In 1935, Einstein, Podolsky, and Rosen proposed a thought experiment that, in their view, exposed the incompleteness of quantum mechanics. Two particles fly apart after interacting, and measuring the momentum of one instantly gives exact knowledge of the other's momentum. It would seem you could also learn the position of the second particle without disturbing it, thereby bypassing Heisenberg's uncertainty principle. For decades, the EPR paradox served as a touchstone for interpretations of quantum theory.

The resolution hides in the logic of prediction. The prediction for the second particle's momentum after measuring the total momentum is not a number but an operator-valued function: its value at each measurement outcome is itself an operator, not an ordinary quantity. This quantum conditional expectation, introduced in the paper, carefully accounts for the fact that measuring one observable does not turn another into a classical variable. It is like the shadow of a measured value: it mirrors the shape, but lives in the quantum world of operators.

The author compares two approaches: quantum conditional expectation and the von Neumann post-measurement state. The second method projects the initial state onto the subspace corresponding to the measured value and normalizes the result. In the limit where the measurement accuracy interval shrinks to zero, both approaches give the same prediction. For a Gaussian entangled state in momentum representation, the conditional mean and variance are computed analytically, and the prediction for the second particle's momentum equals p − p1 with probability 1. Here p is the measured total momentum, and p1 is the first particle's momentum.

When the complete set of observables is measured exactly, the statistical nature of the system vanishes: the probability density becomes a delta function at the observation point. However, such a limit does not correspond to any state vector in Hilbert space. For finite measurement accuracy, Heisenberg's uncertainty relation holds: the product of the uncertainties in position and momentum of the second particle is no less than ħ/2. The EPR paradox is resolved not by a trick but by a rigorous probabilistic scheme showing that exact knowledge of one particle's momentum does not yield exact knowledge of the other's position.

The paper clarifies the logical structure of quantum mechanics. It shows that the EPR paradox arises from ignoring the operator nature of conditional expectations. Practical applications are not yet apparent, but the result matters for interpreting measurements in entangled systems and for quantum information theory. Future work may generalize to more complex observables and imprecise measurements, and apply quantum conditional expectations to problems in quantum parameter estimation and quantum control.