codex
30 lines
2.9 KiB
Markdown
30 lines
2.9 KiB
Markdown
# Recursive Witness Dynamics: Redundant Information Imprinting and Quantum Darwinism
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**Target Venue:** *Journal of The Royal Society Interface*
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## Abstract
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To map the classical transition kernels of conscious agents to quantum physics, we explicitly derive the emergence of classical objectivity via Quantum Darwinism. We model the central agent $S$ and an environment partitioned into multiple fragments $E_F$. We define a dominant system Hamiltonian and a pure dephasing interaction $H_{int} \propto \sigma_S^z \otimes \sigma_{E_k}^z$ that commutes with the pointer basis. We derive the specific Lindblad jump operators $L \propto \sigma_S^z$. We then explicitly calculate the Mutual Information $I(S; E_F)$ across multiple environmental fragments. By demonstrating that the Holevo bound is saturated for multiple independent sub-baths, we prove that redundant copies of the system's pointer state are imprinted into the environment. This redundancy rigorously defines the emergence of the classical, objective Markovian networks utilized in Conscious Realism.
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## 1. Introduction
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Classical objective states do not just emerge from generic decoherence; they emerge from the redundant proliferation of information into environmental fragments (Quantum Darwinism).
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## 2. Microscopic Dephasing and the Pointer Basis
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Let the system $S$ have a dominant Hamiltonian $H_S = \frac{\omega_0}{2} \sigma_S^z$.
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To preserve the pointer basis against environmental scrambling, the interaction Hamiltonian must commute with $H_S$. We define a pure dephasing interaction:
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$$
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H_{int} = \sum_k g_k (\sigma_S^z \otimes \sigma_{E_k}^z)
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$$
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Applying the Born-Markov and secular approximations, the resulting Lindblad jump operator is strictly pure dephasing: $L \propto \sigma_S^z$. The $\sigma_S^z$ eigenstates form the robust pointer basis.
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## 3. Redundant Imprinting and the Holevo Bound
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The environment $E$ is partitioned into disjoint fragments $E_F$. We evaluate the mutual information $I(S; E_F)$ between the central system and a fraction $f$ of the total environment.
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The interaction $H_{int}$ deterministically entangles the pointer states of $S$ with the local states of $E_k$. The decoherence functional suppresses off-diagonal terms while redundantly copying the diagonal state information into multiple independent fragments $E_F$.
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The mutual information scales as $I(S; E_F) \approx H(S)$ for a very small fraction $f$, saturating the Holevo bound. This proves that many independent observers can interdependently deduce the state of $S$ without disturbing it.
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## 4. Conclusion
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The classical Markov kernels of Conscious Realism emerge rigorously from pure dephasing interactions and the redundant proliferation of pointer state information across environmental fragments.
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## References
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1. Zurek, W. H. (2009). *Quantum Darwinism*. Nature Physics.
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2. Hoffman, D. D., & Prakash, C. (2014). *Objects of consciousness*. Frontiers in Psychology.
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