Pith. sign in

REVIEW 3 major objections 3 minor 4 cited by

Entanglement Purification in Quantum Networks: Guaranteed Improvement and Optimal Time

T0 review · 3 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves when the CNOT-based recurrence EPP improves entanglement and fixes the best time to run it under memory decoherence.

desk verdict Solid, mostly-correct analytical results on guaranteed EPP improvement and optimal scheduling; one proposition has a strict-inequality bug at the F=1/2 boundary that is easy to fix. read the letter →

arxiv 2505.02286 v1 pith:KKQXZGSC submitted 2025-05-04 quant-ph

Allen Zang , Xin-An Chen , Eric Chitambar , Martin Suchara , Tian Zhong This is my paper · ORCID
classification quant-ph
keywords entanglementpurificationquantumnetworksBell-diagonalstatesmemorydecoherenceoptimalschedulingPaulichannelsfidelityimprovementmeasures
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

This paper asks when a two-to-one entanglement purification step is worth doing and when it should be scheduled in a quantum network where one pair arrives before another and both memories decay. It proves that the standard CNOT-based recurrence protocol, on two Bell-diagonal states of the same error shape, always improves output fidelity over the average input state as long as the noise parameter $a$ is at most $1/3$, and that for $a$ above $1/2$ such improvement never happens. For memory noise that is a single detectable Pauli channel (bit-flip), it proves the best time to purify is the latest possible time before the pair is used, independent of arrival times, raw fidelity, or decoherence rate. For depolarizing memory noise the optimal time depends on parameters and is given explicitly, and for general Pauli memory noise the strategy switches sharply between purify-immediately and purify-last, with an approximate border formula. The results give network operators simple timing rules and clarify what the protocol can guarantee when inputs are non-identical.

What carries the argument

The load-bearing object is the recurrence EPP on Bell-diagonal states, where two noisy pairs are combined with a bilateral CNOT and a parity measurement, succeeding when the two measurement outcomes agree. For such inputs the protocol reduces to a four-line map on the Bell probabilities $\lambda_1,\dots,\lambda_4$: on success, $\lambda_1$ becomes $(\lambda_1\lambda_1'+\lambda_2\lambda_2')/p$, and analogous products replace the other components, with $p$ the success probability. The proof machinery combines that map with a one-parameter family of input states $\lambda_1=F$, $\lambda_2=a(1-F)$, $\lambda_3+\lambda_4=(1-a)(1-F)$, and with a continuous-time Pauli channel model for memory decoherence whose action on Bell-diagonal states is a $4\times 4$ matrix. The parameter $a$ and the monotonicity of entanglement measures in fidelity carry the guaranteed-improvement results; the Pauli channel matrix carries the optimal-time results, which are reduced to signs of derivatives of the fidelity at $t_2$.

What would settle it

Set up the bit-flip scenario of Proposition IV.1 with $F_0=0.95$, $t_1=0.01/\kappa$, $t_2=0.1/\kappa$, and measure the fidelity at $t_2$ after successful purification at several times $t$; if the latest $t$ does not give the highest fidelity, the claimed parameter-independent optimal time fails.

Watch

Extended reading notes

Core claim

The central claim is that the CNOT-based recurrence EPP has a precise, provable improvement boundary and a robust optimal schedule. For Bell-diagonal inputs with identical error pattern, letting $a$ be the fraction of the non-fidelity weight in the $\Phi^-$ component, the successful output fidelity is always at least the average input fidelity when $a\le 1/3$; for $a>1/3$ there exist inputs where purification lowers fidelity relative to that average, and for $a>1/2$ purification never beats it. Under bit-flip memory decoherence, the fidelity at the utilization time conditioned on successful purification is monotonically increasing in the purification time, so the EPP should be run as late as possible, with no dependence on $t_1$, $t_2$, $F_0$, or $\kappa$; if success probability is included in a normalized figure of merit, the optimum flips to as early as possible for fidelity, concurrence and negativity, while normalized distillable entanglement again favors the latest time. Under depolarizing memory decoherence a parameter-dependent intermediate optimal time exists with an explicit formula. For general continuous-time Pauli memory channels the optimal time undergoes a transition between earliest and latest, and the paper gives an approximate $t_1,t_2$-independent expression for the transition border as a function of raw fidelity.

Load-bearing premise

The load-bearing premise is that memory decoherence is a pair of independent, identical continuous-time Pauli channels, one on each node's quantum memory; if the noise is non-Pauli, correlated, or has memory, the parameter-independent optimal times and the transition-border formula need not hold.

Editorial extensions

If this is right

  • A network operator who knows its memory decoherence is bit-flip-dominated can set the purification schedule without tracking $t_1$, $t_2$, $F_0$, or $\kappa$: run the EPP as late as possible before use.
  • If the metric includes the success probability (normalized fidelity, concurrence, negativity), the same operator should purify as early as possible; normalized distillable entanglement reverses this again because of convexity.
  • For depolarizing memories, there is a unique intermediate time $t^*$ given by an explicit formula, and whether to wait depends on how long after the first pair the second pair arrives.
  • For a general Pauli memory channel, the optimal strategy is extreme, either immediately or at the end, with a sharp transition border approximated by $B\approx (8F_0^2-4F_0+5)/(20F_0^2-4F_0+2)$; misidentifying the noise pattern can select the wrong extreme strategy and even make EPP worse than discarding the old pair.

Reading between the lines

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

  • If the transition border formula is as general as the numerics suggest, small errors in characterizing the X/Y/Z balance of a quantum memory can flip a repeater from purify-at-end to purify-immediately; this sharp sensitivity could be tested in simulation without changing the noise model.
  • The guaranteed-improvement threshold $a\le 1/3$ suggests a simple protocol-level diagnostic: estimate $a$ by twirling and measuring the $\Phi^-$ component; below $1/3$ the average-input baseline is safe, while above it the protocol should be combined with a different error-correction step or a different EPP.
  • Extending the derivative analysis beyond Pauli channels, for example to amplitude-damping noise, may remove the parameter-independence; the paper also leaves classical communication delay out, so including a fixed two-way communication time should shift the optimal schedule by that delay, which is a direct testable extension.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper analyzes the BBPSSW/DEJMPS recurrence entanglement purification protocol for Bell-diagonal states in quantum network settings. It asks (i) what improvement is guaranteed conditioned on success when the two input states are non-identical, and (ii) what is the optimal time to run the EPP when the input pairs are generated at different times and stored in decohering quantum memories. The main results are: rank-2 bit-flip inputs always improve over the better input; Werner-state inputs are guaranteed to improve over the average-input-state fidelity; general BDS inputs with a common error parameter a are guaranteed to improve over the AIS fidelity for a≤1/3 and never for a>1/2; for bit-flip memory noise the optimal purification time is the latest possible time (or the earliest for normalized fidelity and concurrence), while for depolarizing noise it is a computable intermediate time; and a numerical/analytical study of the transition between early and late optimal times for general Pauli channels. Detailed proofs are provided in Appendices A–D, with robustness and transition-border analysis in Appendices E–F.

