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arxiv: 1906.08326 · v1 · pith:U7JQFDHPnew · submitted 2019-06-19 · 🪐 quant-ph

Coherence Fraction

Sumana Karmakar , Ajoy Sen , Indrani Chattopadhyay , Amit Bhar , Debasis Sarkar This is my paper

Pith reviewed 2026-05-25 20:09 UTC · model grok-4.3

classification 🪐 quant-ph
keywords coherence fractionquantum coherencel1-norm coherencecoherence distillabilitydecohering powerquantum channelsbipartite statesX states
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The pith

Coherence fraction quantifies the proximity of a quantum state to a maximally coherent state and serves as a coherence measure.

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

The paper generalizes the concept of entanglement fraction to define coherence fraction for quantum states. This quantity measures how close a state is to being maximally coherent and functions as a coherence measure. It connects directly to the l1-norm coherence and supplies a criterion for whether coherence can be distilled from the state. For quantum channels, the optimal coherence fraction obeys a complementary relation with the channel's decohering power. The work also extends the idea to bipartite states, showing bounds between local and global coherence fractions, and examines how the quantity evolves under channel applications.

Core claim

Coherence fraction is defined by generalizing entanglement fraction to quantify the proximity of a quantum state to a maximally coherent state. This new quantity acts as a measure of coherence, exhibits a connection with l1-norm coherence, and provides criteria for coherence distillability. The optimal coherence fraction for a channel complements its decohering power, and for bipartite pure states and X states the connection to l1-norm holds, with local coherence fractions bounded by a linear function of the global one.

What carries the argument

Coherence fraction, the generalization of entanglement fraction that quantifies proximity to maximally coherent states.

If this is right

  • Coherence fraction connects with l1-norm coherence for single and bipartite states.
  • Coherence fraction provides criteria for coherence distillability.
  • Optimal coherence fraction of a channel obeys a complementary relation with its decohering power.
  • Local coherence fractions of a bipartite state are bounded by a linear function of its global coherence fraction.
  • Dynamics of optimal coherence fraction can be tracked under single-sided and both-sided channel applications.

Where Pith is reading between the lines

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

  • If the measure holds up, it may simplify calculations of coherence in tasks like quantum communication compared to other known quantifiers.
  • Verification on X states could test whether the bipartite bounds on local and global fractions hold in practice.
  • The complementary relation with decohering power might guide design of channels that balance coherence generation and loss.
  • Extension to more complex states could reveal whether the linear bound generalizes beyond pure and X states.

Load-bearing premise

The assumption that the newly defined coherence fraction provides a meaningful and independent measure of coherence whose connections to l1-norm coherence and distillability criteria are valid and useful beyond the definitions themselves.

What would settle it

A quantum state where coherence fraction fails to align with l1-norm coherence values or incorrectly predicts distillability outcomes would falsify the claimed connections.

Figures

Figures reproduced from arXiv: 1906.08326 by Ajoy Sen, Amit Bhar, Debasis Sarkar, Indrani Chattopadhyay, Sumana Karmakar.

Figure 1
Figure 1. Figure 1: FIG. 1: (Color online) Fig.( [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
read the original abstract

The concept of entanglement fraction is generalized to define coherence fraction of a quantum state. Precisely, it quantifies the proximity of a quantum state to maximally coherent state and it can be used as a measure of coherence. Coherence fraction has a connection with $l_1$-norm coherence and provides the criteria of coherence distillability. Optimal coherence fraction corresponding to a channel, defined from this new idea of coherence fraction, obeys a complementary relation with its decohering power. The connection between coherence fraction and $l_1$-norm coherence turns to hold for bipartite pure states and $X$ states too. The bipartite generalization shows that the local coherence fractions of a quantum state are not free and they are bounded by linear function of its global coherence fraction. Dynamics of optimal coherence fraction is also studied for single sided and both sided application of channels. Numerical results are provided in exploring properties of optimal coherence fraction.

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

2 major / 2 minor

Summary. The manuscript generalizes the entanglement fraction to define the coherence fraction of a quantum state as the maximum overlap with a maximally coherent state. It claims this quantity functions as a coherence measure, connects to l1-norm coherence, supplies criteria for coherence distillability, and that the optimal coherence fraction of a channel obeys a complementary relation with its decohering power. Extensions to bipartite pure states and X-states are presented, along with bounds on local coherence fractions in terms of the global one, dynamics under channels, and supporting numerical results.

