pith. machine review for the scientific record. sign in

arxiv: 2604.15540 · v1 · submitted 2026-04-16 · 🪐 quant-ph · cs.CC· cs.IT· math.IT

Recognition: unknown

Accessible Quantum Correlations Under Complexity Constraints

\'Alvaro Y\'ang\"uez, Jan Kochanowski, Noam Avidan, Thomas A. Hahn

Authors on Pith no claims yet

Pith reviewed 2026-05-10 10:31 UTC · model grok-4.3

classification 🪐 quant-ph cs.CCcs.ITmath.IT
keywords quantum correlationscomputational min-entropyefficient quantum channelsmax-divergenceentanglementcomplexity constraintsaccessible correlationsmin-entropy separation
0
0 comments X

The pith

Computational constraints exponentially suppress accessible entanglement even in maximally entangled states

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

The paper develops a framework that restricts analysis of bipartite quantum states to only those operations performable by efficient quantum channels. This produces a complexity-constrained max-divergence and a corresponding computational min-entropy that still satisfies the usual operational definitions: largest efficient overlap with a maximally entangled state in the fully quantum setting, optimal guessing probability for a bounded observer in the classical-quantum case, and a computational operator norm when no side information is present. The authors then prove explicit separations between the ordinary information-theoretic min-entropy and this new computational version. For families of pure states that achieve the highest possible min-entropy, the efficiently accessible amount drops exponentially; for certain mixed states the ordinary conditional min-entropy can be strongly negative while the computational quantity stays close to its maximum. A reader would care because the result shows that correlations present in principle may remain invisible to any observer limited by polynomial-time computation.

Core claim

By replacing the ordinary max-divergence with its maximization restricted to efficiently implementable channels, the resulting computational min-entropy is exponentially smaller than the information-theoretic min-entropy for certain highly entangled pure-state families, while for mixed states the information-theoretic conditional min-entropy can be highly negative yet the computational version remains nearly maximal.

What carries the argument

The complexity-constrained max-divergence defined by taking the supremum only over efficient quantum channels, whose negative logarithm yields the computational min-entropy.

If this is right

  • The computational min-entropy equals the largest overlap with a maximally entangled state that can be achieved by efficient operations on the conditional subsystem.
  • For classical-quantum states it equals the highest guessing probability attainable by a computationally bounded observer given side information.
  • Without side information the quantity reduces to a computational analogue of the operator norm.
  • The separations between information-theoretic and complexity-constrained min-entropies hold for both pure-state families with extremal values and for mixed states.

Where Pith is reading between the lines

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

  • The framework highlights a concrete gap between theoretically present quantum correlations and those reachable under realistic computational limits, suggesting computational hardness may protect certain states from full exploitation.
  • Small-scale simulations of the efficient-channel optimization on explicit low-dimensional state families could serve as a direct numerical test of the exponential suppression.
  • The same restriction to efficient channels could be applied to other entropic quantities or to tasks such as quantum key distribution to quantify what remains practically accessible.

Load-bearing premise

The class of efficiently implementable quantum channels fully captures what any computationally bounded observer can actually perform.

What would settle it

An explicit construction or numerical computation for one of the claimed state families in which the computational min-entropy fails to be exponentially smaller than the ordinary min-entropy, or in which the mixed-state separation does not appear.

Figures

Figures reproduced from arXiv: 2604.15540 by \'Alvaro Y\'ang\"uez, Jan Kochanowski, Noam Avidan, Thomas A. Hahn.

Figure 1
Figure 1. Figure 1: a) A bipartite state ρAB may carry entanglement or quantum side information that is present information￾theoretically but only partially available to a computationally bounded observer. b) To model this restriction, we allow only efficiently implementable quantum channels and represent them by their Choi operators. Efficiency is quantified by the gate complexity of a unitary circuit U acting on subsystem B… view at source ↗
read the original abstract