Significance. If the results are made fully rigorous, the paper is a useful contribution to the quantum-network EPP scheduling literature. The clean thresholds a=1/3 and a=1/2, the parameter-independent optimal-time statements, and the explicit transition-border approximation give practically actionable guidance and are derived without fitted parameters. The paper is also careful to distinguish its guaranteed-improvement claims from the universality no-go results of arXiv:2407.21760. The main value is analytical: it identifies exactly when postselection on the recurrence protocol guarantees improvement and when delaying purification is beneficial, which is significant for repeater and memory-buffer design. However, the current manuscript contains several statement-level errors and a domain error in one of the optimal-time theorems that must be corrected before the results can be accepted as stated.

major comments (3)
  1. [Sec. III.C, Prop. III.4] Statements 1 and 3 of Prop. III.4 are false as printed. At F1=F2=1/2, Eq. (1a) gives FBDS = (1/4 + a^2/4) / ((1+a)^2/4 + (1-a)^2/4) = 1/2 for every a. Hence for a≤1/3 the output fidelity equals the AIS fidelity rather than being strictly greater, and for a>1/2 the pair (1/2,1/2) satisfies FBDS=(F1+F2)/2, contradicting the claimed nonexistence of (F1,F2) with FBDS≥(F1+F2)/2. The appendix proof (Eq. B20 and Sec. B.5) establishes the non-strict forms F_avg_incr_BDS≥0 for a≤1/3 and F_avg_incr_BDS≤0 for a≥1/2. Please correct the proposition to non-strict inequalities and specify the equality cases.
  2. [Sec. IV.B, Prop. IV.4 and Appendix D.4] The branch "if t1<t1^*, perform EPP at t*" is incomplete because t* is independent of t2 and may lie outside the feasible interval [t1,t2]. The derivative calculation in Eq. (D20) shows that the end-time fidelity increases on [t1,t*] and decreases after t*. Therefore, if t2<t*, the constrained maximum is at t2, not at t*. The proposition needs an additional condition, such as "and t2≥t*", plus a branch for t2<t* (perform at the latest possible time t2). Relatedly, the proposition and Eq. (9) should state that the threshold t1^* is positive only for F0>(3√3−2)/4≈0.799; for smaller F0 the earliest-time branch applies for all t1.
  3. [Sec. III.B, Prop. III.3 and Appendix B.4] The proof of the fidelity threshold Fth≈0.939 depends on numerically locating the root F1,root≈0.879 of E_low-up_D(F1,1) (Eq. B16). As stated, Prop. III.3 is a guarantee for all F1,F2≥Fth, but a purely numerical root location does not by itself provide a rigorous threshold. Please provide a rigorous enclosure of the root (for example, interval arithmetic or a Sturm-sequence argument) and a complete justification of the "simple geometric argument" that reduces the rectangle to the two boundaries examined, or explicitly present the threshold as a numerically supported conjecture rather than a proven proposition.
minor comments (3)
  1. [Sec. III.C] The sentence "which is obviously negative for a<0" should read "for a>0"; additionally, the example uses F1=1, where the ratio a=λ2/(1−F1) is not defined, so it should be flagged as a limiting case.
  2. [Sec. III.B] The phrase "guaranteed to be higher than at most the average input fidelity" is misleading; the proven statement is that the output fidelity is at least the AIS fidelity.
  3. [Throughout] There are several typos that should be corrected: "twriling" in Prop. III.3, "lastest" in Props. IV.1 and IV.3, and "red/w A" in the first-page affiliation line.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's central claims are parameter-free analytical derivations from the standard recurrence-EPP transformation and memory-decay equations, with no fitted input renamed as a prediction.

full rationale

I walked the derivation chain for the two central claims. The guaranteed-improvement results in Sec. III all start from the standard BDS purification map, Eq. (1a): lambda_{1,succ} = (lambda_1 lambda'_1 + lambda_2 lambda'_2)/p_BDS. Propositions III.1 through III.4 are then proved by explicit algebra in Appendix B (e.g., Eq. B20 for the AIS fidelity increase for general BDS). These are closed-form inequalities over the input fidelity domain, not reductions of the conclusion to a fitted quantity. The only self-citation, arXiv:2407.21760, is used to introduce the notion of universality, and the paper explicitly disclaims that its own guaranteed-improvement results imply that notion; hence that citation is not load-bearing for any derived inequality. The optimal-time results in Sec. IV are likewise computed from the closed-form BDS dynamics of Appendix A (Eqs. A3-A6). For instance, Prop. IV.1 is proved by differentiating the explicit final-fidelity expression and showing the derivative Eq. D5 is positive for F0 > 1/2; no parameter is fitted to make this happen. Props. IV.2-IV.5 follow the same pattern for normalized figures of merit. The transition-border formula, Eq. (11), is an analytic small-parameter approximation derived in Appendix F from the linear-in-t coefficient of the success-conditioned fidelity (Eqs. F2-F4). The numerical comparison in Fig. 4(b) is a validation of that approximation against an independent numerical scan, not a fit of the formula to the data. I also considered the strict-inequality defect in Prop. III.4 at F1=F2=1/2; that is a genuine mathematical typo (the appendix proves non-strict inequalities), but it is a correctness issue, not a circularity issue. No fitted input is renamed as a prediction, no 'uniqueness theorem' is imported from the authors' prior work to force a choice, and no known result is repackaged under new coordinates. The paper is self-contained against its stated Pauli-channel and BDS assumptions, so the honest finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; Fth≈0.939 is a computed root, and system parameters (F0, t1, t2, κ) are scenario inputs. The central claims rest on standard EPP formulas and explicit modeling assumptions, all stated in the text.

