REVIEW 3 major objections 6 minor 67 references
Superadditivity for Entanglement-Assisted Communication
T0 review · 3 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash
Pith's one-line read This paper proves that the Petz–Rényi channel information governing entanglement-assisted random-coding error exponents can be strictly superadditive: joint coding across channel uses improves reliability even though the entanglement-assist
desk verdict First analytic proof that Petz–Rényi channel information is strictly superadditive for entanglement-assisted communication; the math looks solid, but the advertised error-exponent payoff leans on an external achievability bound. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the Petz–Rényi channel information I_α(N), written through the trace functional Q_α(ρ)=Tr[(Tr_A[ρ_A^{1−α}(√ρ_A Γ_N √ρ_A)^α])^{1/α}], with I_α=(α/(α−1))log Q_α. The machinery consists of four steps: (1) Proposition 1, which shows ρ↦Q_α(ρ) is convex for 0<α<1, making I_α a convex optimization and allowing twirling or pinching to diagonal optimizers; (2) a one-parameter diagonal reduction of the one-copy problem; (3) a two-copy ansatz ρ(κ)=ρ_*⊗ρ_*+κΔ, where Δ is a diagonal, traceless, classically correlated perturbation that preserves the one-copy marginals; and (4) an exact linear-response identity showing the first derivative c_1(α)<0 at κ=0, so by Taylor expansion
What would settle it
Compute I_α(M^⊗2) and 2I_α(M) for the single-heavy Fourier measurement with d=3, λ=1/2, α=1/2 by numerically optimizing Q_α over diagonal two-copy states; the theorem predicts a strictly negative gap I_α(M^⊗2)−2I_α(M). A nonnegative gap would directly refute Theorem 3. Alternatively, exhibit a code for an amplitude-damping channel whose error probability decays faster than 2^{−E_r(nR;N^⊗n)} at a rate below capacity, which would disprove the tightness transfer used in the operational claim.
Extended reading notes
Core claim
The central claim is strict superadditivity of the Petz–Rényi channel information I_α(N): there exist channels for which I_α(N^⊗2)>2I_α(N) for every 0<α<1, and in particular in the data-processing-safe range [1/2,1). The paper proves this analytically for the single-heavy Fourier measurement M with d≥3 and 1/d<λ<1, and for the qubit amplitude-damping channel with 0<γ<1. Because the random-coding error exponent is E_r(R;N)=max_{1/2≤α<1} (1−α)/α [I_α(N)−R], strict superadditivity of I_α gives E_r(2R;N^⊗2)>2E_r(R;N) for every rate R below capacity—a genuine multi-copy enhancement of reliability. The proof reduces I_α to a convex optimization, shows the one-copy optimizer can be chosen diagonal,
Load-bearing premise
The operational conclusion that strict superadditivity of I_α translates into improved error probability rests on the cited achievability bound ε*(n,R)≤1.5·2^{−E_r(nR;N^⊗n)} and on the expectation that E_r is tight; the mathematical superadditivity itself does not depend on that assumption.
Editorial extensions
If this is right
- For the two example channels, I_α(N^⊗2)>2I_α(N) for every 0<α<1, so the random-coding error exponent for two channel uses strictly exceeds twice the single-use exponent at every rate below capacity.
- The strict superadditivity is witnessed by a separable, classically correlated two-copy input, so entanglement between the transmitted systems is not a necessary resource for the reliability enhancement.
- The phenomenon occurs already for entanglement-breaking measurement channels, whose unassisted Holevo information is additive; hence it is not tied to channels that already show capacity nonadditivity.
- The regularized Petz–Rényi information I_α^∞(N)=lim_{n→∞}(1/n)I_α(N^⊗n) admits computable single-letter upper bounds in terms of sandwiched Rényi information, bounding the ultimate multi-copy advantage.
- At α=1 the nonadditivity disappears, matching the additivity of entanglement-assisted capacity, and at α=2 the paper proves strong additivity of I_α.
