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arXiv:2610.13963 · Advanced Quantum Technologies

Detector-Frame Quantum Limits for Heterodyne Timing in Relativistic Optical Links

Salman Sajad Wani, Aatif Kaisar Khan, Saif Al-Kuwari, Mir Faizal

quant-phSubmitted Sep 7, 2026• Announced Sep 9, 2026• 0 citations
arXivHTMLPDF

Abstract

Long-baseline optical links for space interferometry and satellite quantum networks rely on heterodyne phase tracking under Doppler motion and gravitational frequency shifts. These relativistic shifts can generate large detector-frame beat-frequency offsets over finite readout windows and thereby constrain the temporal precision of front-end phase readout. We develop a detector-frame framework for this problem using an effective two-mode signal--local-oscillator model in which relativistic motion rescales the local-oscillator frequency. Within this setting we derive closed-form Mandelstam--Tamm and Margolus--Levitin bounds for representative Gaussian resources and connect global state distinguishability with the operational precision of heterodyne phase tracking. In the bright-local-oscillator regime we identify a geometric--operational mismatch: the joint optical state can evolve on arbitrarily short global timescales while the relevant information rate is set by the measured detector record. Combining this result with an open-loop transient-acquisition analysis, we show that white phase noise and relativistic beat-frequency drift that is approximately constant over the acquisition window generate a cube-root scaling benchmark for the optimal pre-lock open-loop integration window and the minimum residual phase error. These results establish detector-frame benchmarks for quantum-limited heterodyne timing in relativistic optical links and clarify how global dynamical bounds enter task-restricted continuous optical measurements.

Journal reference
Advanced Quantum Technologies 9 (2026), e70446
DOI
10.1002/qute.70446
Comments
36 pages, 1 figure. Published in Advanced Quantum Technologies

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Operational Identity under Boosts - Classical Transparency and a Reduced-Spin Accessibility Tax

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This paper investigates whether relativistic motion imposes an intrinsic cost on certifying a system’s identity. Identity refers here to internal physical parameters, such as a spectroscopic splitting, composition marker, or spin orientation, rather than to Poincaré-invariant labels such as mass and total spin. The central result is that a fixed Lorentz boost does not destroy identity information. A fixed, parameter-independent boost acts unitarily on the complete physical state, so the global quantum Fisher information associated with an internal parameter is exactly preserved. Any apparent loss of identity information must therefore arise from restricted measurements, finite detector resources, discarded degrees of freedom, incomplete knowledge of the boost, or averaging over transformations that are not recorded. The paper develops this conclusion through classical spectroscopy, relativistic flyby estimation, photon frequency encoding, and massive spin-momentum systems. 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Relativistic flyby self-calibration The paper considers a source undergoing a relativistic flyby with unknown velocity, impact parameter, time of closest approach, source amplitude, rest transition frequency, and spectral splitting. The changing line-of-sight velocity generates a Doppler chirp. The same photons can therefore constrain both the trajectory and the source’s internal identity. For a symmetric doublet observed over a window that samples both sides of the encounter, the variance penalty from marginalizing over the unknown trajectory has the form variance inflation = 1 + O[(Δ/ν₀)²], where Δ is the rest-frame splitting and ν₀ is the central rest frequency. The trajectory is determined primarily through the much larger frequency scale ν₀, while the identity parameter is encoded in the smaller separation Δ. For the numerical examples studied, this produces a Fisher-information hierarchy of approximately one million and a trajectory-marginalization penalty below approximately 0.1 percent. The result remains valid when the rest transition frequency is treated as unknown. Two mechanisms make self-calibration possible. First, the ratio of the early and late Doppler asymptotes constrains the source velocity without requiring an external frequency reference. Second, changing the central frequency translates both spectral lines together, while changing the splitting moves them symmetrically apart. For a symmetric doublet, these score directions are approximately orthogonal. The self-calibration result is conditional on adequate signal strength, a resolved spectral structure, a sufficiently long observing window, and a nonsingular nuisance-parameter Fisher matrix. Retarded-time flyby calculation The paper explicitly includes the retarded relation between emission time and observation time. This confirms the small trajectory-marginalization penalty but changes the interpretation of total information accumulation. The blueshifted approach phase is compressed into a short interval of observer time. As a result, an ideal detector can collect less total information