 6 (2dC− 2)(2d12− 1) (2dC− 1)(2d12− 2)−α ≤ Pr V A B1(HC) +V A B2(HC)< 1 3 (2dC− 2)(2d12− 1) (2dC− 1)(2d12− 2)− 2α ≤ Pr V A B1(HC) +V A B2(HC)< E HC ...

  63. [71]

    2− 4√d1(d1d1)1/2 d1X i=1 1/2 i 1/2 i− 1 (d1)3/2−i (d1 + 1)−i # , (C46) v2 = (2dC− 2)(2d12− 1) (2dC− 1)(2d12− 2)

    Additional concentration results on bipartite systems a. Concentration results on V A B1(HC) andV A B2(HC) separately In the previous subsections, we have obtained universal lower bounds for the expectation values of EHC[V A B1(HC)] and EHC[V A B2(HC)]. Using a similar method,...

  64. [72]

    Proposition 8

    (C50) We also consider another function defined as the expectation value of EX(|ψ⟩,|ϕ⟩) over one variable, PX(|ψ⟩) := E |ϕ⟩∼µ(HB1B2 ) [EX(|ψ⟩,|ϕ⟩)], (C51) then we have the following results about the Lipschitz constants of these two functions. Proposition 8. The Lipschitz cons...

  65. [73]

    Preparatory work: ϵ-net and Levy’s lemma To prove the results of this paper, we need to introduce two basic tools

    Concentration result on typical subspaces in the case of a subsystem identity operator a. Preparatory work: ϵ-net and Levy’s lemma To prove the results of this paper, we need to introduce two basic tools. The first tool is the existence of “small” fine nets, which are used to ...

  66. [74]

    Concentration result on typical subspaces in the case of a subspace projector a. Concentration results on the trace distance with respect to a subspace projector Before proving Theorem 5, we first define a function and prove some properties of it, which will be used in the pro...

  67. [75]

    As in the proof of Theorem 5, we also fix a subspace HC0 in the Grassmannian Gr(HB1B2,dC) and transform the trace distance D(Π(B1) C ,e1B1) into a function of U ∈ SU(d12)

    Proof of Proposition 3 Proof. As in the proof of Theorem 5, we also fix a subspace HC0 in the Grassmannian Gr(HB1B2,dC) and transform the trace distance D(Π(B1) C ,e1B1) into a function of U ∈ SU(d12). Then we compute the Lipschitz constant of 29 D(Π(B1) C ,e1B1) =D(TrB2(UΠC0U...

  68. [76]

    Since eη(E,RS) =p dC(u +α), let γ :=k/m satisfy γ <1/2, then the parameter α becomes α = ˜η2 0 dC −u≈ ˜η2 0 dC −d(γ−1/2)m, (F1) where we have used the approximation u≈dk−m/2

    Determining the code distance and the number of physical qudits with inaccuracy being fixed First, let’s consider the case where the QEC inaccuracy eη(E,RS) is a fixed small constant eη0. Since eη(E,RS) =p dC(u +α), let γ :=k/m satisfy γ <1/2, then the parameter α becomes α = ...

  69. [77]

    Since eη(E,RS) = p dC(u +α) with u≈dk−m/2, we assume the inaccuracy equals to da(k−m/2) with a> 0

    Determining the code distance and the number of physical qudits with inaccuracy approaching zero Secondly, we consider the case where eη(E,RS) depends on k,m . Since eη(E,RS) = p dC(u +α) with u≈dk−m/2, we assume the inaccuracy equals to da(k−m/2) with a> 0. In order to avoid ...

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