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Quantum Communication and Fingerprinting — Examples

1 · Prerequisites

2 · Summary

These companion calculations display the difference between public and private random seeds and compute a length-eight fingerprint overlap and swap-test outcome probability.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Public-coin and private-coin SMP are different models

Statement

For each n≥1, a shared uniformly random pair r₁,r₂∈𝔽₂ⁿ lets Alice send (r₁·x,r₂·x) and Bob send (r₁·y,r₂·y); the referee accepts equality iff both bits match. Equal inputs always match and unequal inputs collide with probability 1/4. The shared seed has 2n bits and is a separately declared public resource; independent private seeds do not give this guarantee.

Facts & Assumptions

Given: Equality inputs $x, y \in F_{2}^{n}$ , where $n \geq 1$ .

[F1]

Equality communication problem: Equality requires output $1$ when $x = y$ and $0$ when $x \neq y$ .

[F2]

Finite communication problem and protocol model: Shared randomness is a separately declared resource, and transmitted bits are counted as communication.

Proof

technique · explicit construction and counterexample

1.1givenF1F2choosealgebra

Let public shared seeds $r_{1}, r_{2}$ be independent uniform vectors in $F_{2}^{n}$ . Alice sends $(r_{1} \cdot x, r_{2} \cdot x)$ and Bob sends $(r_{1} \cdot y, r_{2} \cdot y)$ ; the referee outputs equality exactly when both bits match. If $x = y$ , both match surely. If $z = x + y \neq 0$ , choose a coordinate $k$ with $z_{k} = 1$ ; flipping $r_{k}$ pairs seeds bijectively with opposite values of $r \cdot z$ , so each $r_{i} \cdot z$ is uniform, and independence gives collision probability $1 / 4$ . The protocol uses two transmitted bits per sender and a shared seed of $2 n$ bits, which [F2] keeps separate from communication.

2.1step 1.1F1algebra∎

If the parties instead use independent private seed pairs, take the equal input $x = y = e_{1}$ . Each sender's two-bit message is uniform on $F_{2}^{2}$ , independently of the other's, so the messages disagree with probability $3 / 4$ . This violates any bounded-error Equality guarantee such as error at most $1 / 3$ , showing that the common seed is doing real work.

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Finite length-eight fingerprint overlap

Statement

Let four inputs be assigned codewords 00000000, 00001111, 11110000, and 11111111. Distinct codewords have distance at least four. For the first two, the fingerprint overlap is 1/2 and one swap test declares unequal with probability 3/8; equal inputs yield anti probability zero. Each fingerprint uses four qubits.

Facts & Assumptions

Given: Assign $00, 01, 10, 11$ respectively to $u_{0} = 00000000$ , $u_{1} = 00001111$ , $u_{2} = 11110000$ , and $u_{3} = 11111111$ .

[F1]

Fingerprint states are normalized and encode agreement as overlap: The fingerprint overlap is the fraction of agreeing code coordinates.

[F2]

Swap-test measurement and pure-state outcome law: The antisymmetric probability on pure unit states is $(1 - ∣ ⟨ ψ ∣ ϕ ⟩ ∣^{2}) / 2$ .

Proof

technique · direct finite calculation

1.1givenF1algebra

Direct coordinate comparison gives the six pairwise distances $d (u_{0}, u_{1}) = 4$ , $d (u_{0}, u_{2}) = 4$ , $d (u_{0}, u_{3}) = 8$ , $d (u_{1}, u_{2}) = 8$ , $d (u_{1}, u_{3}) = 4$ , and $d (u_{2}, u_{3}) = 4$ . In particular $u_{0}, u_{1}$ agree in their first four positions and differ in the last four; [F1] gives $⟨ h_{00} ∣ h_{01} ⟩ = 4 / 8 = 1 / 2$ .

2.1step 1.1F1F2algebra∎

By [F2], one swap test declares unequal with probability $(1 - (1 / 2)^{2}) / 2 = 3 / 8$ . For any equal codeword pair the overlap is one and the same formula gives probability zero. The index needs $⌈ \log_{2} 8 ⌉ = 3$ qubits and the data bit adds one, so each fingerprint uses four qubits.

Quantum Communication and Fingerprinting — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Public-coin and private-coin SMP are different models

Statement

Facts & Assumptions

Proof

Finite length-eight fingerprint overlap

Statement

Facts & Assumptions

Proof

Sources