pith. sign in

arxiv: 2606.23592 · v1 · pith:GUZOG6IXnew · submitted 2026-06-22 · 🪐 quant-ph

Random dimension reduction and learning symmetric properties of quantum states

Angus Lowe , Xinyu Tan This is my paper

Pith reviewed 2026-06-26 08:15 UTC · model grok-4.3

classification 🪐 quant-ph
keywords random dimension reductionsymmetric quantum propertiessample complexitySchur transformstate tomographyrandom purificationquantum circuitsisometry invariants
0
0 comments X

The pith

Random dimension reduction replaces the ambient dimension with the maximum rank in sample complexity bounds for learning symmetric properties of quantum states.

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

The paper presents random dimension reduction as a procedure that applies random isometries to shrink the dimensions of multiple, possibly distinct quantum states while preserving any property invariant under simultaneous isometries on tensor powers of each state. This serves as a black-box replacement that swaps the full Hilbert-space dimension for the states' maximum rank inside sample-complexity expressions for symmetric learning tasks. When the reduced states are fed into standard tomography, the approach yields tighter upper bounds on the number of copies needed to estimate distances, fidelities, and relative entropies between pairs of states. The procedure admits an efficient circuit realization via the Schur transform and is shown to be distinct from random purification channels, which cannot simultaneously purify multiple different input states.

Core claim

Random dimension reduction simultaneously reduces the dimensions of many, potentially distinct quantum states while preserving properties invariant under the tensor power action of an isometry. This provides a black-box method to replace the dimension with the maximum rank in the sample complexity of learning symmetric properties, even those depending on multiple input states. Dimension reduction followed by full state tomography yields improved upper bounds for estimating distances, fidelities, and relative entropies between pairs of states. An efficient quantum circuit implementation exists using the Schur transform. Expressing the procedure through the Choi-Jamiolkowski isomorphism connec

What carries the argument

random dimension reduction, the procedure that applies random isometries to reduce dimensions of multiple states while preserving tensor-power isometry invariants

If this is right

  • Improved upper bounds on copies needed to estimate distances between pairs of states
  • Tighter sample requirements for fidelity and relative-entropy estimation via post-reduction tomography
  • Efficient circuit realization of the reduction using the Schur transform
  • A separation showing random purification cannot handle multiple distinct states simultaneously

Where Pith is reading between the lines

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

  • The technique may extend to other quantum estimation tasks whose cost depends on ambient dimension rather than rank
  • Analyses previously requiring explicit Schur-polynomial machinery could be simplified by composing the reduction with standard tomography
  • If approximate invariance can be established, the method might apply to nearly symmetric properties

Load-bearing premise

The properties to be learned must stay unchanged when the same isometry is applied to all copies of each input state.

What would settle it

An explicit symmetric property whose minimal sample complexity for learning still grows with the full dimension after random dimension reduction is applied.

read the original abstract

We introduce a procedure called random dimension reduction that simultaneously reduces the dimensions of many, potentially distinct quantum states while preserving properties invariant under the tensor power action of an isometry. This provides a black-box method to replace the dimension with the maximum rank in the sample complexity of learning symmetric properties, even those depending on multiple input states. We show that dimension reduction followed by full state tomography yields improved upper bounds for estimating distances, fidelities, and relative entropies between pairs of states. We also give an efficient quantum circuit implementation of the procedure using the Schur transform. Expressing the action of our procedure through the Choi-Jamiolkowski isomorphism reveals an intimate connection with the recently introduced random purification channel by Tang, Wright, and Zhandry. This perspective also completes an end-to-end analysis of sample-optimal tomography without requiring a reference to the Schur transform or Schur polynomials. Finally, we prove that there does not exist a random purification channel that simultaneously purifies copies of multiple, potentially different input states. Hence, random dimension reduction is related to, but distinct from, random purification.

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.

Referee Report

0 major / 2 minor

Summary. The paper introduces a procedure called random dimension reduction that simultaneously reduces the dimensions of many, potentially distinct quantum states while preserving properties invariant under the tensor-power action of an isometry. This supplies a black-box method to replace dimension d with maximum rank r in the sample complexity of learning symmetric properties, including joint properties of multiple states. The manuscript shows that the procedure followed by full state tomography yields improved upper bounds for estimating pairwise distances, fidelities, and relative entropies; provides an efficient quantum circuit implementation via the Schur transform; establishes a connection to the random purification channel through the Choi-Jamiolkowski isomorphism that completes an end-to-end sample-optimal tomography analysis without Schur polynomials; and proves that no random purification channel exists for simultaneously purifying copies of multiple distinct input states.

