The entropic coherence is a necessary resource for non-energy preserving gates

Marti Perarnau-Llobet; Pavel Sekatski; Riccardo Castellano; Vasco Cavina; Vittorio Giovannetti

REVIEW 3 major objections 4 minor 62 references

The entropic coherence is a necessary resource for non-energy preserving gates

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves that any battery implementing a non-energy-preserving quantum gate must carry a minimum amount of entropic coherence, growing logarithmically with the demanded precision.

desk verdict The scale-invariant coherence bound is likely right and worth publishing, but the proof leans on an unproved companion theorem and has several local defects that must be fixed before I'd trust the details. read the letter →

arxiv 2509.01515 v1 pith:G477D5WQ submitted 2025-09-01 quant-ph

Riccardo Castellano , Vasco Cavina , Marti Perarnau-Llobet , Vittorio Giovannetti , Pavel Sekatski This is my paper

classification quant-ph

keywords entropiccoherencenon-energy-preservinggatesquantumbatteryresourcetheoryofrelativeentropyFisherinformationenergyconservationgateimplementationerror

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

To implement a unitary that changes the energy of a quantum system, an external battery must be coupled through an energy-conserving interaction. This paper establishes that the battery's initial state must contain a minimum amount of 'entropic coherence'—superposition spread across energy eigenspaces—and that the minimum grows as (r/8) log(σ/ε) as the allowed error ε shrinks. The constants r and σ are determined by the gate and the system's energy spectrum, and σ>0 precisely when the gate does not conserve energy. The immediate consequence is that any finite-dimensional battery has an irreducible error floor, so a perfect energy-changing gate demands an unbounded battery. Under a mild density-of-states assumption the same bound also forces battery energy and quantum Fisher information to diverge, in some regimes faster than previously known universal limits, revealing entropic coherence as a fundamental and independent resource for quantum control.

What carries the argument

The central object is the entropic coherence C(ρ_B,H_B)=S(G_{H_B}[ρ_B])−S(ρ_B), where G_H is the twirling map that dephases the state in the energy eigenbasis; it is the relative entropy between the state and its energy-dephased version, measuring superposition across energy eigenspaces. It is the resource: C is invariant under energy-preserving unitaries, non-increasing under partial trace, sub-additive, and continuous in trace distance. The load-bearing identity is the resource inequality C(β_B,H_B) ≥ C(ν_SA,H_SA)−C(ρ_SA,H_SA), which converts gate accuracy into a required amount of battery coherence; the production term C(ν)−C(ρ) is identified, for product energy-eigenstate inputs, with th

What would settle it

Compute the entropy of S_N = X_1+⋯+X_N for a non-lattice three-valued distribution such as X∈{0,1,√2} with probabilities {1/2,1/3,1/6} for N up to 10^6 and compare with (r(χ)/2) log(2πe N λ); if the entropy ever falls below the bound, Eq. (D25) fails and the main result loses its foundation.

Watch

Extended reading notes

Core claim

Main Result (Eq. (7)): any battery family implementing a non-energy-preserving gate V_S to infidelity ε obeys C(β_B,H_B) ≥ (r2/8) log(σ/ε) − o(1), with r2 ∈ {1,…,d_S²−1} and σ>0 exactly when [V_S,H_S]≠0. Qubit gates give r2=2; random gates give r2 ≥ d_S−1 almost surely. The proof runs 2m system copies through alternating V and V^* interactions, turning resource production into the entropy of a sum of 2m iid energy random variables, bounded below by (r/2) log(2πe m λ). A proportionate battery improves the prefactor by 2. Corollary 3: battery dimension diverges as (σ'/ε)^{r2/4}/log(1/ε).

Load-bearing premise

The main inequality assumes a theorem quoted from a companion paper—that the entropy of a sum of many identical independent discrete random variables grows at least as (r/2) log N—which this paper states but does not prove.

