Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://theory.sinp.msu.ru/~tarasov/PDF/IJNS2009.pdf Äàòà èçìåíåíèÿ: Fri Mar 5 22:00:17 2010 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 21:26:39 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: molecules

International Journal of Nanoscience Vol. 8, Nos. 4 & 5 (2009) 337­344 c World Scientific Publishing Company

QUANTUM NANOTECHNOLOGY
VASILY E. TARASOV Skobeltsyn Institute of Nuclear Physics Moscow State University, Moscow 119991, Russia tarasov@theory.sinp.msu.ru Revised 10 Novemb er 2008
Nanotechnology is based on manipulations of individual atoms and molecules to build complex atomic structures. Quantum nanotechnology is a broad concept that deals with a manipulation of individual quantum states of atoms and molecules. Quantum nanotechnology differs from nanotechnology as a quantum computer differs from a classical molecular computer. The nanotechnology deals with a manipulation of quantum states in bulk rather than individually. In this paper, we define the main notions of quantum nanotechnology. Quantum analogs of assemblers, replicators and self-repro ducing machines are discussed. We prove the possibility of realizing these analogs. A self-cloning (self-repro ducing) quantum machine is a quantum machine which can make a copy of itself. The impossibility of ideally cloning an unknown quantum state is one of the basic rules of quantum theory. We prove that quantum machines cannot be self-cloning if they are Hamiltonian. There exist quantum non-Hamiltonian machines that are self-cloning machines. Quantum nanotechnology allows us to build quantum nanomachines. These nanomachines are not only small machines of nanosize. Quantum nanomachines should use new (quantum) principles of work. Keywords : Nanomachines; nanotechnology; assemblers; replicators; self-repro ducing machines; quantum cloning; self-reproducing quantum nanomachine.

1. Introduction
Nanotechnology was first recognized after Richard Feynman presented his talk titled "There's Plenty of Room at the Bottom" to the American Physical Society in 1959, when discussing the changing nature of the field of physics: "The principles of physics, as far as I can see, do not sp eak against the p ossibility of maneuvering things atom by atom". In 1988, Eric Drexler taught the first course in nanotechnology while a visiting scholar at Stanford University. He suggested1 the p ossibility of nanosized ob jects that were self-replicating nanomachines. Nanotechnology deals with a manipulation of individual atoms and molecules to build complex atomic structures. We think that the development
337

of nanotechnology allows us to realize manipulations of individual quantum states of atoms and molecules. The technology for manipulating individual quantum states can b e called "quantum nanotechnology". Note that quantum nanotechnology differs from nanotechnology as a quantum computer differs from a classical molecular computer. Nanotechnology deals with a manipulation of atoms and molecules, which are definitely quantum machines. The technology is realized by the molecules not in their quantum states. This technology is classical, b ecause one cannot find a sup erp osition of "b eing hydrogen" and "b eing carb on". The nanotechnology is based on manipulations of quantum states in bulk rather than individually. Manipulations of individual quantum states give us new p ossibilities.


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in of m m

In this pap er, we prove the p ossibility of buildg a quantum nanotechnology. Quantum analogs assemblers, replicators and self-reproducing achines are discussed. The following quantum achines are considered in this pap er:

(a) A self-cloning (self-reproducing) quantum machine is a quantum machine which can make a copy of itself. (b) A quantum replicator is a quantum machine which can realize a sequence of self-cloning (selfreproducing) quantum op erations. (c) A quantum assembler is defined as a quantum machine which can b e used to build a quantum state structure from given states. Quantum assemblers can b e considered as quantum factories. In this pap er, we concentrate on a model of selfreproducing quantum machines. A theory of selfreproducing classical automata has b een suggested by von Neumann.2 With regard to kinematic selfreplicating classical machines, see Ref. 3. Theoretical approaches to the problem of self-reproduction of molecular machines were attempted by considering the origin of life as an organization of molecules through some catalytic actions, on the analogy of physical interaction. Such pioneering work was done by Eigen, Schuster, and Dyson.4 ­ 7 The discovery of the p olymerase activity of the selfsplicing rib osomal RNA (rib onucleic acid) intervening sequence of Tetrahymena thermophila told us that life started from self-replicative RNA sequences called replicases.8 , 9 Although the replicase is a hyp othetical RNA molecule, the presence of b oth the "information" and "function" of self-replication in the same RNA molecule simplifies the problems of self-reproduction of molecular machines. The well-known self-cloning processes are realized in nature by self-reproducing molecular machines. Information is encoded by molecules, which are definitely quantum machines, but it is encoded in the nature of the molecules not in their quantum state. Such an encoding is classical, b ecause one cannot find a molecule that is a sup erp osition of "b eing hydrogen" and "b eing carb on". If information is represented by sequences of molecules, it can b e self-reproducing and this process can b e called self-cloning. Wigner was probably the first, to consider the problem of self-reproduction within the quantum