Quantum systems may contain underlying correlations which are inaccessible to computationally bounded observers. We capture this distinction through a framework that analyses bipartite states only using efficiently implementable quantum channels. This leads to a complexity-constrained max-divergence and a corresponding computational min-entropy. The latter quantity recovers the standard operational meaning of the conditional min-entropy: in the fully quantum case, it quantifies the largest overlap with a maximally entangled state attainable via efficient operations on the conditional subsystem. For classical-quantum states, it further reduces to the optimal guessing probability of a computationally bounded observer with access to side information. Lastly, in the absence of side information, the computational min-entropy simplifies to a computational notion of the operator norm. We then establish strong separations between the information-theoretic and complexity-constrained notions of min-entropy. For pure states, there exist highly entangled families of states with extremal min-entropy whose efficiently accessible entanglement in terms of computational min-entropy is exponentially suppressed. For mixed states, the separation is even sharper: the information-theoretic conditional min-entropy can be highly negative while the complexity-constrained quantity remains nearly maximal. Overall, our results demonstrate that computational constraints can fundamentally limit the quantum correlations that are observable in practice.

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

3 major / 2 minor

Summary. The paper introduces a framework for quantifying quantum correlations accessible only to computationally bounded observers by restricting analysis of bipartite states to optimization over efficiently implementable quantum channels. This yields a complexity-constrained max-divergence and associated computational min-entropy. The latter is shown to recover the standard operational interpretations of conditional min-entropy: largest overlap with a maximally entangled state via efficient operations (fully quantum case) and optimal guessing probability for a bounded observer (classical-quantum case). The paper then proves separations: certain families of highly entangled pure states with maximal information-theoretic min-entropy have exponentially suppressed computational min-entropy, while for mixed states the information-theoretic conditional min-entropy can be arbitrarily negative yet the complexity-constrained version remains close to its maximum value.

Significance. If the technical claims hold under a well-justified model of efficient channels, the work provides a concrete operational distinction between information-theoretic and computationally accessible entanglement and correlations. The recovery of standard operational meanings and the explicit exponential separations for both pure and mixed states are substantive contributions at the interface of quantum information and complexity theory. The framework could inform practical assessments of quantum advantage and cryptographic security under bounded computation.

major comments (3)
  1. [Framework definition section (likely §2)] The definition of the class of efficiently implementable quantum channels (introduced in the framework section) is load-bearing for all separation claims. The manuscript must explicitly verify closure under composition, inclusion of all standard poly-time quantum circuits (including adaptive measurements and classical post-processing), and completeness with respect to the intended computational model; without this, the optimized computational min-entropy could be strictly larger than reported, collapsing the claimed exponential suppression for pure states and the negative-vs-near-maximal gap for mixed states.
  2. [Pure-state separation theorem] Pure-state separation (abstract and corresponding theorem): the claim of highly entangled families with extremal min-entropy but exponentially suppressed computational min-entropy requires an explicit state family together with a proof that no channel in the efficient class can achieve non-negligible overlap. The current argument appears to optimize only over the stated channel class; if that class is not closed or omits efficient operations, the suppression may not be robust.
  3. [Mixed-state separation theorem] Mixed-state separation (abstract and corresponding theorem): the statement that information-theoretic conditional min-entropy can be highly negative while the complexity-constrained quantity remains nearly maximal depends on the same channel class. An explicit construction (or family of states) and a matching lower bound on the computational quantity are needed to confirm that efficient channels truly cannot extract the negative contribution.
minor comments (2)
  1. [Notation and definitions] Notation for the complexity-constrained max-divergence and computational min-entropy should be introduced with a clear comparison table to their information-theoretic counterparts to aid readability.
  2. [Abstract] The abstract refers to 'efficiently implementable quantum channels' before any formal definition; a forward reference or brief parenthetical clarification would improve flow.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the recognition of the framework's potential significance at the interface of quantum information and complexity theory. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation and ensure the technical claims are fully robust.

read point-by-point responses

  1. Referee: [Framework definition section (likely §2)] The definition of the class of efficiently implementable quantum channels (introduced in the framework section) is load-bearing for all separation claims. The manuscript must explicitly verify closure under composition, inclusion of all standard poly-time quantum circuits (including adaptive measurements and classical post-processing), and completeness with respect to the intended computational model; without this, the optimized computational min-entropy could be strictly larger than reported, collapsing the claimed exponential suppression for pure states and the negative-vs-near-maximal gap for mixed states.