Reading between the lines
- If the random-coding exponent E_r is indeed tight for general channels, then the true error exponent of entanglement-assisted communication is not single-letter, even though capacity is single-letter; quality of communication would require regularization while quantity does not.
- The same linear-response mechanism likely extends to any channel with a diagonal symmetry and a non-uniform one-copy optimizer; a testable conjecture is that superadditivity appears whenever the square (ξ−θ_α)^2 in the Fourier-measurement calculation is nonzero.
- Because the witness is classically correlated, the effect may carry over to channel discrimination and Rényi channel entropy settings, where product strategies would be suboptimal—an implication the paper only touches numerically.
- The gap vanishes continuously as α→1, matching the additive limit; quantifying the α-dependence of the gap could indicate how large the reliability gain is for practical finite block lengths.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the Petz–Rényi channel information I_α(N) for entanglement-assisted communication. It first proves a convex-optimization reduction for I_α, gives additivity results for special channel classes, and then establishes strict superadditivity I_α(N^{⊗2}) > 2 I_α(N) for every α∈(0,1) for two explicit families: a single-heavy Fourier measurement channel (Theorem 3) and amplitude-damping channels (Theorem 4). The proofs proceed by reducing the one-copy problem to a diagonal single-parameter convex optimization, then perturbing the product of one-copy optimizers along a classically correlated diagonal two-copy path and showing the linear-response coefficient is strictly negative. The paper also provides single-letter upper bounds on the regularized quantity I_α^∞(N). The advertised operational conclusion is that this strict superadditivity yields a genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent for rates below capacity, even though the capacity itself is additive.
Significance. If correct, the mathematical result is significant: it shows that the Petz–Rényi channel information, unlike the sandwiched Rényi information, is not additive in the data-processing range, and that communication reliability can improve under joint use of a channel even when the rate/capacity is strictly additive. A particularly clean feature is that the superadditivity witness is a separable, classically correlated two-copy input, so the effect is not attributed to input entanglement. The analytic proofs are detailed and largely self-contained; the identities (C.42) and (D.82) have the correct signs, and the diagonal reductions and convexity arguments appear sound. The explicit finite-dimensional examples and the numerical certificates strengthen the presentation. The main weakness is not in the superadditivity proof itself but in the operational bridge to the error-exponent claim, which rests on an external achievability bound.
major comments (3)
- [Error Exponent and Petz–Rényi Information, Eq. (12)] The abstract and Discussion claim a 'genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent.' This conclusion is obtained by combining the new strict superadditivity theorems with the bound ε*(n,R) ≤ 1.5·2^{−E_r(nR;N^{⊗n})} cited to Ref. [20]. The manuscript does not state the hypotheses or exact theorem of [20] that gives this bound. For the operational claim to be established as stated, [20] must apply to entanglement-assisted codes and must define the exponent via exactly the Petz–Rényi information I_α of Eq. (14). Please quote the relevant statement from [20], or at minimum state explicitly which theorem and assumptions are being imported. The mathematical superadditivity theorems stand independently, but the 'error exponent' interpretation is load-bearing for the title and abstract, so this dependency needs to be made verifiable.
- [Discussion, Table I, and Remark C.4] Table I and the Discussion state strict non-additivity of I_α for α∈(1,2) as well as for α∈(0,1), and Remark C.4 says the proof of Theorem C.3 'naturally extends' to α∈(1,2). The supplied Appendix C proves Theorem C.3 only for 0<α<1: Lemma C.1, the endpoint sign check (C.26), and the minimizer argument all use 2−α>1 and the perspective convexity valid for α<1; the sign of the linear-response coefficient (C.42) is negative precisely because 0<α<1. Since the claimed extension to α∈(1,2) is not proved and is not needed for the main [1/2,1) result, the authors should either supply the missing proof or restrict the claims in Table I and Discussion. As written, the scope statement overreaches the supplied evidence.
