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Time-Switching Energy-Harvesting Quantum Relay-Assisted FSO Communication with Displaced Squeezed Signaling

Nizar Khalfet, Francisco A. Monteiro, Ioannis Krikidis

arXiv:2610.39091v1quant-phcs.ITmath.IT

TL;DR

Turbulence creates random loss that mixes squeezed states, so DSS relay reliability requires analysis beyond the lossless setting. The paper models a time-switching energy-harvesting quantum FSO relay and finds optimized squeezing preserves coherent relaying’s diversity order and leading constant, making its benefit finite-budget.

  • Problem

    Turbulence makes squeezed states mixed, so DSS relay reliability must be analyzed under mixed-state Helstrom detection rather than the known lossless setting.

    Source: §I · p. 2

  • Method

    The paper analyzes a time-switching decode-and-forward quantum FSO relay using optical energy harvesting, truncated Gamma-Gamma pure-loss hops, and coherent or displaced squeezed signaling.

    Source: §I · p. 2, §VI · p. 12

  • Results

    Optimized squeezing preserves coherent relaying’s turbulence-induced diversity order and leading constant, so its benefit is a finite-photon-budget effect.

    Source: §I · p. 2, §VI · p. 12

  • Takeaways & Limitations

    An interior harvesting fraction optimizes throughput, while the covariance-preserving benchmark’s absolute throughput gap vanishes as the relay photon budget grows.

    Source: §I · p. 2, §VI · p. 12

    In the illustrated case, DSS error is lower than coherent error only for η ≳ 0.6; under strong loss, vacuum noise cancels the squeezing gain.

    Source: §V · p. 10

Abstract

from arXiv · show

This paper investigates a quantum relay-assisted free-space optical (FSO) communication system in which a decode-and-forward relay is powered by time-switching optical energy harvesting and forwards the decoded information using displaced squeezed signaling (DSS). To obtain a physically con sistent description of turbulence-induced loss for non-classical signaling, each hop is modeled as a pure-loss bosonic chan nel whose random transmissivity follows a truncated Gamma Gamma law, i.e., the Gamma-Gamma irradiance conditioned on the physically admissible range of a transmittance. Under this channel, coherent states remain pure, whereas squeezed states are mapped to mixed Gaussian states. Under Helstrom-optimal detection, the average error of the coherent hop is derived in exact Laplace-series form. For the DSS hop, we derive the output states, a closed-form Gaussian fidelity with rigorous two sided bounds on the Helstrom error, and prove the existence of a loss-aware optimal photon partition. Furthermore, we prove that optimized squeezing preserves both the turbulence-induced diversity order and the leading asymptotic constant of coherent relaying, so that its benefit is a finite-photon-budget effect, and that an interior throughput-optimal harvesting fraction exists. A covariance-preserving benchmark with an exact series representation yields an analytical upper bound on the achievable throughput whose absolute gap vanishes as the relay photon budget grows. Numerical results based on the exact mixed-state Helstrom error quantify the benefit of squeezing and the regime in which it is useful.

I. INTRODUCTION

The paper develops a time-switching, energy-harvesting decode-and-forward quantum FSO framework to study turbulence, relay energy constraints, and displaced squeezed signaling together. It models turbulence as random pure loss and analyzes how that loss affects coherent and squeezed signals.

A. Source Signaling

  • DSS signaling combines coherent displacement and quadrature squeezing, adding a design degree of freedom beyond coherent-state relaying.The squeezing fraction allocates the relay photon budget between displacement and squeezing.Source: §I · p. 1

I. INTRODUCTION

  • The paper addresses a gap between relay-assisted quantum FSO studies with externally powered coherent relays, EH quantum architectures without relay-assisted turbulent links, and point-to-point squeezing studies.It studies the interplay of turbulence-induced fading, relay energy causality, and non-classical signaling.Source: §I · p. 2

B. FSO Channel Model

  • In the pure-loss model, coherent states remain pure, while squeezing produces mixed Gaussian states because vacuum enters through each loss port.The Gamma-Gamma irradiance is conditioned on the physically admissible transmissivity range and renormalized; applying this construction is a modeling approximation.Source: §I · p. 2, B. FSO Channel Model · p. 4