at higher velocity, while a resolution-limited detector can benefit when blueshift moves the doublet above its instrumental resolution floor. Whether relativistic motion helps or hinders estimation is therefore protocol-dependent. Quantum frequency encoding For a single photon whose identity parameter is encoded in its frequency wavefunction, a collinear Lorentz boost acts as a unitary frequency dilation. The global quantum Fisher information about the encoded parameter is exactly invariant. This result applies to the complete frequency state. It does not imply that every reduced photonic measurement remains invariant, particularly when momentum and polarization are coupled or when the detector accesses only a restricted measurement algebra. Massive carriers and reduced-spin accessibility For a massive spin-1/2 carrier, a Lorentz boost generates momentum-dependent Wigner rotations. The complete spin-momentum state preserves its global quantum Fisher information. However, a detector that measures spin while ignoring momentum accesses only the reduced spin state. Tracing over momentum can therefore reduce the spin-only quantum Fisher information. For canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and the spin-angle encoding studied in the paper, the leading fractional accessibility loss is proportional to [2/(1 + cos²θ)] × (σₚ/mc)² × tanh²(ξ/2), where θ is the encoding angle, σₚ is the momentum spread, m is the carrier mass, and ξ is the boost rapidity. The accessibility tax has three important properties. It depends on the spin-encoding direction. It increases with momentum spread. It saturates at high rapidity because tanh²(ξ/2) approaches one. The reduced-spin information loss therefore does not grow without bound as the relative velocity approaches the speed of light. The missing information is recoverable in principle. It has been transferred into momentum dependence and spin-momentum correlations. A joint measurement of spin and momentum restores access to the invariant global information. The effect is therefore a subsystem-accessibility cost, not an irreversible destruction of identity. Localization and action-count scaling For a minimum-uncertainty wave packet, σₓσₚ = ℏ/2. The Compton wavelength is λ_C = ℏ/(mc). Using the localization scale R = σₓ, the rest-frame action-count is Ξ = ER/(ℏc) = mcσₓ/ℏ = σₓ/λ_C. At large rapidity, the reduced-spin accessibility tax scales as Ξ⁻², with an additional factor determined by the spin-encoding angle. A carrier localized over a distance much larger than its Compton wavelength therefore pays a negligible reduced-spin accessibility cost. For packets that do not saturate the uncertainty relation, the same argument gives a bound rather than an equality. Energy and identity information The paper asks whether the kinetic energy required to match the source frame can be interpreted as the price of recovering identity information. Three arguments reject a universal proportionality. First, frame-matching energy grows without bound with the Lorentz factor, while the reduced-spin accessibility loss saturates. Second, frame matching is not necessary. Classical Doppler self-calibration can recover a spectral identity parameter without accelerating the detector into the source frame. In the quantum case, expanding the measured algebra to include momentum can recover the lost spin-only information. Third, kinetic energy invested in frame matching can in principle be recovered by later deceleration, while the acquired measurement record remains. Energy constrains storage capacity, evolution speed, and thermodynamic reset costs, but the paper finds no universal energy price per unit of Fisher information acquired. Main conclusions The paper establishes that: Fixed, parameter-independent Lorentz boosts preserve global quantum Fisher information. Per-photon spectral identity information is invariant under ideal Doppler dilation. Total information remains detector- and protocol-dependent. Relativistic flyby chirps can self-calibrate unknown source kinematics. Trajectory marginalization produces only a small identity-estimation penalty for well-resolved symmetric doublets. Reduced-spin measurements can lose access to information through momentum-dependent Wigner rotations. The reduced-spin accessibility tax depends on encoding angle and saturates with rapidity. Joint spin-momentum measurements restore the invariant global information. For minimum-uncertainty packets, the reduced-spin tax scales as Ξ⁻². No universal proportionality exists between frame-matching energy and identity information. Scope The global invariance theorem applies to fixed Lorentz transformations that do not depend on the identity parameter. The paper does not analyze unknown boosts as joint quantum-estimation parameters or random boost distributions that are averaged without being recorded. The reduced-spin result is derived for a single massive spin-1/2 carrier, canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and a momentum-insensitive spin measurement. Broad or anisotropic packets, composite carriers, alternative relativistic spin observables, generic reduced photonic channels, and sequential non-collinear boosts remain open extensions. The classical flyby conclusions apply to high-signal, sufficiently resolved, symmetric spectral doublets observed over an adequate time window. Relation to the companion paper The companion paper examines the accelerated rather than inertial case: Action-Count, Observer Integrity, and Horizon Saturation: A Unified Framework from Ξ = ER/(ℏc)https://doi.org/10.5281/zenodo.19599288 Together, the two papers distinguish the operational effects of velocity and acceleration. Uniform velocity preserves global identity information but can redistribut

