Details of calculations from Posner qubits: spin dynamics of entangled Ca₉(PO₄)₆ molecules and their role in neural processing

It has been suggested that ³¹P nuclear spins in Ca₉(PO₄) ⁶ molecules could form the basis of a quantum mechanism for neural processing in the brain. A fundamental requirement of this proposal is that spins in different Ca₉(PO₄) ⁶ molecules can become entangled and remain so for periods (estimated at many hours) that hugely exceed typical ³¹P spin relaxation times. Here, we consider the coherent and incoherent spin dynamics of Ca₉(PO₄) ⁶ arising from dipolar and scalar spin–spin interactions and derive an upper bound of 37 min on the entanglement lifetime under idealized physiological conditions. We argue that the spin relaxation in Ca₉(PO₄) ⁶ is likely to be much faster than this estimate.