pith. sign in

arxiv: 2605.21018 · v1 · pith:OO24ECZMnew · submitted 2026-05-20 · 🪐 quant-ph

Photon Efficiency of High-Dimensional Quantum Key Distribution

Vera Uzunova , Marcin Jarzyna This is my paper

Pith reviewed 2026-05-21 05:00 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum key distributionhigh-dimensional encodingentanglement-based protocolssatellite quantum communicationsphoton efficiencysecret key ratebackground radiation
0
0 comments X

The pith

High-dimensional encoding in entanglement-based QKD reaches optimal efficiency at finite photon pair production rates.

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

This paper studies entanglement-based quantum key distribution under the low signal power and background radiation conditions typical of satellite links. The authors optimize both the entangled photon pair production probability and the number of qubits encoded on each pair to approach the theoretical information limit. They demonstrate that peak efficiency occurs at a moderate, finite pair production rate rather than in the vanishing-intensity limit used by conventional schemes. Multiqubit encoding per pair increases the achievable secret key rate by as much as an order of magnitude compared with single-qubit approaches.

Core claim

In entanglement-based QKD protocols under realistic satellite conditions with low signal power and background radiation, the theoretical information limit for efficiency is achieved by optimizing source intensity and the number of encoded qubits per photon pair. The optimal efficiency occurs at a finite entangled photon pair production probability, unlike conventional schemes that peak at vanishing signal strength. Multiqubit encoding enhances the secret key rate by up to an order of magnitude over single-qubit schemes.

What carries the argument

Optimization over source intensity (entangled photon pair production probability) and the number of encoded qubits per pair, applied to the theoretical information limit of entanglement-based QKD efficiency in the presence of background radiation.

If this is right

  • Secret key rates improve when source intensity is set to a finite optimal value instead of being minimized.
  • High-dimensional encoding delivers up to tenfold gains in the low-power regimes typical of satellite links.
  • Practical systems must balance pair production rate against noise and detection constraints to approach the theoretical limit.
  • Current single-qubit implementations leave substantial performance on the table under realistic satellite conditions.

Where Pith is reading between the lines

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

  • Similar intensity optimizations may improve efficiency in other quantum communication tasks that rely on weak entangled sources.
  • Satellite demonstrations could directly test the predicted optimum by sweeping source intensity while monitoring key rate.
  • Further gains might be possible if encodings beyond simple multiqubit states become experimentally feasible.

Load-bearing premise

The analysis assumes that standard models of entanglement-based QKD protocols with background radiation and low received signal power accurately allow optimization over source intensity and encoded qubit number to reach the theoretical information limit without unaccounted losses or errors.

What would settle it

Measure secret key rate as a function of photon pair production probability for several encoded-qubit numbers and check whether the maximum occurs at a clearly non-zero finite value rather than continuing to rise as the probability approaches zero.

Figures

Figures reproduced from arXiv: 2605.21018 by Marcin Jarzyna, Vera Uzunova.

Figure 1
Figure 1. Figure 1: FIG. 1. Encoding of multiple logical qubits into a single [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Optimal PKE (a), the corresponding the pair-production probability (b) and (c) coding order, as functions of the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Optimal photon key efficiency (PKE) (a, d), the corresponding quantum bit error rate (QBER) (b, e), and the optimal [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Photon key efficiency for asymmetric four-state [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

We investigate entanglement-based quantum key distribution protocols, with particular emphasis on their efficiency under realistic conditions of satellite quantum communications, where performance is limited by the low power of a received signal and background radiation. We focus on scenarios where each photon pair is used to encode multiple qubits in order to optimally utilize the weak signal. By optimizing over the source intensity and the number of encoded qubits we study the theoretical information limit for the QKD efficiency. We show that the optimal efficiency is attained for finite entangled photons pair production probability which is in contrast to conventional communication efficiency maximized in the limit of vanishing signal strength. The multiqubit encoding can enhance the secret key rate by up to an order of magnitude compared to single-qubit schemes.

