Beacon Qubits in Satellite Optical Links

• Beacon qubits are quantum optical encodings that use weak coherent pulses in specific time bins to represent classical bits for satellite identification.
• They leverage joint detection receivers to achieve near quantum limit performance, enhancing photon efficiency and communication reliability.
• The ELROI system applies on–off keying of coherent states, significantly improving active time windows even in high-loss or adverse atmospheric conditions.

A beacon qubit is a quantum optical encoding concept employed in satellite-to-ground free-space optical links, exemplified by the ELROI (Extremely Low Resource Optical Identifier) system, which broadcasts satellite identity via weak optical pulses. Each beacon qubit represents the quantum state associated with a single time-bin (pulse slot), carrying a classical bit through quantum state encoding suitable for discrimination by state-of-the-art quantum receivers. The beacon qubit paradigm enables significant enhancements in photon efficiency, communication rates, and robustness to adverse channel conditions when quantum-enabled ground receivers are utilized (Munar-Vallespir et al., 9 Oct 2025).

1. Quantum-State Encoding in Satellite Beacon Links

In the ELROI system, each bit of the satellite’s 20-bit beacon ID is mapped to an individual time bin implemented by optical pulses. The encoding is realized via on–off keying (OOK) of coherent states in a bosonic mode:

• Binary “0”: vacuum state $\lvert 0 \rangle$
• Binary “1”: coherent state $\lvert \alpha \rangle$ with mean photon number $E = \lvert \alpha \rvert^2$, typically $E \ll 1$ at the receiver

Each time-bin thus corresponds to a two-dimensional subspace of the full bosonic mode space, carrying one qubit's worth of information, although the physical implementation leverages weak coherent light rather than true two-level systems. For practical deployment, satellites emit $2\,\mu$s laser pulses (peak power 1 W) broadcast isotropically ($\approx 2\pi$ solid angle). After propagation over 1,000 km and collection by a 36 cm ground telescope, the received photon rate is approximately 3 photons/s (with $E/B \approx 3$ photons/s for $B = 1$ MHz). Narrowband filtering ($\Delta \lambda' = 10^{-4}$ nm) suppresses background noise to $N \approx 0.01$ photons/s (Munar-Vallespir et al., 9 Oct 2025).

2. Communication Limits: Classical Versus Quantum

The ultimate rate of information extraction from beacon qubit links is governed by the physical model of the optical channel and the receiver architecture.

• Classical AWGN Channel (Shannon/Homodyne): The channel capacity per time-bin, with homodyne detection, is

$$C_S = \frac{1}{2}\log_2\left(1 + \frac{4\gamma E}{2N+1}\right)$$

• Classical AWGN Channel (Heterodyne):

$$C_{S'} = \log_2\left(1 + \frac{\gamma E}{N+1}\right)$$

In the photon-limited ($E \ll 1$) regime, $C_S > C_{S'}$.

• Quantum Channel (Holevo/Gordon Capacity): The Holevo limit for the lossy-thermal bosonic channel is

$C_H(\gamma, E, N) = g(\gamma E + N) - g(N)$

where $g(x) = (x+1)\log_2(x+1) - x\log_2 x$ is the von Neumann entropy of a thermal state. Achieving $C_H$ requires quantum collective measurements.

• Error-Probability Bound: For slot-wise discrimination between $|0\rangle$ and $|\alpha\rangle$, the Helstrom minimum error probability is

$$P_e^{\rm min} = \frac{1}{2} \left(1 - \sqrt{1 - e^{-\gamma E}}\right)$$

Block-coded transmissions (length $n$, rate $R < C$) enable exponential suppression of error rates using quantum coding.

3. Joint Detection Receiver (JDR) and Quantum Advantage

The Joint Detection Receiver (JDR) architecture realizes the Holevo limit by employing collective quantum measurements:

• Blocks of $n$ successive time-bins ($n \sim 10^6$ for 1 s reads) are stored in a short-term quantum memory.
• A collective unitary $U$ is applied to the $n$-mode register, mapping the $2^n$ codewords

$$|\psi_m\rangle = \bigotimes_{i=1}^{n} |\alpha b_i^m \rangle$$

(where $b_i^m \in \{0,1\}$ encodes the $i$th bit of codeword $m$) to an (approximately) orthonormal basis.