Editorial extensions

If this is right

Perfect implementation of any non-energy-preserving gate requires an unbounded battery; every finite-dimensional battery has a nonzero minimal worst-case error.
The required entropic coherence grows only logarithmically in 1/ε, with prefactor r2/8 (r2/4 for proportionate batteries), so the coherence cost is mild but cannot be avoided.
For qubits the exponent r2=2 makes the bound tight against the known O(ε^{-1/2})-level battery construction, pinning the scaling of battery size.
For batteries with at most linearly growing energy-level counts, energy must scale as ε^{-r2/8} and quantum Fisher information as ε^{-r2/4}; when r2>2 (or r2>4) these beat the previous universal bounds.
Rescaling the battery Hamiltonian changes energy and QFI without changing entropic coherence, so the new bound is not a restatement of the energy or QFI bounds—it captures an independent resource.

Reading between the lines

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

A direct experimental probe: for a fixed gate the minimal worst-case infidelity of a d-dimensional battery should fall as ε_min(d) ∼ const·d^{−4/r2}; measuring that power law would test the bound.
Because the entropy-of-sums theorem is imported from a companion preprint, the prefactor r2/8 stands or falls with that theorem; an independent proof, or a counterexample for a non-lattice three-valued distribution, would settle the prefactor question.
The density-of-states assumption is the only place energy enters, so a battery with a sparse, incommensurate spectrum could in principle satisfy the coherence bound while keeping energy and QFI small—suggesting the coherence cost is the more fundamental of the three.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a real result, likely correct in its central claim, but the manuscript as written is not self-contained enough for me to certify the details. Worth serious referee time, but expect a major revision.

The genuinely new thing is the observation that the relative entropy of coherence used in [13,25] is the wrong quantity for energy-exchanging interactions, and that the relative entropy of superposition (defined via the joint twirl) has the right monotonicity properties. The main inequality (Lemma 7) is clean, and the qubit calculation in Result 2 is explicit and convincing. The incommensurability-rank lower bound for random gates is a nice touch. If the main theorem holds, the corollaries on energy and QFI scaling go beyond the earlier universal bounds in a way people will care about.

The soft spots are real. First and most important, the resource-production step rests on Theorem 22/Corollary 23 of the companion paper arXiv:2508.05348, which is stated but not proved here. That theorem is doing the work: it gives the (r/2) log N growth that makes the bound diverge. If it has an error, Result 1 and Eq. (7) collapse. The paper needs either a self-contained proof or a clear statement of why this is an accepted result. Second, the proof of Lemma 11 contains a false equivalence: [V1⊗V2†, H1+H2]=0 iff [V,H]=V. That is not true in general, and while the lemma's conclusion looks salvageable, the proof as written does not establish it. Third, Corollary 3's proof does not deliver its stated power law: it cites Eq. (7) rather than the proportionate bound, and the 'C ≤ 2 dim' step is wrong (the coherence is at most log dim). The dimension scaling is plausible, but the derivation needs to be redone.

The reader's conditional verdict is about right. None of these defects by themselves makes me doubt the central physical claim; they make me doubt the current text. I would send it to peer review with a request for a thorough revision, and I would want the companion theorem either proved in an appendix or clearly attributed with a version the referee can check.

Referee Report

3 major / 4 minor

Summary. The paper studies the task of implementing a non-energy-preserving unitary V_S on a finite-dimensional system S via an energy-preserving joint unitary with a battery B, with worst-case infidelity ε. It defines the entropic coherence C(β,H)=S(G_H[β])−S(β) and proves (Main Result, Eq. (7)) that every suitable battery family must satisfy C(β_B^(ε),H_B^(ε)) ≥ (r_2/8) log(σ/ε) − o(1), where r_2 and σ depend only on V_S and H_S; for “proportionate” batteries the coefficient improves to r_2/4 with a log-squared correction (Eq. (8)). The proof combines a resource inequality (Lemma 7) with lower bounds on the coherence produced by m applications of V_S (Result 1), using an entropy-of-sums theorem from a companion paper, and with continuity estimates (Result 3). Corollaries claim unbounded battery dimension for vanishing error, and lower bounds on energy and quantum Fisher information.

