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Classicalinformationtheory(e.g.,[49])is basedonthe classicalbitasfundamentalatom. This
classical bit, henceforth called cbit, is in one of two classical states t (often interpreted as “true”)
and f (often interpreted as “false”). It is customary to code the classical logical states by ptq=1
and pfq = 0 (psq stands for the code of s). The states can, for instance, be realized by some
condenser who is discharged ( cbit state 0) or charged ( cbit state 1).
≡ ≡
In quantum information theory (cf. [1, 31, 41, 74, 6, 68, 32, 33]), the most elementary unit of
information is the quantum bit, henceforth called qbit. Qbits can be physically represented by a
coherent superposition of the two orthonormal8 states t and f. The qbit states
x =αt+βf (1)
α,β
form a continuum, with α2+ β 2 =1, α,β C.
| | | | ∈
3.1 Coding
Cbits can then be coded by
px q=(α,β)=eiϕ(sinω,eiδcosω) , (2)
α,β
with ω,ϕ,δ R. Qbits can be identified with cbits as follows
∈
(a,0) 1 and (0,b) 0 , a, b =1 , (3)
≡ ≡ | | | |
wherethecomplexnumbersaandbareofmodulusone. Thequantummechanicalstatesassociated
with the classical states 0 and 1 are mutually orthogonal.
Notice that, provided that α,β = 0, a qbit is not in a pure classical state. Therefore, any
6
practical determination of the qbit x amounts to a measurement of the state amplitude of t or
α,β
f. Any such single measurement will be indeterministic (provided again that α,β = 0). That is,
6
theoutcomeofasinglemeasurementoccursunpredictably. Yet, accordingtotherulesofquantum
mechanics,theprobabilitiesthattheqbitx ismeasuredinstatestandf isP (x )= (x ,t)2
α,β t α,β α,β
| |
and P (x )= (x ,f)2 =1 P ( ), respectively.
f α,β α,β t α,β
| | −
The classical and the quantum mechanical concept of information differ from each other in
several aspects. Intuitively and classically, a unit of information is context-free. That is, it is
independent of what other information is or might be present. A classical bit remains unchanged,
nomatterby whatmethods itis inferred. Itobeysclassicallogic. Itcanbe copied. Nodoubts can
be left.
By contrast, quantum information is contextual [55] A quantum bit may appear different,
depending on the method by which it is inferred. Quantum bits cannot be copied or “cloned”
[97, 35, 67, 71, 44, 26]. Classical tautologies are not necessarily satisfied in quantum information
theory. Quantum bits obey quantum logic. And, as has been argued before, they are coherent
superpositions of classical information.
8 (t,t)=(f,f)=1and(t,f)=0.
5
3.2 Reading the book of Nature—a short glance at the prediction cat-
alog
To quote Landauer [59], “What is measurement? If it is simply information transfer, that is done
all the time inside the computer, and can be done with arbitrary little dissipation.” And, one may
add, without destroying coherence.
Indeed,ashasbeenshortlymentionedin(III),thereisreasontobelievethat—atleastuptoa
certain magnitude of complexity—any measurement can be “undone” by a proper reconstruction
of the wave-function. A necessary condition for this to happen is that all information about the
original measurement is lost. In Schr¨odinger’s terms, the prediction catalog (the wave function)
can be opened only at one particular page. We may close the prediction catalog before reading
this page. Then we can open the prediction catalog at another, complementary, page again. By
nowaywecanopenthepredictioncatalogatonepage,readand(irreversibly)memorizethepage,
close it; then open it at another, complementary, page. (Two non-complementary pages which
correspond to two co-measurable observables can be read simultaneously.)
Canwetheninsomesense“undo”knowledgefromconsciousobservation? Thisquestionrelates
to a statement by Wheeler (cf. [95], p. 184) that “no elementary phenomenon is a phenomenon
until it is a[[n irreversibly]] registered (observed) phenomenon.” Where does this irreversible ob-
servation take place? Since the physical laws (with the possible exception of the weak force) are
time-reversible, the act of irreversible observation must, according to Wigner [96], occur in the
consciousness, thereby violating quantum mechanics.
4 Quantum recursion theory
4.1 Reversible computation and deletion of (q)bits
As a prelude to quantum computation, we briefly review classical reversible computation [56, 7,
42,9,61]. This typeofcomputationis characterizedbyasingle-valuedinversetransitionfunction.
That is, logical functions are performed which do not have a single-valued inverse, such as AND or
OR; i.e., the input cannotbe deduced fromthe output. Also deletion ofinformationor other many
(states)-to-one (state) operations are irreversible. Reversible calculation requires every single step
to be reversible. Figure 1 (cf. [61]) draws the difference between one-to-one and many-to-one
computation. This logical irreversibility is associated with physical irreversibility and requires a
minimal heat generation of the computing machine.
Itispossibletoembedanyirreversiblecomputationinanappropriateenvironmentwhichmakes
it reversible. For instance, the computing agent could keep the inputs of previous calculations in
successiveorder. Itcouldsavesavealltheinformationitwouldotherwisethrowaway. Or,itcould
leave markers behind to identify its trail, the Ha¨nsel and Gretel strategy described by Landauer
[61]. That, of course, might amount to tremendous overhead in dynamical memory space (and
time) and would merely postpone the problem of throwing away unwanted information. But, as
was pointed out by Bennett [7], this overhead could be circumvented by making the computer to
erase all intermediate results, leaving behind only the desired output and the originally furnished
input. Bennett’s trick is to do a computation reversibly, then copy its output9 and then, with
one output as input for the reversible computation, run the computation backwards. In order not
to consume exceedingly large intermediate storage resources, this strategy could be applied after
everysinglestep. Thepriceisadoublingofcomputationtime,sinceitrequiresoneadditionalstep
for the back-computation.10
4.2 Selected features of quantum computation
The following features are necessary but not sufficient qualities of quantum computers.
(i) Input, output, programand memory are qbits;
9Copyingcanbedonereversiblyinclassicalphysics,ifthememoryusedforthecopyisinitiallyblank. Quantum
mechanically,thiscannotbedoneonqbits;cfbelow.
10 Ifanirreversiblecomputingagent exists whichcomputes theinputfromagivenoutput, then itispossibleto
translate an irreversiblecomputation from input to output into one which is reversible and erases everything else
except the final output, including the original input; i.e., that simply maps inputs into outputs. For details, see
Bennett[7,9].