Quantum mechanics is intrinsically random. If we look at a radioactive element, it will decay with a specific half-life time, but there is no way to predict exactly when it is going to decay. It is a random process.

On the other hand, the Schrödinger equation that describes the time evolution of a quantum system is fully deterministic. Given well defined initial conditions for the wave-function at some point in time, we can determine the wave-function in any other time, in the future or in the past

.

How can these two statements coexist in a single consistent theory?

The answer is that it is just a question of perspective. If you look at the entire wave-function, you would realize that the theory is deterministic. But if you write down the wave-function as a combination of eigen-states, then each state has a probability associated with it depending on its amplitude, according to the Born rule

,

and therefore from the perspective of the eigen-state the system is random.

For example, take a spin which is in a superposition of up state and down state

.

There is a 50% chance that it is an up spin and 50% chance that it is a down spin. This randomness/probability is intrinsic in quantum mechanic. The randomness is there even without measurement, but it is easier to describe when we talk about a detector that made a measurement. Before the measurement the spin and the detector are uncorrelated.

.

When our detector measures the spin, it also gets into a superposition of states. There is one state in which the spin is up and the detector measured spin up. There is a second state in which the spin is down and the detector measured spin down

.

This measurement process is unitary. The wave-function of the entire system after the measurement can be calculated deterministically. Yet from the point of view of the detector, a purely random event occurred. It is impossible to predict what spin will be detected, because in the final state both spins are detected. But from the point of view of the detector, only one spin is detected, and therefore from its perspective it is a purely random event.

Moreover, random processes are irreversible. If a ball bounces of a wall at a random angle, there is no way to trace back in time the motion of the ball before it hit the wall. Indeed, from the point of view of the detector, the measurement is irreversible. If you start from a state of a detector that measured spin up, there is no way to trace back time to the original state of the spin before measurement. On the other hand, if you start with the full final state that includes the superposition of the two states, you can reverse time and recover the original spin state.

This picture is also consistent in the context of thermodynamics. In thermodynamics, entropy is not an intrinsic property of the physical system. Entropy is a measure of how much we know about the system, it depends on our perspective.

For example, take a classical ideal gas in a box. What happens if you double the volume of the box? If you know the exact position and momentum of every particle in the gas at a certain time, you can predict their future position. Therefore the entropy of the system does not change when the volume increases. But, if you are not aware of the microscopical state, your conclusion would be that the entropy increased with the change in volume.

Back to our quantum system, from the perspective of the entire wave-function, the entropy is constant. As long as you look at the microscopic details of a closed system, the entropy does not increase over time. This is consistent with the fact that the process is reversible.

From the perspective of the detector, there was one bit of random information in the measurement, and therefore the entropy increased exactly by log(2). This is again consistent with the fact that from the point of view of the detector the process is irreversible.

We described the exact same physical system from the point of view of two different actors with different knowledge about the system. One description is deterministic and one is random. Both are correct at the same time, it is just a question of what is your perspective.