arxiv: 2605.09878 · v1 · submitted 2026-05-11 · 🪐 quant-ph · cond-mat.stat-mech

Recognition: no theorem link

Clifford Ergotropy

Somnath Maity , Ryusuke Hamazaki

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:59 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech

keywords Clifford ergotropyquantum magicstabilizer Rényi entropyquantum thermodynamicsClifford operationsresource theorymany-body systemssecond law

0 comments

The pith

Clifford ergotropy is bounded above by a quantity that decreases as the quantum magic of the state increases.

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

The paper defines Clifford ergotropy as the maximum energy extractable from a quantum state when only Clifford operations are allowed. It derives universal upper bounds on this quantity expressed through the infinite-order filtered stabilizer Rényi entropy, so that greater magic tightens the bound and reduces possible extraction. A reader would care because Clifford operations are efficient and physically realizable, turning the result into a concrete limit on work extraction in any process restricted to those operations. The bounds are demonstrated explicitly for one- and two-qubit cases and then used to obtain a form of the second law that holds for typical many-body states under the same restriction.

Core claim

Clifford ergotropy, the extractable energy under the restriction to Clifford operations, admits universal upper bounds that decrease with increasing magic as quantified by the infinite-order filtered stabilizer Rényi entropy. These bounds are illustrated for one- and two-qubit systems, where the two-qubit case exhibits a transition in the control landscape, and they imply a second law of thermodynamics under Clifford operations for typical states in many-body systems.

What carries the argument

The universal upper bound that directly relates Clifford ergotropy to the infinite-order filtered stabilizer Rényi entropy, which quantifies magic and thereby limits extractable energy.

If this is right

The bound is tight enough to be useful for one- and two-qubit systems and reveals a transition in the two-qubit Clifford control landscape.
Typical many-body states obey a second law of thermodynamics when energy extraction is restricted to Clifford operations.
The upper bounds hold universally and become stricter as the state's magic, measured by the infinite-order filtered stabilizer Rényi entropy, increases.
The analysis connects resource-theoretic magic directly to thermodynamic extractability under efficient gate restrictions.

Where Pith is reading between the lines

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

In any quantum device limited to Clifford gates, extra magic resources would be required to reach the highest possible energy extraction in a thermodynamic process.
The same bounding technique could be adapted to other restricted gate sets that arise in near-term hardware, yielding analogous thermodynamic limits.
Numerical checks on small simulators could directly test whether the predicted transition in the two-qubit control landscape appears in physical devices.

Load-bearing premise

That restricting allowed operations to the Clifford group captures the thermodynamic cost of magic and that the filtered stabilizer Rényi entropy is the appropriate quantifier of that cost for systems of every size.

What would settle it

A calculation or measurement on any two-qubit state in which the actual Clifford ergotropy exceeds the numerical value of the upper bound supplied by its infinite-order filtered stabilizer Rényi entropy.

Figures

Figures reproduced from arXiv: 2605.09878 by Ryusuke Hamazaki, Somnath Maity.

read the original abstract

We discuss the interplay between thermodynamics and magic resources in closed quantum dynamics by introducing Clifford ergotropy, the amount of extractable energy under the restriction to Clifford operations. We provide universal upper bounds on Clifford ergotropy, which decrease with increasing magic as quantified by the infinite-order filtered stabilizer R\'enyi entropy. We demonstrate the utility of this bound for one- and two-qubit systems, with the latter exhibiting a notable transition in the control landscape of Clifford ergotropy. Finally, we show that our analysis has nontrivial consequences even for many-body systems, including a form of the second law of thermodynamics under Clifford operations for typical quantum states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines Clifford ergotropy and ties its upper bounds to a magic monotone, but the universality of those bounds may still depend on how the Hamiltonian spectrum aligns with Clifford orbits.

read the letter

The core contribution is a definition of Clifford ergotropy—the maximum energy drop achievable by Clifford unitaries alone—plus explicit upper bounds expressed through the infinite-order filtered stabilizer Rényi entropy. These bounds are presented as universal and decreasing with magic. The authors also report a transition in the optimal Clifford control for two qubits and sketch a second-law style statement for typical many-body states under the same restriction.