formalism.10 The imp ossibility of ideally copying (or cloning) an unknown quantum state is one of the basic rules of quantum mechanics.11 The no-cloning theorem of Wootters and Zurek12 says that there is no quantum copy machine which can copy any unknown quantum pure state. They have proven the no-cloning theorem for pure states and for just unitary transformations. The result of the Wootters­Zurek no-cloning theorem has b een extended to mixed states by Barnum, Caves, Fuchs, Jozsa, and Schumacher.14 The theorem for mixed states14 proves that quantum machines that realize broadcasting of two noncommuting mixed states are imp ossible. The no-cloning theorem tells us that cloning quantum machines cannot work ideally. There is a problem of how well they can copy quantum states, i.e., how close the copy state can b e to the original state. This problem was solved by Buzek and Hillery in Ref. 13, where an approximate cloning system was presented. They suggested13 a quantum cloning machine which is input-stateindep endent (universal). The probabilistic cloning quantum machine was prop osed by Duan and Guo.15 , 16 Note that a quantum model of a replicator dynamics of p opulations6 by using the game theory for evolution of mixed states of a quantum system is considered in Ref. 17. A p opulation is represented by a quantum system in which each subp opulation is represented by a pure state with some probability. In some sense, a self-cloning quantum machine is a quantum analog of a simple mathematical model of rib osomes, i.e., molecular machines which build proteins molecules according to the instruction (program) read from DNA molecules. In this pap er, we concentrate on a model of selfreproducing quantum machines, when information is encoded in states of the quantum machines. Each self-cloning quantum machine is defined by a state of the machine and a transformation (quantum op eration) such that , where is an unknown state and is a "prop er" state of the machine. The prop er state is a state of the machine. In our lab oratory we do not know this quantum state. Therefore this state cannot b e cloning by an external copying device. An ideally copying device cannot b e constructed for the state that is unknown to us. This is the no-cloning theorem. Note that this theorem cannot describ e the p ossibility of selfcloning of this state. A quantum self-cloning process is a copying of the quantum state of a machine by


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the machine instead of a copying by an external device. In order to ideally copy this state by our device, the state must b e known to us. We prove that there exist quantum machines that can b e selfreproducing machines, replicators and assemblers. These quantum machines are non-Hamiltonian. In Sec. 2, a brief review of quantum states and op erations of quantum machines is made to fix notations and provide a convenient reference. In Sec. 3, a model of self-cloning quantum machines is suggested. In Sec. 4, a quantum replicator is considered as a quantum machine that realizes a sequence of self-reproducing quantum op erations. In Sec. 5, quantum assemblers, quantum disassemblers, and quantum viruses are discussed. Finally, a short conclusion is given in Sec. 6.

The element |µ) is called the generalized computational basis.18 , 19 Using the fact that (t) is a self-adjoint, nonnegative op erator of unit trace, we obtain 1 1 0 (t) = (0|(t)) = Tr[(t)] = , 2n 2n 1 2n
µ N -1 µ=0

(3) 2 (t) 1, µ

2. States and Operations of Quantum Machines
Let us give a brief review of quantum states and op erations of quantum machines to fix notations and provide a convenient reference (see for example Refs. 18 and 19). In general, states of quantum machines are describ ed by density op erators. A density op erator is a self-adjoint ( = ), nonnegative ( 0) op erator with unit trace (Tr = 1). Pure states can b e characterized by the condition 2 = . The state |(t)) of an n-qubit machine can b e represented by
N -1

and (t) = µ (t). The most general change of quantum state is a quantum op eration (see for example Refs. 18­21). A quantum op eration is a map (sup erop erator) of a set of density op erators. A quantum op eration is a ^ transformation E that maps a density op erator |) of a quantum machine into a density op erator | ) of the machine. If |) is a density op erator, then ^ E|) should also b e a density op erator. Therefore ^ we have the following requirements for E . A general quantum op eration is a real p ositive (or com^ pletely p ositive) trace-preserving sup erop erator E (n) on a Liouville space H . A linear quantum op era^ tion E can b e represented18 , 19 by the equation
N -1 N -1