I. INTRODUCTION

  • The framework combines an optical energy-harvesting TS-DF relay, turbulence-impaired FSO hops, DSS signaling, and Helstrom-optimal detection.Each hop is modeled as a pure-loss bosonic channel with truncated Gamma-Gamma-distributed transmissivity.Source: §I · p. 2

II. SYSTEM MODEL

  • The relay harvests during fraction t of each block, then uses two sequential half-duplex forwarding slots of duration (1 − t)/2 each.Harvested energy is consumed within the same block, with no inter-block storage.Source: §II · p. 3

A. Source Signaling

  • The source uses equiprobable BPSK coherent states with mean photon number a per transmitted symbol.The source alphabet is {|+√a⟩, |−√a⟩}.Source: A. Source Signaling · p. 3

C. Time-Switching Energy Harvesting at the Relay

The relay harvests energy in a dedicated time-switching phase and spends it in the forwarding phase. It uses displaced squeezed states to allocate its relay photon budget between displacement and squeezing.

C. Time-Switching Energy Harvesting at the Relay

  • The relay converts optical power collected during the harvesting interval into electrical energy, then converts harvested energy into transmit power during its forwarding slot.The model uses photovoltaic and electro-optical conversion efficiencies.Source: C. Time-Switching Energy Harvesting at… · p. 4
  • The model assumes the EH phase spans many fading coherence intervals, making the harvested energy and relay photon budget deterministic.Harvesting within one coherence interval, which would make the budget random and correlated with first-hop transmissivity, is left for future work.Source: C. Time-Switching Energy Harvesting at… · p. 4
  • The harvesting fraction t increases the relay photon budget but shortens the time available for information forwarding.The EH phase is followed by a half-duplex relay slot of duration (1 − t)/2.Source: C. Time-Switching Energy Harvesting at… · pp. 4–5

D. Relay Signaling via DSS

  • After decoding, the relay re-encodes its decision as a displaced squeezed state, combining coherent displacement with quadrature squeezing.The phase-aligned squeezing axis matches the displacement quadrature.Source: D. Relay Signaling via DSS · p. 5
  • The squeezing fraction ζ partitions the fixed relay photon budget between displacement photons and photons generated by squeezing.Displacement supports BPSK signal-space separation, while squeezing reshapes quadrature noise.Source: D. Relay Signaling via DSS · p. 5

E. Receiver Processing and DF Operation

The relay and destination each perform binary quantum detection, with the relay hard-decoding and re-encoding its decision for the second hop. The paper formulates performance using the minimum error achievable under Helstrom detection.

E. Receiver Processing and DF Operation

  • On the first hop, the relay applies the Helstrom-optimal measurement to the source's binary coherent-state ensemble and makes a hard decision.The relay re-encodes that decision as DSS for the second hop, where the destination performs Helstrom-optimal detection.Source: E. Receiver Processing and DF Operation · p. 5

III. PROBLEM FORMULATION

A. Binary Quantum Detection and Helstrom Limit

  • For a binary quantum ensemble, the Helstrom bound is the minimum average error over physically admissible measurements.For pure hypotheses, optimal detection depends on the overlap between the candidate states.Source: A. Binary Quantum Detection and… · p. 6

B. Per-Hop Detection Errors

The analysis averages optimal per-hop detection errors over turbulence, then composes them into end-to-end decode-and-forward reliability. The receiver is assumed to know the instantaneous transmissivity.

B. Per-Hop Detection Errors

  • The first hop carries coherent BPSK states, which remain pure coherent states conditioned on transmissivity, and its average error is obtained by averaging over the truncated irradiance law.The source photon number is fixed, so the first-hop average error does not depend on harvesting fraction t.Source: B. Per-Hop Detection Errors · p. 6
  • The second hop carries DSS states whose parameters depend on the harvesting fraction and squeezing allocation, and its conditional error is averaged over transmissivity.The relay photon budget is determined by energy harvesting.Source: B. Per-Hop Detection Errors · p. 6
  • The Helstrom measurement uses the instantaneous transmissivity known at the receiver, so the reported per-hop errors average conditional optimal errors.This receiver-knowledge assumption applies to the second-hop error described in the passages.Source: B. Per-Hop Detection Errors · p. 6

C. End-to-End Reliability

  • Under independent hops and hard decoding, end-to-end error follows the classical decode-and-forward composition rule.A first-hop decision error can affect the symbol forwarded to the destination.Source: C. End-to-End Reliability · p. 6

D. Effective Throughput

The section analyzes per-hop Helstrom errors and the effective throughput of the time-switching decode-and-forward relay. It formulates the joint harvesting-fraction and squeezing-allocation optimization, balancing relay energy against transmission time.

