On Interchangeability of Probe–Object Roles in Quantum–Quantum Interaction-Free Measurement

Abstract

In this paper we examine Interaction-free measurement where both the probe and the object are quantum particles. We argue that in this case the description of the measurement procedure must by symmetrical with respect to interchange of the roles of probe and object. A thought experiment is being suggested that helps to determine what does and what doesn’t happen to the state of the particles in such a setup. It seems that unlike the case of classical object, here the state of both the probe and the object must change. A possible explanation of this might be that the probe and the object form an entangled pair as a result of non-interaction.

Appendix: A Formal Proof of Aligning Bloch Spheres Representing Interacting Particles

In words this proof shows that:

given two particles with two degrees of freedom each and those DoFs being coupled to each other

for every superposition of those DoFs in one particle there is a corresponding superposition in the DoFs of the other particle that will result in an event of absorbtion in 100% of the cases.

We start with postulating what we know.

$\exists$ an atomic state $\left| {?}\right\rangle _{at}$ s.t. given an idealized atom, it absorbs photon in state $\left| {x}\right\rangle _{ph}$ in 100% of the cases. Let us call such a state $\left| {x}\right\rangle _{at}$

This state $\left| {x}\right\rangle _{at}$ absorbs $\left| {y}\right\rangle _{ph}$ in 0% of the cases. And $\left\langle {x}\right| _{ph}\left| {y}\right\rangle _{ph}=0$

Then

$\exists$$U_{x\rightarrow y}^{at}$ s.t. $U_{x\rightarrow y}^{at}\left| {x}\right\rangle _{at}$ absorbs $\left| {y}\right\rangle _{ph}$ in 100% of the cases. Let’s call $U_{x\rightarrow y}^{at}\left| {x}\right\rangle _{at} = \left| {y}\right\rangle _{at}$

State $\left| {x}\right\rangle _{at}$ absorbs $(\left| {x}\right\rangle _{ph}+i\left| {y}\right\rangle _{ph})/\sqrt{2} = \left| {\sigma {+}}\right\rangle _{ph}$ in 50% of cases.

$\exists$$U_{x\rightarrow \sigma {+}}^{at}$ s.t. $(U_{x\rightarrow \sigma {+}}^{at})^2\left| {x}\right\rangle _{at} = U_{x\rightarrow y}^{at}\left| {x}\right\rangle _{at} = \left| {y}\right\rangle _{at}$ and $U_{x\rightarrow \sigma {+}}^{at}\left| {x}\right\rangle _{at}$ absorbs $\left| {\sigma {+}}\right\rangle _{ph}$ in 100% of the cases.

Same for the state $\left| {\sigma {-}}\right\rangle _{at}$ that absorbs in 100% of the cases the state $\left| {\sigma {-}}\right\rangle _{ph}$