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In the original paper (Quanum Mechanical Interaction-Free Measurements - Elitzur, Vaidman, p.991), they make an algebraic substitution for the 'appearance' or 'scattering' of the bomb (green arrow):

enter image description here

Can anyone shed any light on what this substitution is? The "up" path (where the bomb lays) is $|2\rangle$. I do not see how (algebraically) the 'scattered' state is being introduced.

Thanks!

Abstract of paper

PDF of paper

Wikipedia page

Mark Spinelli
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algo_rithm
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1 Answers1

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$|\text{scattered}\rangle$ refers to the state where the photon is no longer present (because it was absorbed/scattered by the detector). More generally, you can think of it as any state where the state of the photon has become entangled with things outside the system.

The simplest way to model this state is to introduce an ancilla qubit to represent "did the bomb explode", and coherently toggle this qubit conditioned on the photon being in the bomb path. In that case the normal states all get a $\otimes |0_\text{exploded}\rangle$ added on, and the scattered state looks like $|\text{PhotonIsOnBombPath}\rangle \otimes |1_\text{exploded}\rangle$.

Craig Gidney
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  • So there is another Hilbert space with ${|\text{bomb is live}>, |\text{bomb is defective}>}$ as a basis and we take the tensor product with that space? I still don't see how the state in green is arrived at. The bomb and its possibility of exploding are classical probability concepts. How do we incorporate that with the system of mirrors, which is a tensor product of 2 (or perhaps 3) qubits? – algo_rithm Dec 29 '21 at 21:08
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    The bombs function isn't under superposition. You can just figure out the bomb working case on it's own, and the not working case separately. – Craig Gidney Dec 29 '21 at 21:26