Toss a dozen coins on a table. The number of heads-and-tails combinations is just over a thousand. For two dozen coins, it's already a million. The number grows as 2 to the power of the number of coins, and that's fast. But quantum systems are far richer.
Each particle here is not a coin with two sides but an arrow that can point in any direction. A single arrow has many such directions, and when there are N particles, the total number of independent directions is q to the power of N. This quantity is called dimensionality. The authors are interested in something else: how many of these directions can be chosen so that they are almost perpendicular to each other. "Almost" means that the angle between them deviates from a right angle by a tiny fraction, and the scalar product (a measure of overlap) does not exceed a given small tolerance. It turns out that such almost orthogonal directions are not just numerous: their number grows as an exponential of the dimensionality. And the dimensionality itself is exponential in N. This gives an exponential of an exponential. If you imagine a room with a million light bulbs, the number of possible light patterns is not a million but a number that would not fit in the entire Universe.
Such an explosion of almost orthogonal directions is not just a mathematical curiosity. It explains how entire families of branches that barely interfere with each other emerge in the quantum world. When a system interacts with its environment, its state splits into many classical alternatives — much like voices in a noisy room begin to sound separately. The authors applied these estimates to decoherence branches — the process by which quantum entanglement with the environment turns superpositions into mixtures. They showed that even for finite systems and mixed states, one can guarantee the simultaneous almost orthogonality of entire families of branches. This requires no preferred basis or information recording, just typical environmental dynamics.
This approach does not solve the measurement problem, but it provides a quantitative language for describing how the quantum multiverse becomes practically indistinguishable from a set of independent classical histories. Instead of philosophical debates — numbers, tolerances, and probabilities. And again, the room with light bulbs comes to mind: the number of almost orthogonal patterns is so monstrous that no classical analogy can contain it.
If every particle in the Universe were a qubit, the number of almost orthogonal directions would exceed the number of all atoms raised to the power of the number of atoms, and that would still be a tiny fraction of the total number of possible states.
The idea of almost orthogonal branches evokes David Brin's 'Uplift Saga,' where many intelligent species exist in nearly non-overlapping planes of being, only occasionally catching glimpses of each other's shadows.