Generation of entangled photonic states

1. A method of generating an m photon entangled state, the method comprising:

inputting a photon into a respective input mode of each of a plurality of sets of modes, wherein each of the plurality of sets of modes is coupled to an input mode of a different set of modes of the plurality of sets of modes;

performing a first detection operation that includes detecting zero or more photons in the plurality of sets of modes; and

in accordance with a determination, based on a number of photons detected in the first detection operation, that more than m photons remain in the plurality of sets of modes after the first detection operation:

performing a second detection operation that includes detecting zero or more photons in the plurality of sets of modes;

determining, based at least in part on a number of photons detected in the second detection operation, whether the photons remaining in the plurality of sets of modes after the second detection operation are in the m photon entangled state; and

in accordance with a determination that the photons remaining in the plurality of sets of modes after the second detection operation are in the m photon entangled state, outputting the photons remaining in the plurality of sets of modes.

2. The method of claim 1 , wherein each set of modes of the plurality of sets of modes includes at least two modes.

3. The method of claim 1 , wherein respective modes of each set of modes of the plurality of sets of modes are coupled to produce a photonic state for the plurality of sets of modes.

4. The method of claim 1 , further including:

determining, based at least in part on a number of photons detected in the second detection operation, whether the photons remaining in the plurality of sets of modes after the second detection operation are in a state convertible to the m photon entangled state; and

in accordance with a determination that the photons remaining in the plurality of sets of modes after the second detection operation are in the state convertible to the m photon entangled state, outputting the photons remaining in the plurality of sets of modes to a circuit for converting the state to the m photon entangled state.

5. The method of claim 4 , wherein determining whether the photons remaining in the plurality of sets of modes after the second detection operation are in a state convertible to the m photon entangled state includes determining that the photons remaining in the plurality of sets of modes are in a state that is deterministically convertible to the m photon entangled state.

6. The method of claim 4 , wherein determining whether the photons remaining in the plurality of sets of modes after the second detection operation are in a state convertible to the m photon entangled state includes determining that the photons remaining in the plurality of sets of modes are convertible to the m photon entangled state with a success probability that is less than 1.

7. The method of claim 1 , further including:

in accordance with a determination, based on a number of photons detected in the second detection operation, that more than m photons remain in the plurality of sets of modes after the second detection operation:

performing a third detection operation that includes detecting zero or more photons in the plurality of sets of modes;

determining, based at least in part on a number of photons detected in the third detection operation, whether the photons remaining in the plurality of sets of modes after the third detection operation are in the m photon entangled state; and

in accordance with a determination that the photons remaining in the plurality of sets of modes after the third detection operation are in the m photon entangled state, outputting the photons remaining in the plurality of sets of modes.

8. The method of claim 1 , wherein the plurality of sets of modes consists of m sets of modes.

9. The method of claim 1 , wherein the m photon entangled state is a Bell state.

10. The method of claim 1 , wherein the m photon entangled state is an m-photon GHZ state.

11. A device for generating an m photon entangled state, the device comprising:

a plurality of sets of modes, wherein each set of modes of the plurality of sets of modes includes an input mode;

a first set of couplers coupling the input mode of each of the plurality of sets of modes to the input mode of a different set of modes of the plurality of sets of modes; and

a controller configured to:

perform a first detection operation that includes detecting zero or more photons in the plurality of sets of modes; and

in accordance with a determination, based on a number of photons detected in the first detection operation, that more than m photons remain in the plurality of sets of modes after the first detection operation:

perform a second detection operation that includes detecting zero or more photons in the plurality of sets of modes;

determine, based at least in part on a number of photons detected in the second detection operation, whether the photons remaining in the plurality of sets of modes after the second detection operation are in the m photon entangled state; and

in accordance with a determination that the photons remaining in the plurality of sets of modes after the second detection operation are in the m photon entangled state, output the photons remaining in the plurality of sets of modes.

12. The device of claim 11 , wherein each set of modes of the plurality of sets of modes includes at least two modes.

13. The device of claim 11 , further comprising a second set of couplers that couples respective modes of each set of modes of the plurality of sets of modes to produce a photonic state for the plurality of sets of modes.

14. The device of claim 11 , wherein the controller is configured to:

determine, based at least in part on a number of photons detected in the second detection operation, whether the photons remaining in the plurality of sets of modes after the second detection operation are in a state convertible to the m photon entangled state; and

in accordance with a determination that the photons remaining in the plurality of sets of modes after the second detection operation are in the state convertible to the m photon entangled state, output the photons remaining in the plurality of sets of modes to a circuit for converting the state to the m photon entangled state.

15. The device of claim 14 , wherein determining whether the photons remaining in the plurality of sets of modes after the second detection operation are in a state convertible to the m photon entangled state includes determining that the photons remaining in the plurality of sets of modes are in a state that is deterministically convertible to the m photon entangled state.

16. The device of claim 14 , wherein determining whether the photons remaining in the plurality of sets of modes after the second detection operation are in a state convertible to the m photon entangled state includes determining that the photons remaining in the plurality of sets of modes are convertible to the m photon entangled state with a success probability that is less than 1.

17. The device of claim 11 , wherein the controller is configured to:

in accordance with a determination, based on a number of photons detected in the second detection operation, that more than m photons remain in the plurality of sets of modes after the second detection operation:

perform a third detection operation that includes detecting zero or more photons in the plurality of sets of modes;

determine, based at least in part on a number of photons detected in the third detection operation, whether the photons remaining in the plurality of sets of modes after the third detection operation are in the m photon entangled state; and

in accordance with a determination that the photons remaining in the plurality of sets of modes after the third detection operation are in the m photon entangled state, output the photons remaining in the plurality of sets of modes.

18. The device of claim 11 , wherein the plurality of sets of modes consists of m sets of modes.

19. The device of claim 11 , wherein the m photon entangled state is a Bell state.

20. The device of claim 11 , wherein the m photon entangled state is an m-photon GHZ state.