Significance. The results, if fully established, would add entropic coherence to the known necessary resources (energy, QFI) for gate implementation, with the distinctive feature that entropic coherence is invariant under rescaling of the battery Hamiltonian. The proof template is well structured: the separation into resource production (Result 1) and resource regularity (Result 3), the qubit analysis via a discretized normal approximation, and the continuity refinements using the refined Fannes–Audenaert inequality are commendable. The paper also gives explicit falsifiable predictions (Eqs. (7), (8), (15), (16)) and introduces the useful notion of a proportionate battery family. However, the central lower bound currently rests on an unproved companion theorem and on a defective lemma; these gaps need to be repaired before the claims are fully supported.

major comments (3)

[Appendix F, Eq. (D25)] The resource-production lower bound (Result 1, Eq. (D39)) depends on Corollary 23 of the companion paper arXiv:2508.05348. Appendix F states Theorem 22 and Corollary 23 but explicitly says it “relies entirely on the results published in [27]” and gives no proof. This is a nontrivial result: the prefactor r(χ)/2 and the constant λ_1 control the logarithmic growth in m, and they feed directly into r_2, λ_2, σ, and σ′. If Corollary 23 is incorrect or inapplicable, Eq. (D25) collapses and the main result has no foundation. A self-contained proof, or a published peer-reviewed version of [27], is required.
[Lemma 11, Appendix D.2.a] The proof asserts that [V_1⊗V_2^†, H_1+H_2]=0 iff [V,H]=V. From Eq. (D28), the vanishing condition is [V,H]⊗V^† = V⊗[V,H]^†, which implies [V,H]=cV with c real; it does not imply c=1. The subsequent trace argument only rules out c=1. Thus the proof does not establish the existence of the state |φ⟩ with |χ_ψ|≥2. Since Lemma 11 is used to guarantee r_2≥1 and λ_2>0, this is a load-bearing gap. The lemma may be true by a different argument, but the proof as written must be fixed.
[Corollary 3 (main text)] The proof says that combining Eq. (7) with C(β_B^(ε),H_B^(ε)) ≤ 2dim[H_B^(ε)] yields dim[H_B^(ε)] ≥ [(1−o(1))/log(ε^{-1})] (r_2/2)(σ′/ε)^{r_2/4}. This does not follow: Eq. (7) is logarithmic in ε^{-1}, so with C≤2d one obtains at most d ≥ (r/16) log(σ/ε), and with the sharper C≤log d one obtains d ≥ (σ/ε)^{r/8}. Furthermore, “not proportionate” implies superpolynomial growth of N, not exponential growth of dim. The qualitative conclusion (dim→∞) follows from Eq. (7) via C≤log d, but the quantitative bound stated in Corollary 3 is unsupported and should be corrected or removed.

minor comments (4)

[Eq. (D100), Step 3(a)] With m(ε) ∝ ε^{-1/4}, the term −16 log(d_S)√ε m² is a non-vanishing constant (−2r_2), not o(1). The displayed “−r_2/4 − o(1)” is therefore inaccurate. The leading logarithmic scaling is unaffected, but the constants in σ should be recomputed consistently and the o(1) usage fixed.
[Main Result (iii), footnote [26]] The statement “r_2(V_S)=2” for a qubit appears to conflict with footnote [26], which says the quantity defined in Def. 12 cannot equal 2 for a two-dimensional system. Please clarify that in (iii) r_2 denotes the coefficient achieved by the explicit qubit construction, not the value of the Definition 12 quantity.
[Appendix B, proof of (C2)] The notation “∆_A = tr_B[∆_AB]” is easy to misread; it would be clearer to write Tr_B[∆_AB] = ∆_A (by the same argument as Lemma 6). This is a presentation issue only.
[Throughout] Minor typographical issues: “referred to as a battery” (abstract), “istances”, “Fischer Information”, “incommeasurability”, “caracterization”, “worse case”. The reference list also contains duplicates ([19]=[35], [28]=[38]).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation is independent, with the only external input being a parameter-free companion theorem that does not presuppose the target result.