Significance. If the derivations and relations hold, the work supplies a geometrically motivated coherence quantifier with potential utility for distillability criteria and channel analysis. The complementary relation and bipartite bounds, if parameter-free and rigorously derived, would add to the resource theory of coherence; the numerical exploration of dynamics provides concrete illustrations.

major comments (2)
  1. The central claim that coherence fraction provides an independent measure with a connection to l1-norm coherence (and distillability criteria) requires explicit verification that the relation is not tautological by construction; the abstract presents these as following directly from the definition without indicating an independent test or counter-example check.
  2. For the channel result, the complementary relation between optimal coherence fraction and decohering power is stated to hold; the manuscript should identify the precise equation or theorem establishing this (e.g., whether it follows from the definition alone or requires additional assumptions on the channel class).
minor comments (2)
  1. Notation for the maximally coherent state and the precise optimization in the definition of coherence fraction should be standardized across sections to avoid ambiguity when extending to bipartite cases.
  2. The numerical results section would benefit from explicit statements of the Hilbert-space dimensions and channel parameters used, to allow reproduction of the reported dynamics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and indicate the revisions planned.

read point-by-point responses

  1. Referee: The central claim that coherence fraction provides an independent measure with a connection to l1-norm coherence (and distillability criteria) requires explicit verification that the relation is not tautological by construction; the abstract presents these as following directly from the definition without indicating an independent test or counter-example check.

    Authors: The coherence fraction is defined geometrically as the maximum overlap with a maximally coherent state. The connection to l1-norm coherence is obtained by deriving an explicit inequality relating the two quantities from the properties of maximally coherent states and the definition of the l1-norm; this step is not immediate from the definition alone. The distillability criterion is likewise obtained by showing that states with coherence fraction above a threshold admit a protocol that increases the fraction. The manuscript body contains these derivations. To address the concern about clarity, we will revise the abstract to note that the relations are derived rather than definitional and add a short remark with a simple two-qubit example illustrating that the connection requires the intermediate steps. revision: yes

  2. Referee: For the channel result, the complementary relation between optimal coherence fraction and decohering power is stated to hold; the manuscript should identify the precise equation or theorem establishing this (e.g., whether it follows from the definition alone or requires additional assumptions on the channel class).

    Authors: The complementary relation follows directly from the definitions of optimal coherence fraction (the maximum, over input states, of the coherence fraction of the channel output) and decohering power, with no further assumptions on the channel beyond the standard completely positive trace-preserving map. The proof proceeds by relating the two quantities through the action on the maximally coherent basis states. We will add an explicit theorem statement (with equation reference) and a short proof sketch in the revised manuscript to make the origin of the relation transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definition and relations are independent

full rationale

The paper introduces coherence fraction as a new generalization of entanglement fraction, quantifying overlap with a maximally coherent state. Connections to l1-norm coherence, distillability criteria, and the complementary relation with decohering power are stated as derived consequences rather than tautologies or self-referential fits. No load-bearing self-citations, uniqueness theorems from prior author work, or renamings of known results appear in the provided text. The derivation chain remains self-contained against external benchmarks with no steps reducing by construction to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities detailed beyond the new definition itself. The coherence fraction is treated as a postulated measure without independent evidence provided.

invented entities (1)
  • coherence fraction no independent evidence
    purpose: Measure of proximity to maximally coherent state and coherence distillability
    Newly introduced quantity whose validity rests on the paper's definitions and claimed connections.

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Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages · 8 internal anchors

  1. [1]

    Considering the task of quantifying coherence fraction of a state we study its properties and explain its resource theoretic connections

    In this paper, we have developed the idea of coherence fraction of a quantum state in the theory of quantum coherence that quantifies the maximum overlap of a quantum state with a maximally coherent state [24, 37]. Considering the task of quantifying coherence fraction of a state we study its properties and explain its resource theoretic connections. Whene...

    work page

  2. [2]

    bound coherence

    Thus, we can say that every pure coherent state is distillable in qubit system. This result establishes coherence fraction as a faithful quantifier of the resource theory of coherence. This result can also be verified from Theorem 1 in [44] and the result of Winter et al.[15] for qubit system which states that there is no “bound coherence” from which no coh...