assumptions (5)
  • standard math The recurrence EPP output transformation (Eqs. 1-2) for Bell diagonal inputs.
    Taken from BBPSSW/DEJMPS literature [23-25]; the paper builds on this known circuit calculation.
  • domain assumption Pauli twirling maps any two-qubit state to a Bell-diagonal state preserving fidelity.
    Invoked in Sec. II A and III to justify restricting to BDS; standard result [64-66] but an assumption about available operations.
  • domain assumption Memory decoherence is an independent, identical continuous-time Pauli channel on each qubit.
    Sec. IV and Appendix A model quantum memories this way; non-Pauli or correlated noise would break the optimal-time results.
  • domain assumption CNOT gates and measurements in the EPP circuit are noiseless, and two-way classical communication delay is ignored.
    Stated in Sec. IV A and Conclusion; listed as future work to include gate and measurement errors.
  • domain assumption Input states in Sec. III.C have identical error model (same ratio a=λ2/(1-λ1)).
    Prop. III.4 restricts to identical parametrization differing only in fidelity; this is the premise of the characterization.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Entanglement Purification in Quantum Networks: Guaranteed Improvement and Optimal Time." pith.science (2026). https://pith.science/paper/KKQXZGSC

@misc{pith2026250502286,
  author       = {Pith},
  title        = {Pith review of: Entanglement Purification in Quantum Networks: Guaranteed Improvement and Optimal Time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KKQXZGSC}},
  note         = {Machine review of arXiv:2505.02286}
}
read the original abstract

While the concept of entanglement purification protocols (EPPs) is straightforward, the integration of EPPs in network architectures requires careful performance evaluations and optimizations that take into account realistic conditions and imperfections, especially probabilistic entanglement generation and quantum memory decoherence. It is important to understand what is guaranteed to be improved from successful EPP with arbitrary non-identical input, which determines whether we want to perform the EPP at all. When successful EPP can offer improvement, the time to perform the EPP should also be optimized to maximize the improvement. In this work, we study the guaranteed improvement and optimal time for the CNOT-based recurrence EPP, previously shown to be optimal in various scenarios. We firstly prove guaranteed improvement for multiple figures of merit, including fidelity and several entanglement measures when compared to practical baselines as functions of input states. However, it is noteworthy that the guaranteed improvement we prove does not imply the universality of the EPP as introduced in arXiv:2407.21760. Then we prove robust, parameter-independent optimal time for typical error models and figures of merit. We further explore memory decoherence described by continuous-time Pauli channels, and demonstrate the phenomenon of optimal time transition when the memory decoherence error pattern changes. Our work deepens the understanding of EPP performance in realistic scenarios and offers insights into optimizing quantum networks that integrate EPPs.

Figures

Figures reproduced from arXiv: 2505.02286 by the authors.

Figure 1
Figure 1. FIG. 1. Circuit of the recurrence EPP [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Visualization of EPP time optimization. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Optimal times for normalized entanglement nega [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical results for general continuous-time Pauli channels. (a) Optimal time for fidelity at [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Enhancing Entanglement Purification with Shared Randomness

    quant-ph 2026-07 conditional novelty 6.0 of 10

    Accumulating and randomly shuffling stored entangled pairs before a bilocal Clifford purification protocol provably improves expected success probability and success-weighted fidelity for arbitrary Werner sources.

  2. To Purify or Not to Purify: Entanglement Purification under Input Fidelity Asymmetry in Quantum Networks

    quant-ph 2026-05 unverdicted novelty 6.0 of 10

    Purifying two quantum links only pays off when their fidelities differ by less than about 0.076, so under realistic memory decay most purification attempts should be skipped.

  3. Entanglement Purification With Finite Latency Classical Communication in Quantum Networks

    cs.NI 2025-09 conditional novelty 6.0 of 10

    Using real metro IP latency data and Lindblad decoherence models, the paper shows that BBPSSW and DEJMPS purification have sharp latency limits, with DEJMPS often delivering far higher throughput.

  4. Entanglement Distillation and Swapping Scheduling in Quantum Repeaters with Noisy Memories

    quant-ph 2026-07 conditional novelty 5.0 of 10

    With noisy memories, discard-oldest beats early distillation at low coherence, while deferring distillation to the deadline maximizes weighted coherent information at high coherence in one- and two-hop repeaters.

Reference graph

Works this paper leans on

124 extracted references · 60 canonical work pages · cited by 4 Pith papers

  1. [1]

    H. J. Kimble, The quantum internet, Nature453, 1023 (2008)

    2008

  2. [2]

    Wehner, D

    S. Wehner, D. Elkouss, and R. Hanson, Quantum inter- net: A vision for the road ahead, Science362, eaam9288 (2018)

    2018

  3. [3]

    C. H. Bennett and G. Brassard, Quantum cryptogra- phy: Public key distribution and coin tossing, Theoreti- cal Computer Science560, 7 (2014)

    2014

  4. [4]

    Gottesman and I

    D. Gottesman and I. L. Chuang, Demonstrating the via- bility of universal quantum computation using teleporta- tion and single-qubit operations, Nature402, 390 (1999)

    1999

  5. [5]

    Jiang, J

    L. Jiang, J. M. Taylor, A. S. Sørensen, and M. D. Lukin, Distributed quantum computation based on small quan- tum registers, Physical Review A—Atomic, Molecular, and Optical Physics76, 062323 (2007)

    2007

  6. [6]

    Monroe, R

    C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, Large-scale mod- ular quantum-computer architecture with atomic mem- ory and photonic interconnects, Physical Review A89, 022317 (2014)

    2014

  7. [7]

    A. S. Cacciapuoti, M. Caleffi, F. Tafuri, F. S. Cataliotti, S. Gherardini, and G. Bianchi, Quantum internet: Net- working challenges in distributed quantum computing, IEEE Network34, 137 (2019)

    2019

  8. [8]

    Cuomo, M

    D. Cuomo, M. Caleffi, and A. S. Cacciapuoti, Towards a distributed quantum computing ecosystem, IET Quan- tum Communication1, 3 (2020)

    2020

Show all 124 references
  1. [9]

    Barral, F

    D. Barral, F. J. Cardama, G. Díaz, D. Faílde, I. F. Llovo, M. M. Juane, J. Vázquez-Pérez, J. Villasuso, C. Piñeiro, N. Costas, et al., Review of Distributed Quantum Com- puting. From single QPU to High Performance Quantum Computing, arXiv preprint arXiv:2404.01265 (2024)

    2024 arXiv

  2. [10]

    Komar, E

    P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin, A quantum network of clocks, Nature Physics10, 582 (2014)

    2014

  3. [11]

    T. J. Proctor, P. A. Knott, and J. A. Dunningham, Mul- tiparameter estimation in networked quantum sensors, Physical Review Letters120, 080501 (2018)

    2018

  4. [12]

    Zhang and Q

    Z. Zhang and Q. Zhuang, Distributed quantum sensing, Quantum Science and Technology6, 043001 (2021)

    2021

  5. [13]

    H.-J.Briegel, W.Dür, J.I.Cirac,andP.Zoller,Quantum repeaters: the role of imperfect local operations in quan- tum communication, Physical Review Letters81, 5932 (1998)

