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internal operation. Otherwise it remains in the coherent 50:50-superposition.
Alternatively, the computer could be initially prepared in the non-halting state f. After com-
pletion of the task, the halting bit is again switched to the halting state t.
In analogy to the fully classical case [28, 84, 24], the quantum halting amplitude15 Ω can be
defined as a weighted expectation over all computations of C with classical input p (p stands
i i
| |
for the length of p )
i
Ω 2−|pi|/2(t,C(p i)) . (34)
≡
C(Xpi) ∈H
14ThequantumomegawasinventedinameetingofG.Chaitin,A.ZeilingerandtheauthorinaViennesecoffee
house (Caf´e Br¨aunerhof) in January 1991. Thus, the group should be credited for the original invention, whereas
anyblameshouldremainwiththeauthor.
15 ThedefinitionofΩandΥdifferslightlyfromtheonesintroducedbytheauthor previously[89].
14
Likewise, the halting amplitude for a particular output state s,
Υ(s) 2 −|pi|/2(t,C(p i)) . (35)
≡
C(Xpi)=s
ForasetofoutputstatesS = s ,s ,s ,...,s whichcorrespondtomutually orthogonalvectors
1 2 3 n
{ }
in Hilbert space,
Υ(S) 2 −|pi|/2(t,C(p i)) . (36)
≡
C(Xpi) ∈S
Terms corresponding to different programs and states have to be summed up incoherently.
Thus, the corresponding probabilities are
Ω2 = 2 −|pi| (t,C(p i))2 (37)
| | | |
C(Xpi) ∈H
P(s) Υ(s)2 = 2−|pi| (t,C(p i))2 (38)
≡ | | | |
C(Xpi)=s
P(S) Υ(s)2 = 2 −|pi| (t,C(p i))2 . (39)
≡ | | | |
C(Xpi) ∈S C(Xpi) ∈S
The following relations hold,
Υ(S) = Υ(s ) , (40)
i
s Xi∈S
Ω = Υ(H)= Υ(s ) . (41)
i
s Xi∈H
For s S H,
⊂ ⊂ 0 P(s) P(S) Ω2 1 . (42)
≤ ≤ ≤| | ≤
Alternatively, the quantum halting probability andthe quantum algorithmic information by the
quantum algorithmic information content. That is,
P∗(s) = 2−|s∗| =2−H(s) (43)
P∗(S) = P∗(s)= 2−H(s) (44)
s Xi∈S Xs ∈S
P (H)= Ω 2 = 2 H(n) . (45)
∗ ∗ −
| |
n H
X∈
Ω 2 Ω2 , (46)
∗
| | ≤ | |
P∗(s) P(s) , (47)
≤
P (S) P(S) . (48)
∗
≤
The following relations are either a direct consequence of the definition (43)or follow from the
fact that for programs in prefix code, the algorithmic probability is concentrated on the minimal
size programs, or alternatively, that there are few minimal programs:
H(s) = −log 2P∗(s) ; (49)
H(s) = log P(s)+O(1) . (50)
− 2
Notice again that, because of complementarity, single qbits cannot be determined precisely.
They just appear experimentally as some clicks in a counter. What we can effectively do is
to observe a successive number of such qbits, one after the other, from “similar” computation
processes (same preparation, same evolution). By performing these measurements on “similar”
qbits, one can “determine” this qbit within an ε-neighborhood only.
For nontrivial choices of the quantum computer C, several remarks are in order. (In what
follows, we mention only Ω, but the comments apply to Υ as well.) If the program is also coded
inqbits, the abovesumbecomes anintegralovercontinuouslymany states per code symbolofthe
15
programs. In this case, the Kraft sum needs not converge. Just as for the classical analogue it is
possible to “compute” Ω as a limit from below by considering in the t’th computing step (time
τ) all programs of length τ which have already halted. (This “computation” suffers from a radius
of convergence which decreases slower than any recursive function.) The quantum Ω is complex.
Ω2 can be interpreted as a measure for the halting probability of C; i.e., the probability that an
| |
arbitrary (prefix-free) program halts on C.
Finally, anyirreversiblemeasurementof Ω2 causesa statecollapse. Since C(τ,p )maynotbe
i
| |
in a pure state, the series in (34) and (35) will not be uniquely defined even for finite times. Thus
the nondeterministic character of Ω is not only based on classical recursion theoretic arguments
[28] but also on the metaphysical assumption that God plays the quantum dice.
Appendices
A Two-state system
Having set the stage of the quantum formalism, an elementary twodimensional example of a two-
state system shall be exhibited ([39], p. 8-11). Let us denote the two base states by 1 and
2. Any arbitrary physical state ψ is a coherent superposition of 1 and 2 and can be written as