# Quantum-Statistical Dynamics of Cryptocurrency Trajectories: A Centroid-Based Analysis in Hilbert Space

### **1. Quantum States and Superposition**:

Each cryptocurrency logic trajectory can be represented as a quantum state \( | \psi_i \rangle \) in a Hilbert space \( \mathcal{H} \). The entire cryptocurrency universe at any time \( t \) is a superposition of these states:

\[ | \Psi(t) \rangle = \sum_{i=1}^{n} a_i(t) | \psi_i \rangle \]

with \( \sum_{i=1}^{n} |a_i(t)|^2 = 1 \).

### **2. Centroid & Informational Centrality**:

Given our universe of logic trajectories, we can define a centroid or central state \( | \Phi \rangle \) in the Hilbert space \( \mathcal{H} \):

\[ | \Phi \rangle = \frac{1}{n} \sum_{i=1}^{n} | \psi_i \rangle \]

This centroid state can be thought of as the average or "central" logic of the cryptocurrency universe. Deviations from this central logic can be interpreted as informational inefficiencies.

### **3. Quantum Entropy & Distance from Centroid**:

The distance \( d_i \) of a state \( | \psi_i \rangle \) from the centroid \( | \Phi \rangle \) can be given as:

\[ d_i = \sqrt{\langle \psi_i | \psi_i \rangle - 2 \langle \psi_i | \Phi \rangle + \langle \Phi | \Phi \rangle} \]

This distance can be related to the entropy \( S_i \) of a particular state, indicating its informational inefficiency:

\[ S_i = -k \cdot \text{Tr}(\rho_i \ln \rho_i) \]

Where \( \rho_i \) is the density matrix for state \( | \psi_i \rangle \) and \( k \) is a constant. The larger the distance from the centroid, the higher the entropy and vice versa.

### **4. Evolution Operator & Dynamic Centroid**:

The evolution operator \( U(t, t_0) \) ensures our system evolves over time:

\[ | \Psi(t) \rangle = U(t, t_0) | \Psi(t_0) \rangle \]

This implies that our centroid \( | \Phi \rangle \) will also dynamically evolve. Hence, over time, the central logic of the cryptocurrency universe might shift, representing changing market dynamics and trends.

### **5. Quantum Interference & Centroid Displacement**:

Interference can cause certain logic trajectories to amplify or diminish. The overall effect can cause a shift in the centroid position:

\[ \Delta | \Phi \rangle = | \Phi(t_2) \rangle - | \Phi(t_1) \rangle \]

This displacement can be interpreted as the net effect of all interfering trajectories over the time interval \( [t_1, t_2] \).

### **Conclusion**:

Incorporating centroids gives us a geometric and informational perspective on the quantum-like behavior of the cryptocurrency universe. States (cryptocurrencies or strategies) can be viewed in relation to a central tendency, and their informational efficiencies gauged based on their proximity to this center. This mathematical framework offers deeper insights into the evolving dynamics of the crypto market.