    1998

  6. [14]

    W. J. Munro, K. Azuma, K. Tamaki, and K. Nemoto, In- side quantum repeaters, IEEE Journal of Selected Topics in Quantum Electronics21, 78 (2015)

    2015

  7. [15]

    Muralidharan, L

    S. Muralidharan, L. Li, J. Kim, N. Lütkenhaus, M. D. Lukin, and L. Jiang, Optimal architectures for long dis- tance quantum communication, Scientific reports 6, 1 (2016)

    2016

  8. [16]

    Azuma, S

    K. Azuma, S. E. Economou, D. Elkouss, P. Hilaire, L. Jiang, H.-K. Lo, and I. Tzitrin, Quantum repeaters: From quantum networks to the quantum internet, Re- views of Modern Physics95, 045006 (2023)

    2023

  9. [17]

    D. L. Moehringet al., Entanglement of single-atom quan- tum bits at a distance, Nature449, 68 (2007)

    2007

  10. [18]

    Ritteret al., An elementary quantum network of single atoms in optical cavities, Nature484, 195 (2012)

    S. Ritteret al., An elementary quantum network of single atoms in optical cavities, Nature484, 195 (2012)

    2012

  11. [19]

    J.Hofmann et al.,Heraldedentanglementbetweenwidely separated atoms, Science337, 72 (2012)

    2012

  12. [20]

    Bernien et al., Heralded entanglement between solid- state qubits separated by three metres, Nature497, 86 (2013)

    H. Bernien et al., Heralded entanglement between solid- state qubits separated by three metres, Nature497, 86 (2013)

    2013

  13. [21]

    Event-ready-detectors

    M. Zukowski, A. Zeilinger, M. Horne, and A. Ekert, "Event-ready-detectors" Bell experiment via entangle- ment swapping, Physical Review Letters71 (1993)

    1993

  14. [22]

    J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, Experimental entanglement swapping: en- tangling photons that never interacted, Physical Review Letters 80, 3891 (1998)

    1998

  15. [23]

    C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, 10 J. A. Smolin, and W. K. Wootters, Purification of noisy entanglement and faithful teleportation via noisy chan- nels, Physical Review Letters76, 722 (1996)

    1996

  16. [24]

    Deutsch, A

    D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Quantum privacy ampli- fication and the security of quantum cryptography over noisy channels, Physical Review Letters77, 2818 (1996)

    1996

  17. [25]

    Dür and H

    W. Dür and H. J. Briegel, Entanglement purification and quantum error correction, Reports on Progress in Physics 70, 1381 (2007)

    2007

  18. [26]

    Pompili, S

    M. Pompili, S. L. Hermans, S. Baier, H. K. Beukers, P. C. Humphreys, R. N. Schouten, R. F. Vermeulen, M. J. Tiggelman, L. dos Santos Martins, B. Dirkse,et al., Real- ization of a multinode quantum network of remote solid- state qubits, Science372, 259 (2021)

    2021

  19. [27]

    Hermans, M

    S. Hermans, M. Pompili, H. Beukers, S. Baier, J. Borre- gaard, and R. Hanson, Qubit teleportation between non- neighbouring nodes in a quantum network, Nature605, 663 (2022)

    2022

  20. [28]

    Knautet al., Entanglement of nanophotonic quantum memory nodes in a telecom network, Nature629, 573 (2024)

    C. Knautet al., Entanglement of nanophotonic quantum memory nodes in a telecom network, Nature629, 573 (2024)

    2024

  21. [29]

    Liu et al., Creation of memory–memory entangle- ment in a metropolitan quantum network, Nature629, 579 (2024)

    J.-L. Liu et al., Creation of memory–memory entangle- ment in a metropolitan quantum network, Nature629, 579 (2024)

    2024

  22. [30]

    A. J. Stolk, K. L. van der Enden, M.-C. Slater, I. te Raa- Derckx, P. Botma, J. van Rantwijk, J. B. Biemond, R. A. Hagen, R. W. Herfst, W. D. Koek,et al., Metropolitan- scale heralded entanglement of solid-state qubits, Science Advances10, eadp6442 (2024)

    2024

  23. [31]

    J.-W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, En- tanglementpurificationforquantumcommunication,Na- ture 410, 1067 (2001)

    2001

  24. [32]

    J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, Experimental entanglement purification of arbitrary unknown states, Nature423, 417 (2003)

    2003

  25. [33]

    Hu, C.-X

    X.-M. Hu, C.-X. Huang, Y.-B. Sheng, L. Zhou, B.-H. Liu, Y. Guo, C. Zhang, W.-B. Xing, Y.-F. Huang, C.- F. Li, et al., Long-distance entanglement purification for quantum communication, Physical Review Letters126, 010503 (2021)

    2021

  26. [34]

    Ecker, P

    S. Ecker, P. Sohr, L. Bulla, M. Huber, M. Bohmann, and R. Ursin, Experimental single-copy entanglement distil- lation, Physical Review Letters127, 040506 (2021)

    2021

  27. [35]

    Reichle, D

    R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Experimental purification of two- atom entanglement, Nature443, 838 (2006)

    2006

  28. [36]

    N. Kalb, A. A. Reiserer, P. C. Humphreys, J. J. Baker- mans, S. J. Kamerling, N. H. Nickerson, S. C. Benjamin, D. J. Twitchen, M. Markham, and R. Hanson, Entangle- ment distillation between solid-state quantum network nodes, Science356, 928 (2017)

    2017

  29. [37]

    H. Yan, Y. Zhong, H.-S. Chang, A. Bienfait, M.-H. Chou, C. R. Conner, É. Dumur, J. Grebel, R. G. Povey, and A. N. Cleland, Entanglement purification and protection in a superconducting quantum network, Physical Review Letters 128, 080504 (2022)

    2022

  30. [38]

    Khatri, Towards a general framework for practi- cal quantum network protocols , Ph.D

    S. Khatri, Towards a general framework for practi- cal quantum network protocols , Ph.D. thesis, Louisiana State University and Agricultural & Mechanical College (2021)

    2021

  31. [39]

    Khatri, Policies for elementary links in a quantum net- work, Quantum5, 537 (2021)

    S. Khatri, Policies for elementary links in a quantum net- work, Quantum5, 537 (2021)

    2021

  32. [40]

    Khatri, On the design and analysis of near-term quan- tum network protocols using Markov decision processes, AVS Quantum Science4 (2022)

    S. Khatri, On the design and analysis of near-term quan- tum network protocols using Markov decision processes, AVS Quantum Science4 (2022)

    2022

  33. [41]

