Group transformations and entangled-state quantum gates with directionally unbiased linear-optical multiports

The concept of directionally unbiased optical multiports is introduced, in which photons may reflect back out the input direction. A linear-optical implementation is described, and the simplest three-port version studied. Symmetry arguments demonstrate potential for unusual quantum information processing applications. The devices impose group structures on two-photon entangled Bell states and act as universal Bell-state processors to implement probabilistic quantum gates acting on state symmetries. These multiports allow optical scattering experiments to be carried out on arbitrary undirected graphs via linear optics and raise the possibility of linear-optical information processing using group structures formed by optical qudit states.

Figure 1

Schematic of the directionally unbiased three-port. Beam splitters connect input ports to two internal edges and a mirror unit $U$. $U$ is composed of a mirror $M$ and phase plate $P$ (inset). Reflection at a mirror unit imparts to each photon a total phase factor of $-i$.

Figure 2

Two-photon input enters ports $A/B$, with the control state entering $A/C$. Output exits ports $B/C$.

Figure 3

(a) The multiport produces the Abelian four-group multiplication table. (b) Cyclic $Z_2$ subgroups describe reflections about axes shown in (c).

Figure 4

Directionally unbiased four-port. Generalizations to any number of ports $\ge 3$ are possible.

Figure 5

Paths of length up to $N=10$ that enter port $A$ and then exit at the same port.

Figure 6

Paths of length up to $N=10$ that enter port $A$ and then exit at port $B$.

Figure 7

Paths of length up to $N=8$ that enter port $A$ and then exit at port $C$.