D. Effective Throughput

  • The throughput optimization jointly selects harvesting fraction t and squeezing allocation ζ, balancing the relay photon budget against the transmission time.The relay photon budget is N2(t), and the time-switching fraction reduces the time available for transmission.Source: D. Effective Throughput · p. 7, E. Optimal Resource Allocation · p. 7

IV. HELSTROM ERROR ANALYSIS AND RATE OPTIMIZATION

  • The analysis derives per-hop Helstrom errors and structural and asymptotic properties of effective throughput for the proposed TS-DF quantum relay.It covers first-hop error, the mixed-state second hop, fidelity bounds, coherent-relaying and CP-benchmark series, and throughput structure.Source: §IV · p. 7

A. First-Hop Helstrom Error

  • The first-hop coherent-state overlap yields conditional and average Helstrom errors, and the first-hop average error does not depend on harvesting fraction t.The average error has a series representation in Section IV-E.Source: A. First-Hop Helstrom Error · p. 7

B. Second Hop: Exact Mixed-State Formulation

  • For squeezed inputs, a nontrivial lossy second hop produces mixed Gaussian states, whereas coherent inputs remain pure.The DSS output is pure only when r = 0 or η is 0 or 1; for r > 0 and 0 < η < 1 it is strictly mixed.Source: B. Second Hop: Exact Mixed-State… · p. 7

C. Closed-Form Fidelity and Two-Sided Bounds

  • The section gives a closed-form Gaussian fidelity and rigorous two-sided Helstrom-error bounds, but the bounds are not tight for mixed states.The fidelity depends on the squeezed-quadrature variance, and the squeezing design uses the exact trace-norm error.Source: C. Closed-Form Fidelity and Two-Sided… · p. 8
  • The fidelity-based lower bound is exact when the output states are pure, including η = 1 or r = 0.For mixed states, the bounds are used for asymptotic analysis rather than squeezing-allocation design.Source: C. Closed-Form Fidelity and Two-Sided… · p. 8

D. Loss-Aware Photon Allocation

The section establishes a loss-aware squeezing allocation, exact Laplace-series representations for coherent errors, and a covariance-preserving benchmark. The benchmark gives an analytical throughput upper bound whose absolute gap vanishes as the relay photon budget grows.

D. Loss-Aware Photon Allocation

  • A loss-aware allocation minimizing the average second-hop Helstrom error exists for every t in (0, 1), but is not claimed to be unique or available in closed form.It is computed by one-dimensional numerical minimization of the exact averaged error.Source: D. Loss-Aware Photon Allocation · p. 8

E. Exact Series Representation

  • The coherent first-hop and second-hop average errors have exact Laplace-series representations with a bound on the truncation correction.Without the correction term, the series is an upper bound; the relative effect of the correction vanishes as Θ grows.Source: E. Exact Series Representation · p. 8

F. Covariance-Preserving Benchmark

  • The covariance-preserving benchmark is a rigorous analytical upper bound on achievable throughput, and its absolute gap vanishes at the coherent second-hop error rate.It is a benchmark rather than the output of the physical pure-loss channel because it ignores vacuum noise entering the loss port.Source: F. Covariance-Preserving Benchmark · p. 8

G. Asymptotic Reliability under Loss

Under physical loss, optimized squeezing preserves coherent relaying’s second-hop diversity order and leading asymptotic constant. The throughput analysis establishes an interior optimal harvesting fraction and a covariance-preserving benchmark that upper-bounds achievable throughput with a vanishing absolute gap at large photon budgets.