full rationale

The paper's derivation chain is not circular. The entropic coherence C(β_B,H_B) is defined independently (Eq. 3) and is lower-bounded through Lemma 7 and the resource axioms (C1)-(C4); the target bound (7) is not used as an input. The quantities r2(V_S,H_S), λ2(V_S,H_S), σ(V_S,H_S), and σ′(V_S,H_S) are defined as mathematical optimizations over initial states (Definition 12, Eqs. D102, D110), not as constants fitted to the bound. The only load-bearing external input is Theorem 22/Corollary 23 from the authors' companion paper [27], which is a parameter-free statement about sums of i.i.d. discrete random variables; its assumptions concern probability distributions and do not include the gate-implementation result or the entropic-coherence lower bound. Thus, even though the theorem is unproved in the present manuscript and its failure would undermine the main result, this is a reproducibility/correctness risk rather than a circularity. The paper also explicitly distinguishes its entropic coherence from the relative entropy of coherence used in earlier work (Appendix C), so it is not simply renaming a known result. No step reduces by construction to its own input, and no fitted quantity is relabeled as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central result rests on standard quantum-information axioms, the resource axioms C1-C4 (proved in the paper), a refined entropy-continuity inequality from the literature, and an unproved companion theorem on entropy of iid sums. No data-fitting parameters or invented physical entities appear. The two domain assumptions (proportionate families, linear level density) are clearly labeled and restrict the corollaries, not the main result (i).

assumptions (6)

domain assumption Theorem 22 of companion preprint [27]: entropy of sums of iid discrete random variables is lower-bounded via the incommensurability rank, used in Result 1 and Eq. (D25).
The theorem is stated but not proved in this paper; it is the engine that yields the (r/2) log m resource-production bound. Entering at Eq. (D25) and App. F.
standard math Refined Fannes-Audenaert inequality (Lemma 14, ref [29]) for entropy continuity.
Used in Step 2 to bound the resource regularity term; cited from literature.
standard math Resource axioms C1-C4 for entropic coherence (invariance under energy-preserving unitaries, monotonicity under partial trace, subadditivity, regularity), proved in App. B.
These are derived from properties of relative entropy and the twirling map; they underpin Lemma 7 and the whole resource-inequality method.
domain assumption Definition 2: a battery family is 'proportionate' if N(2Emax(β), H_B) ≤ poly(ε^{-1}).
This assumption is needed for the stronger bound in Main Result (ii) and for the refined rank bound in Eq. (D94).
domain assumption Corollary 4 assumes the battery spectral volume satisfies N(H_B,E) ≤ 1 + ηE (linear level density).
Used to convert the entropic coherence bound into an average-energy bound via the harmonic-oscillator extremal spectral density; stated at Corollary 4.
domain assumption Corollary 5 assumes the battery is a harmonic oscillator of frequency ω.
Used to compute the variance bound; stated at Corollary 5.

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Pith. "Pith review of The entropic coherence is a necessary resource for non-energy preserving gates." pith.science (2026). https://pith.science/paper/G477D5WQ

@misc{pith2026250901515,
  author       = {Pith},
  title        = {Pith review of: The entropic coherence is a necessary resource for non-energy preserving gates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G477D5WQ}},
  note         = {Machine review of arXiv:2509.01515}
}

read the original abstract

We consider the task of implementing non-energy preserving gates (NEPG) on a finite-dimensional system S via an energy-preserving interaction with an external battery B. We prove that the entropic coherence of the battery (an instance of the relative entropy of resource) is a necessary resource for this task, and find a lower bound on its minimum amount that has to be present in the battery to be able to implement NEPGs with a fixed desired precision. An immediate corollary is that any finite-dimensional battery is doomed to a certain minimal error in the gate implementation task. Moreover, under assumptions on the density of energy levels in the battery Hamiltonian, our main results imply additional lower bounds on the minimal amount of energy and quantum Fisher information required to implement any gate. We show that these bounds can be stronger than the universal bounds previously established in the literature.

