Equation 36 · Part 5 · A Horizon Is a Toll Booth, Not a Loophole

Symbol E_0

\tau_{\rm ML}(\tau) = \pi\hbar/[2(\langle H(\tau)\rangle-E_0(\tau))]

E_0

What this part means

the ground-state energy of H [ 2 , 1 ].

Its job in the formula

$E_0$ is one of the signed contributions combined to compute the quantity on the left.

Full expressionSymbol E_0Article meaning

Where the article explains it

For a system with a fixed, time-independent Hamiltonian H evolving from a state $\lvert$ $\psi$ $\rangle$ toward any state orthogonal to it, the minimum time required is bounded below by the tight, unified Margolus-Levitin/Mandelstam-Tamm result of Levitin and Toffoli: $\tau_\perp$ $\geq$ $\frac{\pi\hbar}{2\left(\langle H\rangle - E_0\right)}$ , where $\langle$ H $\rangle$ is the mean energy in the evolving state and $E_0$ is the ground-state energy of H [ 2 , 1 ] .

The passage around this formula

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( $\tau$ ) 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 $N_\perp$ [ $\Gamma$ ] as a legitimate partition into instantaneously static Levitin-Toffoli steps is a quasi-static one: over one local orthogonalization time $\tau_{\rm ML}(\tau)$ = $\pi$ $\hbar$ /[2( $\langle$ H( $\tau$ ) $\rangle$ - $E_0(\tau)$ )] ,…

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the article section

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