Entanglement by Path Identity

Quantum entanglement is one of the most prominent features of quantum mechanics and forms the basis of quantum information technologies. Here we present a novel method for the creation of quantum entanglement in multipartite and high-dimensional systems. The two ingredients are (i) superposition of photon pairs with different origins and (ii) aligning photons such that their paths are identical. We explain the experimentally feasible creation of various classes of multiphoton entanglement encoded in polarization as well as in high-dimensional Hilbert spaces—starting only from nonentangled photon pairs. For two photons, arbitrary high-dimensional entanglement can be created. The idea of generating entanglement by path identity could also apply to quantum entities other than photons. We discovered the technique by analyzing the output of a computer algorithm. This shows that computer designed quantum experiments can be inspirations for new techniques.

Figure 1

(a) The simplest example which uses the overlapping modes has been discussed first in [1]. The two crystals (gray squares) can produce one pair of photons (blue lines), either in the first or in the second crystal with the pump beam depicted with black lines. If the two processes are coherent and the photons have the same frequency and polarization, it is not known in which crystal the photons are created. In that case, the resulting photon pair is in the state $|\psi ⟩=(1/\sqrt{2})(|a⟩+|c⟩)|d⟩$. (b) A simple sketch of the same experiment. For simplicity, we will use this more abstract representation of physical experiments in the rest of the manuscript.

Figure 2

Multiphoton entanglement with polarization. (a) In four crystals, two photon-pairs are produced. Crystals 1 and 2 produce horizontally polarized photons, while crystals 3 and 4 produce vertically polarized ones. Four-photon coincidences can only happen when crystals 1 and 2 fire together or when crystals 3 and 4 fire. This leads to a four-particle GHZ state $|\psi ⟩=(1/\sqrt{2})(|H,H,H,H⟩+|V,V,V,V⟩)$. (b) Entangled states with more numbers of particles can be created in an analogous way—here an $n$-photon GHZ state $|\psi ⟩=(1/\sqrt{2})(|H,H,H,\dots ⟩+|V,V,V,\dots ⟩)$ is shown.

Figure 3

$W$ states, such as $|\psi ⟩=\frac{1}{2}(|VHHH⟩+|HVHH⟩+|HHVH⟩+|HHHV⟩)$, represent a different type of multiphoton entanglement. While the GHZ state is considered as the most nonclassical state, a $W$ state is the most robust entangled state because the loss of one particle leaves an entangled state. Here, fourfolds can only happen if both crystals 1 and 2 produce a pair of photons, or crystals 3 and 4, or crystals 5 and 6, or 6 and 7. Interestingly, in this setup, one photon from crystal 1 can stimulate an emission in crystal 3. However, this will not lead to fourfold coincidences, as there is no photon in path $b$. Thus, this event can be neglected.

Figure 4

Multiphoton entanglement with high-dimensional degrees of freedom (in this example: orbital angular momentum). (a) The setup produces two photon pairs. Four-photon coincidences can only occur when crystals 1 and 2 fire together, or crystals 3 and 4, or crystals 5 and 6. The produced photon pairs are all in the lowest mode (such as $\mathrm{OAM}=0$). After each layer of crystals, a hologram increases the OAM of the photons (depicted as a red line). After the third layer, photons from crystals 1 and 2 have an $\mathrm{OAM}=2$ and photons from the middle layer have an $\mathrm{OAM}=1$, which leads to a four-particle three-dimensional entangled GHZ state [$|\psi ⟩=(1/\sqrt{3})(|0,0,0,0⟩+|1,1,1,1⟩+|2,2,2,2⟩)$]. (b) The same technique can also be applied to two-particle states to produce general high-dimensional states. All crystals produce pairs of Gaussian photons. The red lines indicate OAM holograms, and the green lines indicate phase shifters. For example, if the holograms are all $\mathrm{OAM}=1$ and the phases are ignored, then the final output state is $|\psi ⟩=\frac{1}{2}(|0,0⟩+|1,1⟩+|2,2⟩+|3,3⟩)$. However, if one changes phases and the OAM in photon $b$, arbitrary four-dimensional states can be created—for example all 16 four-dimensional Bell states. By increasing the number of crystals, more dimensions can be added.