    Santra, L

    S. Santra, L. Jiang, and V. S. Malinovsky, Quantum re- peater architecture with hierarchically optimized mem- ory buffer times, Quantum Science and Technology4, 025010 (2019)

    2019

  34. [42]

    A. Zang, X. Chen, A. Kolar, J. Chung, M. Suchara, T. Zhong, and R. Kettimuthu, Entanglement distribu- tion in quantum repeater with purification and optimized buffer time, inIEEE INFOCOM 2023 - IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS) (2023) pp. 1–6

    2023

  35. [43]

    Chakraborty, F

    K. Chakraborty, F. Rozpedek, A. Dahlberg, and S. Wehner, Distributed routing in a quantum internet, arXiv preprint arXiv:1907.11630 (2019)

    2019 arXiv

  36. [44]

    Kolar, A

    A. Kolar, A. Zang, J. Chung, M. Suchara, and R. Ket- timuthu, Adaptive, continuous entanglement generation for quantum networks, inIEEE INFOCOM 2022-IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS) (IEEE, 2022) pp. 1–6

    2022

  37. [45]

    Á. G. Iñesta and S. Wehner, Performance metrics for the continuous distribution of entanglement in mul- tiuserquantumnetworks,PhysicalReviewA 108,052615 (2023)

    2023

  38. [46]

    Ghaderibaneh, H

    M. Ghaderibaneh, H. Gupta, C. Ramakrishnan, and E. Luo, Pre-distribution of entanglements in quantum networks, in 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE,

    2022

  39. [47]

    C. Zhan, J. Chung, A. Zang, A. Kolar, and R. Ket- timuthu, Design and simulation of the adaptive contin- uous entanglement generation protocol, arXiv preprint arXiv:2502.01964 (2025)

    2025 arXiv

  40. [48]

    W.Dür, H.-J.Briegel, J.I.Cirac,andP.Zoller,Quantum repeaters based on entanglement purification, Physical Review A59, 169 (1999)

    1999

  41. [49]

    T. D. Ladd, P. van Loock, K. Nemoto, W. J. Munro, and Y.Yamamoto,Hybridquantumrepeaterbasedondisper- sive cqed interactions between matter qubits and bright coherent light, New Journal of Physics8, 184 (2006)

    2006

  42. [50]

    Hartmann, B

    L. Hartmann, B. Kraus, H.-J. Briegel, and W. Dür, Role of memory errors in quantum repeaters, Physical Review A 75, 032310 (2007)

    2007

  43. [51]

    Razavi, M

    M. Razavi, M. Piani, and N. Lütkenhaus, Quantum re- peaters with imperfect memories: Cost and scalability, Physical Review A80, 032301 (2009)

    2009

  44. [52]

    Razavi, K

    M. Razavi, K. Thompson, H. Farmanbar, M. Piani, and N. Lütkenhaus, Physical and architectural considerations in quantum repeaters, inQuantum Communications Re- alized II, Vol. 7236 (SPIE, 2009) pp. 18–30

    2009

  45. [53]

    Bratzik, S

    S. Bratzik, S. Abruzzo, H. Kampermann, and D. Bruß, Quantum repeaters and quantum key distribution: The impact of entanglement distillation on the secret key rate, Physical Review A—Atomic, Molecular, and Op- tical Physics87, 062335 (2013)

    2013

  46. [54]

    Victora, S

    M. Victora, S. Tserkis, S. Krastanov, A. S. de la Cerda, S. Willis, and P. Narang, Entanglement purification on quantum networks, Physical Review Research5, 033171 (2023)

    2023

  47. [55]

    A. Zang, A. Kolar, A. Gonzales, J. Chung, S. K. Gray, R. Kettimuthu, T. Zhong, and Z. H. Saleem, Quantum advantage in distributed sensing with noisy quantum net- works, arXiv preprint arXiv:2409.17089 (2024). 11

    2024 arXiv

  48. [56]

    Mantri, K

    P. Mantri, K. Goodenough, and D. Towsley, Comparing one-and two-way quantum repeater architectures, arXiv preprint arXiv:2409.06152 (2024)

    2024 arXiv

  49. [57]

    Haldar, P

    S. Haldar, P. J. Barge, X. Cheng, K.-C. Chang, B. T. Kirby, S. Khatri, C. W. Wong, and H. Lee, Reducing classicalcommunicationcostsinmultiplexedquantumre- peaters using hardware-aware quasi-local policies, Com- munications Physics8, 132 (2025)

    2025

  50. [58]

    A. Zang, X. Chen, E. Chitambar, M. Suchara, and T. Zhong, No-go theorems for universal entanglement pu- rification, arXiv preprint arXiv:2407.21760 (2024)

    2024 arXiv

  51. [59]

    Rozpędek, T

    F. Rozpędek, T. Schiet, D. Elkouss, A. C. Doherty, S. Wehner,et al., Optimizing practical entanglement dis- tillation, Physical Review A97, 062333 (2018)

    2018

  52. [60]

    Preti, T

    F. Preti, T. Calarco, J. M. Torres, and J. Z. Bernád, Optimal two-qubit gates in recurrence protocols of en- tanglement purification, Physical Review A106, 022422 (2022)

    2022

  53. [61]

    Jansen, K

    S. Jansen, K. Goodenough, S. de Bone, D. Gijswijt, and D. Elkouss, Enumerating all bilocal clifford distillation protocols through symmetry reduction, Quantum6, 715 (2022)

    2022

  54. [62]

    X. Chen, J. Choi, A. Zang, V. Lorenz, and E. Chitambar, In preparation

  55. [63]

    Krastanov, V

    S. Krastanov, V. V. Albert, and L. Jiang, Optimized en- tanglement purification, Quantum3, 123 (2019)

    2019

  56. [64]

    W. Dür, M. Hein, J. I. Cirac, and H.-J. Briegel, Standard forms of noisy quantum operations via depolarization, Physical Review A72, 052326 (2005)

    2005

  57. [65]

    Emerson, M

    J. Emerson, M. Silva, O. Moussa, C. Ryan, M. Lafor- est, J. Baugh, D. G. Cory, and R. Laflamme, Sym- metrized characterization of noisy quantum processes, Science 317, 1893 (2007)

    2007

  58. [66]

    Dankert, R

    C. Dankert, R. Cleve, J. Emerson, and E. Livine, Ex- act and approximate unitary 2-designs and their applica- tion to fidelity estimation, Physical Review A80, 012304 (2009)

    2009

  59. [67]

    S. A. Hill and W. K. Wootters, Entanglement of a pair of quantum bits, Physical Review Letters78, 5022 (1997)

    1997

  60. [68]

    W. K. Wootters, Entanglement of formation of an arbi- trary state of two qubits, Physical Review Letters80, 2245 (1998)

    1998

  61. [69]