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Works this paper leans on

62 extracted references · 50 canonical work pages

[27]

However, it is not true that for a 2-dimensional system the quantity defined in 12 can be equal to 2, it just hap- pens that the bound can be proven independently with that coefficient

work page
[1]

discrete random variables

Then, we identify the quantity C νSA, HSA with the entropy of a sum of 2 m i.i.d. discrete random variables. A lower bound on this entropy has recently been derived in [27], and implies the following bound C νSA,HSA − C ρSA, HSA ≥ r2(VS) 2 log 2πe m g(VS) − o(1), (11) where g(VS) quantifies the asymmetry of the gate and enters in both σ(VS) and σ′(VS) lat...
[2]

Miyadera and H

T. Miyadera and H. Imai, Wigner-araki-yanase theorem on distinguishability, Phys. Rev. A 74, 024101 (2006)

work page 2006
[3]

Navascu´ es and S

M. Navascu´ es and S. Popescu, How energy conservation limits our measurements, Phys. Rev. Lett. 112, 140502 (2014)

work page 2014
[4]

Kimura, B

G. Kimura, B. K. Meister, and M. Ozawa, Quantum lim- its of measurements induced by multiplicative conserva- tion laws: Extension of the wigner-araki-yanase theorem, Phys. Rev. A 78, 032106 (2008)

work page 2008
[5]

Kuramochi and H

Y. Kuramochi and H. Tajima, Wigner-araki-yanase the- orem for continuous and unbounded conserved observ- ables, Phys. Rev. Lett. 131, 210201 (2023)

work page 2023
[6]

Busch and L

P. Busch and L. Loveridge, Position measurements obey- ing momentum conservation, Phys. Rev. Lett. 106, 110406 (2011)

work page 2011
[7]

Tajima, R

H. Tajima, R. Takagi, and Y. Kuramochi, Universal trade-off structure between symmetry, irreversibility, and quantum coherence in quantum processes, arXiv (2025), arXiv:2206.11086 [quant-ph]

arXiv 2025

Show all 62 references

[8]

Marvian and R

I. Marvian and R. W. Spekkens, Asymmetry properties of pure quantum states, Phys. Rev. A 90, 014102 (2014)

2014
[9]

Shitara and H

T. Shitara and H. Tajima, The i.i.d. state convertibility in the resource theory of asymmetry for finite groups and lie groups, arXiv (2023), arXiv:2312.15758 [quant-ph]

2023
[10]

Marvian, Operational interpretation of quantum fisher information in quantum thermodynamics, Phys

I. Marvian, Operational interpretation of quantum fisher information in quantum thermodynamics, Phys. Rev. Lett. 129, 190502 (2022)

2022
[11]

Chitambar and G

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

2019
[12]

G. Gour, I. Marvian, and R. W. Spekkens, Measuring the quality of a quantum reference frame: The relative entropy of frameness, Phys. Rev. A 80, 012307 (2009)

2009
[13]

R. W. Marvian I, Extending noether’s theorem by quan- tifying the asymmetry of quantum states, Nature Com- munications 10.1038/ncomms4821 (2014)

2014 doi
[14]

Chiribella, Y

G. Chiribella, Y. Yang, and R. Renner, Fundamental en- ergy requirement of reversible quantum operations, Phys. Rev. X 11, 021014 (2021)

2021
[15]

Y. Yang, R. Renner, and G. Chiribella, Energy re- quirement for implementing unitary gates on energy- unbounded systems, J. Phys. A 55, 494003 (2022)

2022
[16]

