## 3.1 Physics against logic, via beamsplitters

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 be viewed as a device which has two input and two output ports which we label as

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 from input port

It may seem obvious that, at the very least, the photon is *either* in the transmitted beam *or* in the reflected beam

Now, the axiom of additivity in probability theory says that 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

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 0 (and the photon fired from input port *always* strikes detector 0 and *never* detector 1).
Thus a beam-splitter acts as the square root of

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.^{44}

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:

Recalling Chapter 2, we see that beam-splitters give a physical way of constructing the square root of

bit-flip |
||

beam-splitter |

Note that gate

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