Is quantum mechanics deterministic or random?

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

\Psi(t) = e^{-\frac{i}{\hbar} \int_0^t H(t) dt} \Psi(0) .

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

P(x) = \left|\Psi(x)\right| ,

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

\frac{1}{\sqrt{2}} \left ( \left|\uparrow\right> + \left|\downarrow\right> \right) .

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.

\Psi_{\text{initial}} = \frac{1}{2} \left ( \left|\uparrow\right> + \left|\downarrow\right> \right) \left ( \left|\text{detect}\uparrow\right> + \left|\text{detect}\downarrow\right> \right) .

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

\Psi_{\text{final}} = \frac{1}{\sqrt{2}} \left ( \left|\uparrow\right> \left|\text{detect}\uparrow\right> + \left|\downarrow\right> \left|\text{detect}\downarrow\right> \right) .

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.


Ideal gas, about to double 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.

Understanding quantum mechanics

For almost a century now, there is an on going debate on the interpretations of quantum mechanics. Two such interpretations are the Copenhagen Interpretation (CI) and the Many Worlds Interpretation (MWI).

Beyond that, there is a meta-debate going on for almost as long about whether there is any value in debating these interpretations. Physicists that think this is a pointless endeavor consist of the Shut Up And Calculate (SUAC) camp.

In a recent article in the New York Times, Sean Carroll writes that Even Physicists Don’t Understand Quantum Mechanics. The New York Times is not a scientific journal, so normally I would not cite it as an authoritative scientific source. But it is a good source to demonstrate sentiments. This is an example of a physicist studying the foundation of quantum mechanics who feels that too many physicist do not care that they do not understand quantum mechanics, as long as they know how to make calculations.

Much of what is written here was inspired by Sabine Hossenfelder’s blog. Specifically, by her post on quantum measurement and the long comment thread that followed. This is also where I was referred to a paper by Lev Vaidman on the Ontology of the wave function and the many-worlds interpretation. My views are closely aligned with the views he presents in this paper. Vaidman’s entire career was researching the foundations of quantum mechanics. It would be hard to blame him that he does not care about the subject.

Let me try to convince you that at least some of us in the SUAC camp do care about understanding quantum mechanics. The point is that we actually understand quantum mechanics fairly well. While we might not understand everything, our understanding is deep enough that we are convinced that we do not need to worry about quantum interpretations. We also understand the measurement process. The “measurement problem” is a solved problem.

As a baseline, all parties in this debate actually agree about almost everything. We all agree that the theoretical formulation of quantum mechanics is based on Schrodinger’s equation (and all its extensions) together with the Born rule that interprets the wave-function as a probability amplitude. We also agree about all the experimental results that confirm the validity of quantum mechanics.

The only disagreement is about the process of measurement, which you could say, is somewhat crucial for connecting theory with experiment. So lets delve deeper into the Measurement Postulate.

A measurement involves an observable operator which is an Hermitian operator. Examples of Hermitian operators are position, momentum, energy, angular momentum and spin. An Hermitian operator has real eigen-values that correspond to the real values that we measure. Each eigen-value has a corresponding eigen-state.

This is where the problem begins. The Measurement Postulate states that a measurement would result in the system being in the eigen-state corresponding to the eigen-value that was measured. This is a non-unitary, non-local, non-deterministic, irreversible process. Therefore, it can not be described using the standard Hamiltonian time evolution of quantum mechanics.

The Copenhagen Interpretation (CI) claims that the wave-function collapses during the measurement. Therefore, the measurement process cannot be described using the standard Hamiltonian time evolution of quantum mechanics. The problem with this interpretation is that no one ever came up with a mechanism for this collapse that is consistent with existing experiments and that makes any new unique predictions.

Many Worlds Interpretation (MWI) claims that all branches of the measurement keep on existing, there is no collapse. One could say that the world “splits” during measurement to several worlds, but what MWI really says is that everything is encoded in the wave-function, so there is no special event during measurement. This solves the unitarity problem. We now have a reversible, deterministic, local model.

