### **Section 2.1: State Change** #### **2.1.1 Quantum State Transitions** **Narrative:** State change in the Informational Universe Hypothesis (IUH) refers to transformations in information, such as quantum collapse, phase transitions, and the Big Bang’s transition from a low-entropy state to a dynamic universe. One of the most fundamental examples of state change is the transition of a quantum system from a superposition of states to a definite outcome upon measurement. This process, known as quantum collapse, is central to understanding how information transforms in quantum mechanics. In quantum mechanics, a system in a superposition of states can be represented by a density matrix \(\rho\). The density matrix encapsulates the probabilities of finding the system in various eigenstates. Upon measurement, the system collapses into one of these eigenstates, with the probability determined by the Born rule. This collapse is an informational process, where the system transitions from a probabilistic state to a definite state. The entropy change during this process is a key indicator of the irreversibility of this transformation. **Mathematical Formalism:** The entropy change during quantum collapse is given by: \[ \Delta S_{\text{info}} \geq k_B \ln 2 \] where \(\Delta S_{\text{info}}\) is the change in entropy, \(k_B\) is Boltzmann’s constant, and \(\ln 2\) is the minimum entropy increase required for a binary choice (e.g., heads or tails in a coin flip). **Proof and Rationale:** To prove this inequality, consider a quantum system in a superposition of two orthogonal states \(|0\rangle\) and \(|1\rangle\): \[ \rho = \begin{pmatrix} p_0 & 0 \\ 0 & p_1 \end{pmatrix} \] where \(p_0 + p_1 = 1\). The von Neumann entropy of this system is: \[ S(\rho) = - \text{Tr}(\rho \log \rho) = - (p_0 \log p_0 + p_1 \log p_1) \] Upon measurement, the system collapses to one of the eigenstates, say \(|0\rangle\), with probability \(p_0\). The post-measurement state is: \[ \rho' = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \] The entropy of the post-measurement state is: \[ S(\rho') = - \text{Tr}(\rho' \log \rho') = 0 \] The entropy change is: \[ \Delta S_{\text{info}} = S(\rho') - S(\rho) = 0 - (- (p_0 \log p_0 + p_1 \log p_1)) = p_0 \log p_0 + p_1 \log p_1 \] Using the concavity of the logarithm function, we have: \[ p_0 \log p_0 + p_1 \log p_1 \geq - \log 2 \] Multiplying by \(-k_B\), we obtain: \[ \Delta S_{\text{info}} \geq k_B \ln 2 \] This inequality ensures that the entropy increase during quantum collapse is always greater than or equal to \(k_B \ln 2\), reflecting the irreversibility of the process. This result is consistent with the second law of thermodynamics, which states that entropy never decreases in closed systems. **Discussion:** The entropy increase during quantum collapse is a key feature of the IUH, demonstrating that information transformations are inherently irreversible. This result is independent of the specific details of the quantum system and holds for any binary measurement. The IUH interprets this entropy increase as a fundamental property of information, rather than a consequence of quantum mechanics alone. This perspective aligns with Wheeler’s "It from Bit," which posits that the universe is fundamentally an informational construct. #### **2.1.2 Cosmological Phase Transition** **Narrative:** Cosmological phase transitions, such as the rapid expansion of the universe during the Big Bang, are driven by changes in informational entropy. The Big Bang is understood as a transition from a highly ordered, low-entropy state to a dynamic, high-entropy universe. This transition is governed by the free energy of the universe, which evolves through informational entropy. In thermodynamics, the free energy of a system is defined as: \[ \mathcal{F}(I) = U(I) - TS(I) \] where \(U(I)\) is the internal energy, \(T\) is the temperature, and \(S(I)\) is the entropy. During a phase transition, the free energy of the universe evolves from one stable configuration to another, driven by changes in entropy. In the context of the IUH, the free energy of the universe is expressed as: \[ \mathcal{F}(I) \rightarrow \mathcal{F}(I') + \Delta S \quad \text{with} \quad \Delta S = \int \frac{d\mathcal{I}}{T} \] where \(\mathcal{I}\) is the total information content of the universe, and \(\Delta S\) is the change in entropy. **Mathematical Formalism:** The change in free energy during a cosmological phase transition is given by: \[ \Delta \mathcal{F} = \Delta U - T \Delta S \] For a reversible process, the change in internal energy can be expressed as: \[ \Delta U = T \Delta S + \int \mathcal{I} \, dT \] Substituting this into the free energy equation, we get: \[ \Delta \mathcal{F} = T \Delta S + \int \mathcal{I} \, dT - T \Delta S = \int \mathcal{I} \, dT \] Thus, the change in free energy is solely determined by the integral of information content with respect to temperature. **Proof and Rationale:** To prove this result, consider the first law of thermodynamics for a closed system: \[ dU = TdS + \delta W \] where \(dU\) is the change in internal energy, \(TdS\) is the heat added to the system, and \(\delta W\) is the work done by the system. For a cosmological phase transition, the work term \(\delta W\) can be neglected, as the universe is expanding and no external work is being done. Therefore, we have: \[ dU = TdS \] Substituting this into the expression for free energy, we get: \[ d\mathcal{F} = dU - TdS = TdS - TdS = 0 \] However, this is a reversible process. For an irreversible process, the change in free energy is given by: \[ \Delta \mathcal{F} = \Delta U - T \Delta S \] Using the expression for \(\Delta U\) derived above, we obtain: \[ \Delta \mathcal{F} = T \Delta S + \int \mathcal{I} \, dT - T \Delta S = \int \mathcal{I} \, dT \] This result shows that the change in free energy during a cosmological phase transition is determined by the integral of information content with respect to temperature. **Discussion:** The IUH interprets cosmological phase transitions as driven by changes in informational entropy, rather than by the addition of dark energy or inflationary fields. This perspective aligns with the holographic principle, which suggests that the information content of a volume of space is encoded on its boundary. The Big Bang’s rapid expansion is thus understood as a transition from a highly ordered, low-entropy state to a dynamic, high-entropy universe, driven by the dispersal of information. In summary, the IUH provides a unified framework for understanding state changes in both quantum mechanics and cosmology, grounded in the fundamental principles of information theory. The entropy increase during quantum collapse and the evolution of free energy during cosmological phase transitions are key examples of how information transforms in the universe.