Homodyne vs. Heterodyne Reception in CV-QKD

In the original GG02 protocol, Alice does not choose between an (x)-basis or a (p)-basis. Instead, she performs symmetric Gaussian modulation. For every single pulse, she generates two independent random numbers from a Gaussian distribution—one for (x) and one for (p)—and modulates the laser pulse with both values simultaneously.

Here is exactly how the data tracking and “discarding” works step-by-step under the hood.

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1. The Asymmetry in Data Generation

Let’s look at what Alice holds in her database versus what Bob can physically extract using a homodyne receiver.

Alice's Database (Sent)                     Bob's Hardware (Homodyne Receiver)
-----------------------                     ----------------------------------
Pulse 1: (x1, p1)  =====================>   Measures ONLY x1' (p1 is physically destroyed)
Pulse 2: (x2, p2)  =====================>   Measures ONLY p2' (x2 is physically destroyed)
Pulse 3: (x3, p3)  =====================>   Measures ONLY p3' (x3 is physically destroyed)plaintext

Because Bob uses a homodyne detector, physics forces him to choose. To measure (x), he must lock the phase of his local oscillator (a reference laser) to (0^{\circ }). To measure (p), he must shift it to (90^{\circ }). He cannot do both.

When Bob locks to (0^{\circ }) to read (x_{1}), the wave components carrying (p_{1}) are completely erased by quantum measurement noise. Bob does not have a value for (p_{1}), and he can never get it back.

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2. The Sifting Process (Aligning the Databases)

After sending a block of thousands of pulses, Alice and Bob must align their data over a public internet channel:

1. Bob publishes his measurement choices: He tells Alice, “For Pulse 1 I measured x, for Pulse 2 I measured p, for Pulse 3 I measured p.” (He does not reveal the actual numbers he read).
2. Alice filters her database: Alice looks at her records.
   • For Pulse 1, she keeps (x_{1}) and deletes/ignores (p_{1}) from her key-generation file.
   • For Pulse 2, she keeps (p_{2}) and deletes/ignores (x_{2}) from her key-generation file.

At the end of this step, Alice and Bob finally have correlated, one-dimensional lists of numbers.

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3. Why this is different from “Discrete Variable” QKD (like BB84)

If you are familiar with traditional single-photon protocols like BB84, you might expect a 50% total loss of pulses during sifting. Here is why GG02 homodyne handles loss differently:

In a hypothetical “Discrete-Choice” CV Setup:

If Alice actively picked either an (x)-state or a (p)-state randomly before sending it:

• Alice sends: Pulse 1 modulated only in (x).
• Bob randomly chooses to measure: (p).
• Result: Total basis mismatch. Bob measured pure vacuum noise. The entire pulse is completely useless and thrown away. This happens 50% of the time.

In the Real GG02 Homodyne Setup:

Because Alice always modulates both variables symmetrically:

• Bob can choose (x) or (p) entirely at his own whim.
• No pulse is ever completely thrown away due to a mismatch. Every single pulse yields exactly one useful number because Alice always has a matching record for whatever choice Bob made.

The Real “Loss” in Homodyne

When we say Alice “discards” data, we mean information throughput loss, not pulse loss.

For every pulse sent, Alice expends energy to encode 2 variables’ worth of information, but Bob’s homodyne receiver can only harvest 1 variable’s worth of information. The other half of Alice’s encoded data is deleted during post-processing because Bob’s hardware physically couldn’t read it.

• Heterodyne avoids this by reading both, meaning Alice deletes nothing.
• Homodyne accepts this 50% informational deletion because the single quadrature Bob does choose to measure is read with maximum possible clarity (zero splitter noise).