- [Proposition 1 and Proposition B.1] The title 'Convex optimization reduction' is inaccurate for α∈(1,2): Proposition B.1 proves convexity of the relevant map for α∈(0,1) and concavity for α∈(1,2]. Maximizing a concave objective is not a convex optimization in the standard sense. This does not affect the paper's main α<1 results, but the wording should be corrected to avoid a false general statement.
minor comments (6)
- [Throughout] There are several typographical issues: 'R WTH Aachen' in the affiliation, 'qusi-norms' in Ref. [19], and the heading 'QUALITY OF COMMUNICATION' in the Introduction.
- [Eq. (13)] The maximum is taken over the half-open interval [1/2,1). If the supremum is not attained, the notation should be 'sup' rather than 'max.'
- [Eq. (12)–(13)] The paper uses E_r(nR;N^{⊗n}) in the achievability bound and E_r(R;N) in the definition. This is correct, but a brief sentence explaining that the exponent is evaluated on the n-fold product channel would help readers.
- [Appendix E] Proposition E.2 relies on additivity of the sandwiched Rényi information eI_β for β∈[1/2,1) from Ref. [19]. Since [19] is a preprint, it would strengthen the paper to cite a published version if one becomes available, or at least to state that this particular additivity result is imported from an unpublished source.
- [Eq. (21)–(23)] The range of κ for which ρ_{A1A2}(κ) is a valid density operator is not stated. It is clear from the examples, but an explicit condition would improve readability.
- [Section C.7] The numerical table reports c_1(α) and κ_quad(α) for a four-dimensional instance. It would help to state explicitly that these numbers are illustrations and that the proof uses only the exact sign of c_1(α), not the numerical values.
Circularity Check
No significant circularity: strict superadditivity is proved by self-contained analytic calculations; the [20] bound is background for the operational framing, not an input to the superadditivity proof.
full rationale
The central claims (Theorems 3 and 4) are established by direct computation, not by assuming superadditivity. Appendix C reduces the one-copy optimization to a convex one-parameter problem (C.16-C.20), proves the minimizer is interior and not uniform (C.26-C.29), then computes the two-copy linear response along an explicit classically-correlated diagonal path. The sign of c1(alpha) is fixed by the closed-form identity (C.42) together with t_alpha != 1/d. Appendix D does the same for amplitude damping using the factorized variation identity (D.61) and stationarity (D.80), giving dQ/dkappa = (alpha-1)/(alpha(2-alpha)) x_alpha^2 (D.82), with x_alpha != 0 proved by (D.84)-(D.87). No fitted parameter is renamed as a prediction, and no theorem in the chain is equivalent to its own conclusion. External inputs (Lieb-Ando convexity, additivity of the sandwiched Renyi information from [17,19]) are standard results from independent papers. Equation (12), attributing the achievability bound to [20], is a self-citation by the first author and is used to translate superadditivity into an error-exponent statement; however, it is not used in the proof of strict superadditivity, and the exponent comparison E_r(2R;N^2)>2E_r(R;N) follows algebraically once I_alpha(N^2)>2I_alpha(N) is established. The conjectured tightness of E_r is explicitly described as an expectation in the text, not a needed input. Hence the only caveat is minor, non-load-bearing self-citation in background material.
Assumptions & free parameters
assumptions (6)
- standard math Lieb's Concavity Theorem and Ando's Convexity Theorem: (X,Y) ↦ X^p ⊗ Y^{1-p} is jointly convex for p∈(-1,0)∪(1,2) and jointly concave for p∈(0,1).
- standard math Data-processing/contractivity of Petz Rényi (α∈[0,2]) and sandwiched Rényi (α≥1/2) relative entropies, plus the ordering D_α ≥ ᄑD_α.
- standard math Sion's minimax theorem can be applied to the inf-sup exchange in (B20)-(B23) after a compactness regularization.
- standard math Fekete's lemma for superadditive sequences: lim_{n→∞} a_n/n = sup_n a_n/n.
- domain assumption Finite-dimensional Hilbert spaces and CPTP maps model quantum channels; measurement channels are described by POVMs.
