Quantum Physics Still Necessary Absent Entanglement

Non-local phenomena associated with the entanglement of two particles cannot be interpreted with any classical physics theory and requires the non-deterministic, often non-intuitive, form of reasoning used by quantum mechanics. But, entanglement isn't the only place where it has been definitively shown by experiment that quantum mechanical reasoning is necessary. Even when entanglement is specifically excluded from the experiment, quantum mechanical methods are still necessary:

[A] team of quantum physicists led by Anton Zeilinger from the Faculty of Physics at the University of Vienna and from the IQOQI of the Austrian Academy of Sciences . . . used a "qutrit" -- a quantum system consisting of a single photon that can assume three distinguishable states. "We were able to demonstrate experimentally that quantum mechanical measurements cannot be interpreted in a classical way even when no entanglement is involved," Radek Lapkiewicz explains. The findings relate to the theoretical predictions by John Stewart Bell, Simon B. Kochen, and Ernst Specker.

Underlying source: Radek Lapkiewicz, Peizhe Li, Christoph Schaeff, Nathan K. Langford, Sven Ramelow, Marcin Wieśniak, Anton Zeilinger. Experimental non-classicality of an indivisible quantum system. Nature, 2011; 474 (7352): 490 DOI: 10.1038/nature10119

While the experiment merely confirms what was widely believed to be true already, the more ways that a conclusion is confirmed, the less room there is for an alternative theoretical explanation.