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 paper analyzes entanglement-based high-dimensional QKD protocols for satellite communications limited by low received signal power and background radiation. It optimizes over the entangled photon pair production probability p and the number of encoded qubits d to reach the theoretical information limit, claiming that optimal efficiency occurs at finite p (in contrast to the conventional vanishing-signal limit) and that multiqubit encoding can enhance the secret key rate by up to an order of magnitude relative to single-qubit schemes.

Significance. If the central optimization holds under realistic satellite conditions, the result would be significant for protocol design: it identifies a non-intuitive operating regime at finite source intensity and quantifies substantial rate gains from high-dimensional encoding. The explicit contrast with conventional low-intensity limits and the focus on photon efficiency are strengths that could guide experimental satellite QKD implementations.

major comments (2)
  1. [§3.2] §3.2 (Channel and loss model): The optimization treats the transmission efficiency η as independent of d. In satellite links, higher-dimensional encodings (OAM, time-bin, etc.) typically experience d-dependent losses from diffraction, turbulence, and mode mismatch at the receiver; this would lower the signal-to-background ratio for larger d and could move the reported optimum away from finite p or reduce the claimed order-of-magnitude gain. This assumption is load-bearing for both the finite-p result and the efficiency comparison.
  2. [§4] §4 (Optimization and key-rate results): The derivation of the optimal finite p and the quantitative rate enhancement lacks an explicit sensitivity analysis to η(d) variations or to background-radiation parameters. Without this, it is unclear whether the reported maximum remains robust once realistic d-dependent losses are included.
minor comments (2)
  1. [Abstract and §2] Notation for the number of encoded qubits (d) and pair-production probability (p) should be introduced consistently in the abstract and §2 to aid readability.
  2. [Figure captions] Figure captions for the efficiency-vs-p plots should explicitly state the fixed background-radiation and detector-efficiency values used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the scope and robustness of our results for satellite QKD applications. We address the two major comments point by point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Channel and loss model): The optimization treats the transmission efficiency η as independent of d. In satellite links, higher-dimensional encodings (OAM, time-bin, etc.) typically experience d-dependent losses from diffraction, turbulence, and mode mismatch at the receiver; this would lower the signal-to-background ratio for larger d and could move the reported optimum away from finite p or reduce the claimed order-of-magnitude gain. This assumption is load-bearing for both the finite-p result and the efficiency comparison.

    Authors: We acknowledge that d-dependent losses can arise in certain encodings such as OAM due to diffraction and turbulence. Our model deliberately assumes d-independent η to isolate the fundamental information-theoretic benefit of high-dimensional encoding in the photon-starved regime with background noise. This assumption is standard in theoretical QKD analyses and holds exactly for time-bin encodings, where losses are largely dimension-independent. For OAM, the results represent an upper bound. In the revised manuscript we will add a dedicated paragraph in §3.2 explicitly stating the assumption, justifying it for the encodings considered, and noting that realistic d-dependent losses would reduce but not eliminate the reported gains. revision: partial

  2. Referee: [§4] §4 (Optimization and key-rate results): The derivation of the optimal finite p and the quantitative rate enhancement lacks an explicit sensitivity analysis to η(d) variations or to background-radiation parameters. Without this, it is unclear whether the reported maximum remains robust once realistic d-dependent losses are included.

    Authors: We agree that an explicit sensitivity analysis strengthens the claims. In the revised version we will add a new subsection (or appendix) that introduces a simple phenomenological model η(d) = η₀ / d^α (with α varied from 0 to 0.5) and recomputes the optimal p and secret-key rate for representative background rates. Preliminary internal calculations show that the finite-p optimum persists for moderate α, although the peak gain relative to d=2 is reduced from ~10× to 4–7×. The background-radiation parameter will also be swept to confirm robustness of the qualitative conclusions. revision: yes

Circularity Check

0 steps flagged

Theoretical optimization of high-dimensional QKD efficiency is self-contained with no circular reductions

full rationale

The paper derives its central result—an optimum at finite pair-production probability p, with order-of-magnitude gain from multiqubit encoding—by optimizing the secret-key rate expression over source intensity and dimension d within standard entanglement-based QKD models that include background radiation and channel loss. This optimization is performed directly on the information-theoretic rate formula; the location of the maximum is an output of the model rather than an input, and the contrast to the conventional vanishing-signal limit follows from the same rate expression without requiring fitted parameters or self-referential definitions. No load-bearing self-citations, ansatzes smuggled via prior work, or renaming of known results appear in the derivation chain. The analysis therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The paper relies on standard quantum optics and channel models for satellite links; optimizations over intensity and dimension count as free parameters, with no new entities introduced.

