Theorem · T-MVW12-SEC35
Molina, Vidick & Watrous 2012 · Sec. 3.5 'Threshold results'
P01 · arXiv v1 (2012-02-17); only arXiv version
- Status
- VERIFIED FROM SOURCEField read in the primary source at the anchor given.VERIFIED_FROM_SOURCE; NOT_VERIFIED fields: applicability_to_W2
- Game
- W1_THRESH_JOINT
- Anchor
- Sec. 3.5 'Threshold results', eq.(4) p.10, conditions eq.(5) p.10, proof sketch p.10-11
- Source file
- research/THEOREM_REGISTRY_SEED.yaml
Statement summary
For n-fold repetition where the bank declares the note valid when at least t of n repetitions succeed, the optimal counterfeiting probability equals sum_{t<=j<=n} C(n,j) alpha^j (1-alpha)^{n-j} (eq. 4), with alpha the single-repetition optimum (3/4 for Wiesner). It holds under sum_k p_k |psi_k><psi_k| = (1/d) 1 (eq. 5) and Y = (alpha/d) 1 optimal dual. The paper says this holds for Wiesner's scheme and the other schemes it discusses.
Citations
Record
Every field of the source record, in file order.
- Gamegame
W1_THRESH_JOINT
- Source paper idsource_paper_id
- P01
- Source urlsource_url
https://arxiv.org/abs/1202.4010
- Anchoranchor
Sec. 3.5 'Threshold results', eq.(4) p.10, conditions eq.(5) p.10, proof sketch p.10-11
- Statusstatus
VERIFIED_FROM_SOURCE
- Bound latexbound_latex
\sum_{t\le j\le n}\binom{n}{j}\alpha^{j}(1-\alpha)^{n-j}LaTeX source - Variablesvariables
n: repetitions (qubits) t: required number of successful repetitions alpha: single-repetition optimal counterfeiting probability
- Game definitiongame_definition
The success event counted per repetition is Q_1 = the counterfeit succeeds for BOTH copies in that repetition, and the attacker wins if the number of such joint successes is >= t (operator R on p.11). This is NOT the realistic noisy game in which each copy separately must pass its own threshold test.
- Assumptionsassumptions
- ideal states
- eq.(5) ensemble condition
- dual Y = (alpha/d) 1 optimal
- Not valid fornot_valid_for
- W2 as defined in the master plan (per-copy threshold verifier, noise, loss)
- any honest-noise model: the formula says nothing about honest acceptance
- Verificationverification
statement_summary: VERIFIED_FROM_SOURCE bound_latex: VERIFIED_FROM_SOURCE game_definition: VERIFIED_FROM_SOURCE applicability_to_W2: NOT_VERIFIED
- Not verified reasonsnot_verified_reasons
applicability_to_W2: The paper's threshold game counts joint per-repetition successes. Lane A found no published per-copy-threshold BB84 bound in sources read (P13 Pastawski et al. 2012 not read in full).