The Hawking Projector uses black hole physics as inspiration for quantum data compression. The fundamental concept relates to how information behaves near a black hole's event horizon, particularly drawing from Hawking radiation theory.

$$ \mathcal{H}(\rho) = -\text{Tr}(\rho \log \rho) $$

Where $\mathcal{H}(\rho)$ represents the von Neumann entropy of a quantum state $\rho$, analogous to the entropy of a black hole.

The compression algorithm addresses concepts from the black hole information paradox, which presents a fundamental conflict between quantum mechanics and general relativity. According to quantum mechanics, information cannot be destroyed (quantum unitarity), while general relativity suggests that information crossing the event horizon becomes permanently inaccessible to outside observers.

Stephen Hawking proposed that information is preserved through quantum entanglement between the radiation emitted from black holes and the interior matter. This quantum information scrambling transforms but conserves the information in a highly complex encoded form rather than destroying it.

The algorithm implements this principle by computing quantum state entropy:

$ S = -\sum_{i} p_i \log_2(p_i) $

Where $p_i$ represents the probability of measuring the quantum system in state $i$.

The Shannon entropy is then calculated as a mathematical approximation of the von Neumann entropy ($S(\rho) = -\text{Tr}(\rho\log\rho)$), which quantifies quantum information content. This entropy calculation measures how distributed the information becomes across different quantum states after applying the compression transformations, analogous to how black holes distribute information across radiation and interior degrees of freedom.

The encoding process uses quantum circuits that preserve structural information, similar to how information might be encoded on a black hole's event horizon:

$$ |\psi\rangle = \sum_{i=0}^{2^n-1} \alpha_i |i\rangle $$

Where $|\psi\rangle$ is the quantum state representing encoded image data, and $\alpha_i$ are complex amplitudes.

A form of the holographic principle is implemented, where the information content of a region is encoded on its boundary:

$$ S \propto \frac{A}{4G\hbar} $$

This equation from black hole thermodynamics relates entropy ($S$) to the area ($A$) of the event horizon. In the code, this is implemented as spatial locality preservation in the quantum encoding:

sqrt_n = int(np.sqrt(self.num_qubits))

The quantum circuits use controlled rotation gates to simulate the scrambling of information that occurs near a black hole:

$$ R_y(\theta) = e^{-i\theta\sigma_y/2} = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix} $$

Where $\theta$ is the `ROTATION_ANGLE` parameter that controls information scrambling. This is implemented as:

```python # Controlled scrambling with configurable rotation angle for i in range(self.num_qubits):

  qc.ry(self.config['ROTATION_ANGLE'], i)

```

The reconstruction algorithm uses a weighted combination of quantum states, inspired by how information might be retrieved from Hawking radiation:

$$ \text{reconstructed\_bit}_i = \begin{cases} 1 & \text{if } \sum_{j} p_j \cdot \text{bit}_j^i \geq \text{threshold} \\ 0 & \text{otherwise} \end{cases} $$

Where $p_j$ are the probabilities of different quantum states, and $\text{bit}_j^i$ is the $i$-th bit of the $j$-th state.

The code calculates an entropy ratio that measures how close the system is to maximum entropy:

$$ \text{entropy\_ratio} = \frac{\text{entropy}}{\text{max\_entropy}} = \frac{\text{entropy}}{n} $$

Where $n$ is the number of qubits. This relates to the idea that black holes represent maximum entropy configurations.

The spatial relationships in the image are preserved using a context-aware reconstruction method that treats adjacent blocks as entangled systems:

$$ \text{reconstructed\_array}[y:y+h, x:x+w] += \text{block\_array} \cdot \text{weight} $$

This is implemented in the `reconstruct_with_context` method.

The reinforcement learning environment uses a reward function that incorporates metrics inspired by black hole information theory:

$$ \text{reward} = w_1 \cdot \text{pixel\_match\_rate} + w_2 \cdot \text{edge\_preservation} + w_3 \cdot \text{detail\_preservation} + \ldots $$

These metrics measure how well the information is preserved through the compression and reconstruction process, similar to how information might theoretically be preserved despite passing through a black hole.