3/4-efficient Bell measurement with passive linear optics and unentangled ancillae

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Compared to Grice [ 10 ] who uses two extra entangled photons (one extra Bell pair) to reach 75% BM efficiency, in our scheme we will need four extra unentangled photons to obtain a value of 75%. The main practical advantage of our scheme over Grice’s [ 10 ] then becomes manifest when deterministic single-photon sources are employed [ 22 ] , as opposed to a probabilistically generated Bell pair. In order to benefit from our approach, such unconditional single-photon sources do not have to produce ideal pure Fock states. For instance, purities of more than 90% for detector efficiencies greater than 95% would suffice in principle, as we will show in the second-to-last section of this paper. Of course, four unconditionally prepared ancilla photons can also be turned into one ancilla Bell pair by using the methods of linear-optics quantum computation [ 1 ] . However, this transformation is again non-deterministic, depending on the detection of two photons at the output of two non-deterministic, nonlinear sign shift gates. For the case of heralded single photons, our scheme would need a four-photon detection to herald four ancilla photons, whereas the standard linear-optics approach [ 1 ] for a Bell-pair creation would require a six-photon detection

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Important differences of our generalized scheme from Grice’s include: our setup is divided into to halves, where each half gives a new unambiguous-state discrimination problem for three 2-mode states, and this problem is addressed with ancillae having 0 or 2 photons in each mode

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For an illustration of N = 1 → N = 2 N=1\rightarrow N=2 , see the supplementary material

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A proof of this Lemma can be found in the supplementary material

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We made various numerical tests with mathematica to obtain the needed ancillae for N = 2 N=2 from single-photon states | 1111 ⟩ \mathinner{|{1111}\delimiter 86414091} and also tried feed-forward techniques using an ancillary photon, but none were successful

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A more detailed explanation on how to reach 25 32 \frac{25}{32} and possibilities to generalize this result are discussed in the supplementary material

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For a derivation of this statement see the supplementary

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