Figure 1
An ancilla state-based setup for three-dimensional control gates , , and . (a) A genuinely three-dimensional action of a second-order parity sorter (). On the left is the physical implementation as an interferometer with Dove prisms that introduces a reflection and a mode-dependent phase of or , introduced by Leach et al. [36]. Right, the abstract representation of the element. It can separate photons deterministically with mode (transmitted) from photons with mode number (reflected). Crucial for our requirements is that photons with odd mode numbers are randomly separated into two output paths, just as in a conventional beam splitter. Thus, the is a two-input, two-output element, which can perform entirely different transformations on three orthogonal modes. (b) The state of a three-photon state determines the transformation of the target photon. The target photon is overlapped with the three ancilla photons at a each. One output mode is connected to a detector, which heralds a correct transformation. If all detectors see a photon, the transformation was successful, and the output photon has the correct state. Importantly, transmits modes and probabilistically splits odd mode numbers. The state is encoded in the odd number space . The desired transformation can be achieved by adjusting the ancilla state. stands for projections into a two-dimensional subspace. (c)–(e) We show the transformation in detail. The ancilla state for the transformation is , which is a three-photon W-state [37] (up to local transformations), and its generation has been discussed in [38, 39]. If the target photon is , the only way to have all detectors see a photon (after the filter) is to have the photon in path being which leaves to the output port and the photons in path and both being in the state and going to detectors D2 and D3. This happens with a probability of because odd modes need to reflect in a specific way four times. The other two terms of will not be able to cause all three detectors, D1–D3, to click. The state will not be able to create a click in detector D2, and will not produce a click in either D1 or D3. Therefore, the pattern in the image is the only possible combination. Similar reasoning leads to the conclusions for the target photons being in and . Exactly the same logic holds for the other two types of transformations: with and with . For equal transformation probability, the ancillary states need to be weighted, as shown in Fig. 3 and the Supplemental Material (SM) [29].
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