Is quantum mechanics on a measurement level a deterministic theory or a probability theory?

Probability theory. Evidence: when physicists do quantum measurements they find the results of individual runs are unpredictable. Only frequencies of multiple runs are predictable and match the theoretical results of quantum mechanics.

How can this possibly be consistent with unitarity as described above?

During a quantum measurement (measuring a system S by an apparatus A) the complete system S+A viewed at the microscopic level undergoes unitary evolution. During that evolution the system S become entangled with the apparatus A. However, by experimental design, this entanglement when viewed as a macroscopic approximation is seen to have some simplifying features:

a. The apparatus is in a mixed state of pointer states

b. The possible eigenvectors of some observable of S have coupled to the pointer states

c. Off-diagonal "interference" terms have become suppressed by decoherence due to the many internal degrees of freedom of A.

Owing to the special nature of these pointer states of A (from OP *"some many-particle systems may well be approximated as classical and can store the information of measurement outcomes"*) we now have an objective fact about our universe.

Only one of the pointer states has in fact actually occurred in our universe (we can make this statement whether on not a physicist actually reads the pointer and discovers which universe we are actually in).

We can then make the inference that for this particular run of the S+A interaction, the system S in fact belongs to the subensemble giving rise to the occurance of this pointer state. We can make this reduction of the original ensemble based on this objective information about our universe. Restricted to this subensemble, we still have unitary evolution when viewed at the exact microscopic level.

Disclaimer: I don't know whether this really makes any sense, but this is what the reference referred to by OP seems to be saying.

Follow up question: so can we say QM is a probability theory for practical purposes but deterministic in principle?

No I think not. Here is the confusion: having banished the need for explicit wave function collapse from the QM formalism it seems that all we are left with is deterministic unitary evolution of the wavefunction of our closed system. Hence surely QM is deterministic. But no. The indeterminism in the outcome of measurements is still present in the wavefunction.

In fact the QM formalism tells us precisely when it is able to be deterministic and when not: it is deterministic whenever the quantum state is an eigenvector of the operator related to the measurement in question. Remarkably from this one postulate it is possible to *derive* that quantum mechanics is probabilistic (i.e. we can derive the Born Rule).

Explicitly, we can show that it is *deterministic* that if the evolution of S+A is run $N$ times (with $N \rightarrow \infty$) then the frequencies of different results will follow precisely the Born rule probabilities. However for a single run there is no such determinism. For a single run it is only determined that there will be *an* outcome.

This approach to QM is described by Arkani-Hamed here.

**Edit**

For a more advanced discussion of these ideas I recommend Is the statistical interpretation of Quantum Mechanics dead?

## Best Answer

1) Classical is not equivalent to determinism. You could use probabilities in classical problems (Statistical mechanics, for instance)

2) It makes more sense, in fact, to consider the difference between a classical probabilistical problem, and a quantum probabilistical problem.

3) Quantum correlations are stronger than classical correlations, this is because, in quantum mechanics, we work with probabilites complex amplitudes $\psi$, instead of working directly with probabilites $p$ (The relation is $p = |\psi|^2$). Some experiment results cannot be explained by classical correlations.

4) If you consider, for instance, a superposition 1-spin quantum state like $\psi = |+_z>$ + $|-_z>$, a measurement of the spin on the $z$ axis will gives you always $+1$

OR$-1$. So, from the point of view of the measurement, it is anOR, it is not a AND. You will have 50% probability to measure +1, and 50% probability to measure -1.