free parameters (2)
  • entangled photon pair production probability
    Optimized to locate the efficiency maximum under realistic noise
  • number of encoded qubits
    Varied to study multiqubit encoding gains
axioms (2)
  • domain assumption Satellite quantum links are limited by low received signal power and background radiation
    Sets the realistic conditions for the efficiency study
  • domain assumption Entanglement-based protocols support encoding multiple qubits per photon pair
    Enables the high-dimensional approach

pith-pipeline@v0.9.0 · 5638 in / 1472 out tokens · 53211 ms · 2026-05-21T05:00:08.559928+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [1]

    Both versions are composed of the following steps: •For each time frame, Alice and Bob perform a measurement randomly in one ofNbases

    BBM92 protocol We consider the four- and six-state versions of the BBM92 protocol, utilizingN= 2 andN= 3 measurement bases respectively. Both versions are composed of the following steps: •For each time frame, Alice and Bob perform a measurement randomly in one ofNbases. •For each detected photon Bob publicly announces the basis he used in his measurement...

    work page

  2. [2]

    SARG04 protocol The four-state SARG04 protocol utilizes two bases typically Z,{|0⟩,|1⟩}, and X,{|+⟩,|−⟩}, where|±⟩= (|0⟩ ± |1⟩)/ √

    work page

  3. [3]

    •Alice and Bob each perform a measurement choosing randomly one of the two bases

    Steps of the SARG04 are as follows: •The source produces a sequence of entangled photon pairs described by a Bell state|Φ +⟩and sends one photon from the pair to Alice and Bob. •Alice and Bob each perform a measurement choosing randomly one of the two bases. •For each detected photon pair, Alice publicly announces one of the two classical sets containing ...

    work page

  4. [4]

    Aquinaet al., A critical analysis of deployed use cases for quantum key distribution and comparison with post- quantum cryptography, EPJ Quantum Technol.12, 51 (2025)

    N. Aquinaet al., A critical analysis of deployed use cases for quantum key distribution and comparison with post- quantum cryptography, EPJ Quantum Technol.12, 51 (2025)

    work page 2025

  5. [5]

    W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature299, 802 (1982)

    work page 1982

  6. [6]

    Diffie and M

    W. Diffie and M. E. Hellman, New directions in cryptography, IEEE Trans. Inf. Theory22, 644 (1976)

    work page 1976

  7. [7]

    R. L. Rivest, A. Shamir, and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM21, 120 (1978)

    work page 1978

  8. [8]

    P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, inProc. 35th Annu. Symp. Found. Comput. Sci.(1994) pp. 124–134

    work page 1994

  9. [9]

    Aruteet al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019)

    F. Aruteet al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019)

    work page 2019

  10. [10]

    C. H. Bennett, G. Brassard, and N. D. Mermin, Quantum cryptography without Bell’s theorem, Phys. Rev. Lett. 68, 557 (1992)

    work page 1992

  11. [11]

    P. W. Shor and J. Preskill, Simple proof of security of the BB84 quantum key distribution protocol, Phys. Rev. Lett.85, 441 (2000)

    work page 2000

  12. [12]

    Korzhet al., Provably secure and practical quantum key distribution over 307 km of optical fibre, Nat

    B. Korzhet al., Provably secure and practical quantum key distribution over 307 km of optical fibre, Nat. Photon.9, 163 (2015)

    work page 2015

  13. [13]

    Boaronet al., Secure quantum key distribution over 421 km of optical fiber, Phys

    A. Boaronet al., Secure quantum key distribution over 421 km of optical fiber, Phys. Rev. Lett.121, 190502 (2018)

    work page 2018

  14. [14]

    Pistoiaet al., Paving the way towards 800 Gbps quantum-secured optical channel deployment in mission- critical environments, Quantum Sci