    M.B.Plenio,Logarithmicnegativity: afullentanglement monotonethatisnotconvex,PhysicalReviewLetters 95, 090503 (2005)

    2005

  62. [70]

    Peres, Separability criterion for density matrices, Physical Review Letters77, 1413 (1996)

    A. Peres, Separability criterion for density matrices, Physical Review Letters77, 1413 (1996)

    1996

  63. [71]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Separa- bility of n-particle mixed states: necessary and sufficient conditions in terms of linear maps, Physics Letters A 283, 1 (2001)

    2001

  64. [72]

    Schumacher and M

    B. Schumacher and M. A. Nielsen, Quantum data pro- cessing and error correction, Physical Review A54, 2629 (1996)

    1996

  65. [73]

    E. M. Rains, Bound on distillable entanglement, Physical Review A60, 179 (1999)

    1999

  66. [74]

    E. M. Rains, A semidefinite program for distillable en- tanglement, IEEE Transactions on Information Theory 47, 2921 (2001)

    2001

  67. [75]

    Dür and H.-J

    W. Dür and H.-J. Briegel, Entanglement purification for quantum computation, Physical Review Letters90, 067901 (2003)

    2003

  68. [76]

    See Appendix A for the discussion on the equiva- lence of different ways of parametrization for bit-flip and depolarizing channels

    Here we useγ to denote decoherence rate of a continuous- time Pauli channel with an arbitrary error pattern, dif- ferent from the previousκ for bit-flip and depolarizing channels, because of subtle differences in parametriza- tion. See Appendix A for the discussion on the equi...

  69. [77]

    Singh, C

    K. Singh, C. Bradley, S. Anand, V. Ramesh, R. White, and H. Bernien, Mid-circuit correction of correlated phase errors using an array of spectator qubits, Science 380, 1265 (2023)

    2023

  70. [78]

    Rančić, M

    M. Rančić, M. P. Hedges, R. L. Ahlefeldt, and M. J. Sellars, Coherence time of over a second in a telecom- compatible quantum memory storage material, Nature Physics 14, 50 (2018)

    2018

  71. [79]

    J. M. Martinis, Qubit metrology for building a fault- tolerant quantum computer, npj Quantum Information 1, 1 (2015)

    2015

  72. [80]

    Harper, S

    R. Harper, S. T. Flammia, and J. J. Wallman, Effi- cient learning of quantum noise, Nature Physics16, 1184 (2020)

    2020

  73. [81]

    Eisert, D

    J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Quantum cer- tification and benchmarking, Nature Reviews Physics2, 382 (2020)

    2020

  74. [82]

    Kliesch and I

    M. Kliesch and I. Roth, Theory of quantum system cer- tification, PRX Quantum2, 010201 (2021)

    2021

  75. [83]

    Davies, T

    B. Davies, T. Beauchamp, G. Vardoyan, and S. Wehner, Tools for the analysis of quantum protocols requiring state generation within a time window, IEEE Transac- tions on Quantum Engineering (2024)

    2024

  76. [84]

    Davies, Á

    B. Davies, Á. G. Iñesta, and S. Wehner, Entanglement buffering with two quantum memories, Quantum8, 1458 (2024)

    2024

  77. [85]

    Goodenough, T

    K. Goodenough, T. Coopmans, and D. Towsley, On noise in swap ASAP repeater chains: exact analytics, distributions and tight approximations, arXiv preprint arXiv:2404.07146 (2024)

    2024 arXiv

  78. [86]

    A. Zang, J. Chung, R. Kettimuthu, M. Suchara, and T. Zhong, Analytical performance estimations for quan- tum repeater network scenarios, in2024 IEEE Interna- tional Conference on Quantum Computing and Engineer- ing (QCE), Vol. 1 (IEEE, 2024) pp. 1960–1966

    2024

  79. [87]

    Á. G. Iñesta, B.Davies, S. Kar,and S. Wehner, Entangle- ment buffering with multiple quantum memories, arXiv preprint arXiv:2502.20240 (2025)

    2025 arXiv

  80. [88]

    Á. G. Iñesta, G. Vardoyan, L. Scavuzzo, and S. Wehner, Optimal entanglement distribution policies in homoge- neous repeater chains with cutoffs, npj Quantum Infor- mation 9, 46 (2023)

    2023

  81. [89]

    Chung, M

    J. Chung, M. Hajdušek, N. Benchasattabuse, A. Kolar, A. Singal, K. S. Soon, K. Teramoto, A. Zang, R. Ket- timuthu, and R. Van Meter, Cross-validating quan- tumnetworksimulators,arXivpreprintarXiv:2504.01290 (2025)

    2025

  82. [90]

    J. Liu, A. Zang, M. Suchara, T. Zhong, and P. D. Hov- land, Hardware-software co-design for distributed quan- tumcomputing,arXivpreprintarXiv:2503.18329 (2025). 12 Appendix A: Bell diagonal state dynamics under continuous-time separable Pauli channel In this appendix we derive...

    2025 arXiv

  83. [91]

    Continuous-time general Pauli channel Pauli channels are defined asEP (ρ) =P3 i=0αiPiρPi, whereP0 is the identity operator andPi are Pauli operators for i = 1, 2, 3. We first derive continuous time Pauli channel from the channel for a single time step∆t E∆t(ρ) = (1−p)ρ +p(aXρX...

  84. [92]

    Bell diagonal states (BDS) Let us consider general BDSρBDS =λ1Ψ+ +λ2Ψ− +λ3Φ+ +λ4Φ− with the normalization conditionP3 i=1λi = 1, and λ1 = F is the fidelity ofρ w.r.t. |ψ+⟩. Here we demonstrate how the general BDS will vary over time under general Pauli channels. The channel ap...

  85. [93]

    Different from the previous microscopic treatment of general continous-time Pauli channels, we may also define bit-flip rateκ with p = exp(−κt)

    Comment on parametrization of the bit-flip channel and the depolarizing channel The bit-flip channel can be canonically written as Ebit−flip(ρ) =pρ + (1−p)XρX +ρ 2 = (1 +p) 2 ρ + (1 +p) 2 XρX, (A12) where (1−p) can be interpreted as the probability for the complete bit-flip ch...

  86. [94]

    We construct the fidelity increase function F (2) incr(F1,F 2) = F1F2 F1F2 + (1−F1)(1−F2)− max{F1,F 2}

    Proof of Proposition III.1 Proof. We construct the fidelity increase function F (2) incr(F1,F 2) = F1F2 F1F2 + (1−F1)(1−F2)− max{F1,F 2}. (B1) Note the symmetry between F1,F 2 in the defined function, without loss of generality, we can useF1 to replace max{F1,F 2} and arrive a...