^ E=
µ=0 =0

Eµ |µ)( |,

(4)

where N = 4n . The matrix Eµ has the form E
µ

=

1 2n

^ Tr[µ E ( )],

|(t)) =
µ=0

|µ)µ (t),

(1)

where N = 4n and µ (t) = (µ|(t)) are real-valued functions. Here, we denote an element A of a Liouville space by a ket vector |A). The inner product of two elements |A) and |B ) of the Liouville space is defined as (A|B ) = Tr[A B ]. Regarding the concepts of Liouville space and sup erop erators, see for example Refs. 18­21. The basis for a Liouville space (n) H of an n-qubit machine is defined by 1 |µ) = |µ1 ,... ,µn ) = |µ ) 2n 1 = |µ1 ··· µn ), 2n

where µ = µ1 ··· µn . As a result, quantum states and quantum op erations can b e describ ed by matrices µ and Eµ .

3. Self-Cloning Quantum Machines
A self-cloning (self-reproducing) machine is considered as a system which can make a copy of itself.1 Let us define self-cloning quantum machines.10
Definition. A self-cloning (SCQM) is the pair

quantum

machine

^ SCQM = {E ; |)}, (2) ^ where the quantum op eration E transforms the input unknown data | ) according to some given program |) such that ^ E |) | ) = |) |). (5)

where we use µ in the representation µ = µ1 4n-1 + ··· + µn-1 4+ µn , µi {0, 1, 2, 3}, is a tensor product, and (µ|µ ) = µµ . Here µk are Pauli matrices.


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In general, |) can b e a pure or mixed state. Let us define the following two typ es of quantum machines.
Definition . A quantum ^ the quantum op eration E U , where ^ form E = U states of the machine, non-Hamiltonian.

machine is Hamiltonian if can b e represented in the U U = UU = I for all otherwise it is said to b e

A self-cloning quantum machine is defined by a state |) of the machine and a transformation ^ ^ (quantum op eration) E such that E|) | ) = |) |), where | ) is an unknown state, and |) is a "prop er" state of the machine. The prop er state is a fixed state of the machine. In our lab oratory we do not know this quantum state. Therefore this state cannot b e cloning by the external copying machines. The ideally copying device cannot b e constructed for the state that is unknown to us.11 , 12 This is the no-cloning theorem. Note that this theorem cannot describ e the p ossibility of self-cloning of this state. A quantum self-cloning process is a copying of a quantum state of a machine by the machine itself instead of a copying by an external device. In order to ideally copy this state by our device, the state must b e known to us. It is not hard to prove that no Hamiltonian self-cloning quantum machines exist.
Theorem . There exist non-Hamiltonian quantum machines which can make copies of themselves.

we can assume that a basis to which belongs is (n) (n) known and , H , where dim(H ) = 4n , i.e. the representation (1) is used. Note that the ^ ^ quantum op eration R(n) gives R(n) | ) = |). This is the self-cloning quantum machine that makes a copy of itself, not copying any changes in time. Let us consider a self-cloning quantum machine that can give itself a copy including any changes in time. This typ e of self-cloning quantum machine is defined by a state (t) of the machine at t > 0 ^ and the time-dep endent quantum op eration E (t, t ) such that ^ E (t, t )|(t )) | (t )) = |(t +t)) |(t )), (7) where is an unknown state, and t = t - t is a time of cloning. Here t is an instant of the b eginning of cloning, and t is an instant of finishing of quantum cloning. The original state [source (t )] of the machine is changed during the time of copying. The process (7) can b e realized by the selfcloning quantum machine ^ SCQMt = {E (t, t ),}, where the self-reproducing quantum op eration is ^ ^ ^ E (t, t ) = S (n) (t, t ) R Here ^ R
(n) (n)

(t, t ).