•Jul 13, 2026•0 citations
Long-baseline optical links for space interferometry and satellite quantum networks rely on heterodyne phase tracking under Doppler motion and gravitational frequency shifts. These relativistic shifts can generate large detector-frame beat-frequency offsets over finite readout windows and thereby constrain the temporal precision of front-end phase readout. We develop a detector-frame framework for this problem using an effective two-mode signal--local-oscillator model in which relativistic motion rescales the local-oscillator frequency. Within this setting we derive closed-form Mandelstam--Tamm and Margolus--Levitin bounds for representative Gaussian resources and connect global state distinguishability with the operational precision of heterodyne phase tracking. In the bright-local-oscillator regime we identify a geometric--operational mismatch: the joint optical state can evolve on arbitrarily short global timescales while the relevant information rate is set by the measured detector record. Combining this result with an open-loop transient-acquisition analysis, we show that white phase noise and relativistic beat-frequency drift that is approximately constant over the acquisition window generate a cube-root scaling benchmark for the optimal pre-lock open-loop integration window and the minimum residual phase error. These results establish detector-frame benchmarks for quantum-limited heterodyne timing in relativistic optical links and clarify how global dynamical bounds enter task-restricted continuous optical measurements.
Timing and phase resolution in satellite QKD, kilometre-scale gravitational-wave detectors, and space-borne clock networks hinge on quantum-speed limits (QSLs), yet benchmarks omit relativistic effects for coherent and squeezed probes. We derive first-order relativistic corrections to the Mandelstam-Tamm and Margolus-Levitin bounds. Starting from the Foldy-Wouthuysen expansion and treating −p4/(8m3c2)-p^{4}/(8 m^{3} c^{2})−p4/(8m3c2) as a harmonic-oscillator perturbation, we propagate Gaussian states to obtain closed-form QSLs and the quantum Cramér-Rao bound. Relativistic kinematics slow evolution in an amplitude- and squeezing-dependent way, increase both bounds, and introduce an ε2t2ε^{2} t^{2}ε2t2 phase drift that weakens timing sensitivity while modestly increasing the squeeze factor. A single electron (ε≈1.5×10−10ε\approx 1.5\times 10^{-10}ε≈1.5×10−10) in a 5.4 T5.4\,\mathrm{T}5.4T Penning trap, read out with 149 GHz149\,\mathrm{GHz}149GHz quantum-limited balanced homodyne, should reveal this drift within ∼15 min\sim 15\,\mathrm{min}∼15min -- within known hold times. These results benchmark relativistic corrections in continuous-variable systems and point to an accessible test of the quantum speed limit in high-velocity or strong-field regimes.
Summary form only given. Quantum key distribution (QKD) has raised increased attention over the past years as one of the most attractive quantum technologies for practical implementation. QKD has already been implemented in intra-city networks all around the world. But up to now, bridging global distances with quantum communication remains an outstanding challenge. A promising candidate to provide this link is via optical satellite communication. As space-to-ground communication is already well developed for classical applications, one can make use of the already existing technology for QKD, i.e. modern Laser Communication Terminals (LCTs) may be adapted for quantum communication. An important first step to achieve this goal is a precise characterization of the system and the channel with regard to their quantum noise behaviour. We present quantum-limited measurements of signals from an LCT located on the satellite Alphasat in geostationary Earth orbit. The LCT generates binary phase shift keyed (BPSK) coherent states originally used for classical data communication. Our receiver is a Transportable Adaptive Optical Ground Station (T-AOGS) located at the Teide observatory on Tenerife. It corrects for phase front distortions in order to interfere the incoming signal with a local oscillator. This allows for heterodyne detection or, by phase-locking the local oscillator, for homodyne detection. The detector itself is quantum noise limited such that it can resolve the quantum uncertainty of the detected states (see Fig. 1). Evaluating the signal variance for homodyne detection, we are able to show that the detected states are quantum limited coherent states. Thus, we conclude that the coherence of the quantum states is preserved after propagation over a distance of 38600 km including the passage through turbulent atmosphere and we give an upper bound for any excess noise that the states might have acquired [1]. Moreover, we extent our analysis to heterodyne detection, characterizing arbitrary field quadratures. Our results indicate that satellite quantum communication is feasible in principle in such a scenario.