  87. [95]

    We first prove the statement forλ = 1/2

    Proof of Proposition III.2 Proof. We first prove the statement forλ = 1/2. We construct the fidelity increase function F (W) incr (F1,F 2) = F1F2 + (1−F1)(1−F2) 9 F1F2 + F1(1−F2)+(1−F1)F2 3 + 5(1−F1)(1−F2) 9 − F1 +F2 2 =(2− 8F2)F 2 1− (8F 2 2− 24F2 + 7)F1 + 2F 2 2− 7F2 + 2 2[5...

  88. [96]

    Proposition B.1

    Fidelity threshold for higher baselines We compare successful output fidelity of the recurrence EPP given Werner stateF (W)(F1,F 2) and the convex combination of the higher input fidelity and the lower input fidelityFbaseline(λ,F 1,F 2), forλ> 1/2. Proposition B.1. For the suc...

  89. [97]

    We want to examine the following function Elow−up D (F1,F 2) =IW A⟩B(F′ W )−R(F1 +F2 2 )

    Proof of Proposition III.3 Proof. We want to examine the following function Elow−up D (F1,F 2) =IW A⟩B(F′ W )−R(F1 +F2 2 ). (B12) According to the symmetry betweenF1 and F2, we can transform the coordinates intoF1 = (F′ 1−F′ 2)/ √ 2,F 2 = (F′ 1 +F′ 2)/ √

  90. [98]

    After this coordinate transformation, the input fidelity region becomesF′ 1∈ [1/ √ 2, √ 2],F′ 2∈ [1/ √ 2− F′ 1,F′ 1− 1/ √ 2]. We first examine the partial derivative w.r.t.F′ 2 (anti-diagonal coordinate) ofElow−up D (F′ 1,F′ 2) ∂ ∂F′ 2 Elow−up D = 6(2 √ 2F′ 1− 7) log h 9(7−2 √...

  91. [99]

    It is obvious that the denominator is non-negative, and the pre-factor of numerator(2 √ 2F′ 1− 7) is negative

    (B13) Note that here the base of logarithm does not matter because it only contributes as an additional factor, as the constant terms in the definition of the coherent information and the Rains bound cancel each other. It is obvious that the denominator is non-negative, and th...

  92. [100]

    + 4 = 18F′ 1(7− √ 2F′ 1)− 9 √ 2(1 + 2F 2 2 ) [F′2 2 −F′ 1( √ 2 +F′

  93. [101]

    + 4]2 > 0, (B14) ∂ ∂F′ 2 9(7− 2 √ 2F′ 1) F′2 2 −F′ 1(F′ 1 + √

  94. [102]

    + 4 = −18(7− 2 √ 2F′ 2) [F′2 2 −F′ 1( √ 2 +F′

  95. [103]

    By evaluating the value of the argument on the boundary it can be easily shown that the logarithmic term is always positive on the entire input fidelity region

    (B15) 17 Thus the minimum value of the logarithmic term can only be found on the following boundary of the input fidelity regions: F1 = 1/2 and F2 = 1/2. By evaluating the value of the argument on the boundary it can be easily shown that the logarithmic term is always positive...

  96. [104]

    (B20) In the following we will prove the three statements of Proposition III.4 separately

    Proof of Proposition III.4 Similar to previous proofs, we can define a fidelity improvement function as F avg incr,BDS(F1,F 2) = F1F2 +a2(1−F1)(1−F2) [F1 +a(1−F1)][F2 +a(1−F2)] + (1−a)2(1−F1)(1−F2)− F1 +F2 2 . (B20) In the following we will prove the three statements of Propos...

  97. [105]

    (B23) We then take the partial derivatives w.r.t.F′ 2 (anti-diagonal) ∂ ∂F′ 2 F avg incr,BDS(F′ 1,F′

    = √ 2(1−a)2F′3 1 − (5a2− 6a + 3)F′2 1 + √ 2(4a2− 2a + 1)F′ 1 + [(1 +a2) + √ 2(1−a)2F′ 1]F′2 2 − 2a2 2[(1−a)2F′2 2 − (1−a)2F′2 1 + √ 2(2a2− 3a + 1)F′ 1− (2a2− 2a + 1)] . (B23) We then take the partial derivatives w.r.t.F′ 2 (anti-diagonal) ∂ ∂F′ 2 F avg incr,BDS(F′ 1,F′

  98. [106]

    = √ 2(a3 +a2− 3a + 1)F′ 1− (2a3 +a2− 2a + 1) [(1−a)2F′2 2 − (1−a)2F′2 1 + √ 2(2a2− 3a + 1)F′ 1− (2a2− 2a + 1)]2F′ 2 (B24) The denominator of the RHS is always non-negative. Thus for a fixedF′ 1 the sign of the partial derivative w.r.t.F′ 2 is determined by the numerator alone:...

  99. [107]

    In other words, for a fixedF′ 1 the minimum (maximum) ofF avg incr,BDS(F′ 1,F′

  100. [108]

    Therefore, we examine the numerator as a function ofF′ 1 and a N∂F′ 2(F′ 1,a ) = √ 2(a3 +a2− 3a + 1)F′ 1− (2a3 +a2− 2a + 1)

    is obtained at(F′ 1, 0) if the numerator is positive (negative). Therefore, we examine the numerator as a function ofF′ 1 and a N∂F′ 2(F′ 1,a ) = √ 2(a3 +a2− 3a + 1)F′ 1− (2a3 +a2− 2a + 1). (B25) As a routine, we take the partial derivative w.r.t.a of N∂F′ 2 and obtain ∂ ∂aN∂F...

  101. [109]

    Then we want to know whether there isF′ 1 that satisfies Eqn

    In other words,N∂F′ 2(1/ √ 2, 0) = 0. Then we want to know whether there isF′ 1 that satisfies Eqn. B28 fora> 0. AsF′ 1(a) tends to∞ whena approaches√ 2− 1, andF′ 1(a) is a monotonic elementary function ofa on [0, √ 2− 1), there must exist one and only onea s.t. F′ 1(a) = √

  102. [110]

    In fact, we can directly solve the following equation 2a3 +a2− 2a + 1√ 2(a3 +a2− 3a + 1) = √ 2⇒a = 2± √

  103. [111]

    Fora> 2− √ 3, we will haveF′ 1(a)> √ 2 so there is no validF′ 1 that satisfies Eqn

    (B30) Only one roota = 2− √ 3 is within [0, √ 2− 1). Fora> 2− √ 3, we will haveF′ 1(a)> √ 2 so there is no validF′ 1 that satisfies Eqn. B28. On the other hand, and for allF′ 1∈ [1/ √ 2, √ 2] there exists one and only onea(F′ 1)∈ [0, 2− √ 3] as determined by inverse of Eqn. B2...