Tajima, N

H. Tajima, N. Shiraishi, and K. Saito, Uncertainty rela- tions in implementation of unitary operations, Phys. Rev. Lett. 121, 110403 (2018)

2018
[17]

Tajima, N

H. Tajima, N. Shiraishi, and K. Saito, Coherence cost for violating conservation laws, Phys. Rev. Res. 2, 043374 (2020)

2020
[18]

Piccione, M

N. Piccione, M. Maffei, A. N. Jordan, K. W. Murch, and A. Auff` eves, Saturating a fundamental bound on quantum measurements’ accuracy, arXiv (2024), arXiv:2404.12910 [quant-ph]

2024 arXiv
[19]

Aberg, Quantifying superposition, arXiv preprint quant-ph/0612146 (2006)

J. Aberg, Quantifying superposition, arXiv preprint quant-ph/0612146 (2006)

2006 arXiv
[21]

Horodecki, J

M. Horodecki, J. Oppenheim, and R. Horodecki, Are the laws of entanglement theory thermodynamical?, Phys. Rev. Lett. 89, 240403 (2002)

2002
[22]

HORODECKI and J

M. HORODECKI and J. OPPENHEIM, (quantumness in the context of) resource theories, International Journal of Modern Physics B 27, 1345019 (2012)

2012
[23]

Breuer and F

H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford Uni- versity Press, 2007)

2007
[24]

This minimal quantity is sometimes called the “cost” of implementation of the gate, however, we avoid using this terminology, as only a small fraction of the resource is lost after one use of the battery, implying (as underlined for istance in [13, 33]) that the same battery s...
[25]

Kudo and H

D. Kudo and H. Tajima, Fisher information matrix as a resource measure in the resource theory of asymmetry with general connected-lie-group symmetry, Phys. Rev. A 107, 062418 (2023)

2023
[26]

Takagi and H

R. Takagi and H. Tajima, Universal limitations on im- plementing resourceful unitary evolutions, Phys. Rev. A 101, 022315 (2020)

2020
[28]

Castellano and P

R. Castellano and P. Sekatski, On the entropy growth of sums of iid discrete random variables, arXiv preprint arXiv:2508.05348 (2025)

2025 arXiv
[29]

K. M. R. Audenaert, A sharp continuity estimate for the von neumann entropy, J. Phys. A 40, 8127 (2007)

2007
[30]

Audenaert, B

K. Audenaert, B. Bergh, N. Datta, M. G. Jabbour, ´Angela Capel, and P. Gondolf, Continuity bounds for quantum entropies arising from a fundamental entropic inequality, arXiv (2024), arXiv:2408.15306 [quant-ph]

2024 arXiv
[31]

Castellano, V

R. Castellano, V. Cavina, M. Perarnau-Llobet, P. Sekatski, and V. Giovannetti, Exact require- ments for battery-assisted qubit gates, arxiv (2025), arXiv:2506.11855 [quant-ph]

2025
[32]

Since S e−γH tr e−γH is strictly decreasing in γ, the solution is always unique
[33]

Yu, Quantum fisher information as the convex roof of variance, arXiv (2013), arXiv:1302.5311 [quant-ph]

S. Yu, Quantum fisher information as the convex roof of variance, arXiv (2013), arXiv:1302.5311 [quant-ph]

2013 arXiv
[34]

˚Aberg, Catalytic coherence, Phys

J. ˚Aberg, Catalytic coherence, Phys. Rev. Lett. 113, 150402 (2014)

2014
[35]

Fuchs and J

C. Fuchs and J. van de Graaf, Cryptographic distin- guishability measures for quantum-mechanical states, IEEE Transactions on Information Theory 45, 1216 6 (1999)

1999
[36]

Baumgratz, M

T. Baumgratz, M. Cramer, and M. B. Plenio, Quantify- ing coherence, Phys. Rev. Lett. 113, 140401 (2014)

2014
[37]

Wehrl, General properties of entropy, Rev

A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50, 221 (1978)

1978
[38]