Referee Report

2 major / 2 minor

Summary. The paper introduces Clifford ergotropy as the maximum energy extractable from a quantum state via Clifford unitaries alone. It derives universal upper bounds on this quantity that decrease monotonically with the infinite-order filtered stabilizer Rényi entropy (a magic monotone), demonstrates the bounds and a control-landscape transition for one- and two-qubit systems, and extends the analysis to many-body systems to obtain a form of the second law that holds for typical states under Clifford-restricted dynamics.

Significance. If the universal bounds and the many-body second-law statement are rigorously established, the work supplies a concrete resource-theoretic link between magic and thermodynamic work extraction under a computationally relevant restriction (Clifford operations). The explicit small-system calculations and the observation that the bound tightens with magic provide falsifiable predictions that could be tested in near-term devices; the many-body extension, if free of hidden assumptions on the Hamiltonian spectrum, would be a notable addition to the literature on restricted-operation thermodynamics.

major comments (2)

[Abstract / §3] Abstract and the derivation of the universal bound (presumably §3): the claim that the upper bound on Clifford ergotropy is independent of the specific eigenvalues of H and holds for arbitrary many-body Hamiltonians is load-bearing for the universality statement. The skeptic correctly notes that Clifford orbits may not sample the spectrum uniformly unless the Hamiltonian commutes with the Clifford group action or its eigenspaces are Clifford-invariant; the manuscript must explicitly state the spectral assumption (or prove independence) rather than relying on the post-Clifford energy evaluation alone.
[§4 / many-body section] Two-qubit transition and many-body extension: the reported change in optimal Clifford control with increasing magic is interesting, but the manuscript must show that the same magic-monotone bound continues to upper-bound the extractable work when the system size grows and the Hamiltonian is generic (e.g., a random Ising model). Without an explicit check that the filtered Rényi entropy still controls the overlap with the Clifford orbit for non-stabilizer initial states, the many-body second-law claim risks being limited to Hamiltonians whose spectrum aligns with stabilizer subspaces.

minor comments (2)

Notation: the infinite-order filtered stabilizer Rényi entropy is introduced without a compact symbol; adopting a short notation (e.g., M_∞) would improve readability when the bound is stated repeatedly.
[Abstract] The abstract states that the bounds 'decrease with increasing magic'; the manuscript should include a short remark clarifying whether the inequality is strict or non-increasing, and whether equality cases are characterized.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments below and will revise the manuscript accordingly to strengthen the presentation of the universal bound and the many-body extension.

read point-by-point responses

Referee: [Abstract / §3] Abstract and the derivation of the universal bound (presumably §3): the claim that the upper bound on Clifford ergotropy is independent of the specific eigenvalues of H and holds for arbitrary many-body Hamiltonians is load-bearing for the universality statement. The skeptic correctly notes that Clifford orbits may not sample the spectrum uniformly unless the Hamiltonian commutes with the Clifford group action or its eigenspaces are Clifford-invariant; the manuscript must explicitly state the spectral assumption (or prove independence) rather than relying on the post-Clifford energy evaluation alone.

Authors: We agree that an explicit statement of the spectral assumption is needed for rigor. The bound is derived from the fact that the infinite-order filtered stabilizer Rényi entropy constrains the maximum overlap between a state and any Clifford orbit, which in turn limits the achievable energy difference after a Clifford unitary; this difference is then evaluated with respect to the eigenvalues of H. Because the entropy bound is independent of H, the resulting upper bound on Clifford ergotropy inherits that independence provided the Clifford group acts sufficiently ergodically on the energy eigenspaces. We will add a clarifying paragraph in §3 that states this assumption explicitly (namely, that either H commutes with the Clifford action or its spectrum is sufficiently non-degenerate that the orbit samples the eigenvalues uniformly) and note that the bound remains valid for any H satisfying this condition. This addresses the concern without altering the derivation. revision: yes
Referee: [§4 / many-body section] Two-qubit transition and many-body extension: the reported change in optimal Clifford control with increasing magic is interesting, but the manuscript must show that the same magic-monotone bound continues to upper-bound the extractable work when the system size grows and the Hamiltonian is generic (e.g., a random Ising model). Without an explicit check that the filtered Rényi entropy still controls the overlap with the Clifford orbit for non-stabilizer initial states, the many-body second-law claim risks being limited to Hamiltonians whose spectrum aligns with stabilizer subspaces.

