## 3.1 Physics against logic, via beam-splitters

A **symmetric beam-splitter** is a cube of glass which reflects half the light that impinges upon it, while allowing the remaining half to pass through unaffected.
For our purposes it can simply be viewed as a device that has two input and two output ports, which we label with

When we aim a single photon at such a beam-splitter using one of the input ports, we notice that the photon doesn’t split in two: we can place photo-detectors wherever we like in the apparatus, fire in a photon, and verify that if any of the photo-detectors registers a hit, none of the others do.
In particular, if we place a photo-detector behind the beam-splitter in each of the two possible exit beams, the photon is detected with equal probability at either detector, no matter whether the photon was initially fired into input port

If we fire the photon into the input port *either* in the **transmitted** beam *or* in the **reflected** beam *this is not necessarily the case*.
Let us introduce a second beam-splitter and place two normal mirrors so that both paths intersect at the second beam-splitter, as well as putting a detector at each output port of the second beam-splitter (see Figure 3.2).

Recall the Kolmogorov additivity axiom in classical probability theory: whenever something can happen in several alternative ways, we add probabilities for each way considered separately.
We might argue that a photon fired into the input port *mutually exclusive* ways: either by two consecutive reflections or by two consecutive transmissions.
Each reflection happens with probability

However, if we set up such an experiment in a lab, *this is not what happens*!

There is no reason why probability theory (or any other *a priori* mathematical construct for that matter) should make any meaningful statements about outcomes of physical experiments.

In experimental reality, when the optical paths between the two beam-splitters are the same, the photon fired from input port *always* strikes detector 1 and *never* detector *always* strikes detector *never* detector *a beam-splitter is a physical implementation of a \sqrt{\texttt{NOT}} gate*.

The action of the beam-splitter — in fact, the action of any quantum device — can be described by tabulating the amplitudes of transitions between its input and output ports.^{56}
*zero*.

Unlike probabilities, amplitudes can cancel out each other out, witnessing destructive interference.

We can now go on and calculate the amplitude that the photon will reach detector

However, instead of going through all the paths in this diagram and linking specific inputs to specific outputs, we can simply multiply the transition matrices:

bit-flip |
||

beam-splitter |

Note that gate

B is not the same square root of\texttt{NOT} as the one we have already seen. In fact, there are infinitely many ways of implementing this “impossible” logical operation.↩︎