- domain assumption The error-exponent achievability bound ε*(n,R)≤1.5·2^{-E_r(nR;N⊗n)} from Cheng–Liu [20] correctly describes the entanglement-assisted random-coding exponent.
Cite this review
Pith. "Pith review of Superadditivity for Entanglement-Assisted Communication." pith.science (2026). https://pith.science/paper/AOTGQO4M
@misc{pith2026260715151,
author = {Pith},
title = {Pith review of: Superadditivity for Entanglement-Assisted Communication},
year = {2026},
howpublished = {\url{https://pith.science/paper/AOTGQO4M}},
note = {Machine review of arXiv:2607.15151}
}
abstract
The entanglement-assisted capacity of a quantum channel admits an additive single-letter characterization, implying that joint encodings across channel uses cannot increase the ultimate communication rate. Here, we show that this additive picture does not extend to communication reliability. Specifically, we prove that the Petz-R\'enyi channel information can be strictly superadditive for every $\alpha\in[1/2,1)$, yielding a genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent, even though the entanglement-assisted capacity remains additive. We establish this phenomenon analytically already for measurement channels, which are entanglement-breaking and have additive unassisted capacity. Remarkably, this strict superadditivity is witnessed by a separable, classically correlated two-copy channel-input marginal, demonstrating that no entanglement between the transmitted systems is required. Our results show that, although correlations across channel uses cannot increase the ultimate rate of entanglement-assisted communication, they can enhance its reliability.
Figures
Figures from the paper (1 more)
Reference graph
Works this paper leans on
-
[20]
M. K. Gupta and M. M. Wilde, Multiplicativity of completely boundedp-norms implies a strong converse for entanglement-assisted capacity, Communications in Mathematical Physics334, 867 (2014)
2014
-
[1]
C. E. Shannon, A mathematical theory of communica- tion, The Bell System Technical Journal27, 379 (1948)
1948
-
[2]
projective measurement channels,
-
[3]
covariant quantum channels,
-
[4]
We defer the detailed proof to Appendix B
and classical-quantum channels. We defer the detailed proof to Appendix B. STRICT SUPERADDITIVITY In quantum information, a favorable convex optimiza- tion landscape typically ensures that fundamental quan- tities remain computable via single-letter formulas. Un- expectedly, the Petz–R´ enyi informationIα(N) possesses fundamentally distinct behaviors from...
-
[5]
Holevo, The capacity of the quantum channel with general signal states, IEEE Transactions on Information Theory44, 269 (1998)
A. Holevo, The capacity of the quantum channel with general signal states, IEEE Transactions on Information Theory44, 269 (1998)
1998
-
[6]
Schumacher and M
B. Schumacher and M. D. Westmoreland, Sending clas- sical information via noisy quantum channels, Physical Review A56, 131 (1997). 6 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 110−8 10−7 10−6 10−5 10−4 10−3 10−2 α gap (bits) eI1/(2−α)(M)−I α(M⊗3)/3 Iα(M⊗3)/3−I α(M) Iα(M⊗2)/2−I α(M) FIG. 4. The single-letter sandwiched upper gap decreases to zero asα↗1
1997
-
[7]
M. B. Hastings, Superadditivity of communication capac- ity using entangled inputs, Nature Physics5, 255 (2009)
2009
Show all 67 references
-
[8]
Aubrun, S
G. Aubrun, S. Szarek, and E. Werner, Hastings’s addi- tivity counterexample via Dvoretzky’s theorem, Commu- nications in Mathematical Physics305, 85 (2011)
2011
-
[9]
Lloyd, Capacity of the noisy quantum channel, Phys- ical Review A55, 1613 (1997)
S. Lloyd, Capacity of the noisy quantum channel, Phys- ical Review A55, 1613 (1997)
1997
-
[10]
P. W. Shor, The quantum channel capacity and coherent information, Lecture Notes, MSRI Workshop on Quan- tum Computation (2002)
2002
-
