If this "collapse of the wavefunction" really happens (so-called "objective collapse") that would make the physics nondeterministic. The present state of the system only determines the probabilities of the various possible outcomes of the measurement.
But before we conclude that "objective collapse" is the way to go, I'd advise caution. This whole "collapse" business is introduced for reasons that are philosophical, not physical. What the physics (the math, actually) tells us is a different story. The full present state of the system is not knowable on the basis of classical observables. Quantities that characterize the present state cannot be localized in space or time. This implies that some of these "hidden" properties of the system may, in fact, be constrained by future events. Seemingly paradoxically, this does not actually violate causality: Despite being fundamentally nonlocal, the quantum theory (at least in the form of quantum field theory) cannot be used to send signals from the future to the past. Nonetheless, this nonlocal business is disturbing. It is resolved if we assume that no, the future does not constrain the present, rather, the actual future happens because the act of measurement causes the system to collapse! (Again, objective collapse.) Unfortunately, that seems like a cure that is worse than the disease: This collapse has to happen simultaneously in the entire universe (after all, we replace one nonlocal description of the state of the system with a different one) including not just all locations but all times!
OK, so what if we don't go so far? Sure, "collapse" is a useful concept to deal with practical scenarios, e.g., when the instrument is obviously a macroscopic object (say, a cat) and its state is really never in doubt (no one has ever seen, or will ever see, a cat that is both alive and dead.) But then we have an important point to ponder: Sure, we use "collapse" as a practical tool, but we know that in reality, things really never collapse, and the wavefunction simply evolves towards a near-eigenstate because it is constrained by a future measurement. In other words, we'd be taking the nonlocality of quantum physics literally.
In this case, we have a theory that is deterministic but nonlocal. Yet, in the form of quantum field theory, it would still be a theory that respects causality, with no faster-than-light or backwards-in-time influences, ever.
I'm personally in favor of this viewpoint so my immediate reaction to the question is that yes, quantum physics is deterministic. I recognize though that there are many other popular interpretations, but in the end, all this interpretation business is firmly in the realm of philosophy, not physics: The equations are the same, the predictions are the same, the results of experiments are the same no matter what philosophical baggage we attach to them, mostly just to resolve the cognitive dissonance that the weirdness of the quantum world can produce in our minds.
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