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Fundamental limits on polarization entanglement distribution in optical fiber

Stefano Pirandola

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Phys. Rev. Research 8, 033222 – Published 24 August, 2026

DOI: https://doi.org/10.1103/xqrp-xrby

Abstract

Characterizing the ultimate rates of entanglement distribution is essential for both foundational research and the practical deployment of quantum technologies. To investigate these limits, we introduce an erasure-Pauli channel model describing the distribution of polarization entanglement in optical fiber. For this channel, we derive bounds on the rates of entanglement distribution and related quantum resources under optimal local operations and two-way classical communication (two-way assisted capacities). This framework allows us to determine the optimal repeaterless performance achievable over realistic optical fibers affected by polarization mode dispersion, thereby providing a rigorous benchmark for long-distance polarization-based quantum communication. Finally, we show that both our model and capacity bounds remain robust under the inclusion of detector dark counts.

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References (34)

  1. C. H. Bennett and G. Brassard, Quantum cryptography: Public key distribution and coin tossing, in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (IEEE, Bangalore, India, 1984), pp. 175–179.
  2. A. K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67, 661 (1991).
  3. D. Bruß, Optimal eavesdropping in quantum cryptography with six states, Phys. Rev. Lett. 81, 3018 (1998).
  4. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74, 145 (2002).
  5. S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, et al., Advances in quantum cryptography, Adv. Opt. Photonics 12, 1012 (2020).
  6. C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  7. S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Advances in photonic quantum sensing, Nat. Photon. 12, 724 (2018).
  8. J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, Distributed quantum computation over noisy channels, Phys. Rev. A 59, 4249 (1999).
  9. H. J. Kimble, The quantum internet, Nature (London) 453, 1023 (2008).
  10. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed. (Cambridge University Press, Cambridge, 2010).
  11. C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects, Phys. Rev. A 89, 022317 (2014).
  12. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  13. S. Pirandola, End-to-end capacities of a quantum communication network, Commun. Phys. 2, 51 (2019).
  14. A. Treiber, A. Poppe, M. Hentschel, D. Ferrini, T. Lorünser, E. Querasser, T. Matyus, H. Hübel, and A. Zeilinger, A fully automated entanglement-based quantum cryptography system for telecom fiber networks, New J. Phys. 11, 045013 (2009).
  15. S. Wengerowsky, S. K. Joshi, F. Steinlechner, J. R. Zichi, S. M. Dobrovolskiy, R. van der Molen, J. W. N. Los, V. Zwiller, M. A. M. Versteegh, A. Mura, D. Calonico, M. Inguscio, H. Hübel, L. Bo, T. Scheidl, A. Zeilinger, A. Xuereb, and R. Ursin, Entanglement distribution over a 96-km-long submarine optical fiber, Proc. Natl. Acad. Sci. USA 116, 6684 (2019).
  16. Note that our model is necessarily more general than the “dephrasure channel” [17, 18, 19, 20, 21], which is restricted to erasure combined with dephasing noise. In contrast, our erasure-Pauli construction allows for a generic Pauli noise process on the nonerased polarization component. Moreover, as we discuss afterward, the model is further extended beyond the standard erasure setting by including detector dark counts, which transform some no-photon events into effective noisy polarization outputs.
  17. F. Leditzky, D. Leung, and G. Smith, Dephrasure channel and superadditivity of the coherent information, Phys. Rev. Lett. 121, 160501 (2018).
  18. S. Pirandola, R. Laurenza, and L. Banchi, Conditional channel simulation, Ann. Phys. 400, 289 (2019).
  19. V. Siddhu, Entropic singularities give rise to quantum transmission, Nat. Commun. 12, 5750 (2021).
  20. V. Siddhu and R. B. Griffiths, Positivity and nonadditivity of quantum capacities using generalized erasure channels, IEEE Trans. Inf. Theory 67, 4533 (2021).
  21. A. Nema, A. G. Maity, S. Strelchuk, and D. Elkouss, Noise is resource-contextual in quantum communication, Phys. Rev. Res. 6, 033089 (2024).
  22. C. D. Poole and R. E. Wagner, Phenomenological approach to polarisation dispersion in long single-mode fibres, Electron. Lett. 22, 1029 (1986).
  23. P. K. A. Wai and C. R. Menyuk, Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence, J. Lightwave Technol. 14, 148 (1996).
  24. G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, New York, 2012).
  25. S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, Fundamental limits of repeaterless quantum communications, Nat. Commun. 8, 15043 (2017).
  26. The two-way assisted capacities represent the optimal rates that are achievable for tasks such as qubit transmission (Q2), entanglement distribution (D2), and key generation (K2). More precisely, these are the maximum rates at which qubits, ebits, or key bits can be distributed over a quantum channel, assuming that Alice and Bob do not have preshared entanglement but can perform the most general local operations assisted by two-way classical communication. For example, in a QKD scenario, this means that Bob can send classical information to Alice on how to prepare the quantum systems at the input of the channel, and Alice can send classical information to Bob on how to measure them at the output. These exchanges of information are potentially unlimited. The one described represents a repeaterless scenario, i.e., a communication configuration where Alice and Bob are directly connected by the intermediate quantum channel without any third party in the middle. In contrast, in a repeater-based scenario, the communication between Alice and Bob would be mediated by a middle relay or repeater, say Charlie, connected to Alice via channel E1 and to Bob via channel E2. In this case, the end-to-end rate between Alice and Bob could exceed the value of a repeaterless rate achieved over a direct quantum channel E=E2∘E1, where Charlie does not intervene.
  27. V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Quantifying entanglement, Phys. Rev. Lett. 78, 2275 (1997).
  28. V. Vedral and M. B. Plenio, Entanglement measures and purification procedures, Phys. Rev. A 57, 1619 (1998).
  29. O. Simeone, Classical and Quantum Information Theory: Uncertainty, Information, and Correlation (Cambridge University Press, Cambridge, 2026).
  30. Note that we may also write the lower bound for the erasure-depolarizing channel. This is simply given by the formula C2(EEPD)≥η[1−H2(3p/4)−3p4log23].
  31. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  32. Write a polarization qubit in the {|H〉,|V〉} basis as ρ=ρHHρHVρVHρVV, (20)where ρHV=〈H|ρ|V〉 is the off-diagonal polarization coherence. Under first-order PMD, polarization coherence decays as ρHV→e−d/LρHV. A depolarizing channel acts on the off-diagonal term as ρHV→(1−p)ρHV, while a dephasing channel acts as ρHV→(1−2p)ρHV. Equating these coherence-reduction factors to the PMD case above, we find 1−p/p∞=e−d/L, (21)where p∞=1(1/2) for depolarizing (dephasing).
  33. S. Massar and S. Popescu, Reducing polarization mode dispersion with controlled polarization rotations, New J. Phys. 9, 158 (2007).
  34. From first-order PMD theory, the mean squared differential group delay (DGD) 〈(ΔT)2〉 scales as 〈(ΔT)2〉=DPMD2d, where d is the fiber length [22, 24]. Requiring the rms DGD to be comparable to the photon coherence time τ leads to τ2≃DPMD2LDH, where LDH denotes the decoherence length. The additional factor of 2 appearing in Eq. (13) depends on the precise definition of the coherence time and on the chosen visibility threshold. Under an effective exponential approximation, we can write the single-photon coherence visibility as V≃exp[−〈(ΔT)2〉/2τ2], which leads to V≃exp[−d/LDH], where LDH=2τ2/DPMD2.

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Phys. Rev. Research 8, 033222– Published 24 August, 2026
  • Received 4 March 2026
  • Accepted 4 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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