Lindblad, Completely positive maps and entropy inequalities, Communications in Mathematical Physics (1975)

G. Lindblad, Completely positive maps and entropy inequalities, Communications in Mathematical Physics (1975)

1975
[39]

K. M. R. Audenaert, A sharp continuity estimate for the von Neumann entropy, J. Phys. A 40, 8127 (2007), arXiv:quant-ph/0610146 [quant-ph]

2007 arXiv
[40]

A. A. Mele, Introduction to Haar Measure Tools in Quan- tum Information: A Beginner’s Tutorial, Quantum 8, 1340 (2024)

2024
[41]

M. L. MEHTA, 5 - gaussian unitary ensemble, in Random Matrices (Revised and Enlarged Second Edition), edited by M. L. MEHTA (Academic Press, San Diego,
[42]

Golowich, A new berry-esseen theorem for expander walks (2023), arXiv:2212.08134 [math.PR]

L. Golowich, A new berry-esseen theorem for expander walks (2023), arXiv:2212.08134 [math.PR]

2023 arXiv
[43]

L. H. Y. Chen, L. Goldstein, and Q.-M. Shao, Normal approximation by Stein’s method, Probability and its Applications (New York) (Springer, Heidelberg,
[44]

E. H. Lieb and M. B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, in Inequalities: Selecta of Elliott H. Lieb, edited by M. Loss and M. B. Ruskai (Springer Berlin Heidelberg, Berlin, Heidelberg, 2002) pp. 63–66

2002
[45]

L. H. Y. Chen, L. Goldstein, and Q.-M. Shao, Discretized normal approximation, in Normal Approximation by Stein’s Method (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011) pp. 221– 232

2011
[46]

Dowson and A

D. Dowson and A. Wragg, Maximum-entropy distribu- tions having prescribed first and second moments (cor- resp.), IEEE Transactions on Information Theory19, 689 (1973)

1973
[47]

T. M. Cover, Maximum entropy, in Elements of Information Theory (John Wiley & Sons, Ltd, 2005) Chap. 12, pp. 409–425, https://onlinelibrary.wiley.com/doi/pdf/10.1002/047174882X.ch12

2005 doi
[48]

A. V. Lazo and P. Rathie, On the entropy of continu- ous probability distributions (corresp.), IEEE Trans. Inf. Theor. 24, 120–122 (2006)

2006
[49]

Since S exp−ζ(H−µ)2 tr h exp−ζ(H−µ)2 i is decreasing in ζ, the function ζC (µ) is well difined
[50]

Gavalakis and I

L. Gavalakis and I. Kontoyiannis, Entropy and the dis- crete central limit theorem, Stochastic Processes and their Applications 170, 104294 (2024). 7 CONTENTS References 5 A. Channel distances and their relationships 7 B. The entropic coherence is a resource for the gate imple...

2024
[51]

Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications 10, 285 (1975)

M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications 10, 285 (1975)

1975
[53]

Resource inequalities and proof outline 10
[54]

Resource production for general gates 13 b

Step 1: Bounding the resource production 11 a. Resource production for general gates 13 b. Resource production for a qubit gate 15
[55]

Bounding the rank of GHSA [ν(m) SA ] 19 b

Step 2: Refining resource regularity 18 a. Bounding the rank of GHSA [ν(m) SA ] 19 b. Bounding the rank of eν(m) SA 19 c. The final bounds 20
[56]

Step 3: Choosing the best initial state. 21 a. Main result (i) i.e. any battery 21 b. Main result (ii), proportionate battery 22 E. Proving corollaries 4 and 5 22 F. Entropy of sums of discrete i.i.d. random variables 25 Appendix A: Channel distances and their relationships To...
[57]

For this sake, let us consider a battery B in contact with a system S and an auxiliary system A (this will be convenient later on)

Resource inequalities and proof outline In this section we will present a generalization of the method used in [13, 25] and show how to use a resource satisfying the properties (C1)-(C4) to create a bound on the precision in the implementation of NEPGs. For this sake, let us c...
[58]