    work page

  3. [3]

    ThusFc(I±iσ1,2) = 1,Fc[±i(σ1,2+σ3)] = 1

    or (0,± 1√ 2 ,± 1√ 2), φ =±π . ThusFc(I±iσ1,2) = 1,Fc[±i(σ1,2+σ3)] = 1. Depolarizing channel. Evolution of a qubit ρ under depolarizing channel (Λ dep) is given by Λdep(ρ) = (1− p)ρ + p I 2 , 0≤ p≤ 1. (23) 7 Simple calculations give optimal coherence fraction as Fc(Λdep) = 1− p 2 . (24) Therefore, 1 2≤F c(Λdep)≤ 1. Corresponding to the completely depolari...

    work page

  4. [4]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81, 865-942 (2009)

    work page 2009

  5. [5]

    M. A. Nielsen, Isaac L. Chuang, Quantum Computation and Quantum Information , Cambridge University Press (2011)

    work page 2011

  6. [6]

    Lloyd, J

    S. Lloyd, J. Phys. : Conf. Series 302, 012037 (2011)

    work page 2011

  7. [7]

    C. M. Li, N. Lambert, Y. -N. Chen, G. -Y. Chen and F. Nori, Sci. Rep. 2, 885 (2012)

    work page 2012

  8. [8]

    S. F. Huelga and M. B. Plenio, Contemp. Phys. 54, 181 (2013)

    work page 2013

  9. [9]

    Skrzypczyk, A

    P. Skrzypczyk, A. J. Short and S. Popescu, Nature Communications 5, 4185 (2014)

    work page 2014

  10. [10]

    ˚Aberg, Phys

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

    work page 2014

  11. [11]

    Lostaglio, D

    M. Lostaglio, D. Jennings and T. Rudolph, Nat. Commun. 6, 6383 (2015)

    work page 2015

  12. [12]

    Lostaglio, K

    M. Lostaglio, K. Korzekwa, D. Jennings and T. Rudolph, Phys. Rev. X 5, 021001 (2015)

    work page 2015

  13. [13]

    The extraction of work from quantum coherence

    K. Korzekwa, M. Lostaglio, J. Oppenheim and D. Jennings, arXiv:1506.07875v1 (2015)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  14. [14]

    Streltsov, E

    A. Streltsov, E. Chitambar, S. Rana, M. N. Bera, A. Winter, M. Lewenstein, Phys. Rev. Lett. 116, 240405 (2016)

    work page 2016

  15. [15]

    Girolami, Phys

    D. Girolami, Phys. Rev. Lett. 113, 170401 (2014)

    work page 2014

  16. [16]

    Baumgratz, M

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

    work page 2014

  17. [17]

    L. H. Shao, Z. Xi, H. Fan and Y. Li, Phys. Rev. A 91, 042120 (2015)

    work page 2015

  18. [18]

    Winter1 and D

    A. Winter1 and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)

    work page 2016

  19. [19]

    Chitambar and M

    E. Chitambar and M. -H. Hsieh, Phys. Rev. Lett. 117, 020402 (2016)

    work page 2016

  20. [20]

    X. Yuan, H. Zhou, Z. Cao and X. Ma, Phys. Rev. A 92, 022124 (2015)

    work page 2015

  21. [21]

    Chitambar, A

    E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso and M. Lewenstein, Phys. Rev. Lett. 116, 070402 (2016)

    work page 2016

  22. [22]

    C Napoli, T. R. Bromley, M Cianciaruso, M. Piani, N. Johnston and G. Adesso, Phys. Rev. Lett. 116, 150502 (2016)

    work page 2016

  23. [23]

    Streltsov, U

    A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera and G. Adesso, Phys. Rev. Lett. 115, 020403 (2015)

    work page 2015

  24. [24]

    Singh, M

    U. Singh, M. N. Bera, H. S. Dhar, and A. K. Pati, Phys. Rev. A 91, 052115 (2015)

    work page 2015

  25. [25]

    Z. Xi, Y. Li, and H. Fan, Scientific Reports 5, 10922 (2015). 11

    work page 2015

  26. [26]

    T. R. Bromley, M. Cianciaruso and G. Adesso, Phys. Rev. Lett. 114, 210401 (2015)

    work page 2015

  27. [27]