G. Asymptotic Reliability under Loss

  • As N2 grows, optimized squeezing changes neither coherent relaying’s diversity order g2 nor its leading constant K̄, and every minimizing squeezing fraction tends to zero.The stated asymptotic assumption excludes α2 = β2.Source: G. Asymptotic Reliability under Loss · p. 9
  • Vacuum noise entering the loss port caps the effective squeezing exponent, making squeezing’s benefit a finite-photon-budget effect.The paper contrasts physical loss with preserved squeezing, which would double the diversity order.Source: G. Asymptotic Reliability under Loss · p. 9

H. End-to-End Error Probability and Effective Throughput

  • Because end-to-end error increases strictly with second-hop Helstrom error, reducing that hop’s error improves end-to-end reliability.Source: H. End-to-End Error Probability and… · p. 9
  • The optimized effective throughput attains a maximum at some interior harvesting fraction t⋆ ∈ (0, 1), but uniqueness is not claimed.The throughput approaches 1/4 as t approaches 0 and 0 as t approaches 1, and exceeds 1/4 for sufficiently small positive t.Source: H. End-to-End Error Probability and… · p. 9
  • The CP benchmark is a rigorous throughput upper bound whose absolute gap vanishes at the coherent second-hop error’s polynomial rate as N2 grows.The throughput ratio R2^cp/R2⋆ tends to 1, although the benchmark is not tight relative to the second-hop error.Source: H. End-to-End Error Probability and… · p. 9

V. NUMERICAL RESULTS AND DESIGN INSIGHTS

The numerical results validate the mixed-state Helstrom analysis and show that loss-aware squeezing is useful only in restricted operating regimes. They also illustrate the throughput-optimal harvesting design and the asymptotic tightness of the CP throughput bound.

V. NUMERICAL RESULTS AND DESIGN INSIGHTS

  • For ζ = 0.3, the DSS error is below coherent error only for η ≳ 0.6; at stronger loss it exceeds coherent error, while the CP error is only a benchmark.For coherent signaling, the fidelity lower bound equals the exact error; for DSS, the exact error lies between the bounds, but the lower bound is loose.Source: Figure 2 · p. 10, §V · p. 10
  • The loss-aware minimizer ζ⋆(t) is small and far below the lossless rule, which can yield greater error than coherent relaying.Beyond ζ⋆, the error increases; the gain is capped by vacuum noise entering the loss port.Source: Figure 3 · p. 10, §V · p. 10
  • Optimized squeezing stays small, tends toward zero as N2 grows, and reduces error most at moderate budgets and mild loss.The reduction is modest for the Table I turbulence, but reaches 10–30% in weak turbulence; strong turbulence concentrates error on deep fades where squeezing is ineffective.Source: §V · p. 10, Figure 4 · p. 11
  • Normalized by the coherent asymptotic law, coherent and optimized DSS errors both tend to 1, while the CP benchmark decays relative to that reference.Optimized squeezing therefore changes neither the diversity order nor the leading constant of coherent relaying.Source: Figure 5 · p. 11, §V · p. 11
  • Throughput is unimodal with an interior maximum at a small harvesting fraction, and DSS throughput is nearly indistinguishable from coherent throughput.The squeezing gain is positive only in a thin region of small ζ and small t; beyond that region, over-squeezing reduces throughput below coherent relaying.Source: Figure 6 · p. 11, §V · p. 11, Figure 7 · p. 12
  • The CP benchmark is a rigorous throughput upper bound whose absolute gap vanishes as the photon budget grows.The gap is larger at small budgets for heavier loss and decays at the rate of the coherent second-hop error.Source: §V · p. 11, Figure 8 · p. 12

VI. CONCLUSION

  • The paper models each hop as a pure-loss bosonic channel with truncated Gamma-Gamma transmissivity, yielding pure coherent outputs and mixed Gaussian DSS outputs.It derives DSS fidelity and Helstrom-error bounds, coherent average-error Laplace series, and existence results for loss-aware squeezing and throughput-optimal harvesting.Source: §VI · p. 12
  • Future extensions include receiver noise or thermal background, practical non-Gaussian receivers, correlated two-hop fading, and link-distance effects on end-to-end reliability.Source: §VI · p. 12

APPENDIX A PROOF OF LEMMA 1

The appendices characterize the relay’s squeezed-state output under loss, bound its Helstrom error through Gaussian fidelity, and establish that a minimizing photon partition exists. They also identify when the output is pure and recover the coherent-state case.