Authors: The many-body second-law statement is formulated for typical (Haar-random or high-entropy) initial states, for which the filtered stabilizer Rényi entropy directly bounds the distance to the nearest stabilizer state and hence the maximum overlap with any Clifford orbit. Because typical states have energy distributions that concentrate around the mean for generic local Hamiltonians (including random Ising models), the Clifford-restricted work extraction is controlled by the same monotone irrespective of the precise spectrum. We will expand the discussion in §4 to include a short argument showing that the bound carries over to generic many-body Hamiltonians via this typicality, without requiring stabilizer-subspace alignment. While a full numerical survey for large random Ising instances lies beyond the present scope, the analytic typicality argument suffices to support the claim for the class of states considered. revision: partial

Circularity Check

0 steps flagged

No significant circularity; bounds derived from definitions without reduction to inputs.

full rationale

The paper introduces Clifford ergotropy directly from the restriction to Clifford unitaries and derives universal upper bounds using the infinite-order filtered stabilizer Rényi entropy as a magic quantifier. These steps follow from the operational definitions and standard properties of Clifford orbits and stabilizer polytopes, without fitting parameters to data subsets, self-citation chains that bear the central claim, or renaming known results as new derivations. The many-body second-law consequence is presented as a consequence for typical states, remaining independent of the specific Hamiltonian spectrum in the stated universality. No load-bearing step reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the newly introduced definition of Clifford ergotropy and the choice of the infinite-order filtered stabilizer Rényi entropy as the magic quantifier; these are domain assumptions standard in quantum resource theories but not independently verified here.

axioms (2)

domain assumption Clifford operations form the relevant restricted set of unitaries for thermodynamic work extraction
The paper defines Clifford ergotropy by restricting to this gate set by construction.
domain assumption The infinite-order filtered stabilizer Rényi entropy quantifies the relevant magic resource for energy extraction
The bounds are stated to decrease with this specific entropy measure.

invented entities (1)

Clifford ergotropy no independent evidence
purpose: Quantify maximum extractable energy under Clifford restriction
Newly defined quantity whose properties are derived in the paper.

pith-pipeline@v0.9.0 · 5395 in / 1416 out tokens · 49763 ms · 2026-05-12T04:59:31.159762+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · 2 internal anchors

[1]

Then, Eq

= 1 such thatM α = 0 for the stabilizer states [32]. Then, Eq. (9) is written as ECl(ˆρ)≤E(ˆρ) +e−M∞/2∥H∥1,(12) where M∞ =−ln[r 2 1] (13) is the infinite-order filtered SRE (α→ ∞). Remarkably, the inequality (12), stating that the up- per bound becomes smaller if the magic becomes larger, indicates that the magic can be an obstacle for work ex- traction. ...

work page
[2]

Vinjanampathy and J

S. Vinjanampathy and J. Anders, Quantum thermody- namics, Contemporary Physics57, 545 (2016)

work page 2016
[3]

Goold, M

J. Goold, M. Huber, A. Riera, L. d. Rio, and P. Skrzypczyk, The role of quantum information in ther- modynamics—a topical review, Journal of Physics A: Mathematical and Theoretical49, 143001 (2016)

work page 2016
[4]

Campaioli, S

F. Campaioli, S. Gherardini, J. Q. Quach, M. Polini, and G. M. Andolina, Colloquium: Quantum batteries, Rev. Mod. Phys.96, 031001 (2024)

work page 2024
[5]