[11]
Devetak, The private classical capacity and quantum capacity of a quantum channel, IEEE Transactions on Information Theory51, 44 (2005)
I. Devetak, The private classical capacity and quantum capacity of a quantum channel, IEEE Transactions on Information Theory51, 44 (2005)
2005
-
[12]
Smith and J
G. Smith and J. Yard, Quantum communication with zero-capacity channels, Science321, 1812 (2008)
2008
-
[13]
T. S. Cubitt, D. Elkouss, W. Matthews, M. Ozols, D. P´ erez-Garc ´ ıa, and S. Strelchuk, Unbounded number of channel uses may be required to detect quantum ca- pacity, Nature Communications6, 6739 (2015)
2015
-
[14]
P. W. Shor, Additivity of the classical capacity of entanglement-breaking quantum channels, Journal of Mathematical Physics43, 4334 (2002)
2002
-
[15]
C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thap- liyal, Entanglement-assisted classical capacity of noisy quantum channels, Phys. Rev. Lett.83, 3081 (1999)
1999
-
[16]
Bennett, P
C. Bennett, P. Shor, J. Smolin, and A. Thapliyal, Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem, IEEE Transactions on Information Theory48, 2637 (2002)
2002
-
[17]
A. S. Holevo, On entanglement-assisted classical capac- ity, Journal of Mathematical Physics43, 4326 (2002)
2002
-
[18]
C. H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein–Podolsky– Rosen channels, Phys. Rev. Lett.70, 1895 (1993)
1993
-
[19]
C. H. Bennett and S. J. Wiesner, Communication via one- and two-particle operators on Einstein–Podolsky– Rosen states, Physical Review Letters69, 2881 (1992)
1992
-
[21]
Li and Y
K. Li and Y. Yao, Strong converse exponent for entanglement-assisted communication, IEEE Transac- tions on Information Theory70, 5017 (2024)
2024
-
[22]
Li and Q
K. Li and Q. Xu, Completely bounded qusi-norms, their mutiplicativity, and new additivity results of quantum channels (2026), arXiv:2603.16722 [quant-ph]
2026
-
[23]
Cheng and P.-C
H.-C. Cheng and P.-C. Liu, Error exponents for quan- tum packing problems via an operator layer cake theorem (2025), arXiv:2507.06232 [quant-ph]
2025 arXiv
-
[24]
K. G. H. Vollbrecht and R. F. Werner, Entanglement measures under symmetry, Physical Review A64, 062307 (2001)
2001
-
[25]
Elkouss and S
D. Elkouss and S. Strelchuk, Superadditivity of private information for any number of uses of the channel, Phys- ical Review Letters115, 040501 (2015)
2015
-
[26]
Note that nonadditivity via classical correlations has also been observed in wiretap classical-quantum channels
-
[27]
M¨ uller-Lennert, F
M. M¨ uller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, On quantum R´ enyi entropies: A new generalization and some properties, Journal of Mathe- matical Physics54, 122203 (2013)
2013
-
[28]
M. M. Wilde, A. Winter, and D. Yang, Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched R´ enyi relative en- tropy, Communications in Mathematical Physics331, 593 (2014)
2014
-
[29]
H. Qi, Q. Wang, and M. M. Wilde, Applications of position-based coding to classical communication over quantum channels, Journal of Physics A: Mathematical and Theoretical51, 444002 (2018)
2018
-
[30]
Cheng, Simple and tighter derivation of achiev- ability for classical communication over quantum chan- nels, PRX Quantum4, 040330 (2023), arXiv:2208.02132 [quant-ph]
H.-C. Cheng, Simple and tighter derivation of achiev- ability for classical communication over quantum chan- nels, PRX Quantum4, 040330 (2023), arXiv:2208.02132 [quant-ph]. 7
2023 arXiv
-
[31]
Shannon, R
C. Shannon, R. Gallager, and E. Berlekamp, Lower bounds to error probability for coding on discrete memo- ryless channels. I, Information and Control10, 65 (1967)
1967
-
[32]
Dalai, Lower bounds on the probability of error for classical and classical-quantum channels, IEEE Transac- tions on Information Theory59, 8027 (2013)
