An honorable mention might go to Bennett, Brassard, Ekert, Smolin, and friends' prototype tabletop machine for implementing the BB84 protocol, built in the late '80s and described in Scientific American in 1992.
In his article on the history of quantum cryptography, Brassard states:
Essentially without any special budget allocated to the project, we were able, in late October 1989, to establish history’s first secret quantum transmission, over a staggering distance of 32.5 centimetres...!
This used a plurality of qubits, way more than two as in the question, but all in a product state. It's more akin to a quantum computer in the same way that the Enigma machine is akin to a (Turing-complete) classical computer.
Nonetheless Brassard comments that the act of building the prototype did cause physicists to start paying attention to quantum cryptography, and I would offer by extension, to quantum computing.
I remember seeing it in my copy of Scientific American when it was published in '92. About two years after the SciAm article, Shor announced his factoring algorithm. (Shor has suggested that he was aware of BB84, and presumably of this machine in particular, prior to the development of the factoring algorithm.) All of this was before the word "qubit" was even coined.
Such a lovely piece of history. I wonder whether any pieces of the prototype still exists.