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and 1 ,0 (field mode 1 filled with one quantum, field mode 2 unfilled).
1 2
| i
An ideal amplifier, denoted by A, should be able to copy a classical bit state; i.e., it should
create an identical particle in the same mode
A 0 ,1 A 0 ,2 , A 1 ,0 A 2 ,0 . (7)
i 1 2 f 1 2 i 1 2 f 1 2
| i→ | i | i→ | i
Here, A and A stand for the initial and the final state of the amplifier.
i f
What about copying a proper qbit; i.e., a coherent superposition of the cbits f = 0 ,1 and
1 2
| i
t = 1 ,0 ? According to the quantum evolution law, the corresponding amplification process
1 2
| i
should be representable by a linear (unitary) operator; thus
A (α0 ,1 +β 1 ,0 ) A (a0 ,2 +b2 ,0 ) . (8)
i 1 2 1 2 f 1 2 1 2
| i | i → | i | i
Yet, the true copy of that qbit is the state
(α |0 1,1 i+β |1 1,0 i)2 =(αa†2+βa†1)2 |0 i=α2 |0 1,2 i+2αβ |0 1,1 i|1 1,0 i+β2 |2 1,0 . (9)
2 2 2 2 2 2
i
Bycomparing(8)with(9)itcanbeseenareasonable(linearunitaryquantummechanicalevolution
for an) amplifier which could copy a qbit exists only if the qbit is classical.
Amoredetailedanalysis(cf. [67,71],inparticular[44,26])revealsthatthecopying(amplifica-
tion) process generates an amplification of the signal but necessarily adds noise at the same time.
This noise can be interpreted as spontaneous emission of field quanta (photons) in the process of
amplification.
One applicationofthis feature is quantum cryptography[8, 10,11]. Thereby,the impossibility
to copy qbits is used for a cryptographic communication via quantum channels.
4.2.2 Context dependence of qbits
This section could be skipped at first reading.
Assume that in an EPR-type arrangement[38] one wants to measure the product
P =m1m2m1m2m1m2
x x y y z z
of the direction of the spin components of each one of the two associated particles 1 and 2 along
the x, y and z-axes. Assume that the operators are normalized such that mj = 1, i x,y,z ,
| i| ∈ { }
j 1,2 . One way to determine P is measuring and, based on these measurements, “counter-
∈ { }
factually inferring” [75, 69] the three “observables” m1m2, m1m2 and m1m2. By multiplying
x y y x z z
them, one obtains+1. Another, alternative,wayto determine P is measuringand,basedonthese
measurements, “counterfactually inferring” the three “observables”m1m2, m1m2 and m1m2. By
x x y y z z
multiplying them, one obtains 1. In that way, one has obtained either P = 1 or P = 1. Asso-
− −
ciate with P = 1 the bit state zero 0 and with P = 1 the bit state 1. Then the bit is either in
−
state zero or one, depending on the way or context it was inferred.
This kind of contextuality is deeply rooted in the non-Booleanalgebraic structure of quantum
propositions. Note also that the above argument relies heavily on “counterfactual reasoning,” be-
cause,for instance, only two of the six observablesmj can actually be experimentally determined.
i
Here,theterm“counterfactualreasoning”[75,69]standsforargumentsinvolvingresultsofincom-
patible experiments, i.e., experiments which could never be performed simultanuously, since the
associated operators do not commute. The results thus have to be inferred rather than measured,
and the existence of such “elements of physical reality” thus have to be tacitly assumed [38].
4.3 Universal quantum computer based on the U(2)-gate
The “brute force” method of obtaining a (universal) quantum computer [5, 31] by quantizing
the “hardware” components of a Turing machine suffers from the same problem as its classical
counterpart–itseemstechnologicallyunreasonabletoactuallyconstructauniversalquantumdevice
with a “scaled down” (to nanometer size) model of a Turing machine in mind.
We therefore pursue a more fundamental approach. Recall that an arbitrary quantum time
evolution in finite-dimensional Hilbert space is given by x(t)=Ux(t ), where U is unitary.
0
It is well known that any n-dimensional unitary matrix U can be composed from elementary
unitarytransformationsintwodimensionalsubspacesofCn. Thisisusuallyshowninthecontextof
9
parameterization of the n-dimensional unitary groups (cf. [72], chapter 2 and [78, 79]). Thereby,
a transformation in n-dimensional spaces is decomposed into transformations in 2-dimensional
subspaces. This amounts to a successive array of U(2) elements, which in their entirety forms an
arbitrary time evolution U(n) in n-dimensional Hilbert space.
It remains to be shown that the universal U(2)-gate is physically operationalizable. This is
done in appendix D in the framework of Mach-Zehnder interferometry.
The number ofelementary U(2)-transformationsis polynomially bounded anddoes not exceed
n
=n(n 1)/2=O(n2).
2 −
(cid:18) (cid:19)
4.4 Other models of universal quantum computation
Deutsch [32] has proposed a model of universal computation based on quantum computation
networks. In a recent paper, Barenco et al. [3] show that a set of gates that consists of all U(2)
(one-bit) quantum gates and the two-bit exclusive-or gate (that maps Boolean values (x,y) to
(x,x y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2n))
⊕
can be expressed as compositions of these gates.
Thereby,the statesin a2n-dimensionalHilbert spaceare constructedas the productstate ofn
particlesin2-dimHilbertspace,whereastheinterferometricapproachusingU(2)-gatesintroduced
before is based on a single particle state in 2n-dimensional Hilbert space. In order to obtain the
mixing between different particle states, xor-gates are needed. The interferometric approach does
not need xor-gates explicitly.
It has been claimed that certain NP-hardproblems such as factoring can be solvedin polyno-
mial time [82] on quantum computers. We shall not pursue these matters further [33, 34, 14, 15,
5, 27, 82].
One of the most common models
4.5 Nomenclature
Considera(notnecessarilyuniversal)quantumcomputerC anditsithprogramp ,which,attime