## On the Brightness of the Thomson Lamp. A Prolegomenon to Quantum Recursion Theory (2009)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Svozil09onthe,

author = {Karl Svozil},

title = {On the Brightness of the Thomson Lamp. A Prolegomenon to Quantum Recursion Theory },

year = {2009}

}

### OpenURL

### Abstract

Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accelerated (hyper-) computers and the recursion theoretic diagonal methods are discussed. As quantum information is not bound by the mutually exclusive states of classical bits, it allows a consistent representation of fixed point states of the diagonal operator. In an effort to reconstruct the self-contradictory feature of diagonalization, a generalized diagonal method allowing no quantum fixed points is proposed.

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Citation Context ... and “off,” respectively, and the switching process with the concatenation of “+1” and “-1” performed so far, then the divergent infinite series associated with the Thomson lamp is the Leibniz series =-=[29,30,3,31]-=- ∞∑ s = (−1) n =1− 1+1− 1+1−··· A = 1 . (3) 2 n=0 Here, “A” indicates the Abel sum [3] obtained from a “continuation” of the geometric series, or alternatively, by s =1−s. As this shows, formal summat... |

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Citation Context ...ices in reverse order in which the quanta pass these elements [53, 54]; i.e., U bs ( ) ( ) ( ) ( iϕ e 0 i sin ω cos ω i(α+β) e 0 1 0 (ω, α, β, ϕ) = 0 1 cos ω i sin ω 0 1 0 eiβ ) ( ) i (α+β+ϕ) i (β+ϕ) =-=(13)-=- i e sin ω e cos ω = . e i (α+β) cos ω i e i β sin ω For this physical setup, the phases ω = π/2, β = λ − π/2 and ϕ = −α can be arranged such that U bs (π/2, α, λ − π/2, −α) = diag(e iλ , e iλ ). Anot... |

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Citation Context ...unitary transformations of the form [U2(ω, α, β, ϕ)] −1 diag(1, eiλ ( )U2(ω, α, β, ϕ) = cos = 2 ω + ei λ sin 2 1 ω 1 2e−i (α+ϕ)(ei λ − 1) sin(2 ω) 2ei (α+ϕ)(ei λ − 1) sin(2 ω) ei λ cos2 ω + sin 2 ) ω =-=(10)-=- for arbitrary λ have fixed points.Applying non-classical operations on quantum bits with no fixed points [U2(ω, α, β, ϕ)] −1 diag(eiµ , eiλ ( )U2(ω, α, β, ϕ) = e = i µ cos2 ω + ei λ sin 2 1 ω 1 2e−i... |

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Citation Context ...al diagonalization procedures. Thereby, one could allow the entire range of two-dimensional unitary transformations [51] −i β U2(ω, α, β, ϕ) = e ( i α −i ϕ e cos ω −e sin ω ei ϕ sin ω e−i α ) cos ω , =-=(8)-=- where −π ≤ β, ω ≤ π, − π π 2 ≤ α, ϕ ≤ 2 , to act on the quantum bit. A typical example of a non-classical operation on a quantum bit is the “square root of not” ( √ X · √ X = X) gate operator √ X = 1... |

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Citation Context ...g. 1 can be described either by the transitions [52] P1 : |0〉 → |0〉e i(α+β) , P2 : |1〉 → |1〉e iβ , S : |0〉 → √ T |1 ′ 〉 + i √ R |0 ′ 〉, S : |1〉 → √ T |0 ′ 〉 + i √ R |1 ′ 〉, P3 : |0 ′ 〉 → |0 ′ 〉e iϕ , =-=(12)-=- where every reflection by a beam splitter S contributes a phase π/2 and thus a factor of eiπ/2 = i to the state evolution. Transmitted beams remain unchanged; i.e., there are no phase changes. Altern... |

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Citation Context ... and “off,” respectively, and the switching process with the concatenation of “+1” and “-1” performed so far, then the divergent infinite series associated with the Thomson lamp is the Leibniz series =-=[29, 30, 3, 31]-=- ∞∑ s = (−1) n = 1 − 1 + 1 − 1 + 1 − · · · A = 1 . (3) 2 n=0 Here, “A” indicates the Abel sum [3] obtained from a “continuation” of the geometric series, or alternatively, by s = 1−s. As this shows, f... |

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Citation Context ...e iλ ,e iλ ). Another example is U bs (π/2, 2λ, −π/2 − λ, 0) = diag(e iλ ,e −iλ ). For the physical realization of general unitary operators in terms of beam splitters the reader is referred to Refs. =-=[55,56,57,58]-=-. 6 Summary In summary we have discussed some physical aspects related to the limit operations of the Thomson lamp. This physical limit, regardless of the formally unbounded and even infinite number o... |

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Citation Context ...ent sums could be valuable approximations of solutions of differential equations which might be even analytically solvable. One of the bestknown examples of this case is Euler’s differential equation =-=[32, 4, 33]-=- z 2 y ′ +y = z, which has both (i) a formal solution as a power series ˆ f(z) = − ∑ ∞ n=0 n! (−z)n , which diverges for all nonzero values of z; as well as (ii) an exact solution ˆf(z) = e 1/z Ei(−1/... |

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Citation Context ...cal case and for the formal Abel sum, they represent a fifty-fifty mixture of the “on” and “off” states. 4 Quantum Fixed Point of Diagonalization Operator In set theory, logic and in recursion theory =-=[21,44,45,46,22,23]-=-, the method of proof by contradiction (reductio ad absurdum) requires some “switch of the bit value,” which is applied in a self-reflexive manner. Classically, total contradiction is achieved by supp... |

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