(t, t ) =

2n |(t ))(0|,

This statement means that the transformation (5), where is an unknown quantum state, can b e realized by non-Hamiltonian quantum machines. The ideally self-reproducing non-Hamiltonian quantum machine can b e constructed. This theorem states that self-reproducing transformations exist. To prove this theorem, a self-cloning quantum op eration will b e presented. Let b e a state of a quantum machine. The transformation (5), where is an unknown quantum state, can b e realized if the self-reproducing ^ quantum op eration E of the quantum machine ^ ,} is defined by SCQM = {E ^ ^ ^ E = I (n) R where ^ R
(n) (n)

^ and S (n) (t, t ) is a quantum op eration that describ es the change of a quantum state of the quantum machine such that ^ S
(n)

(t, t )|(t )) = |(t)).

The state (t ) on the right hand side of Eq. (7) can b e considered as the next "young" generation of the state (t).

4. Quantum Replicator
In the terminology of Dawkins,22 machines that give rise to copies of themselves are called replicators. In this environment, RNA molecules qualify: a single molecule soon b ecomes 2, then 4, 8, 16, 32, and so forth, multiplying exp onentially. Drexler defined a replicator as a nanomachine which can get itself copied,1 including any changes it may have undergone. In a broader sense, a replicator is a machine which can make a copy of itself, not necessarily copying any changes it may have undergone.

,
4n -1

(6)

=



2n

|)(0|,

^ I (n) =

|µ)(µ|.
µ=0

Note that the matr Eµ is a tensor product of ix the matrices Rµ = 2n µ 0 and Iµ = µ . Here


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We consider a quantum replicator as a quantum machine that realizes a sequence of self-reproducing quantum op erations.
Definition. A quantum replicator (QR) is the pair

^ where Ik is an identity sup erop erator on H i.e.,
k ^ ^ Ik = 2 I (n) .

(M (k ))

,

^ QR = {E (tN ,t0 ); (t0 )}, where the quantum op eration ^ ^ ^ E (tN ,t0 ) = EN (tN ,tN -1 ) EN ^ ··· E1 (t1 ,t0 )
-1

For example, we use ^ ^ E2 (t2 ,t1 ) E1 (t1 ,t0 ) ,t
N -2

(t

N -1

) (8)

^ ^ ^ ^ = E2 (t2 ,t1 ) (E1 (t1 ,t0 ) I (n) I (n) ). Substitution of Eq. (9) into this equation gives ^ ^ E2 (t2 ,t1 ) E1 (t1 ,t0 ) ^ ^ ^^^ ^^ ^ = (ST ST R R)(ST R I (n) I (n) ) ^ ^ ^^ ^^ ^^ = ST ST ST R RI (n) RI (n ^2 ^^ ^ ^ = ST ST R R R. As a result, we obtain
)

transforms the input unknown data | ) according to a given program, 0 (t0 ) = (t0 ). The op erators ^ Ek (tk ,tk-1 ) are defined by ^ Ek (tk ,tk-1 )|k-1 (tk-1 )) |k-1 (tk-1 )) = |
k -1

(tk )) |0 (t0 )),
-1

(k )

where we use the notation |k (tk )) = |k |
(k ) 0

(tk ))

(k -1) |0

(t

k -1

)),

^2 ^^ ^ ^ ^ E (t2 ,t0 ) = ST ST R R R. Using Eq. (8) and the definition of multiplication , we obtain ^ ^ ^ E (t1 ,t0 ) = ST R, ^2 ^^ ^ ^ ^ E (t2 ,t0 ) = ST ST R R R, ^ ^3 ^2 ^ ^ ^ ^ ^ ^ E (t3 ,t0 ) = ST ST R ST R ST R R ^^^ R R R, and so on. The self-reproducing op eration for quantum replication is ^ E (t
N +1

(t)) = 2 0 (t) (k = 0,... ,N ).

k

Let us consider the quantum replicator with tN = NT , where T is a time of creation of the duplicate, and N is a p ositive integer numb er (the generation numb er). A set of self-reproducing quantum op erations ^ (tk ,tk-1 ) can b e called a quantum life. These E op erations are defined by ^ ^ ^ E1 (t1 ,t0 ) = ST R, ^ E2 (t2 ,t1 ) = ^ E3 (t3 ,t2 ) = ^ ^ ST ST ^ ^ ST ST ^^ R R, ^^ R R, ^ ^ ^^ ST ST R R (9)

^N ,t0 ) = ST

+1

^N ^ 2 ST R ···
N +1 k =0

2N

^ R.