Lenard, Thermodynamical proof of the gibbs formula for elementary quantum systems, Journal of Statistical Physics19, 575 (1978)

A. Lenard, Thermodynamical proof of the gibbs formula for elementary quantum systems, Journal of Statistical Physics19, 575 (1978)

work page 1978
[6]

Pusz and S

W. Pusz and S. L. Woronowicz, Passive states and kms states for general quantum systems, Communications in Mathematical Physics58, 273 (1978)

work page 1978
[7]

A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhysics Letters67, 565 (2004)

work page 2004
[8]

Perarnau-Llobet, K

M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Ac´ ın, Extractable work from correlations, Phys. Rev. X5, 041011 (2015)

work page 2015
[9]

E. G. Brown, N. Friis, and M. Huber, Passivity and practical work extraction using gaussian operations, New Journal of Physics18, 113028 (2016)

work page 2016
[10]

Francica, F

G. Francica, F. C. Binder, G. Guarnieri, M. T. Mitchi- son, J. Goold, and F. Plastina, Quantum coherence and ergotropy, Phys. Rev. Lett.125, 180603 (2020)

work page 2020
[11]

L. P. Garc´ ıa-Pintos, A. Hamma, and A. del Campo, Fluc- tuations in extractable work bound the charging power of quantum batteries, Phys. Rev. Lett.125, 040601 (2020)

work page 2020
[12]

H.-L. Shi, S. Ding, Q.-K. Wan, X.-H. Wang, and W.-L. Yang, Entanglement, coherence, and extractable work in quantum batteries, Phys. Rev. Lett.129, 130602 (2022)

work page 2022
[13]

Mitsuhashi, K

Y. Mitsuhashi, K. Kaneko, and T. Sagawa, Character- izing symmetry-protected thermal equilibrium by work extraction, Phys. Rev. X12, 021013 (2022)

work page 2022
[14]

Puliyil, M

S. Puliyil, M. Banik, and M. Alimuddin, Thermodynamic signatures of genuinely multipartite entanglement, Phys. Rev. Lett.129, 070601 (2022)

work page 2022
[15]

F. E. Q. Rodriguez, F. A. Mele, S. F. E. Oliviero, V. Giovannetti, L. Lami, and V. Cavina, Extracting energy via bosonic gaussian operations, arXiv preprint arXiv:2503.21748 (2025)

work page arXiv 2025
[16]

Optimal Work Extraction from Finite-Time Closed Quantum Dynamics

S. Sugimoto, T. Sagawa, and R. Hamazaki, Optimal work extraction from finite-time closed quantum dynamics, arXiv preprint arXiv:2508.20512 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025
[17]

Hokkyo and M

A. Hokkyo and M. Ueda, Universal upper bound on er- gotropy and no-go theorem by the eigenstate thermaliza- tion hypothesis, Phys. Rev. Lett.134, 010406 (2025)

work page 2025
[18]

Polo-Rodr´ ıguez, F

B. Polo-Rodr´ ıguez, F. Centrone, G. Adesso, and M. Al- imuddin, Ergotropic characterization of continuous- variable entanglement, Phys. Rev. Lett.136, 050201 (2026)

work page 2026
[19]

Bravyi and A

S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal clifford gates and noisy ancillas, Phys. Rev. A71, 022316 (2005)

work page 2005
[20]

Veitch, S

V. Veitch, S. A. Hamed Mousavian, D. Gottesman, and J. Emerson, The resource theory of stabilizer quantum computation, New Journal of Physics16, 013009 (2014)

work page 2014
[21]

Gottesman, Theory of fault-tolerant quantum compu- tation, Phys

D. Gottesman, Theory of fault-tolerant quantum compu- tation, Phys. Rev. A57, 127 (1998)

work page 1998
[22]

Aaronson and D

S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A70, 052328 (2004)

work page 2004
[23]

Bravyi, G

S. Bravyi, G. Smith, and J. A. Smolin, Trading classical and quantum computational resources, Phys. Rev. X6, 021043 (2016)

work page 2016
[24]

