I'll try my best but it has been a few years since quantum for me.

Basically, over time a quantum system can be transformed by applying an operator to the original state to yield the new state. These operators can be represented as a "transformation" between one quantum space to another (the one at the future time).

Because these transformations preserve probability (which is like a norm of the system), they must have a property known as being unitary.

Unitary matrices are invertible, so this is what they mean by "can be undone."

If all probabilities have to sum to 100% at all times, it limits how you can go from one state to another. I think an additional fact you’re unaware of here is that states transition into each other smoothly. So if you’re at 80% state A and 20% state B, at a very small time later, you can only be at 79% A and 21% B, say. This is related to the fact that the time evolution operator can be written as a power series in dt.

The explanation in the press article is quite confusing, but the probability does not axed 1.0. The idea is that instead of a transformation from a space of dimension 2 into a space of dimension 2, they make a transformation from a space of dimension 2 into a space of dimension 3. In both cases the total probability is conserved and is 1.0.

Something related.

In classical computers you can do:

  X = 4
  X = 5

In quantum computers this sequence of operations is impossible, because the program needs to respect unitarity and need to be able to be undone/run backwards.

But you can't run this program backwards because the second operation (=5) completely erases the information from the first step (=4)

Classically, isn't the reverse program just (?):

X = 5

X = 4

I feel like 2 things are being mixed. In QM/QC, the operators must be unitary, which roughly means they "rotate" states and their inverse "rotation" exists, so (in principle) the reverse operation can be performed; so (in principle), we can construct the reverse operations, execute them, and kinda reverse the time evolution. But it doesn't mean that the quantum state (at time t) of the system encodes the time history (t'<t) of the state.

> But it doesn't mean that the quantum state (at time t) of the system encodes the time history (t'<t) of the state.

it does if you take an everettian perspective on QM, the universal wavefunction at time T can always be unwound to time T'.

I should have been more specific, the 4 and 5 values are dynamic runtime values, not constants known ahead of time.