Significance. If the derivations hold, the work supplies a general, black-box technique for tightening sample complexities of symmetric quantum properties by substituting rank for dimension, with direct applicability to multi-state estimation tasks. The explicit efficient circuit, the end-to-end tomography analysis independent of Schur polynomials, and the impossibility result for multi-state purification are concrete strengths that enhance the contribution.

minor comments (2)
  1. [Abstract] Abstract: the claim of 'improved upper bounds' for distances, fidelities, and relative entropies is stated without explicit sample-complexity expressions or the precise rank-dependent improvement factor; adding these (even in summarized form) would strengthen readability.
  2. The connection to random purification is presented via the Choi-Jamiolkowski isomorphism, but the precise mapping between the two channels and the resulting sample-complexity expressions could be highlighted with a short comparison table or equation reference for clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of its contributions, and recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces random dimension reduction as a new black-box procedure that preserves isometry-invariant properties and replaces dimension d with max rank r in sample complexity bounds. The central claims rest on the definition of the procedure itself, its explicit circuit implementation via Schur transform, and a proof of non-existence for a multi-state random purification channel. No step reduces a claimed prediction or uniqueness result to a prior fit, self-citation chain, or ansatz imported from the authors' own prior work; the connection to Tang et al. is external and the end-to-end tomography analysis is presented as independent of Schur polynomials. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated beyond the definition of the new procedure itself.

pith-pipeline@v0.9.1-grok · 5715 in / 1025 out tokens · 19530 ms · 2026-06-26T08:15:49.515753+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

50 extracted references · 20 canonical work pages

  1. [1]

    2015 , eprint=

    Quantum Spectrum Testing , author=. 2015 , eprint=

    2015

  2. [2]

    The Theory of Quantum Information , publisher=

    Watrous, John , year=. The Theory of Quantum Information , publisher=

  3. [3]

    2025 , eprint=

    A List of Complexity Bounds for Property Testing by Quantum Sample-to-Query Lifting , author=. 2025 , eprint=

    2025

  4. [4]

    Foundations and Trends® in Communications and Information Theory , author=

    Polynomial Methods in Statistical Inference: Theory and Practice , volume=. Foundations and Trends® in Communications and Information Theory , author=. 2020 , pages=. doi:10.1561/0100000095 , number=

    work page doi:10.1561/0100000095 2020

  5. [5]

    2016 , eprint=

    Chebyshev polynomials, moment matching, and optimal estimation of the unseen , author=. 2016 , eprint=

    2016

  6. [6]

    Linear Algebra Done Right , ISBN =

    Axler, Sheldon , year =. Linear Algebra Done Right , ISBN =. doi:10.1007/978-3-031-41026-0 , journal =

    work page doi:10.1007/978-3-031-41026-0

  7. [7]

    and Harrow, Aram W

    Childs, Andrew M. and Harrow, Aram W. and Wocjan, Paweł , pages=. Weak Fourier-Schur Sampling, the Hidden Subgroup Problem, and the Quantum Collision Problem , ISBN=. doi:10.1007/978-3-540-70918-3_51 , booktitle=

    work page doi:10.1007/978-3-540-70918-3_51

  8. [8]

    Gily\' e n, Y

    Gily\'. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics , year =. doi:10.1145/3313276.3316366 , booktitle =

    work page doi:10.1145/3313276.3316366

  9. [9]

    Introduction to

    Mele, Antonio Anna , journal =. Introduction to. doi:10.22331/q-2024-05-08-1340 , url =

    work page doi:10.22331/q-2024-05-08-1340 2024

  10. [10]

    2026 , eprint=

    Random purification channel made simple , author=. 2026 , eprint=

    2026

  11. [11]

    Quantum principal component analysis.Nature Physics, 10(9):631–633, 2014.doi:10.1038/nphys3029