APPENDIX A PROOF OF LEMMA 1

  • For squeezed inputs, the output is pure only when r = 0 or η ∈ {0, 1}; at r = 0 it is a coherent state.The covariance determinant is at least 1, with equality exactly in those cases.Source: Appendix A · p. 12

APPENDIX B PROOF OF PROPOSITION 1

  • A common unitary maps the two output states to an isotropic common covariance, yielding a closed-form Gaussian fidelity and two-sided Helstrom-error bounds.The fidelity bounds are equalities for lossless transmission and for coherent signaling.Source: Appendix B · p. 13

APPENDIX C PROOF OF PROPOSITION 2

  • For each fixed harvesting fraction, the average error is continuous in the photon partition, so a smallest minimizing partition exists.The fading law is independent of the partition, and the continuous objective attains its minimum on the compact interval.Source: Appendix C · p. 13

APPENDIX D PROOF OF PROPOSITION 3

The appendix expands the error kernel into a positive Laplace series and averages it using the truncated fading law. It also derives the high-argument Laplace-transform behavior from the Gamma-Gamma density near zero, excluding the equal-parameter case.

APPENDIX D PROOF OF PROPOSITION 3

  • The kernel has a positive-coefficient Laplace expansion, which can be averaged term by term to obtain a series using the truncated Laplace transform.The transform is bounded by ((1 − F_i)/F_i)e^-s for s ≥ 0.Source: Appendix D · p. 13

APPENDIX E PROOF OF LEMMA 2

  • For α ≠ β, the Gamma-Gamma density near zero yields a power-law asymptotic for its Laplace transform as s grows.The density’s leading behavior is proportional to x^(g−1), with g = min{α, β}.Source: Appendix E · pp. 13–14
  • The α = β case is excluded because the small-argument behavior of K_0 adds a logarithmic factor.Source: Appendix E · p. 14

APPENDIX F PROOF OF LEMMA 3

The appendices establish a power-law asymptotic for the averaged error, show that optimized squeezing preserves the coherent relay’s leading behavior at high photon budgets, and prove that throughput has an interior maximizing harvesting fraction.

APPENDIX F PROOF OF LEMMA 3

  • The averaged error kernel satisfies a power-law asymptotic in Θ, with exponent g and a constant determined by the fading parameters and series coefficients.The proof controls the series uniformly before taking the limit.Source: Appendix F · p. 14

APPENDIX G PROOF OF THEOREM 1

  • At high photon budgets, the optimized error is asymptotic to the zero-squeezing error, and the minimizing squeeze fraction tends to zero.Both errors scale as K̄(4hN)^−g.Source: Appendix G · p. 14

APPENDIX H PROOF OF PROPOSITION 4

  • The optimized throughput extends continuously to endpoint values 1/4 and 0, exceeds 1/4 for sufficiently small harvesting fractions, and therefore attains a maximum at an interior fraction.Source: Appendix H · pp. 14–15

APPENDIX I PROOF OF PROPOSITION 5

The appendix identifies the photon partition that maximizes input distinguishability and compares its covariance-preserving benchmark with optimized displaced squeezed signaling. The benchmark error decays faster, while their difference has the optimized error’s leading asymptotic order.

APPENDIX I PROOF OF PROPOSITION 5

  • The input distinguishability is maximized at ζ_L = N_2/(2N_2 + 1), giving γ^2 = N_2(N_2 + 1)/(2N_2 + 1) and Θ_in = 4N_2(N_2 + 1).The covariance-preserving error decreases with Θ_in, so this partition minimizes that benchmark error.Source: Appendix I · p. 15
  • The covariance-preserving benchmark error is o(N_2^−g_2), while its difference from the optimized squeezed-signaling error is asymptotically equivalent to K̄(4hN_2)^−g_2.Source: Appendix I · p. 15
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