GMCS CV-QKD

Alice’s State Preparation

RNG

The protocol begins in digital domain, with Alice generating a uniform binary bit stream using a QRNG (Quantum Random Number Generator).

Algorithmic mapping to Gaussian

The uniform bit stream is mapped to a bi-variate Gaussian distribution, using Box-Muller Transform:

Let and , then the quadratures are computed as:

Here, is Alice’s modulation variance (in Shot Noise Units (SNUs)). Alice dynamically sets based on the estimated channel transmittance () and excess noise () to maximize secret key capacity.

DAC

The computed pairs are quantized and sent to DACs.

• Quantization Noise: A DAC with bits introduces quantization noise with a variance , where is the voltage step size.
• Amplification: The raw DAC outputs and are passed through analog amplifiers and low-pass anti-aliasing filters to match the half-wave voltage ().

Such noises must be strictly bounded, as Eve can potentially hide within any noise Alice generates.

IQ Modulation

Alice has a continuous-wave laser with a local oscillator field . To modulate the carrier with generated values, she uses an IQ modulator to encode the amplitude () and phase () quadratures. The exact optical transfer function of the IQ modulator biased at the transmission null (carrier suppression) is:

To prevent distortion, and maintain Gaussian statistics, Alice heavily attenuates her driving voltages to operate in the small signal linear region where . Using small angle approximation, , we get the output field as:

Power Scaling

This modulated field physically represents the coherent state , where the complex amplitude is . Alice sets up a variable optical attenuator, to scale the power. The average photon number per pulse injected into the fiber is dictated by her chosen variance.

Over the entire Gaussian ensemble, the expected photon number is directly proportional to her parameter setting: . She tunes the VOA to ensure the photon count matches the required .

Bob’s Reception

Mixing and Extraction

Bob receives degraded quantum signal mode , which has been attenuated by channel transmittance and corrupted by excess noise .

Bob performs coherent detection, because the received signal is too weak to drive the demodulator. He mixes this weak signal with a highly stable, powerful LO laser, , using a 50-50 balanced beam splitter.

The LO is treated as a coherent field with a large amplitude and a controllable phase: .

The beam splitter transforms the input modes into two output spatial modes, and :

Optical to Electronic Homodyne Conversion

The two output beams strike a matched pair of PIN photodiodes. These output current proportional to the photon number operators and .

A differential amplifier subtracts these photocurrents to give the difference signal . This subtraction cancels the common mode noise of the LO.

This difference term isolates the quantum interference between weak signal and strong LO:

Substituting the classical LO definition, the difference amplifier acts as a homodyne multiplier:

By actively modulating the LO phase , Bob randomly selects his measurement basis for each incoming pulse. Setting measures the amplitude quadrature (), while shifting to measures the phase quadrature ().

Detector Noise

The hardware introduces two imperfections:

1. The quantum efficiency of photodiodes and
2. The electronic noise variance of the transimpedance amplifier . The beam splitter only transmits the fraction of the signal and leaks vacuum noise.

The total noise added by Bob’s homodyne detector, referred to his optical input port is:

Digitization

After Bob passes the difference current through ADC, he gets a classical variable , linearly correlated to Alice’s prepared values :

Here represents the total lumped system noise.

In the variance, represents the inevitable vacuum shot noise and is the total noise referred back to Alice’s transmitter.

Distribution of

Alice’s prepared signal, Addition and scaling of independent Gaussian variables:

Then follows a Gaussian distribution

Basis Sifting

Following the quantum transmission, Alice holds twice as much data in her memory as Bob successfully measured.

To align their datasets for post-processing, Bob must broadcast his phases choices to Alice over a public channel.

For every index , Alice checks Bob’s basis and decides to keep either or

After the sifting step is over, Alice is left with half of her initial information.

Post Processing Step 1: Parameter Estimation