One criticism of MWI is that without collapse the measurement has no effect on our system. Choosing an observable Hermitian operator is equivalent to choosing a basis, it does not change the physics. This is a legitimate criticism, but the problem is not in quantum mechanics. The problem is that in the formulation of the Measurement Postulate, people forget to mention that a measurement requires an interaction between the observable operator and our system.

For example, if we want to measure the spin of a particle, we use the Stern–Gerlach experiment as a detector. Such a detector is essentially equivalent to adding an interaction term in the Hamiltonian that describes the coupling between the particle’s spin and a non-uniform magnetic field. A measurement is not just a change of basis. We can model how our detector deflects different spins in different directions.

The Stern-Gerlach experiment

The Stern-Gerlach experiment

To illustrate the effect of the measurement, think about an experiment that starts with a spin pointing up (\left|\uparrow\right>). The spin is first measured in the X axis and then it is measured in the Y axis.


A Stern–Gerlach experiment where the initial state is spin up. First the spin is measured in the X direction and then in the Y direction.

If, like the critics claim, the measurement in the X axis had no effect on the spin, it would stay in the up state. Then the measurement in the Y direction would just be an up state (\left|\uparrow\right>). If the measurement “collapses” the wave-function, there would be a mix of two pure spin states, one pointing left (\left|\leftarrow\right>) and one pointing right (\left|\rightarrow\right>). Then, the measurement in the Y axis would measure each such pure state as a combination of up (\left|\uparrow\right>) and down (\left|\downarrow\right>)  spins, resulting in a mixed state of 4 pure spin states. 


Four different outcome of our experiment where we vary the strength of the measurement in the X direction. (a) there is no X measurement, we only measure spin up. (b) weak X measurement, there is small separation in X and therefore a small chance of measuring spin down. (c) stronger X measurement, better separation in X and therefore a higher change of measuring spin down. (d) Full separation in X and therefore all four possible outcomes have equal probability.

Conceptually, one could calculate the results of this experiment using the Schrödinger equation, modelling explicitly the elusive wave-function collapse. There is no need for any extra measurement postulate. I must admit that I did not perform these calculations. Solving the Schrödinger equation for this case can be done analytically. But matching the boundary conditions and setting initial condition for the wave-packet is a bit more complicated. If I’ll get around to it, I will post a detailed explanation of the calculation. Meanwhile, the plot above illustrates the expected results.

Another criticism of MWI is that we cannot have any evidence about other worlds because these worlds have no effect on our world. But we do have evidence. In the double slit experiment the particle goes through one slit in one world and through the other slit in the other world. The interference pattern in the double slit experiment is the manifestation of the effect that different worlds have on each other.

Some worry that scaling up the measurement to macroscopic scales would require giving up reductionism. Yet the Stern-Gerlach experiment is an explicit example of how a microscopic spin quantity transforms into a macroscopic spatial separation.

Others worry that there is something mysterious going on during decoherence. Again, the Stern-Gerlach experiment shows us exactly how it works. The calculation is time-reversible.  Still, the incoming wave-packet is a combination of energy states that come out of the detector with very specific phases for each state. Reversing the experiment by sending in two spins and trying to get out a pure spin state is clearly not practical.

The last criticism I can think of is that in MWI probabilities are meaningless if we claim that all outcomes exist. I do not see how this different from CI where all outcomes are possible. Moreover, it is not different from classical probabilities.

To summarize, since the introduction of the Schrödinger equation in 1925, and the Born rule in 1926, we have made some progress in our understanding of quantum mechanics. The EPR experiment, Bell’s inequalities and Everett’s many worlds interpretation contributed towards this understanding. This progress convinces us that nature behaves exactly according to the rules quantum mechanics. There is just no shred of evidence for a missing ingredient in our understanding.

As physicists we always hope that nature would challenge us with fascinating riddles. Unfortunately, I am fairly convinced that interpretations of quantum mechanics and the Measurement Postulate are not such riddles.