M. Dalai, Lower bounds on the probability of error for classical and classical-quantum channels, IEEE Transac- tions on Information Theory59, 8027 (2013)
2013
-
[33]
Cheng, M.-H
H.-C. Cheng, M.-H. Hsieh, and M. Tomamichel, Quan- tum sphere-packing bounds with polynomial prefactors, IEEE Transactions on Information Theory65, 2872 (2019)
2019
-
[34]
B.-A. Shi, B. Nakibo˘ glu, and H.-C. Cheng, On the con- verse bounds for entanglement-assisted communication (2026), (appear soon)
2026
-
[35]
Hayashi and M
M. Hayashi and M. Tomamichel, Correlation detection and an operational interpretation of the R´ enyi mutual information, Journal of Mathematical Physics57, 102201 (2016)
2016
-
[36]
Petz, Quasi-entropies for finite quantum systems, Re- ports on Mathematical Physics23, 57 (1986)
D. Petz, Quasi-entropies for finite quantum systems, Re- ports on Mathematical Physics23, 57 (1986)
1986
-
[37]
Cooney, M
T. Cooney, M. Mosonyi, and M. M. Wilde, Strong converse exponents for a quantum channel discrimina- tion problem and quantum-feedback-assisted communi- cation, Communications in Mathematical Physics344, 797 (2016)
2016
-
[38]
Oufkir, M
A. Oufkir, M. Tomamichel, and M. Berta, Error exponent of activated non-signaling-assisted classical-quantum channel coding, Letters in Mathematical Physics116, 10.1007/s11005-026-02061-z (2026)
2026 doi
-
[39]
Girardi, A
F. Girardi, A. Oufkir, B. Regula, M. Tomamichel, M. Berta, and L. Lami, Quantum umlaut information (2025), arXiv:2503.21479 [quant-ph]
2025
-
[40]
Gour and M
G. Gour and M. M. Wilde, Entropy of a quantum channel, Physical Review Research3, 10.1103/physrevre- search.3.023096 (2021)
2021 doi
-
[41]
Araki, On an inequality of Lieb and Thirring, Letters in Mathematical Physics19, 167 (1990)
H. Araki, On an inequality of Lieb and Thirring, Letters in Mathematical Physics19, 167 (1990)
1990
-
[42]
Mosonyi and T
M. Mosonyi and T. Ogawa, Quantum hypothesis test- ing and the operational interpretation of the quantum R´ enyi relative entropies, Communications in Mathemat- ical Physics334, 1617 (2014)
2014
-
[43]
E. H. Lieb, Convex trace functions and the Wigner- Yanase-Dyson conjecture, Advances in Mathematics11, 267 (1973)
1973
-
[44]
Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra and its Applications26, 203 (1979)
T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra and its Applications26, 203 (1979)
1979
-
[45]
We impose the standard condition on the group covariant channels whose representation acting on the input space is irreducible
-
[46]
Mosonyi and T
M. Mosonyi and T. Ogawa, Strong converse exponent for classical-quantum channel coding, Communications in Mathematical Physics355, 373 (2017)
2017
-
[47]
Mosonyi and F
M. Mosonyi and F. Hiai, On the quantum R´ enyi relative entropies and related capacity formulas, IEEE Transac- tions on Information Theory57, 2474 (2011)
2011
-
[48]
Cheng, L
H.-C. Cheng, L. Gao, and M.-H. Hsieh, Properties of non- commutative R´ enyi and Augustin information, Commu- nications in Mathematical Physics390, 501 (2022)
2022
-
[49]
Li and D
K. Li and D. Yang, Reliability function of classical- quantum channels, Physical Review Letters134, 10.1103/physrevlett.134.010802 (2025)
2025 doi
-
[50]
Liu and H.-C
P.-C. Liu and H.-C. Cheng, On Araki-type trace inequali- ties, Linear Algebra and its Applications728, 320 (2026)
2026
-
[51]
M. M. Wilde, Optimized quantumf-divergences and data processing, Journal of Physics A: Mathematical and Theoretical51, 374002 (2018)
2018
-
[52]
Jenˇ cov´ a, R´ enyi relative entropies and noncommuta- tivel p-spaces, Annales Henri Poincar´ e19, 2513 (2018)
A. Jenˇ cov´ a, R´ enyi relative entropies and noncommuta- tivel p-spaces, Annales Henri Poincar´ e19, 2513 (2018). 8 Appendix A: Definitions and Notation We consider finite-dimensional Hilbert space. We denote byρ A andσ B quantum states (i.e. density matrix) on quantum syste...
