Quantum Digital Signatures (2D Phase Space)

Signature Keys

Coherent amplitude |α| 1.20

Copies per signature L 30

Accept threshold s_a 0.10

Protocol

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Live Readouts

State overlap s = e^(-2|α|²)

0.0672

Mean photon number n̄ = |α|²

1.44

Forgery bound s^L

Last run: error rate

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How it works

A quantum digital signature (QDS) lets Alice sign a classical bit so that Bob and Charlie can both verify it later, and neither one — nor an outside forger Eve — can alter it undetected, using only the impossibility of perfectly distinguishing two non-orthogonal quantum states.

Public key for bit 0:  |α⟩   (coherent state, phase 0)
Public key for bit 1:  |−α⟩  (coherent state, phase π)
Overlap:  s = |⟨α|−α⟩| = e^(−2|α|²)

This 2D build plots the two public keys directly on the physical I/Q quadrature plane (in-phase vs. quadrature amplitude) — the native 2D phase space of a coherent state — rather than adding a decorative third axis. Alice sends L independent copies of the correct public key to each recipient before signing. To verify a signed bit, a recipient performs unambiguous state discrimination (USD) on each copy — a measurement that either identifies the state correctly or returns "inconclusive," but never confidently returns the wrong answer. The optimal USD failure (inconclusive) probability for two equally-likely states with overlap s is exactly s.

P(inconclusive per copy) = s = e^(−2|α|²)
P(forge all L copies)    ≈ s^L = e^(−2|α|²·L)
Accept rule: mismatch fraction over L copies ≤ threshold s_a

|α| slider — coherent-state amplitude. A larger α pushes the two phase-space clouds (|α⟩ and |−α⟩, rendered as point clouds with the intrinsic quantum shot-noise radius σ=1/2 on both quadratures) further apart on the I axis, shrinking the overlap s.
Copies L — how many quantum public-key copies back the signature. Security grows exponentially in L because Eve must fool every USD test simultaneously — the bound panel plots s^L against L on a log scale so the exponential drop is visible directly.
Threshold s_a — the mismatch rate a legitimate recipient tolerates (real channels are noisy, so some inconclusive results are expected even from an honest key).
Bob verifies — samples L honest copies of the claimed public key and runs the USD test on each; the mismatch rate should sit near the theoretical overlap s.
Eve attempts forgery — Eve has no valid key, so she must guess; each of her L guesses independently succeeds against USD with probability ≈ s, so her overall forgery only survives if her simulated mismatch rate randomly falls below s_a — increasingly unlikely as L grows.
Drag / scroll on the main plot — pan and zoom the phase-space view, e.g. to inspect how close the clouds sit to the x=0 decision boundary at small α.

This is the coherent-state scheme (Andersson–Curty–Jex / Clarke et al.) — one of the first quantum digital signature protocols realized experimentally without a trusted quantum memory, distinct from quantum key distribution (which shares a secret key) or quantum teleportation (which moves a state): here the security goal is unforgeable authorship of a classical message.

Public key |α⟩ (bit 0) Public key |−α⟩ (bit 1) Last measurement sample

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Forgery bound s^L vs L

Trial history (error rate)