Optical very-long-baseline-interferometry (VLBI) aims to increase angular resolution capabilities beyond those of the radio regime and enable advanced spectroscopic performance. VLBI requires a distributed network of coherence to properly synthesize the apertures together. Fiber optics have revolutionized precision frequency-and-time transfer for applications such as communication, ranging, and timing and offer a platform for distributing coherent optical references for VLBI. However, signals transmitted through optical fibers are subject to the noise environment in which the fiber is laid. This is particularly problematic for precision optical frequency transfer over long distances and for precision timing across large baselines for applications such as VLBI. To combat the added phase noise due to fiber-piston action and refractive index changes in optical fibers, a phase cancellation method is required. In this thesis, we detail the evolution of a one-way optical frequency stabilization technique to increase the stability by limiting the out-of-loop path in a traditional phase-tracking heterodyne interferometer. Each apparatus’s stability was first tested over a 4.09 km SMF-28 fiber spool. The second and third apparatus iterations were individually deployed over a telecom SMF-28 fiber link of 2923 m in round-trip length. The final evolution involved a custom micro-optic component to reduce the out-of-loop to 1.4 cm and achieved in-loop instability of 4x10-19 at 1 ks and a worst-case out-of-loop timing-jitter below 200 attoseconds for up to 1 ks over the telecom link. The performance of the final apparatus is more than sufficient to distribute a coherent optical reference to perform distributed thermal heterodyne measurements.
Accurate timekeeping is essential for scientific and technological advancements, particularly in areas of communication, networking, navigation, and high precision measurements. While many methods of time resolution are already established using classical resources, they fail to combine high precision outcomes with large-scale implementations. Additionally, numerous quantum networking architectures using satellite-assisted methods have demonstrated quantum communication on scales exceeding ground-based methods. For these reasons, we propose the use of a time synchronization method by which pairs of highly time-correlated photons are exchanged between clocks on satellites and clocks on Earth, al- lowing users to reconstruct the time offsets between their clocks. This is known as Quantum Clock Synchronization (QCS). We analyze global time synchronization, via numerical simulation, using a network of satellites equipped with quantum resources, such as entangled photon sources (SPDC) and single photon detectors. We find that for satellites in low-earth orbits, a configuration of 50 evenly-space satellites provides constant visibility and sub-nanosecond time synchronization to any location on Earth. Due to practical constraints, clocks available for use on satellites are often of lesser quality than those available for ground use. However, by allowing the satellites in our network to synchronize among themselves, we overcome this limitation, forming a highly precise “master clock” capable of global, high precision time synchronization. We then examine the amount of quantum security provided by polarization-entangled photon sources as a potential way to protect an experimental implementation of QCS from attacks such as signal spoofing. To do this, we identify regions on Earth where Bell violations between exchanged entangled photons can ensure quantum security at various confidence vii levels. Combined with similar insights gained from determining the network scale of our QCS protocol at the single-link level, we demonstrate the potential for highly precise, global, and potentially quantum-secure clock synchronization. Finally, we modify our QCS protocol to support frequency, or tick-rate, offsets in addition to constant time offsets. We motivate this by presenting some relativistic effects that can introduce frequency differences between clocks in space and on Earth. Provided that relative velocities between clocks are corrected for, we identify constant and variable clock defects through numerical simulations.