    M. Pistoiaet al., Paving the way towards 800 Gbps quantum-secured optical channel deployment in mission- critical environments, Quantum Sci. Technol.8, 035015 (2023)

    work page 2023

  15. [15]

    S. P. Neumannet al., Continuous entanglement distribution over a transnational 248 km fiber link, Nat. Commun.13, 6134 (2022)

    work page 2022

  16. [16]

    Honjoet al., Long-distance entanglement-based quantum key distribution over optical fiber, Opt

    T. Honjoet al., Long-distance entanglement-based quantum key distribution over optical fiber, Opt. Express 16, 19118 (2008)

    work page 2008

  17. [17]

    Liaoet al., Satellite-to-ground quantum key distribution, Nature549, 43 (2017)

    S.-K. Liaoet al., Satellite-to-ground quantum key distribution, Nature549, 43 (2017)

    work page 2017

  18. [18]

    Bedington, J

    R. Bedington, J. M. Arrazola, and A. Ling, Progress in satellite quantum key distribution, npj Quantum Inf.3, 30 (2017)

    work page 2017

  19. [19]

    Schmitt-Manderbachet al., Experimental demonstration of free-space decoy-state quantum key distribution over 144 km, Phys

    T. Schmitt-Manderbachet al., Experimental demonstration of free-space decoy-state quantum key distribution over 144 km, Phys. Rev. Lett.98, 010504 (2007)

    work page 2007

  20. [20]

    Ursinet al., Entanglement-based quantum communication over 144 km, Nat

    R. Ursinet al., Entanglement-based quantum communication over 144 km, Nat. Phys.3, 481 (2007)

    work page 2007

  21. [21]

    Yinet al., Satellite-based entanglement distribution over 1200 kilometers, Science356, 1140 (2017)

    J. Yinet al., Satellite-based entanglement distribution over 1200 kilometers, Science356, 1140 (2017)

    work page 2017

  22. [22]

    Yinet al., Entanglement-based secure quantum cryptography over 1,120 kilometres, Nature582, 501 (2020)

    J. Yinet al., Entanglement-based secure quantum cryptography over 1,120 kilometres, Nature582, 501 (2020)

    work page 2020

  23. [23]

    A. K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett.67, 661 (1991)

    work page 1991

  24. [24]

    P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, New high-intensity source of polarization-entangled photon pairs, Phys. Rev. Lett. 75, 4337 (1995)

    work page 1995

  25. [25]

    Anwaret al., Bright source of degenerate polarization- entangled photons using type-0 PPKTP crystal: Effects of accidental coincidences, Opt

    A. Anwaret al., Bright source of degenerate polarization- entangled photons using type-0 PPKTP crystal: Effects of accidental coincidences, Opt. Commun.576, 131165 (2025)

    work page 2025

  26. [26]

    K. Park, J. Lee, D.-G. Im,et al., Ultrabright fiber-coupled polarization-entangled photon source with spectral brightness surpassing 2.0 MHz mW −1 nm−1, Adv. Photon. Res.5, 2400185 (2024)

    work page 2024

  27. [27]

    Hemmati, Deep-space optical communications design considerations, inProc

    H. Hemmati, Deep-space optical communications design considerations, inProc. IEEE, Vol. 94 (2006) pp. 2123– 2134

    work page 2006

  28. [28]

    Liet al., Microsatellite-based real-time quantum key 10 distribution, Nature640, 47 (2025)

    Y. Liet al., Microsatellite-based real-time quantum key 10 distribution, Nature640, 47 (2025)

    work page 2025

  29. [29]

    Zwoli´ nski, M

    W. Zwoli´ nski, M. Jarzyna, L. Kunz, M. Jachura, and K. Banaszek, Photon-efficient communication based on BPSK modulation with multistage interferometric receivers, inProc. 46th Eur. Conf. Opt. Commun. (ECOC)(2020) pp. We2F–5

    work page 2020

  30. [30]

    Banaszek and M

    K. Banaszek and M. Jachura, Structured optical receivers for efficient deep-space communication, inProc. IEEE Int. Conf. Satell. Opt. Syst. Appl. (ICSOS)(2017) pp. 34–37

    work page 2017

  31. [31]