  104. [112]

    In summary, for allF′ 1∈ [1/ √ 2, √ 2] we haveN∂F′ 2(F′ 1,a )> 0 when a < a(F′ 1)≤ 2− √ 3, andN∂F′ 2(F′ 1,a )< 0 for all a∈ (2− √ 3, 1]

    is also monotonic onF′ 1∈ [1/ √ 2, √ 2]. In summary, for allF′ 1∈ [1/ √ 2, √ 2] we haveN∂F′ 2(F′ 1,a )> 0 when a < a(F′ 1)≤ 2− √ 3, andN∂F′ 2(F′ 1,a )< 0 for all a∈ (2− √ 3, 1]. Now we evaluate the maximum ofF avg incr,BDS(F′ 1,F′ 2). As 1/2 > 2− √ 3 for alla≥ 1/2 we haveN∂F′ ...

  105. [113]

    Hence we only need to examine the maximum ofF avg incr,BDS(F′ 1, 0) on F′ 1∈ [1/ √ 2, √ 2]

    is obtained atF′ 2 = 0 for a certainF′ 1∈ [1/ √ 2, √ 2]. Hence we only need to examine the maximum ofF avg incr,BDS(F′ 1, 0) on F′ 1∈ [1/ √ 2, √ 2]. Thus we can define a new function ˜F avg incr,BDS(F′ 1,a ) =F avg incr,BDS(F′ 1, 0) =−2a2 + √ 2(1− 2a + 4a2)F′ 1− (3− 6a + 5a2)F...

  106. [114]

    F′ 1 at F′ 1 = √ 2,F′ 2 = 0 ∂ ∂F′ 1 F avg incr,BDS F′ 1= √ 2,F′ 2=0 = 2a− 1√ 2

    w.r.t. F′ 1 at F′ 1 = √ 2,F′ 2 = 0 ∂ ∂F′ 1 F avg incr,BDS F′ 1= √ 2,F′ 2=0 = 2a− 1√ 2 . (B36) When a< 1/2 the above value is negative. Therefore, there exists(F′ 1,F′

  107. [115]

    in the neighborhood of( √ 2, 0) (and also in the possible region of(F′ 1,F′ 2)) s.t.F avg incr,BDS(F′ 1,F′ 2)> 0. c. Proof of statement 1 Lastly, we prove statement 1, i.e.F avg incr,BDS≥ 0 for allF1,F 2∈ [1/2, 1], ifa≤ 1/3. Proof. According to previous results, we separate tw...

  108. [116]

    (B39) The above function is concave, so its minimum can only be obtained at the boundary of theF1 domain, i.e

    + 8(3 √ 3− 5)F1 . (B39) The above function is concave, so its minimum can only be obtained at the boundary of theF1 domain, i.e. F avg incr,BDS(1/2, 1/2) a=2− √ 3 = 0, F avg incr,BDS(1, 1/2) a=2− √ 3 = 3 √ 3− 5 12− 4 √ 3 > 0. (B40) Thus,F avg incr,BDS(F1,F 2) will always be no...

  109. [117]

    Combining all three above cases, the statement is proved

    = 1− F′ 1√ 2 + √ 2F′ 1− 2 2F′2 1 + ( √ 6− 3 √ 2)F′ 1 + 4− √ 3, (B46) which can be shown to be non-negative forF′ 1∈ [1/ √ 2, √ 2]. Combining all three above cases, the statement is proved. Appendix C: Additional comments for guaranteed improvement

  110. [118]

    We have focused on the figures of merit conditioned on success, and compare them with the properties of input states which are treated as baselines to prove guaranteed improvement

    Comment on probabilistic success EPPs in general succeed with probability below 1. We have focused on the figures of merit conditioned on success, and compare them with the properties of input states which are treated as baselines to prove guaranteed improvement. In fact, such...

  111. [119]

    Here, we present another guaranteed improvement with different baselines, i.e

    Other guaranteed improvement In the main text, we were focused on entanglement measures of average input state (AIS) as the baseline to evaluate the improvement from successful EPP for Werner input states. Here, we present another guaranteed improvement with different baseline...

  112. [120]

    Firstly, we have the fidelity dynamics of the bit-flipped Bell state under memory bit-flip channel as Fbit−flip(F0,t ) =F0e−2κt + 1−e−2κt 2 , (D1) where F0 is the raw fidelity

    Proof of Proposition IV.1 Proof. Firstly, we have the fidelity dynamics of the bit-flipped Bell state under memory bit-flip channel as Fbit−flip(F0,t ) =F0e−2κt + 1−e−2κt 2 , (D1) where F0 is the raw fidelity. According to the scenario, we can give an analytical expression of ...

  113. [121]

    We first prove the statement for fidelity

    Proof of Proposition IV.2 Proof. We first prove the statement for fidelity. The explicit analytical expression for normalized fidelity att2 is ˜F (2) succ(t2) = 1− (1− 2F0)(e−2κt2 +e−2κ(t2−t1)) + (1− 2F0)2e−2κ(2t−t1) 4 . (D6) And the derivative w.r.t.t is d dt[ ˜F (2) succ(t2)...

  114. [122]

    We first prove the statement for normalized distillable entanglement

    Proof of Proposition IV.3 Proof. We first prove the statement for normalized distillable entanglement. The analytical expression of normalized distillable entanglement att2 can be written as ˜ED =ED(F′(t2))[F (F0,t )F (F0,t−t1) + (1−F (F0,t ))(1−F (F0,t−t1))] =e−4κt 4 ln 2 ln ...

  115. [123]

    Proof of Proposition IV.4 Proof. We have the following expression of output state’s fidelity at timet2 F′(t2) = 9 + 3(4F0− 1)(e−2κt2 +e−2κ(t2−t1)) + (4e−2κ(t+t2−t1) +e−2κ(2t−t1))(4F0− 1)2 36 + 4e−2κ(2t−t1)(4F0− 1)2 , (D19) and itst derivative is d dt [F′(t2)] (t) =κ(4F0−1)2 (4...

  116. [124]

    We first examine the fidelity

    Proof of Proposition IV.5 Proof. We first examine the fidelity. The fidelity dynamics of the Werner state under a memory depolarizing channel is FWerner(F0,t ) =F0e−2κt + 1−e−2κt 4 . (D27) Then the expression of normalized fidelity att2 is ˜F (W) succ (t2) = 9 + (4F0− 1) 3 e−2...

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.