The fields of precision timekeeping and spectroscopy increasingly rely on optical-frequency-comb interferometry. However, comb-based measurements are not described by existing quantum theory because they exhibit both large mode mismatch and finite-strength local oscillators. To establish this quantum theory, we derive measurement operators for homodyne detection with arbitrary mode overlap. These operators are a combination of quadrature and intensity-like measurements, which inform a filter that maximizes the quadrature-measurement signal-to-noise ratio. Furthermore, these operators establish a foundation to extend frequency-comb interferometry to a wide range of scenarios, including metrology with nonclassical states of light.
This paper investigates whether relativistic motion imposes an intrinsic cost on certifying a system’s identity. Identity refers here to internal physical parameters, such as a spectroscopic splitting, composition marker, or spin orientation, rather than to Poincaré-invariant labels such as mass and total spin. The central result is that a fixed Lorentz boost does not destroy identity information. A fixed, parameter-independent boost acts unitarily on the complete physical state, so the global quantum Fisher information associated with an internal parameter is exactly preserved. Any apparent loss of identity information must therefore arise from restricted measurements, finite detector resources, discarded degrees of freedom, incomplete knowledge of the boost, or averaging over transformations that are not recorded. The paper develops this conclusion through classical spectroscopy, relativistic flyby estimation, photon frequency encoding, and massive spin-momentum systems. Classical spectral identity A Doppler transformation uniformly rescales the line center, line width, and internal splitting of a spectrum. For a spectral doublet, the dimensionless ratio between the splitting and the line center remains unchanged. The per-photon Fisher information about the rest-frame splitting is therefore invariant under an ideal Doppler dilation. Relativistic motion can still change the total information collected because photon number, detector resolution, observing duration, and spectral bandwidth depend on the measurement protocol. The paper compares three detector classes: An ideal spectrometer with unlimited bandwidth A spectrometer with fixed observer-frame resolution A resolution-limited detector with a finite observing band These detectors produce different information-versus-velocity laws. Relativity therefore does not impose a universal identity-collapse exponent independently of the detector and resource convention. Relativistic flyby self-calibration The paper considers a source undergoing a relativistic flyby with unknown velocity, impact parameter, time of closest approach, source amplitude, rest transition frequency, and spectral splitting. The changing line-of-sight velocity generates a Doppler chirp. The same photons can therefore constrain both the trajectory and the source’s internal identity. For a symmetric doublet observed over a window that samples both sides of the encounter, the variance penalty from marginalizing over the unknown trajectory has the form variance inflation = 1 + O[(Δ/ν₀)²], where Δ is the rest-frame splitting and ν₀ is the central rest frequency. The trajectory is determined primarily through the much larger frequency scale ν₀, while the identity parameter is encoded in the smaller separation Δ. For the numerical examples studied, this produces a Fisher-information hierarchy of approximately one million and a trajectory-marginalization penalty below approximately 0.1 percent. The result remains valid when the rest transition frequency is treated as unknown. Two mechanisms make self-calibration possible. First, the ratio of the early and late Doppler asymptotes constrains the source velocity without requiring an external frequency reference. Second, changing the central frequency translates both spectral lines together, while changing the splitting moves them symmetrically apart. For a symmetric doublet, these score directions are approximately orthogonal. The self-calibration result is conditional on adequate signal strength, a resolved spectral structure, a sufficiently long observing window, and a nonsingular nuisance-parameter Fisher matrix. Retarded-time flyby calculation The paper explicitly includes the retarded relation between emission time and observation time. This confirms the small trajectory-marginalization penalty but changes the interpretation of total information accumulation. The blueshifted approach phase is compressed into a short interval of observer time. As a result, an ideal detector can collect less total information at higher velocity, while a resolution-limited detector can benefit when blueshift moves the doublet above its instrumental resolution floor. Whether relativistic motion helps or hinders estimation is therefore protocol-dependent. Quantum frequency encoding For a single photon whose identity parameter is encoded in its frequency wavefunction, a collinear Lorentz boost acts as a unitary frequency dilation. The global quantum Fisher information about the encoded parameter is exactly invariant. This result