    Lloyd, Seth and Mohseni, Masoud and Rebentrost, Patrick , year =. Quantum principal component analysis , volume =. Nature Physics , publisher =. doi:10.1038/nphys3029 , number =

    work page doi:10.1038/nphys3029

  12. [12]

    2022 , eprint=

    Improved Quantum Algorithms for Fidelity Estimation , author=. 2022 , eprint=

    2022

  13. [13]

    2025 , eprint=

    Quantum algorithms for Uhlmann transformation , author=. 2025 , eprint=

    2025

  14. [14]

    2011 , isbn =

    Valiant, Gregory and Valiant, Paul , title =. 2011 , isbn =. doi:10.1145/1993636.1993727 , booktitle =

    work page doi:10.1145/1993636.1993727 2011

  15. [15]

    Linear Representations of Finite Groups , year =

    Serre, Jean-Pierre , doi =. Linear Representations of Finite Groups , year =. Graduate Texts in Mathematics , publisher =

  16. [16]

    Symmetry, representations, and invariants , volume =

    Goodman, Roe and Wallach, Nolan R , date-added =. Symmetry, representations, and invariants , volume =

  17. [17]

    Estimating the Unseen: Improved Estimators for Entropy and Other Properties , volume =

    Valiant, Gregory and Valiant, Paul , year =. Estimating the Unseen: Improved Estimators for Entropy and Other Properties , volume =. Journal of the ACM , publisher =. doi:10.1145/3125643 , number =

    work page doi:10.1145/3125643

  18. [18]

    Representation theory: a first course , volume =

    Fulton, William and Harris, Joe , date-added =. Representation theory: a first course , volume =

  19. [19]

    2025 , eprint=

    Beating full state tomography for unentangled spectrum estimation , author=. 2025 , eprint=

    2025

  20. [20]

    2025 , eprint=

    Conjugate queries can help , author=. 2025 , eprint=

    2025

  21. [21]

    IEEE Transactions on Information Theory , volume =

    Wang, Qisheng and Zhang, Zhicheng , year=. Fast Quantum Algorithms for Trace Distance Estimation , volume=. IEEE Transactions on Information Theory , publisher=. doi:10.1109/tit.2023.3321121 , number=

    work page doi:10.1109/tit.2023.3321121 2023

  22. [22]

    2025 , eprint=

    Mixed state tomography reduces to pure state tomography , author=. 2025 , eprint=

    2025

  23. [23]

    2024 , eprint=

    Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer , author=. 2024 , eprint=

    2024

  24. [24]

    John Wright , title =

  25. [25]

    2005 , eprint=

    Applications of coherent classical communication and the Schur transform to quantum information theory , author=. 2005 , eprint=

    2005

  26. [26]

    The Spectra of Quantum States and the Kronecker Coefficients of the Symmetric Group , volume =

    Christandl, Matthias and Mitchison, Graeme , year =. The Spectra of Quantum States and the Kronecker Coefficients of the Symmetric Group , volume =. Communications in Mathematical Physics , publisher =. doi:10.1007/s00220-005-1435-1 , number =

    work page doi:10.1007/s00220-005-1435-1

  27. [27]

    Quantum universal variable-length source coding , author =. Phys. Rev. A , volume =. 2002 , publisher =

    2002

  28. [28]

    Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) , year=2022, chapter =

    Mehdi Soleimanifar and John Wright , title =. Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) , year=2022, chapter =

    2022

  29. [29]

    Nearly Tight Bounds for Testing Tree Tensor Network States , year=

    Lovitz, Benjamin and Lowe, Angus , journal=. Nearly Tight Bounds for Testing Tree Tensor Network States , year=

  30. [30]

    Probing R\'enyi entanglement entropy via randomized measurements , journal =

    Tiff Brydges and Andreas Elben and Petar Jurcevic and Beno. Probing R\'enyi entanglement entropy via randomized measurements , journal =. 2019 , doi =

    2019

  31. [31]

    Cross-Platform Verification of Intermediate Scale Quantum Devices , author =. Phys. Rev. Lett. , volume =. 2020 , publisher =

    2020

  32. [32]