    Y. Peng, Y. Jiang and H. Fan, Phys. Rev. A 93, 032326 (2016)

    work page 2016

  28. [28]

    Hu, Phys

    H. Hu, Phys. Rev. A 94, 012326 (2016)

    work page 2016

  29. [29]

    Mani and V

    A. Mani and V. Karimipour, Phys. Rev. A 92, 032331(2015)

    work page 2015

  30. [31]

    Z. Xi, M. Hu, Y. Li and H. Fan, arXiv:1510.06473 (2015)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  31. [32]

    K. Bu, L. Zhang, and J. Wu, arXiv: 1509.09109v4

    work page internal anchor Pith review Pith/arXiv arXiv

  32. [33]

    C. H. Bennett, D. P. Divincenzo, J. A. Smolin and W. K. Wootters, Phys. Rev. A 54, 3824-3851 (1996)

    work page 1996

  33. [34]

    Deutsch, A

    D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu and A. Sanpera, Phys. Rev. Lett. 77, 2818-2821 (1996)

    work page 1996

  34. [35]

    C. H. Bennett, G. Brassard, S. Popescu B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722-725 (1996)

    work page 1996

  35. [36]

    Horodecki, P

    M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. Lett. 78, 574-577 (1997)

    work page 1997

  36. [37]

    Horodecki, P

    M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. A 60, 1888-1898 (1999)

    work page 1999

  37. [38]

    F.Verstraete and H.Verschelde,Phys. Rev. A 66, 022307 (2002)

    work page 2002

  38. [39]

    Grondalski, D

    J. Grondalski, D. M. Etlinger, D. F. V. James, Phys. Lett. A, 300, 569-576 (2002)

    work page 2002

  39. [40]

    Maximally coherent states

    Z. Bai, and Shuanping Du, arXiv:1503.07103v2 (2015)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  40. [41]

    R. Pal, S. Bandyopadhyay, S. Ghosh, Phys. Rev. A 90, 052304(2014)

    work page 2014

  41. [42]

    Quantum speed limits, coherence and asymmetry

    I. Marvian, R. W. Spekkens, and P. Zanardi, arXiv:1510.06474 (2015)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  42. [43]

    Genuine quantum coherence

    A. Streltsov, arXiv:1511.08346 (2015)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  43. [44]

    Quantum processes which do not use coherence

    B. Yadin, J. Ma, D. Girolami, M. Gu, and V. Vedral, arXiv:1512.02085 (2015)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  44. [45]

    X. Yuan, H. Zhou, Z. Cao, and X. Ma, Phys. Rev. A 92, 022124 (2015)

    work page 2015

  45. [46]

    Yu and J

    T. Yu and J. H. Eberly, Quantum Inform. Comput. 7, 459 (2007)

    work page 2007

  46. [47]

    S. Du, Z. Bai and Y. Guo, Phys. Rev. A 91, 052120 (2015)

    work page 2015

  47. [48]

    M. B. Ruskai, S. Szarek and E. Warner, Lin. Alg. Appl., 347, 159(2002)

    work page 2002

  48. [49]

    Guo and J

    Yu. Guo and J. Hou, arXiv:1206:2159v4(2012)

    work page 2012

  49. [50]

    Selfcomplementary quantum channels

    M. Smaczynski, W. Roga, K. Zyczkowski, arXiv:1602.02140 (2016)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  50. [51]

    Y. Yao, X. Xiao, L. Ge, and C.P. Sun, Phys. Rev. A 92, 022112 (2015)

    work page 2015

  51. [52]

    Radhakrishnan, M

    C. Radhakrishnan, M. Parthasarathy, S. Jambulingam, T. Byrnes, Phys. Rev. Lett., 116, 150504 (2016)

    work page 2016

  52. [53]

    L.M. Duan, M. D. Lukin, J.I. Cirac and P. Zoller, Nature, 414, 413 (2001). Appendix A: Proof of Theorem 1 Coherence fraction of an arbitrary state ρ of dimension d is given by Fc(ρ) = 1 d + 1 d Cl1(ρ). (A1) if and only if θjp + θpk = 2nπ + θjk (A2) where θjk∈ [0, 2π] is the argument of (j, k)-th element of the density matrix ρ. Proof : Any qudit state can...

    work page 2001