Bravyi, D

S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, Simulation of quantum circuits by low- rank stabilizer decompositions, Quantum3, 181 (2019)

work page 2019
[25]

Howard and E

M. Howard and E. Campbell, Application of a resource theory for magic states to fault-tolerant quantum com- puting, Phys. Rev. Lett.118, 090501 (2017)

work page 2017
[26]

Leone, S

L. Leone, S. F. E. Oliviero, and A. Hamma, Stabilizer r´ enyi entropy, Phys. Rev. Lett.128, 050402 (2022)

work page 2022
[27]

Leone and L

L. Leone and L. Bittel, Stabilizer entropies are mono- tones for magic-state resource theory, Phys. Rev. A110, L040403 (2024)

work page 2024
[28]

Haug and L

T. Haug and L. Piroli, Quantifying nonstabilizerness of matrix product states, Phys. Rev. B107, 035148 (2023)

work page 2023
[29]

P. S. Tarabunga, E. Tirrito, T. Chanda, and M. Dal- monte, Many-body magic via pauli-markov chains—from criticality to gauge theories, PRX Quantum4, 040317 (2023)

work page 2023
[30]

Hoshino, M

M. Hoshino, M. Oshikawa, and Y. Ashida, Stabilizer r´ enyi entropy and conformal field theory, Phys. Rev. X 16, 011037 (2026)

work page 2026
[31]

Hoshino and Y

M. Hoshino and Y. Ashida, Stabilizer r´ enyi entropy en- codes fusion rules of topological defects and boundaries, Phys. Rev. Lett.136, 080402 (2026)

work page 2026
[32]

Turkeshi, E

X. Turkeshi, E. Tirrito, and P. Sierant, Magic spread- ing in random quantum circuits, Nature Communications 16, 2575 (2025)

work page 2025
[33]

Turkeshi, A

X. Turkeshi, A. Dymarsky, and P. Sierant, Pauli spec- trum and nonstabilizerness of typical quantum many- body states, Phys. Rev. B111, 054301 (2025)

work page 2025
[34]

Magni, A

B. Magni, A. Christopoulos, A. De Luca, and X. Turkeshi, Anticoncentration in clifford circuits and beyond: From random tensor networks to pseudomagic states, Phys. Rev. X15, 031071 (2025)

work page 2025
[35]

Odavi´ c, M

J. Odavi´ c, M. Viscardi, and A. Hamma, Stabilizer en- tropy in nonintegrable quantum evolutions, Phys. Rev. B112, 104301 (2025)

work page 2025
[36]

Szombathy, A

D. Szombathy, A. Valli, C. u. u. u. u. P. m. c. Moca, L. Farkas, and G. Zar´ and, Asymptotically independent fluctuations of stabilizer r´ enyi entropy and entanglement 6 in random unitary circuits, Phys. Rev. Res.7, 043072 (2025)

work page 2025
[37]

P. S. Tarabunga and E. Tirrito, Magic transition in measurement-only circuits, npj Quantum Information 11, 166 (2025)

work page 2025
[38]

Maity and R

S. Maity and R. Hamazaki, Local spreading of stabi- lizer r´ enyi entropy in a brickwork random clifford circuit, Phys. Rev. Res.8, 013324 (2026)

work page 2026
[39]

Bejan, P

M. Bejan, P. W. Claeys, and J. Yao, Magic spreading under unitary clifford dynamics (2025), arXiv:2511.21487 [quant-ph]

work page arXiv 2025
[40]

S. F. E. Oliviero, L. Leone, A. Hamma, and S. Lloyd, Measuring magic on a quantum processor, npj Quantum Information8, 148 (2022)

work page 2022
[41]

Haug and M

T. Haug and M. Kim, Scalable measures of magic re- source for quantum computers, PRX Quantum4, 010301 (2023)

work page 2023
[42]

T. Haug, S. Lee, and M. S. Kim, Efficient quantum al- gorithms for stabilizer entropies, Phys. Rev. Lett.132, 240602 (2024)

work page 2024
[43]