As a result, we have ^ E (t
k

N +1

,t0 ) =



2k

^N ST

-k

^ R. ,

and so on. The self-reproducing op eration for the k + 1 generation is ^ Ek
+1

Here we use the notation
N +1 k =0

(t

k +1

,tk ) =

2k

^ ST

2k

^ R.

Ak = A0 A1 ··· AN

+1

^ Here ST is a quantum op eration that describ es the change of a quantum state of the quantum machine such that ^ ST |(t)) = |(t + T )). As a sp ecial case, we can consider the identity quan(n) ^ ^ tum op eration ST = I (n) for |0 ) H . (n) ^ If |(t)) H , then Ek (tk ,tk-1 ) is a sup erk (M (k )) , where M (k) = n2 . In the op erator on H definition, we use the comp osition , which means that ^ ^ ^ ^ ^ Ek+1 Ek = Ek+1 (Ek Ik ),

and 2 S means the tensor product 2k times. Self-reproduction is a fundamental feature of all known life. A life can b e considered as phenomena which are non-Hamiltonian (op en) systems able to get themselves copied. A quantum life can b e considered as an evolution of non-Hamiltonian quantum machines that can b e considered as self-cloning machines. Quantum self-cloning processes can b e considered as an analog of some reproduction that is a process by which a machine creates an identical copy of itself without a contribution of information from other states. Note that quantum machines that reproduce through this reproduction tend to grow in numb er exp onentially.


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Wigner found10 that quantum mechanics leads to a practically zero probability for the existence of self-cloning machines. If we assume the p ossibility of the existence of one quantum replicator, then we obtain an exp onential growth in the numb er of these machines. As a result, the probability for the existence of self-reproducing machines grows exp onentially also.

Definition . A quantum cloning assembler (QCA)

is the pair ^ QCA = {Ef ; |); |f )}, where |) H is a state of the quantum machine (m) is a fixed state to b e copied. A and |f ) H ^ QCA is describ ed as a quantum op eration Ef that transforms the input unknown data | ) H (m) H according to a given program |f ) such that ^ Ef |) | ) = |) |f ) |f ). There exists a non-Hamiltonian QA that ideally duplicates a fixed state f . A QA as a nonHamiltonian quantum machine is represented by the quantum op eration ^ ^2 ^ (11) E = I (n) A(m ) ,
f

(n)

(m)

5. Quantum Assembler, Quantum Disassembler and Quantum Virus 5.1. Quantum assembler
An assembler1 is a molecular nanomachine that can b e programmed to build a molecular structure or device from simpler chemical building blocks. We can define a quantum assembler as a quantum machine that can clone a fixed state or a sequence of states.
Definition. A quantum assembler (QA) is the pair

where ^2 A(m ) = and |0) H ^2 A(m ) gives
(m)

2m |f f )(0|,
(m)

2

H

. Note that the op eration

^ QA = {Ef ; |); |f )}, where |) H is a state of the quantum machine (m) is a fixed state to b e copied. A QA and |f ) H ^ can b e describ ed as a quantum op eration Ef that transforms the input unknown data | ) H according to a given program |f ): ^ Ef |) | ) = |) |f ). No quantum Hamiltonian assembler exists. A QA is a non-Hamiltonian quantum machine. It is represented by the quantum op eration ^ ^ ^ Ef = I (n) A(m) , where ^ A(m) =
(m) (m) (n)

^2 A(m ) | ) = |f f ). As a result, the quantum op eration (11) defines a QCA.

5.2. Quantum disassembler
A disassembler1 is a sys take an ob ject apart a recording its structure can define a quantum the following way: tem of nanomachines able to few atoms at a time, while at the molecular level. We analog of a disassembler in

(10)

Definition . A quantum disassembler (QDA) is a quantum machine able to take an unknown sequence,

|k1 ) |k2 ) ··· |km ),

(12)



2n |f )(0|,

and |0) H . In general, the fixed state f can b e a sequence of m quantum states: |f ) =
s k =1

|fk ) = |f1 ) |f2 ) ··· |fm ).