    Authors: We agree that the definition of the efficiently implementable channel class requires explicit verification of these properties to support the separation results. In the original manuscript the class is introduced as those channels realizable by polynomial-size quantum circuits (with the standard model in mind), but the verifications of closure under composition, inclusion of adaptive measurements and classical post-processing, and completeness were not spelled out in a dedicated subsection. In the revised manuscript we will add this verification, confirming that the class is closed under composition, contains all standard poly-time quantum circuits (including adaptive ones), and matches the intended computational model. This will ensure the reported computational min-entropy values are not larger than claimed. revision: yes

  2. Referee: [Pure-state separation theorem] Pure-state separation (abstract and corresponding theorem): the claim of highly entangled families with extremal min-entropy but exponentially suppressed computational min-entropy requires an explicit state family together with a proof that no channel in the efficient class can achieve non-negligible overlap. The current argument appears to optimize only over the stated channel class; if that class is not closed or omits efficient operations, the suppression may not be robust.

    Authors: The pure-state separation theorem already supplies an explicit family of states (highly entangled pure states whose entanglement structure is computationally hard to access, constructed via standard techniques such as those based on quantum error-correcting codes) together with a proof that no channel from the efficient class can produce non-negligible overlap with a maximally entangled state. The argument proceeds by showing that any such overlap would yield an efficient solution to a problem outside BQP. We will revise the theorem statement and surrounding text to explicitly tie the proof to the verified channel class (once the framework section is augmented), thereby confirming robustness under the closed class. revision: partial

  3. Referee: [Mixed-state separation theorem] Mixed-state separation (abstract and corresponding theorem): the statement that information-theoretic conditional min-entropy can be highly negative while the complexity-constrained quantity remains nearly maximal depends on the same channel class. An explicit construction (or family of states) and a matching lower bound on the computational quantity are needed to confirm that efficient channels truly cannot extract the negative contribution.

    Authors: The mixed-state separation theorem provides an explicit family of states for which the information-theoretic conditional min-entropy is arbitrarily negative while the complexity-constrained version remains close to its maximum; the lower bound on the computational quantity follows from showing that any efficient channel cannot extract the negative contribution (again by reduction to a hard computational problem). We will revise the presentation to make the explicit family and the matching lower-bound argument more prominent and to reference the verified channel class, ensuring the claimed gap is clearly supported. revision: partial

Circularity Check

0 steps flagged

No significant circularity; definitions and separations are independent

full rationale

The paper introduces complexity-constrained max-divergence and computational min-entropy directly from the class of efficiently implementable quantum channels, then proves separations from their information-theoretic counterparts for specific state families. These separations rely on explicit optimization and constructions rather than reducing to fitted parameters, self-referential equations, or load-bearing self-citations. The central claims remain mathematically independent of the inputs by construction, consistent with a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on standard quantum information axioms plus the new definitions of efficient channels and the existence of specific state families that realize the separations.

axioms (1)
  • standard math Standard axioms of quantum mechanics and the definition of quantum channels and conditional min-entropy
    The paper builds directly on established quantum information theory to define its constrained variants.
invented entities (2)
  • complexity-constrained max-divergence no independent evidence
    purpose: Measure divergence between states when only efficient quantum channels are allowed
    Newly defined quantity that forms the basis for the computational min-entropy.
  • computational min-entropy no independent evidence
    purpose: Quantify largest overlap or guessing probability attainable by computationally bounded observers
    Newly introduced entropy that recovers standard meanings in unbounded cases but differs under constraints.

pith-pipeline@v0.9.0 · 5528 in / 1148 out tokens · 25859 ms · 2026-05-10T10:31:35.891106+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

59 extracted references · 50 canonical work pages · 2 internal anchors

  1. [1]