Mukherjee, A

A. Mukherjee, A. Roy, S. S. Bhattacharya, and M. Banik, Presence of quantum correlations results in a nonvanish- ing ergotropic gap, Phys. Rev. E93, 052140 (2016)

work page 2016
[44]

Alimuddin, T

M. Alimuddin, T. Guha, and P. Parashar, Bound on er- gotropic gap for bipartite separable states, Phys. Rev. A 99, 052320 (2019)

work page 2019
[45]

Schuster, J

T. Schuster, J. Haferkamp, D. Hangleiter, and J. Eis- ert, Pauli spectrum and nonstabilizerness of typical quan- tum many-body states, Physical Review B111, 054301 (2025)

work page 2025
[46]

Beato, P

N. Beato, P. Patil, and M. Bukov, Topological phase tran- sitions in a constrained two-qubit quantum control land- scape, Phys. Rev. Lett.135, 110803 (2025)

work page 2025
[47]

Beato, P

N. Beato, P. Patil, and M. Bukov, Toward a theory of phase transitions in quantum control landscapes, Phys. Rev. X15, 041014 (2025)

work page 2025
[48]

Pfeuty, The one-dimensional ising model with a trans- verse field, ANNALS of Physics57, 79 (1970)

P. Pfeuty, The one-dimensional ising model with a trans- verse field, ANNALS of Physics57, 79 (1970)

work page 1970
[49]

Tasaki, Statistical mechanical derivation of the second law of thermodynamics, arXiv preprint cond- mat/0009206 (2000)

H. Tasaki, Statistical mechanical derivation of the second law of thermodynamics, arXiv preprint cond- mat/0009206 (2000)

work page arXiv 2000
[50]

Goldstein, T

S. Goldstein, T. Hara, and H. Tasaki, The second law of thermodynamics for pure quantum states, arXiv preprint arXiv:1303.6393 (2013)

work page arXiv 2013
[51]

T. N. Ikeda, N. Sakumichi, A. Polkovnikov, and M. Ueda, The second law of thermodynamics under unitary evolu- tion and external operations, Annals of Physics354, 338 (2015)

work page 2015
[52]

Iyoda, K

E. Iyoda, K. Kaneko, and T. Sagawa, Fluctuation the- orem for many-body pure quantum states, Phys. Rev. Lett.119, 100601 (2017)

work page 2017
[53]

Chiba, Y

Y. Chiba, Y. Yoneta, R. Hamazaki, and A. Shimizu, Sec- ond law of thermodynamics in closed quantum many- body systems, arXiv preprint arXiv:2602.06657 (2026)

work page arXiv 2026
[54]

Chiba and Y

Y. Chiba and Y. Yoneta, Exact thermal eigenstates of nonintegrable spin chains at infinite temperature, Phys. Rev. Lett.133, 170404 (2024)

work page 2024
[55]

Chitambar and G

E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys.91, 025001 (2019)

work page 2019
[56]

T. K. Konar and J. Zakrzewski, Interplay of nonsta- bilizerness and ergotropy in quantum batteries (2026), arXiv:2605.03600 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[57]

Popescu, A

S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nature Physics2, 754 (2006)

work page 2006
[58]

Reimann, Generalization of von neumann’s approach to thermalization, Phys

P. Reimann, Generalization of von neumann’s approach to thermalization, Phys. Rev. Lett.115, 010403 (2015)

work page 2015
[59]

Hamazaki and M

R. Hamazaki and M. Ueda, Atypicality of most few-body observables, Phys. Rev. Lett.120, 080603 (2018)

work page 2018
[60]

E. S. Meckes,The random matrix theory of the classical compact groups, Vol. 218 (Cambridge University Press, 2019). END MA TTER PROOF OF THE EXTENSIVE INFINITE-ORDER FIL TERED SRE FOR TYPICAL ST A TES Here, we show that the extensive infinite-order filtered SREM ∞ for typical states. For this purpose, we evaluate Prob (r1 ≥ϵ) = Prob max µ |ρµ| ≥ϵ ,(21) fr...

work page 2019