2018
-
[53]
Isometric channels: Iα(N) = sup ρA 2H 2−α α (A)ρ = ( 2 logd A α∈(0,2], +∞α >2, (B6) whereH α(A)ρ ≡ 1 1−α log Tr[ρα]is the R´ enyi entropy andd A denotes the dimension of the input Hilbert space
-
[54]
Projective measurement channels with|Y|nonzero projectors:I α(N) = log|Y|forα∈(0,2]. 10
-
[55]
Covariant quantum channels [42]:I α(N) = 1 1−α logd A + α α−1 log Tr h TrA (ΓN AB)α 1/αi forα∈(0,1)∪(1,2]
-
[56]
Another class of channels satisfying (B7) is classical-quantum channelsN X→B, whose input statesρ X are restricted to diagonal matrices
Channels with commuting inputs: Suppose every admissible density operatorρ A in the input algebra commutes withΓ N AB, i.e., ρA ⊗1 B,Γ N AB = 0AB,(B7) we have Iα(N) = sup ρA α α−1 log Tr h (TrA [ρA ·Γ α AB])1/α i , α∈(0,1).(B8) RemarkB.4.A trivial class of channels satisfying ...
-
[57]
Fix d≥3, 1 d < λ <1, b= 1−λ d−1 .(C1) Let a0 =λ, a 1 =· · ·=ad−1 =b,(C2) and letω= e 2πi/d
The single-heavy F ourier measurement Throughout this section, we writed=d A, and shorthandM y =M y A. Fix d≥3, 1 d < λ <1, b= 1−λ d−1 .(C1) Let a0 =λ, a 1 =· · ·=ad−1 =b,(C2) and letω= e 2πi/d. OnA ∼= Cd, define |vy⟩= d−1X j=0 √aj ωjy |j⟩, y= 0, . . . , d−1.(C3) The measureme...
-
[58]
LetZ= Pd−1 j=0 ωj|j⟩⟨j|be the generalized Pauli phase operator and Z m|j⟩=ω mj|j⟩, m= 0,
Diagonal reduction of the one-copy optimization The diagonal reduction follows from covariance and convexity. LetZ= Pd−1 j=0 ωj|j⟩⟨j|be the generalized Pauli phase operator and Z m|j⟩=ω mj|j⟩, m= 0, . . . , d−1.(C10) The phase operator permutes the transposed Fourier effectsM ...
-
[59]
,1−t α d−1 ,(C30) and define the zero-sum vector u= 1,− 1 d−1 ,
A correlated diagonal two-copy path Let p=p(t α) = tα, 1−t α d−1 , . . . ,1−t α d−1 ,(C30) and define the zero-sum vector u= 1,− 1 d−1 , . . . ,−1 d−1 .(C31) For realκ, set πκ(i, j) =pipj +κ·u iuj.(C32) For all sufficiently small|κ|,π κ is a bipartite probability distribution....
-
[60]
Atκ= 0, all four contributions factor into the one-copy termsr α and zα
Linear response For 0< t <1, define ξ(t) = λ−b η(t) (C37) and θα(t) = 1 ζα(t) " λt1−α −b 1−t d−1 1−α# .(C38) Then r′ α(t) rα(t) =φ α(t), φ α(t) = α−1 α ξ(t) + 2−α α θα(t),(C39) and z′ α(t) =− zα(t) α(1−t) .(C40) The stationarity condition (C27) is therefore rα(tα)φα(tα) = zα(t...