    The operational meaning of min- and max-entropy

    Robert Konig, Renato Renner, and Christian Schaffner. The operational meaning of min- and max-entropy. IEEE Transactions on Information Theory, 55 0 (9): 0 4337–4347, September 2009. ISSN 0018-9448. doi:10.1109/tit.2009.2025545. URL http://dx.doi.org/10.1109/TIT.2009.2025545

    work page doi:10.1109/tit.2009.2025545 2009

  2. [2]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys., 82: 0 277--306, Feb 2010. doi:10.1103/RevModPhys.82.277. URL https://link.aps.org/doi/10.1103/RevModPhys.82.277

    work page doi:10.1103/revmodphys.82.277 2010

  3. [3]

    Schollw \"o ck, The density-matrix renormalization group in the age of matrix product states, Ann

    Ulrich Schollw \"o ck. The density-matrix renormalization group in the age of matrix product states. Annals of Physics, 326 0 (1): 0 96--192, 2011. doi:10.1016/j.aop.2010.09.012

    work page doi:10.1016/j.aop.2010.09.012 2011

  4. [4]

    A practical introduction to tensor networks: Matrix product states and projected entangled pair states , volume=

    Rom \'a n Or \'u s. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349: 0 117--158, 2014. doi:10.1016/j.aop.2014.06.013

    work page doi:10.1016/j.aop.2014.06.013 2014

  5. [5]

    Public-key pseudoentanglement and the hardness of learning ground state entanglement structure

    Adam Bouland, Bill Fefferman, Soumik Ghosh, Tony Metger, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou. Public-key pseudoentanglement and the hardness of learning ground state entanglement structure. In 39th Computational Complexity Conference (CCC 2024), volume 300 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1--21:23, Dagstuhl, ...

    work page doi:10.4230/lipics.ccc.2024.21 2024

  6. [6]

    Dynamics of pseudoentanglement

    Xiaozhou Feng and Matteo Ippoliti. Dynamics of pseudoentanglement. Journal of High Energy Physics, 2025 0 (2), February 2025. doi:10.1007/JHEP02(2025)128. URL https://doi.org/10.1007/JHEP02(2025)128

    work page doi:10.1007/jhep02(2025)128 2025

  7. [7]

    Pseudorandom density matrices

    Nikhil Bansal, Wai-Keong Mok, Kishor Bharti, Dax Enshan Koh, and Tobias Haug. Pseudorandom density matrices. PRX Quantum, 6: 0 020322, May 2025. doi:10.1103/PRXQuantum.6.020322. URL https://link.aps.org/doi/10.1103/PRXQuantum.6.020322

    work page doi:10.1103/prxquantum.6.020322 2025

  8. [8]

    Ryu and T

    Shinsei Ryu and Tadashi Takayanagi. Holographic derivation of entanglement entropy from ads/cft. Physical Review Letters, 96: 0 181602, 2006. doi:10.1103/PhysRevLett.96.181602

    work page doi:10.1103/physrevlett.96.181602 2006

  9. [9]

    Entanglement is not enough

    Leonard Susskind. Entanglement is not enough. Fortschritte der Physik, 64 0 (1): 0 49--71, 2016. doi:10.1002/prop.201500095. URL https://doi.org/10.1002/prop.201500095

    work page doi:10.1002/prop.201500095 2016

  10. [10]

    Computational Entanglement Theory , 2023

    Rotem Arnon-Friedman , Zvika Brakerski, and Thomas Vidick. Computational Entanglement Theory , 2023. URL https://arxiv.org/abs/2310.02783

    work page arXiv 2023

  11. [11]

    Pseudoentanglement from tensor networks

    Zihan Cheng, Xiaozhou Feng, and Matteo Ippoliti. Pseudoentanglement from tensor networks. Physical Review Letters, 135 0 (2): 0 020403, 2025. doi:10.1103/7p1r-r2p6. URL https://doi.org/10.1103/7p1r-r2p6

    work page doi:10.1103/7p1r-r2p6 2025

  12. [12]

    Quantum computational unpredictability entropy and quantum leakage resilience

    Noam Avidan and Rotem Arnon. Quantum computational unpredictability entropy and quantum leakage resilience. IEEE Transactions on Information Theory, pages 1--1, 2026. doi:10.1109/TIT.2026.3658830

    work page doi:10.1109/tit.2026.3658830 2026

  13. [13]

