Manuscript Oai283-A@1 · Oai283-A.tex · proposition · 6fb75895

Proposition: Nonuniqueness of a generator

Regularity, dynamics, and implementation non-transfers › A unitary at one time is not a dynamical law · lines 1603–1618 · proof included or linked

Oai283-A@1#prop:generator

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Exact knowledge of a unitary channel at one declared time does not determine its Hamiltonian generator.

Proof. For a fixed $t_{0} > 0$ , let $P$ be a nonzero proper projector. Both

H_{0} = 0, H_{1} = \frac{2 π ℏ}{t _{0}} P

give $e^{- i t_{0} H_{j} /ℏ} = I$ , but they give different evolution at other times. Adding any scalar multiple of identity also changes the Hamiltonian without changing its channel evolution.

Displayed formulas in this passage

unlabeled · line 1612
$H_{0} = 0, H_{1} = \frac{2 π ℏ}{t _{0}} P$

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section: A unitary at one time is not a dynamical law