SargBounds
Correlated bit and phase errors, and the tolerable QBERs they fix
12/12 passed in 850ms
| Test | What it does | Result |
|---|---|---|
test_bracket_ends_at_a_third | 1 - h2(e) - H(Z1|X1) turns upward past e1 = 0.385, so the one-photon bisection brackets on (0, 1/3], with sarg_limit(1) inside it. | ✅ pass |
test_cutoff_leaves_no_trace | Past either cutoff sarg_cost returns one bit and sarg_phase 1/2, the same constants at the edge and far past it, so the value does not say how far past a channel is. | ✅ pass |
test_domain_sees_crossing | sarg_domain's live1 and live2 track sarg_cost below 1 and sarg_phase below 1/2, flipping exactly at the returned 1/3 and 1/6, with e2 = 0 keeping live2 true and e1 or e2 outside [0, 1/2] refused. | ✅ pass |
test_limit_is_a_root | The one-way distillation rate 1 - H(X) - H(Z|X) is zero at sarg_limit(1), positive a tenth below it and negative a tenth above. | ✅ pass |
test_maximiser_moves | Above e1 = 1/3 the independence point p_Y = 1.5 e1^2 enters Eq. (9)'s range and maximises: 0.9097 against 0.8262 at p_Y = e1/2, at e1 = 0.45. | ✅ pass |
test_no_key_above_a_third | No single-photon bit error rate at or above 1/3 certifies key at any f_ec, H(Z1|X1) being one bit and the phase error 1/2 there, while 0.33 still certifies. | ✅ pass |
test_phase_departs_past_a_sixth | sarg_phase reaches 1/2 at e2 = 1/6 and is capped there, departing from Eq. (10)'s 0.6093 at 0.25 and 0.6677 at 0.30, which unclamped would credit 0.0347 bit per two-photon detection at 0.25. | ✅ pass |
test_phase_is_equation_34 | sarg_phase equals Fung, Tamaki & Lo Eq. (34) at a = 0 to 1e-15 across its domain, reaching their | ✅ pass |
test_phase_is_half_again | The one-photon phase error rate is 1.5 e1 at the worst case p_Y = e1/2, Tamaki and Lo's p_ph = 1.5 p_bit, and the charged H(Z1|X1) sits strictly below its h2. | ✅ pass |
test_phase_minimises_g | sarg_phase agrees to 1e-7 with a direct minimisation of x e2 + g(x) over a wide grid of x. | ✅ pass |
test_tolerable_limits | The one- and two-photon tolerable bit error rates are 9.689% and 2.710%, against the 9.68% and 2.71% Tamaki and Lo publish, both below BB84's 11.0%. | ✅ pass |
test_worst_case_weight | sarg_cost equals the searched worst H(Z1|X1) over | ✅ pass |