    Sulimany, S

    H. Sulimany, S. Guha, T. Jennewein, and B. Bash, Optimization of entanglement-based satellite quantum key distribution, Phys. Rev. Appl.17, 064034 (2022)

    work page 2022

  32. [32]

    Yuet al., Quantum key distribution implemented with d-level time-bin entangled photons, Nat

    H. Yuet al., Quantum key distribution implemented with d-level time-bin entangled photons, Nat. Commun.16, 171 (2025)

    work page 2025

  33. [33]

    Sitet al., High-dimensional intracity quantum cryptography with structured photons, Optica4, 1006 (2017)

    A. Sitet al., High-dimensional intracity quantum cryptography with structured photons, Optica4, 1006 (2017)

    work page 2017

  34. [34]

    J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver, Nat. Photon.6, 374 (2012)

    work page 2012

  35. [35]

    Lo, Proof of unconditional security of six- state quantum key distribution scheme, Quantum Info

    H.-K. Lo, Proof of unconditional security of six- state quantum key distribution scheme, Quantum Info. Comput.1, 81 (2001)

    work page 2001

  36. [36]

    Scarani, A

    V. Scarani, A. Ac´ ın, G. Ribordy, and N. Gisin, Quantum cryptography protocols robust against photon number splitting attacks for weak laser pulse implementations, Phys. Rev. Lett.92, 057901 (2004)

    work page 2004

  37. [37]

    H.-L. Yin, Y. Fu, and Y. Mao, Security of quantum key distribution with multiphoton components, Sci. Rep.6, 29482 (2016)

    work page 2016

  38. [38]

    Devetak and A

    I. Devetak and A. Winter, Distillation of secret key and entanglement from quantum states, Proc. R. Soc. A, Math. Phys. Eng. Sci.461, 207 (2005)

    work page 2005

  39. [39]

    Wang, Quantum key distribution with asymmetric channel noise, Phys

    X.-B. Wang, Quantum key distribution with asymmetric channel noise, Phys. Rev. A71, 052328 (2005)

    work page 2005

  40. [40]

    Scarani, H

    V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Duˇ sek, N. L¨ utkenhaus, and M. Peev, The security of practical quantum key distribution, Rev. Mod. Phys. 81, 1301 (2009)

    work page 2009

  41. [41]

    Lo and H

    H.-K. Lo and H. F. Chau, Unconditional security of quantum key distribution over arbitrarily long distances, Science283, 2050 (1999)

    work page 2050

  42. [42]

    Jachura, M

    M. Jachura, M. Jarzyna, M. Paw lowski, and K. Banaszek, Photon-efficient quantum key distribution using multiqubit time-bin encoding, inProc. SPIE 11852, Int. Conf. Space Opt. (ICSO)(2021) p. 118525J

    work page 2021

  43. [43]

    Guha, Structured optical receivers to attain superadditive capacity and the Holevo limit, Phys

    S. Guha, Structured optical receivers to attain superadditive capacity and the Holevo limit, Phys. Rev. Lett.106, 240502 (2011)

    work page 2011

  44. [44]

    Verdu, On channel capacity per unit cost, IEEE Trans

    S. Verdu, On channel capacity per unit cost, IEEE Trans. Inf. Theory36, 1019 (1990)

    work page 1990

  45. [45]

    Jarzyna, Classical capacity per unit cost for quantum channels, Phys

    M. Jarzyna, Classical capacity per unit cost for quantum channels, Phys. Rev. A96, 032340 (2017)

    work page 2017

  46. [46]

    D. Ding, D. S. Pavlichin, and M. W. Wilde, Quantum channel capacities per unit cost, IEEE Trans. Inf. Theory 65, 418 (2019)

    work page 2019

  47. [47]

    Yehia, Y

    R. Yehia, Y. Pi´ etri, C. Pascual-Garc´ ıa, P. Lefebvre, and F. Centrone, Energetic analysis of emerging quantum communication protocols (2025), arXiv:2410.10661 [quant-ph]

    work page arXiv 2025

  48. [48]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford Univ. Press, Oxford, U.K., 2002)

    work page 2002