The key to the above statement is Bell's theorem; that no theory based on local realism can reproduce the results of quantum theory.
Einstein's great insight into this problem was outlined in his famous EPR paper. Therein, he outlined measurements contrary to the Heisenberg uncertainty principle using entangled particles; coming to the conclusion that quantum theory must be incomplete. However, the EPR paper had assumed the property of local realism.
Following the development of Bell's theorem, and associated Bell's inequality, experimental tests have sided with quantum theory, indicating that local realism is not a property of reality.
So Einstein was wrong?! Not so fast. The EPR test is no longer considered a valid proof of incompleteness, as Bell's test results indicate that quantum theory is inherently non-local. While that has ruled out the entire class of local hidden variable theories, a deeper theory based on non-local hidden variable theories is still possible. The biggest outstanding problem with quantum theory, is that it is not strictly a physical theory. Rather it is a measurement theory. However, measurements are not well-described physically. This is known as the measurement problem, which was outlined with outstanding clarity by John Bell in an article titled, Against measurement[1] .
The underlying principle physics is based on is the principle of determinism. In quantum theory, this is reflected in the principle of unitarity. That means the evolution of the quantum state is smooth, and consistent with causal interactions. However, this principle represents only half of the theory, which is the physical part of the theory. The entirety of the theory also includes measurement axioms, which are not physical.
If the physical part of the theory provides a good description of reality, then it is possible that a deeper physical theory that solves the measurement problem might be possible.
The physical part of the theory was put to the test in a 2019 paper that looked into the underlying physics of a quantum jump. Quantum jumps are a manifestation of the measurement problem, representing an irreversible change in a quantum state. Examples are radioactive decay, and more commonly the fluorescent emission of light, referred to as spontaneous emission. In the theory, these processes just occur unpredictably. The 2019 paper was notable for being able to detect the onset of a quantum jump and to reverse the process, thus illustrating that the jump itself was a purely physical process that occurred over a finite time. However, the authors note that the timing of the onset of the jump remains non-deterministic[2] .
Getting back to Einstein. I suspect that Einstein, like many physicists, expected that the principle of determinism should hold. The non-deterministic element of quantum theory associated with measurement outcomes is contrary to this principle. Therefore it is logical to assume some deeper theory. Moreover, the fact that there is a metaphysical element to the description of the measurement process actually supports this contention.
It may just be that the measurement problem is not solvable, because it describes what we observe, rather than the state of the system. This is a philosophical problem regarding the distinction between epistemic and ontic theories. Measurement theories are naturally epistemic, whereas physical theories are ontic. It may be that the ontic theory requires a godâ€™s-eye-perspective. As such, it would be useless for predicting measurement outcomes.
If Einstein were to be reincarnated, I'm sure he'd be fascinated with the modern understanding of quantum theory. Footnotes [1] https://www.informationphilosopher.com/solutions/scientists/bell/Against_Measurement.pdf
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