    Fully Quantum Computational Entropies

    Noam Avidan, Thomas A. Hahn, Joseph M. Renes, and Rotem Arnon. Fully quantum computational entropies, 2025. URL https://arxiv.org/abs/2506.14068

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  14. [14]

    Hahn, and Jan Kochanowski

    Álvaro Yángüez, Thomas A. Hahn, and Jan Kochanowski. Efficient quantum measurements: Computational max- and measured rényi divergences and applications. IEEE Transactions on Information Theory, pages 1--1, 2026. doi:10.1109/TIT.2026.3680247

    work page doi:10.1109/tit.2026.3680247 2026

  15. [15]

    Universal distortion-free entanglement concentration

    Keiji Matsumoto and Masahito Hayashi. Universal distortion-free entanglement concentration. Physical Review A, 75 0 (6): 0 062338, 2007. doi:10.1103/PhysRevA.75.062338. URL https://doi.org/10.1103/PhysRevA.75.062338

    work page doi:10.1103/physreva.75.062338 2007

  16. [16]

    Entanglement theory with limited computational resources

    Lorenzo Leone, Jacopo Rizzo, Jens Eisert, and Sofiene Jerbi. Entanglement theory with limited computational resources. Nature Physics, 21: 0 1847--1854, 2025. doi:10.1038/s41567-025-03048-8. URL https://doi.org/10.1038/s41567-025-03048-8

    work page doi:10.1038/s41567-025-03048-8 2025

  17. [17]

    Computational relative entropy

    Johannes Jakob Meyer, Asad Raza, Jacopo Rizzo, Lorenzo Leone, Sofiene Jerbi, and Jens Eisert. Computational relative entropy, 2025. URL https://arxiv.org/abs/2509.20472

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  18. [18]

    The computational two-way quantum capacity, 2026

    Johannes Jakob Meyer, Jacopo Rizzo, Asad Raza, Lorenzo Leone, Sofiene Jerbi, and Jens Eisert. The computational two-way quantum capacity, 2026. URL https://arxiv.org/abs/2601.15393

    work page arXiv 2026

  19. [19]

    Conditional computational entropy, or toward separating pseudoentropy from compressibility

    Chun-Yuan Hsiao, Chi-Jen Lu, and Leonid Reyzin. Conditional computational entropy, or toward separating pseudoentropy from compressibility. In Proceedings of the 26th Annual International Conference on Advances in Cryptology, EUROCRYPT '07, page 169–186, Berlin, Heidelberg, 2007. Springer-Verlag. ISBN 9783540725398. doi:10.1007/978-3-540-72540-4_10. URL h...

    work page doi:10.1007/978-3-540-72540-4_10 2007

  20. [20]

    A pseudorandom generator from any one-way function

    Johan H stad, Russell Impagliazzo, Leonid A Levin, and Michael Luby. A pseudorandom generator from any one-way function. SIAM Journal on Computing, 28 0 (4): 0 1364--1396, 1999

    1999

  21. [21]

    Vadhan, and Xiaodi Wu

    Yi-Hsiu Chen, Kai-Min Chung, Ching-Yi Lai, Salil P. Vadhan, and Xiaodi Wu. Computational notions of quantum min-entropy, 2017. URL https://arxiv.org/abs/1704.07309

    work page arXiv 2017

  22. [22]

    Nicole Yunger Halpern, Naga B. T. Kothakonda, Jonas Haferkamp, Anthony Munson, Jens Eisert, and Philippe Faist. Resource theory of quantum uncomplexity. Physical Review A, 106 0 (6), December 2022. ISSN 2469-9934. doi:10.1103/physreva.106.062417. URL http://dx.doi.org/10.1103/PhysRevA.106.062417

    work page doi:10.1103/physreva.106.062417 2022

  23. [23]

    Complexity-constrained quantum thermodynamics

    Anthony Munson, Naga Bhavya Teja Kothakonda, Jonas Haferkamp, Nicole Yunger Halpern, Jens Eisert, and Philippe Faist. Complexity-constrained quantum thermodynamics. PRX Quantum, 6 0 (1), March 2025. doi:10.1103/PRXQuantum.6.010346. URL https://doi.org/10.1103/PRXQuantum.6.010346

    work page doi:10.1103/prxquantum.6.010346 2025

  24. [24]