We can consider a QA such that m = 2 and f1 = f2 . This assembler can b e called a quantum cloning assembler.

of known quantum states from a set {k : k = 1,... ,M } and then record (represent) its structure [the sequence (k1 ,k2 ,... ,km )] by the generalized computational states |k]: 1 |0] = |0) , 2n (13) 1 |k] = |0) + |µ)Ck (k = 1,... , 4n - 1), 2n where 1 (14) 0 < Ck 1 - n . 2


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A QDA is the pair ^^ QDA = {E ; Ec ; |c ); |k ); k = 1,... ,M }, where |k ) H k are states of the quantum ^ machine. A QDA consists of a quantum op eration E which transforms the input unknown sequence (12) of known states into the sequence of states |k]: ^ E |k1 ) |k2 ) ··· |km ) = |k1 ] |k2 ] ··· |km ]. The main part of a QDA should b e a quantum com^ puter {Ec ; |c )} which can realize a quantum computation by quantum op eration on mixed states.19 This computer allows one to identify a state |ki ) from a set of known states, i.e. it allows one to define the numb er ki . Note that a quantum algorithm of this identification is an op en question at this moment.
(n )

6. Conclusion
Quantum nanotechnology allows us to build quantum nanomachines. Quantum nanomachines cannot b e considered only as molecular nanomachines,23 ­ 27 just as quantum computers are not only molecular computers. Quantum nanomachines are not only machines of nanosize. They should use new (quantum) principles of work. A quantum computer is an example of a quantum machine for computations. Quantum replicators and quantum assemblers, which are considered in this pap er, are other examples of quantum machines. Quantum machines can b e used for creation of quantum states, and complex structures of quantum states. For example, they can b e used for self-cloning of quantum states. Quantum cloning machines can create (by self-cloning op erations) sup erconducting states of molecular nanowires,28 ­ 30 sup erfluiding states31 ­ 33 of nanomachines motion, or sup erradiance states34 ­ 37 of nanomachines that are molecular nanoantennas.

5.3. Quantum virus
The suggested theorem ab out self-cloning quantum machines means that there exist quantum analogs of RNA. Another p ossible corollary of the theorem is the existence of a quantum analog of viruses. A virus is a molecular machine that is unable to grow or reproduce outside a big molecule that is a "host cell". We can assume the existence of a quantum analog of the virus. A quantum virus is a self-reproducing quantum machine that is unable to grow or reproduce outside a big quantum system. A viral quantum machine, or quantum virion, consists of "genetic material", the quantum replicator (analog of RNA), within a quantum protective coat that can b e called a quantum capsid. The shap e of a quantum capsid can b e varied from simple forms to more complex structures with an envelop e. Functionally, quantum viral envelop es are used to help quantum viruses enter big quantum systems (quantum host cells). The no-cloning theorem has a direct application to secret communications and quantum cryptography. A striking feature of quantum mechanics represented in the no-cloning theorem is that one cannot freely and ideally read out information of a system without affecting the state of the system. It is known that the information can b e approximate and probability copying.13 , 15 , 16 We assume that a quantum machine (such as a quantum computer) with a quantum virus can ideally copy this information (by the self-cloning op eration) without affecting itself.

References
1. K. E. Drexler, Engines of Creation: The Coming Era of Nanotechnology (Oxford University Press, 1990), see also http://www.e-drexler.com/ d/06/00/EOC/EOC Table of Contents.html. 2. J. von Neumann, Theory of Self-Reproducing Automata Source (University of Illinois, 1966). 3. R. A. Freitas, Jr. and R. C. Merkle, Kinematic SelfReplicating Machines (Landes Bioscience, 2004), see also http://www.molecularassembler.com/KSRM. htm. 4. M. Eigen, Die Naturwissenschaften 47, 3465 (1971). 5. M. Eigen and P. Schuster, Die Naturwissenschaften 64, 541 (1977). 6. M. Eigen and P. Schuster, The Hypercycle. A Principle of Natural Self-Organization (Springer, Berlin, New York, 1979). 7. F. J. Dyson, J. Mol. Evol. 18, 344 (1982). 8. T. R. Cech, Proc. Natl. Acad. Sci. USA 83, 4360 (1986). 9. J. D. Watson, N. H. Hopkins, J. W. Roberts, J. A. Steitz and A. M. Weiner, Molecular Biology of the Gene, 3rd edn., Vol. II (Benjamin Cummings, California, 1987), pp. 1103­1124. 10. E. P. Wigner, The Logic of Personal Know ledge: Essays Presented to Michael Polanyi (Routledge and Paul, London, 1961), pp. 231­238; also Symmetries and Reflections (Indian University Press, Blo omington, London, 1970). 11. V. Scarani, S. Iblisdir and N. Gisin, Rev. Mod. Phys. 77, 1225 (2005).


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