Quantum Optical X-Box

In my previous post, I described a paper by Aaronson and Arkhipov entitled “The Computational Complexity of Linear Optics”. In this post, I would like to suggest an implementation of their experiment with optical fibers. Call it a thought experiment (Gedanken experiment) because I’m not an experimentalist so I can’t fill in all the minute experimental details. I’m not an Einstein either, so don’t expect too much from me, although I do believe the device I propose is physically reasonable.

Fig.1

The device uses the following building blocks

• photon source and photon detector (see Fig.1) These are just what their names suggests.
• optical delay and mirror (see Fig.1) These are just what their names suggests.
• quantum rail-switch (see Fig.2) For the purposes of this thought experiment, I will postulate an ideal element called a quantum rail-switch with the following properties. 

  A rail-switch has 3 ports. Two ports are interchangeable, while the third one, the odd-man-out, is different from the other two. 

  One of the interchangeable ports is blocked at all times. There is a switch on the device whereby one can control which of the two interchangeable ports is blocked. In our drawings, we will indicate that an interchangeable port is blocked by putting an “x” next it.
  

  

  Fig.2

  Fig.2 shows 4 different ways of using rail-switches.

  If a mode enters the odd-man-out port, then (1) the same mode exits, unchanged, the non-blocked interchangeable port (2) nothing exits the blocked interchangeable port. 

  If a mode enters the non-blocked interchangeable port and blackness enters the blocked interchangeable port, then the noise from the blackness is blocked by the rail-switch and the mode that enters the rail-switch exits it, unchanged, through the odd-man-out port.

Our full device is shown in Fig.3. It contains a box (shown in orange in Fig.3), which we will call henceforth a quantum X-Box. The x-box contains at its center a single beam splitter (shown in purple in Fig.3). The two modes that enter the beam splitter and the two modes that exit it form an X pattern, hence the name x-box.

Fig.3

For definiteness, Fig.3 shows an x-box with 4 left and 4 right ports. 4 here is the value of the variable m (= total number of modes) in the AA paper. Once I explain Fig.3, you will see that this device can be generalized easily to m equal to any power of 2.

Inside the x-box
As shown in Fig.3, each of the 4 left ports of the x-box leads to an A_j switch. 

Each A_j switch has two fibers exiting it: one (indicated in green) goes directly to one of the right ports of the x-box and the other goes into one of the B_j switches. 

Each of the B_j switches has two fibers exiting it: one goes to a C_j switch located ABOVE the beam splitter, and the other goes to a C_j switch located BELOW the beam splitter.

The two C_j switches above (below, resp.) the beam splitter each sends one fiber to switch D0 (D1, resp.)

D0 (D1, resp.) sends one fiber to the top side (bottom side, resp.) of the beam splitter.  

The x-box is has left-right mirror symmetry.

Outside the x-box
As shown in Fig.3, each left-side(right-side, resp.) port of the x-box leads via a fiber to a rail-switch that in turn leads to(1) a photon source (photon detector, resp.) and (2) a delay/mirror.

How to operate it
Initially, the photon sources emit photons nearly simultaneously. The photons pass through the x-box and exit it. Each time, except the last time, that the photons exit the x-box, they meet delays/mirrors that make them to go through the x-box one more time. The last time that the photons exit the x-box, they meet the detectors.

Fig.4

Each time the photons are outside the x-box, busy traversing the delays, one changes the x-box as follows:

1. two of the 4 input ports are chosen at random. These two chosen input ports are made “active”, and rest of the input ports are made “inactive”
2. the reflection and transmission coefficients of the beam splitter are changed at random. 

Fig.4 shows an example in which the top two input ports were chosen to be the ones that are active. The red lines indicate the path of the photons that enter the active ports. Note that choosing active and inactive ports means blocking a slew of rail-switch ports, in such a way as to guarantee that: 

1. the two modes that enter the active ports reach the beam splitter and mix there, and then re-emerge out of the x-box at the ports which are the mirror images of the ports they entered. 
2. all modes that enter the inactive ports are immediately directed, unchanged, out of the x-box. 

Balancing
It’s possible to build a quantum xbox that is balanced:

Assume that the transit time through all rail-switches is the same. Assume that in Fig.3, one makes

1. the length of all green colored fibers the same
2. the length of all non-green colored fibers from an A_j to a B_j switch the same
3. the length of all fibers from a B_j to a C_j switch the same
4. the length of all fibers from a C_j to a D_j switch the same
5. the length of all fibers from a D_j switch to the beam splitter the same
6. the length of any green colored fiber = the sum of the fiber lengths traversed by an active photon (plus a tiny delay equal to the transit time through 6 switches (2 B_j, 2 C_j, 2 D_j) and the beam splitter)

If these constraints are satisfied, the transit time for a photon to make one xbox pass will become path-independent (the same for all possible active and passive paths). Thus, balanced xboxes have a common transit time.