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τML(τ)=πℏ/[2(⟨H(τ)⟩−E0(τ))]\tau_{\rm ML}(\tau) = \pi\hbar/[2(\langle H(\tau)\rangle-E_0(\tau))]

Why this formula appears here

The dossier for this paper flags something the two equations above do not, by themselves, settle: the periodic-evolution bound above was derived for a Hamiltonian with a genuinely time-independent spectrum. A real computation drives H(τ\tau) through a sequence of different instantaneous Hamiltonians, one per gate, and nothing guarantees in advance that a chain of instantaneously applied bounds sums to a valid bound on the whole chain. The condition that licenses treating N⊥N_\perp[Γ\Gamma] as a legitimate partition into instantaneously static Levitin-Toffoli steps is a quasi-static one: over one local orthogonalization time τML(τ)\tau_{\rm ML}(\tau) = π\piℏ\hbar/[2(⟨\langle H(τ\tau)⟩\rangle-E0(τ)E_0(\tau))] ,…

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τML\tau_{\rm ML}

Symbol tau_rm ML

tauru_rm ML is part of the quantity the equation computes from the expression on the right.

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Published contexts (1)

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τML(τ)=πℏ/[2(⟨H(τ)⟩−E0(τ))]\tau_{\rm ML}(\tau) = \pi\hbar/[2(\langle H(\tau)\rangle-E_0(\tau))]

Equation 36 · Evolutionary Physics

A Horizon Is a Toll Booth, Not a Loophole

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

The dossier for this paper flags something the two equations above do not, by themselves, settle: the periodic-evolution bound above was derived for a Hamiltonian with a genuinely time-independent spectrum. A real computation drives H(τ\tau) through a sequence of different instantaneous Hamiltonians, one per gate, and nothing guarantees in advance that a chain of instantaneously applied bounds sums to a valid bound on the whole chain. The condition that licenses treating N⊥N_\perp[Γ\Gamma] as a legitimate partition into instantaneously static Levitin-Toffoli steps is a quasi-static one: over one local orthogonalization time τML(τ)\tau_{\rm ML}(\tau) = π\piℏ\hbar/[2(⟨\langle H(τ\tau)⟩\rangle-E0(τ)E_0(\tau))] ,…

Meanings in this article

  • τ\tau: a time, the integral has action units.
  • E0E_0: the ground-state energy of H [ 2 , 1 ].
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