-
[61]
RemarkC.4.In this paper, we only focus on the rangeα∈(0,1) forI α(N)
Strict superadditivity Theorem C.3(Strict superadditivity of Fourier measurements).For everyd≥3, everyλ∈(1/d,1), and every 0< α <1, the single-heavy Fourier measurement defined in(C4)satisfies Iα(M⊗2)>2I α(M).(C50) The strict improvement is witnessed by the infinitesimal diago...
-
[62]
In computations one may chooseκby the one-dimensional minimization ofκ7→Q α(ρκ;M ⊗2)
Quadratic choice of the correlation strength For strictness, the signc 1(α)<0 is enough. In computations one may chooseκby the one-dimensional minimization ofκ7→Q α(ρκ;M ⊗2). Equivalently, if Qα(ρα(κ);M ⊗2) =K (1)(α)2 +c 1(α)κ+c 2(α)κ2 +O(κ 3) (C54) andc 2(α)>0, the quadratic ...
-
[63]
However, note that (C57) is not a universal witness; one has to consider at least the second-order derivative bound
The four-dimensional instance The following numerical examples illustrate the scale of the linear response with parameters d= 4, λ= 9 25 , b= 16 75 .(C56) α c 1(α)κ quad(α) leading-order gap 0.50−9.818×10 −3 2.563×10 −3 9.13×10 −5 0.60−6.651×10 −3 1.510×10 −3 2.91×10 −5 0.70−3...
-
[64]
By Proposition B.1, the mapρ7→Q α(ρ;N) is convex for 0< α <1
Diagonal one-copy reduction Let Z:=|0⟩ ⟨0| − |1⟩ ⟨1|.(D8) The amplitude-damping Choi matrix satisfies (ZA ⊗Z B)ΓN AB(ZA ⊗Z B) = ΓN AB.(D9) For convenience, define the positive operator T N α (ρ) := TrA h ρ1−α A √ρA ΓN AB √ρA αi ,(D10) so that Qα(ρ;N) = Tr h T N α (ρ)1/α i .(D1...
-
[65]
A correlated diagonal two-copy path Consider the diagonal two-copy state ρα(κ) =ρ t ⊗ρ t +κdiag(1,−1,−1,1) = diag (p00(κ), p01(κ), p10(κ), p11(κ)),(D35) where p00(κ) =t 2 +κ, p 01(κ) =tu−κ, p10(κ) =tu−κ, p 11(κ) =u 2 +κ.(D36) This is a full-rank density operator whenever −min{...
-
[66]
Hence p ρα(κ) (ΓN)⊗2 p ρα(κ) = 1X r,s=0 |ψrs(κ)⟩ ⟨ψrs(κ)|(D43) is an orthogonal sum of rank-one operators
The four two-copy output weights The decomposition (D3) gives four two-copy branches |ϕr⟩A1B1 ⊗ |ϕs⟩A2B2 , r, s∈ {0,1}.(D41) For the diagonal stateρ α(κ), define |ψrs(κ)⟩:= p ρα(κ)A1A2 ⊗1 B1B2 |ϕr⟩A1B1 |ϕs⟩A2B2 .(D42) The four vectors|ψ rs(κ)⟩have mutually orthogonal supports....
-
[67]
Let r0 :=t, r 1 :=u, d 0 := 1, d 1 :=−1.(D63) Thus d dt rx =d x, p xy(κ) =r xry +κd xdy.(D64) For a single channel use, lets= 0 denote the no-jump branch ands= 1 the jump branch
F actorized linear response Define a0 :=γt 2−αℓα−2 −γ α, a 1 :=γηu 2−αℓα−2 = γw1 ℓ ,(D58) b0 :=t 1−αℓα−1 −γ α, b 1 :=−ηu 1−αℓα−1 =− w1 u .(D59) Lemma D.3(Factorized first variation).The one-copy derivatives satisfy w′ i(t) = (α−1)a i + (2−α)b i, i∈ {0,1}.(D60) Moreover, along ...
Reviewed August 1, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.