• A photon-counting array (multiport arrangement) performs a projective measurement onto the output mode occupation patterns, implementing the POVM $\{\Pi_m\}$ with $\Pi_m = U^\dagger |m\rangle\langle m| U$.

Physically, $U$ is constructed from beam splitters, phase-shifters, and non-destructive photon counters. The JDR's collective nature is required to achieve the Holevo limit $C_H$ rather than the (lower) Shannon capacity $C_S$, raising photon efficiency by a factor of 6–7 in clear weather (Munar-Vallespir et al., 9 Oct 2025).

4. Performance Metrics: Time-To-Read and Active Time Window

The primary operational benchmarks are the time required to successfully read the beacon (“Time-To-Read,” TTR) and the resultant window available for satellite use (“Active Time Window,” ATW):

• Time-To-Read: ${\rm TTR} = \frac{20\ {\rm bits}}{B \cdot C}$

For typical parameters $B = 10^6$ symbols/s, $$(\gamma, E, N) = (5.15 \times 10^{-15}, 6.4\times 10^9\ {\rm ph/s}, 10^{-3}\ {\rm ph/s})$$:

• $C_S \approx 8.65$ bits/s $\implies$ TTR$_S \approx 2.31$ s
• $C_H \approx 59.27$ bits/s $\implies$ TTR$_H \approx 0.34$ s

In adverse weather (extra 22 dB attenuation):

• TTR$_S \approx 1650.35$ s ($\gg$ pass duration $T_s = 1054$ s; no decode possible)
• TTR$_H \approx 166.07$ s
• Active Time Window: ${\rm ATW} = T_s - 1.5\cdot {\rm TTR}$ ($1.5$ is a code alignment factor)
• In adverse conditions:
  • With SSR (conventional detection): ${\rm ATW}_S \leq 0$ (no identification)
  • With JDR: $${\rm ATW}_H \approx 1054 - 1.5\times166.07 \approx 804.9$$ s (about 76% of a low-elevation pass)

This leads to a nearly twenty-fold increase in the usable window for subsequent network operations, as the quantum-enabled link operates almost continuously throughout the satellite pass.

5. Physical and System Parameters

The critical parameters influencing beacon qubit link performance are:

Symbol	Definition	Role in Capacity
$\gamma$	Link transmittance	Scales $C_S$, $C_H$
$E$	Mean received photons per slot	Sets signal energy
$N$	Background photons per slot	Appears in channel noise
$B$	Symbol or time-bin rate	Converts bits/use to bits/s
$A_{\rm tel}$	Telescope area	Contributes to $\gamma$, $N$
$\Delta\lambda$	Filter bandwidth	Enters noise calculation
$T_f$	Filter transmission factor	Affects both $\gamma$, $N$
$\epsilon_{DQE}$	Detector quantum efficiency	Affects both $\gamma$, $N$

Explicitly,

$$\gamma = \frac{A_{\rm tel} \, \tau_{\rm atm} \, T_f \, \epsilon_{DQE}}{4\pi r^2}$$

$$N \approx I_\lambda\,\Delta\lambda\,A_{\rm tel}\,\tau_{\rm atm}\,T_f\,\epsilon_{DQE}/B$$

where $r$ is the slant range; $I_\lambda$ is the sky background spectral radiance.

By shifting system complexity from the satellite transmitter to the ground station JDR — requiring narrowband spectral filters, quantum memory, and non-destructive photon counters — beacon qubit links attain operation at the Holevo limit, rather than the classical Shannon limit. This enables reliable, high-efficiency communication even in extremely low received power or high-loss conditions (Munar-Vallespir et al., 9 Oct 2025).

6. Implications and Applications

Beacon qubit-based links, as realized in the ELROI-JDR architecture, demonstrate a decisive operational advantage in optical satellite identification and low-rate signaling. In practice, the enabled extension of the active time window under adverse atmospheric conditions—from zero to approximately 800 s in cited scenarios—not only increases network utilization but also broadens the range of feasible weather and elevation conditions for satellite-ground coordination. A plausible implication is the potential for similar joint quantum measurement approaches to fundamentally enhance photon-starved optical links in other science and communications contexts (Munar-Vallespir et al., 9 Oct 2025).