    Proceedings of the 48th Annual ACM Symposium on Theory of Computing , pages =

    O'Donnell, Ryan and Wright, John , title =. 2016 , isbn =. doi:10.1145/2897518.2897544 , booktitle =

    work page doi:10.1145/2897518.2897544 2016

  33. [33]

    and Ji, Zhengfeng and Wu, Xiaodi and Yu, Nengkun , title =

    Haah, Jeongwan and Harrow, Aram W. and Ji, Zhengfeng and Wu, Xiaodi and Yu, Nengkun , title =. 2016 , isbn =. doi:10.1145/2897518.2897585 , booktitle =

    work page doi:10.1145/2897518.2897585 2016

  34. [34]

    Efficient quantum to- mography II

    O'Donnell, Ryan and Wright, John , title =. 2017 , isbn =. doi:10.1145/3055399.3055454 , booktitle =

    work page doi:10.1145/3055399.3055454 2017

  35. [35]

    doi:10.1088/0305-4470/31/20/006 , year =

    Masahito Hayashi , title =. doi:10.1088/0305-4470/31/20/006 , year =

    work page doi:10.1088/0305-4470/31/20/006

  36. [36]

    2013 , eprint=

    The Church of the Symmetric Subspace , author=. 2013 , eprint=

    2013

  37. [37]

    Liu and Thatchaphol Saranurak and Aaron Sidford and Zhao Song and Di Wang , editor =

    B. Improved Quantum data analysis , year =. doi:10.1145/3406325.3451109 , booktitle =

    work page doi:10.1145/3406325.3451109

  38. [38]

    2026 , eprint=

    On Estimating the Quantum Tsallis Relative Entropy , author=. 2026 , eprint=

    2026

  39. [39]

    and Harrow, Aram W

    Bacon, Dave and Chuang, Isaac L. and Harrow, Aram W. , title =. Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms , pages =. 2007 , isbn =

    2007

  40. [40]

    and Wagner, Aaron B

    Acharya, Jayadev and Issa, Ibrahim and Shende, Nirmal V. and Wagner, Aaron B. , journal=. Estimating Quantum Entropy , year=

  41. [41]

    Theory of Computing Library , series =

    Montanaro, Ashley and Wolf, Ronald. A Survey of Quantum Property Testing , year =. doi:10.4086/toc.gs.2016.007 , publisher =

    work page doi:10.4086/toc.gs.2016.007 2016

  42. [42]

    Proceedings of the 34th International Conference on Machine Learning , pages =

    A Unified Maximum Likelihood Approach for Estimating Symmetric Properties of Discrete Distributions , author =. Proceedings of the 34th International Conference on Machine Learning , pages =. 2017 , editor =

    2017

  43. [43]

    Proceedings of the 31st Conference On Learning Theory , pages =

    Local moment matching: A unified methodology for symmetric functional estimation and distribution estimation under Wasserstein distance , author =. Proceedings of the 31st Conference On Learning Theory , pages =. 2018 , editor =

    2018

  44. [44]

    , title=

    Ando, T. , title=. Mathematische Zeitschrift , year=

  45. [45]

    and O'Donnell, Ryan , year=

    Flammia, Steven T. and O'Donnell, Ryan , year=. Quantum chi-squared tomography and mutual information testing , volume=. doi:10.22331/q-2024-06-20-1381 , journal=

    work page doi:10.22331/q-2024-06-20-1381 2024

  46. [46]

    Distance measures to compare real and ideal quantum processes

    Gilchrist, Alexei and Langford, Nathan K. and Nielsen, Michael A. , year=. Distance measures to compare real and ideal quantum processes , volume=. Physical Review A , publisher=. doi:10.1103/physreva.71.062310 , number=

    work page doi:10.1103/physreva.71.062310

  47. [47]

    Ricard,. H. Archiv der Mathematik , year=

  48. [48]

    Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing , pages =

    Anshu, Anurag and Landau, Zeph and Liu, Yunchao , title =. 2022 , isbn =. doi:10.1145/3519935.3519974 , booktitle =

    work page doi:10.1145/3519935.3519974 2022

  49. [49]

    Quantum state certification , booktitle =

    B. Quantum state certification , year =. doi:10.1145/3313276.3316344 , booktitle =

    work page doi:10.1145/3313276.3316344

  50. [50]

    2025 , eprint=

    High-dimensional quantum Schur transforms , author=. 2025 , eprint=

    2025