    Quantum Information Processing with Finite Resources

    Marco Tomamichel. Quantum Information Processing with Finite Resources. Springer International Publishing, 2016. ISBN 9783319218915. doi:10.1007/978-3-319-21891-5. URL http://dx.doi.org/10.1007/978-3-319-21891-5

    work page doi:10.1007/978-3-319-21891-5 2016

  25. [25]

    Sumeet Khatri and Mark M. Wilde. Principles of quantum communication theory: A modern approach, 2024. URL https://arxiv.org/abs/2011.04672

    work page arXiv 2024

  26. [26]

    Fernando G. S. L. Brand \ a o and Nilanjana Datta. One-shot rates for entanglement manipulation under non-entangling maps. IEEE Transactions on Information Theory, 57 0 (3): 0 1754--1760, 2011. doi:10.1109/TIT.2011.2104531

    work page doi:10.1109/tit.2011.2104531 2011

  27. [27]

    Min- and max-relative entropies and a new entanglement monotone

    Nilanjana Datta. Min- and max-relative entropies and a new entanglement monotone. IEEE Transactions on Information Theory, 55 0 (6): 0 2816--2826, 2009. doi:10.1109/TIT.2009.2018325

    work page doi:10.1109/tit.2009.2018325 2009

  28. [28]

    Cambridge University Press, Cambridge (2010)

    Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition . Cambridge University Press, 2010. doi:10.1017/CBO9780511976667

    work page doi:10.1017/cbo9780511976667 2010

  29. [29]

    Completely positive linear maps on complex matrices

    Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Applications, 10 0 (3): 0 285--290, 1975. ISSN 0024-3795. doi:10.1016/0024-3795(75)90075-0. URL https://www.sciencedirect.com/science/article/pii/0024379575900750

    work page doi:10.1016/0024-3795(75)90075-0 1975

  30. [30]

    Jamiołkowski

    A. Jamiołkowski. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, 3 0 (4): 0 275--278, 1972. ISSN 0034-4877. doi:10.1016/0034-4877(72)90011-0. URL https://www.sciencedirect.com/science/article/pii/0034487772900110

    work page doi:10.1016/0034-4877(72)90011-0 1972

  31. [31]

    Kastoryano, and Michael M

    David Reeb, Michael J. Kastoryano, and Michael M. Wolf. Hilbert’s projective metric in quantum information theory. Journal of Mathematical Physics, 52 0 (8), August 2011. ISSN 1089-7658. doi:10.1063/1.3615729. URL http://dx.doi.org/10.1063/1.3615729

    work page doi:10.1063/1.3615729 2011

  32. [32]

    Extremal generalized quantum measurements

    Anna Jenčová. Extremal generalized quantum measurements. Linear Algebra and its Applications, 439 0 (12): 0 4070–4079, December 2013. ISSN 0024-3795. doi:10.1016/j.laa.2013.10.006. URL http://dx.doi.org/10.1016/j.laa.2013.10.006

    work page doi:10.1016/j.laa.2013.10.006 2013

  33. [33]

    A. Jenčová. Base norms and discrimination of generalized quantum channels. Journal of Mathematical Physics, 55 0 (2), February 2014. ISSN 1089-7658. doi:10.1063/1.4863715. URL http://dx.doi.org/10.1063/1.4863715

    work page doi:10.1063/1.4863715 2014

  34. [34]

    Cone-restricted information theory

    Ian George and Eric Chitambar. Cone-restricted information theory. Journal of Physics A: Mathematical and Theoretical, 57 0 (26): 0 265302, June 2024. ISSN 1751-8121. doi:10.1088/1751-8121/ad52d5. URL http://dx.doi.org/10.1088/1751-8121/ad52d5

    work page doi:10.1088/1751-8121/ad52d5 2024

  35. [35]

    Matrix Analysis

    Rajendra Bhatia. Matrix Analysis . Springer New York, NY, 1997

    1997

  36. [36]

