7.1 The Discrete Solution to Schrödinger’s Cat: Entanglement and Branching under Unitary Evolution

In this chapter, we face the most troubling and controversial field in physics: The Measurement Problem.

In classical mechanics, observers are “God’s eyes” standing outside; their observation behavior does not interfere with physical system evolution. But in quantum mechanics, observation seems to play an active, even destructive role—it causes wave function collapse. This directly conflicts with unitary evolution of Schrödinger’s equation.

In our discrete QCA ontology, since the universe is a strictly unitary, deterministic computational process (Axiom $\Omega$), what exactly is “collapse”? Where does probability come from?

This chapter will propose a revolutionary view: Wave functions never collapse. So-called “collapse” and “randomness” are merely perspective effects produced when finite local observers perform self-location in the vast holographic entanglement network.

We will prove: Objective Reality is not a priori existence, but consensus reached between different observers through entanglement and communication.

Let’s return to that famous thought experiment: Schrödinger’s cat.

In a closed box, there is a cat, a radioactive atom, and a poison device. The atom is in superposition of decayed and undecayed:

$$|{\psi }_{atom}⟩=\alpha |0⟩+\beta |1⟩$$

According to linearity of quantum mechanics, after some time, the entire system (atom + cat) evolves into a macroscopic superposition:

$$|{\Psi }_{total}⟩=\alpha |0⟩|Live⟩+\beta |1⟩|Dead⟩$$

The problem is: When we open the box, we either see a live cat or a dead cat, never a “half-dead half-alive” cat. Why?

Traditional Explanation (Copenhagen): At the moment of observation, wave function non-unitarily jumps to one branch.

QCA Explanation (Many-branches/Relative State): Wave function continues unitary evolution, but observer is also entangled.

7.1.1 Physicalization of Observers

In QCA framework, observers are not gods, but physical subsystems with memory and processing capabilities.

Let observer’s initial state be $|Ready⟩$.

When observer opens the box to observe, their state unitarily entangles with system state (driven by local interaction Hamiltonian):

$$|{\Psi }_{final}⟩={\hat{U}}_{obs}(|{\Psi }_{total}⟩\otimes |Ready⟩)$$

$$=\alpha |0⟩|Live⟩|SawLive⟩+\beta |1⟩|Dead⟩|Saw$$

Note that the entire wave function $|{\Psi }_{final}⟩$ is still a pure state, no collapse occurred. But this wave function now contains two terms, each describing a self-consistent world history.

• Branch A: Atom undecayed, cat alive, observer sees cat alive.

• Branch B: Atom decayed, cat dead, observer sees cat dead.

7.1.2 Dynamical Isolation of Branches

The key question is: Why can’t observer in Branch A sense existence of Branch B?

This stems from QCA network’s vast degrees of freedom and decoherence.

Cats and observers are macroscopic objects, containing $10^{23}$ order qubits.

Two macroscopic states $|Live⟩$ and $|Dead⟩$ are not only orthogonal in Hilbert space, but almost impossible to overlap.

To make these two branches interfere again (i.e., make observer A sense observer B), we need to precisely reverse states of all $10^{23}$ particles in the entire system.

Under QCA’s complex dynamics, probability of such reversal decays exponentially with time (Poincaré recurrence time is unimaginably long).

Therefore, although mathematically both branches exist in global wave function, physically and dynamically, they are causally disconnected.

For observer in Branch A, Branch B is like falling into a black hole horizon—theoretically exists, but practically inaccessible.

Conclusion:

Collapse is not disappearance of wave function, but inaccessibility of information.

This phenomenon is called “Branching”. The universe does not split into two; rather, observer’s own state splits into two mutually orthogonal, mutually invisible copies.

Each copy believes it sees the only outcome. This is the source of subjective “collapse” illusion.

In the next section, we will solve a more difficult problem: Why is the probability observer sees live cat $|\alpha {|}^2$, not something else? We will give a pure mathematical proof of Born’s rule.