applies to the complete frequency state. It does not imply that every reduced photonic measurement remains invariant, particularly when momentum and polarization are coupled or when the detector accesses only a restricted measurement algebra. Massive carriers and reduced-spin accessibility For a massive spin-1/2 carrier, a Lorentz boost generates momentum-dependent Wigner rotations. The complete spin-momentum state preserves its global quantum Fisher information. However, a detector that measures spin while ignoring momentum accesses only the reduced spin state. Tracing over momentum can therefore reduce the spin-only quantum Fisher information. For canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and the spin-angle encoding studied in the paper, the leading fractional accessibility loss is proportional to [2/(1 + cos²θ)] × (σₚ/mc)² × tanh²(ξ/2), where θ is the encoding angle, σₚ is the momentum spread, m is the carrier mass, and ξ is the boost rapidity. The accessibility tax has three important properties. It depends on the spin-encoding direction. It increases with momentum spread. It saturates at high rapidity because tanh²(ξ/2) approaches one. The reduced-spin information loss therefore does not grow without bound as the relative velocity approaches the speed of light. The missing information is recoverable in principle. It has been transferred into momentum dependence and spin-momentum correlations. A joint measurement of spin and momentum restores access to the invariant global information. The effect is therefore a subsystem-accessibility cost, not an irreversible destruction of identity. Localization and action-count scaling For a minimum-uncertainty wave packet, σₓσₚ = ℏ/2. The Compton wavelength is λ_C = ℏ/(mc). Using the localization scale R = σₓ, the rest-frame action-count is Ξ = ER/(ℏc) = mcσₓ/ℏ = σₓ/λ_C. At large rapidity, the reduced-spin accessibility tax scales as Ξ⁻², with an additional factor determined by the spin-encoding angle. A carrier localized over a distance much larger than its Compton wavelength therefore pays a negligible reduced-spin accessibility cost. For packets that do not saturate the uncertainty relation, the same argument gives a bound rather than an equality. Energy and identity information The paper asks whether the kinetic energy required to match the source frame can be interpreted as the price of recovering identity information. Three arguments reject a universal proportionality. First, frame-matching energy grows without bound with the Lorentz factor, while the reduced-spin accessibility loss saturates. Second, frame matching is not necessary. Classical Doppler self-calibration can recover a spectral identity parameter without accelerating the detector into the source frame. In the quantum case, expanding the measured algebra to include momentum can recover the lost spin-only information. Third, kinetic energy invested in frame matching can in principle be recovered by later deceleration, while the acquired measurement record remains. Energy constrains storage capacity, evolution speed, and thermodynamic reset costs, but the paper finds no universal energy price per unit of Fisher information acquired. Main conclusions The paper establishes that: Fixed, parameter-independent Lorentz boosts preserve global quantum Fisher information. Per-photon spectral identity information is invariant under ideal Doppler dilation. Total information remains detector- and protocol-dependent. Relativistic flyby chirps can self-calibrate unknown source kinematics. Trajectory marginalization produces only a small identity-estimation penalty for well-resolved symmetric doublets. Reduced-spin measurements can lose access to information through momentum-dependent Wigner rotations. The reduced-spin accessibility tax depends on encoding angle and saturates with rapidity. Joint spin-momentum measurements restore the invariant global information. For minimum-uncertainty packets, the reduced-spin tax scales as Ξ⁻². No universal proportionality exists between frame-matching energy and identity information. Scope The global invariance theorem applies to fixed Lorentz transformations that do not depend on the identity parameter. The paper does not analyze unknown boosts as joint quantum-estimation parameters or random boost distributions that are averaged without being recorded. The reduced-spin result is derived for a single massive spin-1/2 carrier, canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and a momentum-insensitive spin measurement. Broad or anisotropic packets, composite carriers, alternative relativistic spin observables, generic reduced photonic channels, and sequential non-collinear boosts remain open extensions. The classical flyby conclusions apply to high-signal, sufficiently resolved, symmetric spectral doublets observed over an adequate time window. Relation to the companion paper The companion paper examines the accelerated rather than inertial case: Action-Count, Observer Integrity, and Horizon Saturation: A Unified Framework from Ξ = ER/(ℏc)https://doi.org/10.5281/zenodo.19599288 Together, the two papers distinguish the operational effects of velocity and acceleration. Uniform velocity preserves global identity information but can redistribut