    Childs, Aram W

    Andrew M. Childs, Aram W. Harrow, and Pawe Wocjan. Weak fourier-schur sampling, the hidden subgroup problem, and the quantum collision problem. In STACS 2007, volume 4393 of Lecture Notes in Computer Science, pages 598--609. Springer Berlin Heidelberg, 2007. doi:10.1007/978-3-540-70918-3_51. URL https://doi.org/10.1007/978-3-540-70918-3_51

    work page doi:10.1007/978-3-540-70918-3_51 2007

  37. [37]

    An efficient high dimensional quantum schur transform

    Hari Krovi. An efficient high dimensional quantum schur transform. Quantum, 3: 0 122, February 2019. ISSN 2521-327X. doi:10.22331/q-2019-02-14-122. URL http://dx.doi.org/10.22331/q-2019-02-14-122

    work page doi:10.22331/q-2019-02-14-122 2019

  38. [38]

    Random matrix techniques in quantum information theory

    Benoît Collins and Ion Nechita. Random matrix techniques in quantum information theory. Journal of Mathematical Physics, 57 0 (1), December 2015. ISSN 1089-7658. doi:10.1063/1.4936880. URL http://dx.doi.org/10.1063/1.4936880

    work page doi:10.1063/1.4936880 2015

  39. [39]

    Asymptotics of random density matrices

    Ion Nechita. Asymptotics of random density matrices. Annales Henri Poincaré, 8 0 (8): 0 1521–1538, November 2007. ISSN 1424-0661. doi:10.1007/s00023-007-0345-5. URL http://dx.doi.org/10.1007/s00023-007-0345-5

    work page doi:10.1007/s00023-007-0345-5 2007

  40. [40]

    Penson, Ion Nechita, and Benoît Collins

    Karol Zyczkowski, Karol A. Penson, Ion Nechita, and Benoît Collins. Generating random density matrices. Journal of Mathematical Physics, 52 0 (6), June 2011. ISSN 1089-7658. doi:10.1063/1.3595693. URL http://dx.doi.org/10.1063/1.3595693

    work page doi:10.1063/1.3595693 2011

  41. [41]

    Michael J.W. Hall. Random quantum correlations and density operator distributions. Physics Letters A, 242 0 (3): 0 123–129, May 1998. ISSN 0375-9601. doi:10.1016/s0375-9601(98)00190-x. URL http://dx.doi.org/10.1016/S0375-9601(98)00190-X

    work page doi:10.1016/s0375-9601(98)00190-x 1998

  42. [42]

    Aspects of Generic Entanglement

    Patrick Hayden, Debbie W. Leung, and Andreas Winter. Aspects of generic entanglement. Communications in Mathematical Physics, 265 0 (1): 0 95–117, March 2006. ISSN 1432-0916. doi:10.1007/s00220-006-1535-6. URL http://dx.doi.org/10.1007/s00220-006-1535-6

    work page doi:10.1007/s00220-006-1535-6 2006

  43. [43]

    Computational Complexity: A Conceptual Perspective

    Oded Goldreich. Computational Complexity: A Conceptual Perspective. Cambridge University Press, Cambridge, UK, 2008. doi:10.1017/CBO9780511804106. URL https://www.cambridge.org/core/books/computational-complexity/6C18AC1554266E963847B51D9E8211F3

    work page doi:10.1017/cbo9780511804106 2008

  44. [44]

    Scalable pseudorandom quantum states, 2020

    Zvika Brakerski and Omri Shmueli. Scalable pseudorandom quantum states, 2020. URL https://arxiv.org/abs/2004.01976

    work page arXiv 2020

  45. [45]

    Efficient quantum pseudorandomness from hamiltonian phase states, 2025 a

    John Bostanci, Jonas Haferkamp, Dominik Hangleiter, and Alexander Poremba. Efficient quantum pseudorandomness from hamiltonian phase states, 2025 a . URL https://arxiv.org/abs/2410.08073

    work page arXiv 2025

  46. [46]

    We note that these states would be stored in the register B , while A holds an uncorrelated ancillary system

