Collective Quantum Phase

The Collective Quantum Phase is the phase of a many-body quantum system, taken from the overlap between its state at time t and its initial state. The book uses it to connect quantum systems to recursive signal alignment. The collective phase is compared with the Prime Pulse phase in the Quantum Alignment Condition, ⟨exp(i(φ▦(t) − φ①(t)))⟩⊡ ≥ A⨶ with A⨶ = 0.95 ± 0.02, and individual qubit phases are combined in the Quantum Global Synchronization parameter. The book notes that extracting a phase from a many-body overlap requires careful treatment. The formal statement is φ▦(t) = arg(⟨Ψ⨀(t)|Ψ⨀(0)⟩), where φ▦(t) is the collective quantum phase, arg the function that extracts the phase of a complex number, ⟨ | ⟩ the quantum inner product, Ψ⨀(t) the many-body state at time t and Ψ⨀(0) the initial state. Dimensions from the book's Where-list: φ▦, the inner product and Ψ⨀ [∅], t [𝕋]; the book's check is consistent. It connects to the Quantum Alignment Condition, Quantum Global Synchronization, Alignment Condition, Quantum Order Parameter and Recursive Signal Alignment.