(D12), which corresponds to the amount of resource produced by the gate V (m) S acting on the state

Step 1: Bounding the resource production Our goal here, is to choose specific initial states ρ(m) SA that ‘nicely’ lower bounds the expression in Eq. (D12), which corresponds to the amount of resource produced by the gate V (m) S acting on the state. Using the definition of th...
[59]

Then, with probability one r2(VS, HS) ≥ dS − 1

on the dS dimensional Hilbert space, (or from absolutely continuous mesures with respect to them). Then, with probability one r2(VS, HS) ≥ dS − 1. (D40) Proof. Following the Def. 12 consider the initial state |φ⟩T := |E1, E1⟩S1S2 ⊗ |E1, E1⟩A1A2 ∈ Eig[HT ]. (D41) By definition ...
[60]

Step 2: Refining resource regularity The next step is to upper bound the difference in the relative entropy produced by the ideal and approximate gates C(ν(m) SA , HSA) − C(eν(m) SA , HSA) = S(GHSA [ν(m) SA ]) − S(GHSA [eν(m) SA ]) + S(eν(m) SA ) − S(ν(m) SA ), (D74) as a func...
[61]

Step 3: Choosing the best initial state. As discussed in the proof outline, to obtain a lower bound on C(β, HB) it now remains to combine the bounds on resource production (Results 1 and 2) and resource regularity (Result 3) derived in the last two sections, and choose the opt...
[62]

χj is a non-empty lattice for j = 1, .., k(definition 16)
[63]

If |χj| = 1 for all j = 1,

{χj}j=1,..,k is incommensurable (definition 18). If |χj| = 1 for all j = 1, . . . , kwe say that the canonical prepartition is degenerate. One notes that if the sets χ1, ..χk all have a single element (i.e. the prepartition is degenerate), one can merge any two of them to obta...
[1991]

revised and enlarged second edition ed., pp. 79–122

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [27]

However, it is not true that for a 2-dimensional system the quantity defined in 12 can be equal to 2, it just hap- pens that the bound can be proven independently with that coefficient

work page

[2] [1]

discrete random variables

Then, we identify the quantity C νSA, HSA with the entropy of a sum of 2 m i.i.d. discrete random variables. A lower bound on this entropy has recently been derived in [27], and implies the following bound C νSA,HSA − C ρSA, HSA ≥ r2(VS) 2 log 2πe m g(VS) − o(1), (11) where g(VS) quantifies the asymmetry of the gate and enters in both σ(VS) and σ′(VS) lat...

[3] [2]

Miyadera and H

T. Miyadera and H. Imai, Wigner-araki-yanase theorem on distinguishability, Phys. Rev. A 74, 024101 (2006)

work page 2006

[4] [3]

Navascu´ es and S

M. Navascu´ es and S. Popescu, How energy conservation limits our measurements, Phys. Rev. Lett. 112, 140502 (2014)

work page 2014

[5] [4]

Kimura, B

G. Kimura, B. K. Meister, and M. Ozawa, Quantum lim- its of measurements induced by multiplicative conserva- tion laws: Extension of the wigner-araki-yanase theorem, Phys. Rev. A 78, 032106 (2008)

work page 2008

[6] [5]

Kuramochi and H

Y. Kuramochi and H. Tajima, Wigner-araki-yanase the- orem for continuous and unbounded conserved observ- ables, Phys. Rev. Lett. 131, 210201 (2023)

work page 2023

[7] [6]

Busch and L

P. Busch and L. Loveridge, Position measurements obey- ing momentum conservation, Phys. Rev. Lett. 106, 110406 (2011)

work page 2011

[8] [7]

Tajima, R

H. Tajima, R. Takagi, and Y. Kuramochi, Universal trade-off structure between symmetry, irreversibility, and quantum coherence in quantum processes, arXiv (2025), arXiv:2206.11086 [quant-ph]

arXiv 2025