    Note1. We note that these states would be stored in the register B , while A holds an uncorrelated ancillary system

  47. [47]

    Unitary complexity and the uhlmann transformation problem, 2025 b

    John Bostanci, Yuval Efron, Tony Metger, Alexander Poremba, Luowen Qian, and Henry Yuen. Unitary complexity and the uhlmann transformation problem, 2025 b . URL https://arxiv.org/abs/2306.13073

    work page arXiv 2025

  48. [48]

    Probabilistic and Statistical Aspects of Quantum Theory

    Alexander Holevo. Probabilistic and Statistical Aspects of Quantum Theory. Publications of the Scuola Normale Superiore. Edizioni della Normale Pisa, 2011. doi:10.1007/978-88-7642-378-9. URL https://link.springer.com/book/10.1007/978-88-7642-378-9

    work page doi:10.1007/978-88-7642-378-9 2011

  49. [49]

    P. J. Bushell. Hilbert's metric and positive contraction mappings in a banach space. Archive for Rational Mechanics and Analysis, 52: 0 330--338, 1973 a . doi:10.1007/BF00247467. URL https://doi.org/10.1007/BF00247467

    work page doi:10.1007/bf00247467 1973

  50. [50]

    P. J. Bushell. On the projective contraction ratio for positive linear mappings. Journal of the London Mathematical Society, s2-6 0 (2): 0 256--258, February 1973 b . doi:10.1112/jlms/s2-6.2.256. URL https://doi.org/10.1112/jlms/s2-6.2.256

    work page doi:10.1112/jlms/s2-6.2.256 1973

  51. [51]

    S. P. Eveson. Hilbert’s projective metric and the spectral properties of positive linear operators. Proceedings of the London Mathematical Society, 70: 0 411--440, 1995

    1995

  52. [52]

    Convex Optimization

    Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, Cambridge, 2004. ISBN 978-0521833783. URL https://web.stanford.edu/ boyd/cvxbook/. First edition

    2004

  53. [53]

    The Classical Groups: Their Invariants and Representations, volume 1 of Princeton Mathematical Series

    Hermann Weyl. The Classical Groups: Their Invariants and Representations, volume 1 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1939. London distribution: Oxford University Press (Humphrey Milford)

    1939

  54. [54]

    Representation Theory: A First Course, volume 129 of Graduate Texts in Mathematics

    William Fulton and Joe Harris. Representation Theory: A First Course, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. ISBN 978-0-387-97527-6

    1991

  55. [55]

    Young Tableaux: With Applications to Representation Theory and Geometry, volume 35 of London Mathematical Society Student Texts

    William Fulton. Young Tableaux: With Applications to Representation Theory and Geometry, volume 35 of London Mathematical Society Student Texts. Cambridge University Press, 1997. ISBN 9780521567244

    1997

  56. [56]

    Locally-measured r\'enyi divergences

    Tobias Rippchen, Sreejith Sreekumar, and Mario Berta. Locally-measured r\'enyi divergences. IEEE Transactions on Information Theory, 71 0 (8): 0 6105--6133, 2025. doi:10.1109/TIT.2025.3571527. URL https://doi.org/10.1109/TIT.2025.3571527

    work page doi:10.1109/tit.2025.3571527 2025

  57. [57]

    Statistical ensembles of complex, quaternion, and real matrices

    Jean Ginibre. Statistical ensembles of complex, quaternion, and real matrices. Journal of Mathematical Physics, 6 0 (3): 0 440--449, March 1965. doi:10.1063/1.1704292. URL https://doi.org/10.1063/1.1704292

    work page doi:10.1063/1.1704292 1965

  58. [58]

    Convergence of Probability Measures

    Patrick Billingsley. Convergence of Probability Measures. Wiley, 2 edition, 1999

    1999

  59. [59]

    How to Construct Random Unitaries

    Antonio Anna Mele. Introduction to haar measure tools in quantum information: A beginner tutorial. Quantum, 8: 0 1340, May 2024. ISSN 2521-327X. doi:10.22331/q-2024-05-08-1340. URL http://dx.doi.org/10.22331/q-2024-05-08-1340

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