Empirical/QM/ElitzurVaidman: the bomb tester (interaction-free measurement)

The Elitzur–Vaidman bomb tester: a balanced Mach–Zehnder interferometer (two 50:50 beam splitters = H) is tuned so a single photon always exits the bright port and never the dark port — perfect destructive interference at the dark port. Placing a live bomb (a which-path absorber that detonates if the photon takes that arm) in one arm destroys the coherence: now the dark port fires with probability 1/4. A dark-port click therefore certifies a live bomb without the photon ever hitting it — an interaction-free measurement.

• bomb_absent_dark_zero — no bomb: dark-port Born probability = 0 (full interference, H·H = I).
• bomb_safe_prob — the photon reaches the second beam splitter without detonating the bomb, = 1/2.
• bomb_dark_given_safe — conditioned on survival, the dark port fires with probability 1/2.
• bomb_present_dark — live bomb: total dark-port probability = 1/2 · 1/2 = 1/4.
• interaction_free — 0 < 1/4: the dark port fires only with a live bomb, so a dark click is a bomb certificate with no interaction (the surviving photon took the empty arm).

Experimental verification: Kwiat, Weinfurter, Herzog, Zeilinger, Kasevich 1995. CSD note: the bomb is a which-path de-isolation that would form a record (detonation) on the occupied arm; the dark-port click is the ontic signature that the record-forming interaction was available on the other arm — information without interaction.

References

Empirical/CSD/MachZehnderVolume.lean (the interferometer as H·D(φ)·H); Empirical/Metrology/Ramsey.lean (the beam-splitter / √2 amplitude machinery).

noncomputable def CSD.Empirical.QM.ElitzurVaidman.rt2 : ℂ

√2 as a complex scalar.

Equations

• CSD.Empirical.QM.ElitzurVaidman.rt2 = ↑√2

theorem CSD.Empirical.QM.ElitzurVaidman.norm_rt2 : ‖rt2‖ = √2

theorem CSD.Empirical.QM.ElitzurVaidman.norm_div_rt2_sq (x : ℂ) : ‖x / rt2‖ ^ 2 = ‖x‖ ^ 2 / 2

‖x/√2‖² = ‖x‖²/2.

def CSD.Empirical.QM.ElitzurVaidman.ket0 : Fin 2 → ℂ

Input path states |0⟩, |1⟩ (the two interferometer arms).

Equations

• CSD.Empirical.QM.ElitzurVaidman.ket0 = ![1, 0]

def CSD.Empirical.QM.ElitzurVaidman.ket1 : Fin 2 → ℂ

Equations

• CSD.Empirical.QM.ElitzurVaidman.ket1 = ![0, 1]

noncomputable def CSD.Empirical.QM.ElitzurVaidman.bs (v : Fin 2 → ℂ) : Fin 2 → ℂ

The 50:50 beam splitter (Hadamard): H(v) = ((v₀+v₁)/√2, (v₀−v₁)/√2).

Equations

• CSD.Empirical.QM.ElitzurVaidman.bs v = ![(v 0 + v 1) / CSD.Empirical.QM.ElitzurVaidman.rt2, (v 0 - v 1) / CSD.Empirical.QM.ElitzurVaidman.rt2]

noncomputable def CSD.Empirical.QM.ElitzurVaidman.darkProb (v : Fin 2 → ℂ) : ℝ

Born probability of the dark port (component 1) of an amplitude vector.

Equations

• CSD.Empirical.QM.ElitzurVaidman.darkProb v = ‖v 1‖ ^ 2

theorem CSD.Empirical.QM.ElitzurVaidman.bomb_absent_dark_zero : darkProb (bs (bs ket0)) = 0

No bomb: the dark port never fires. The balanced interferometer is H·H = I, so the photon returns to |0⟩ and the dark-port probability is 0 — perfect destructive interference.

theorem CSD.Empirical.QM.ElitzurVaidman.bomb_safe_prob : darkProb (bs ket0) = 1 / 2

The photon survives (does not detonate the bomb) with probability 1/2: after the first beam splitter H|0⟩ = (|0⟩+|1⟩)/√2, the amplitude on the empty (non-bomb) arm has weight 1/2.

theorem CSD.Empirical.QM.ElitzurVaidman.bomb_dark_given_safe : darkProb (bs ket1) = 1 / 2

Conditioned on survival, the second beam splitter sends the photon to the dark port with probability 1/2 (H|1⟩ = (|0⟩−|1⟩)/√2).

noncomputable def CSD.Empirical.QM.ElitzurVaidman.pDarkLive : ℝ

Total dark-port probability with a live bomb: survive (1/2) then reach the dark port (1/2).

Equations

• One or more equations did not get rendered due to their size.

theorem CSD.Empirical.QM.ElitzurVaidman.bomb_present_dark : pDarkLive = 1 / 4

Live bomb: the dark port fires with probability 1/4. In contrast to the no-bomb case (0), so a dark-port click certifies a live bomb.

theorem CSD.Empirical.QM.ElitzurVaidman.interaction_free : darkProb (bs (bs ket0)) < pDarkLive

Interaction-free measurement: the dark port fires with probability 0 when there is no bomb but 1/4 when there is — so a dark-port click detects a live bomb without the photon interacting with it.