document: main laws of quantum mechanics forbids the perfect cloning of the quantum states , see corresponding discussion for the pure states in @xcite , @xcite , and for the mixed states in @xcite . 
   but it is possible to carry out an approximate copying of the quantum states @xcite . 
   quantum cloning machines ( qcm ) depend on the conditions accepted at its designing . 
   they can produce identical copies of the initial state ( symmetric qcm ) , nonidentical copies ( non - symmetric qcm ) , the quality of the copying can be either identical for all states ( universal qcm ) or depend on the state ( state - dependente qcm ) . 
   detailed discussion of the different variants of qcm and theirs possible applications in quantum cryptography and quantum informatics can be found in @xcite , @xcite . 
   one possible application of the qcm is an eavesdropping of the quantum channel . 
   the aim of such eavesdropping defines the main properties of the designing qcm . 
   one can design qcm which copies only part of the quantum state , for instance . 
   such qcm can be useful if eavesdropper , usually called eve , intends to catch part of the transmitted quantum information only . 
   some classical analogue of this situation can be classical eavesdropping of the key words in the transmitted classical information . at quantum cloning 
   we can choose the different parts of the quantum signal in which we are interested . 
   in this paper we intend to discuss some `` partial '' qcm , which copies one constituent of the two - partite states . 
   our approach gives the possibility to consider qcm for a mixed states too . 
   it is well known fact , that any mixed state can be considered as a reduction of a pure state , which is called `` purification '' of the mixed state @xcite . 
   so , cloning of the mixed state can be considered as a `` partial '' cloning of the `` purification '' of the mixed state . 
   some difference between the `` partial '' cloning machine and the cloning machine for the mixed states is connected with the corresponding difference of the sets of the initial states , see details below . 
   note , that the main attention in the present literature was devoted to the cloning of the pure states @xcite , @xcite . 
   we consider two - partite qubit states , qubits are elements of two - dimensional hilbert space @xmath0 . in order to construct qcm we need in tensor product of three such spaces on the ancilla space : @xmath1 , here different components are marked by indexes . 
   the first and third qubit components constitute a quantum state which carries information in the quantum channel , and the state of first component is interesting for eve . 
   the second component is a blank state , where we will copy the first component , the last component is necessary for the realization of the qcm . 
   let quantum channel carries the quantum state @xmath2 @xmath3 where normalization condition holds , @xmath4 here and below @xmath5 are base vectors in @xmath6 . 
   we suppose , that eve s goal is a copying of the first component of this state . after tracing one can obtain : @xmath7 @xmath8 so , eve has to realize the cloning to produce the pair of states ( in the first and second components respectively ) closest to @xmath9 we consider here symmetric qcm , so , we suppose , that states in the first and second components have to coincide . then produced state must be symmetric with regard to permutation of the first and second components . 
   let us introduce the orthonormal basis in the subspace of @xmath10 symmetric regarding this permutation : @xmath11 let s assume , that the second component be in state @xmath12 initially . 
   description of the qcm is , in essence , the definition of the corresponding unitary operator @xmath13 . 
   following to @xcite , @xcite , we set @xmath14 @xmath15 where @xmath16 are some vectors , belonging to @xmath17 . 
   symmetry of qcm is provided by the fact , that right - hand part of this relation contains linear combinations of vectors @xmath18 only . 
   taking into account ( [ init ] ) , we obtain : @xmath19 @xmath20 @xmath21 generally speaking , the choice of the unitary operator @xmath13 is very broad and corresponding analysis is quite complex even for the lowest dimensions , so usually one admits some additional restrictions . 
   we suppose as in @xcite , that following conditions ( which guarantee the unitarity of @xmath13 ) are fulfilled : @xmath22 @xmath23 let @xmath24 so as @xmath25 , @xmath26 . in this case qcm 
   produces the next state from @xmath27 : @xmath28 @xmath29 @xmath30 reducing this state on the first component , we obtain : @xmath31 @xmath32 it is necessary to compare the initial state and state which is produced by the qcm , in other words , we have to choose the measure of the closeness of these states . 
   there are different measures , specifically , fidelity . 
   it is defined for the mixed states as @xmath33 ^2 $ ] , this value is not very suitable for the analytical considerations . 
   we use here more convenient measure : @xmath34 ^2 = w(\zeta , \nu , \psi ) , \ ] ] where @xmath35 + \ ] ] @xmath36 @xmath37 .\ ] ] this value estimates the difference between initial and final states with fixed parameters @xmath38 . for the determination of the qcm parameters we average this value respect to the set of all initial states . 
   we use here the next parametrization of the initial state @xmath39 : @xmath40 @xmath41 where @xmath42 here the first component has zero phase due to the corresponding freedom of the choice . for the averaging we need in corresponding measure . 
   supposing that all states @xmath43 are equiprobable , we choose as such a measure @xmath44 @xmath45 simple calculations lead to the conclusion , that @xmath46 takes its minimal value at @xmath47 , @xmath48 . 
   this value @xmath49 implies , that vectors @xmath50 and @xmath51 are parallel . 
   the values of the fidelity @xmath33 ^2 $ ] , calculated at @xmath52 , for the states on the `` real '' part of the bloch sphere , @xmath53 @xmath54 , @xmath55 , are plotted in fig.1 . 
   evidently , that this qcm is state - dependent one , because the quality of the cloning depends on the quantum state . 
   the construction described above can be used for the cloning of the mixed states . 
   note , that final state produced by qcm depend on the reduction of the initial state @xmath56 ( [ red ] ) only . 
   it means , that one can reverse our considerations and take the mixed state @xmath56 as initial one . 
   then pure state @xmath43 defined by relation ( [ init ] ) belongs to the space of the larger dimension and it is a `` purification '' of the state @xmath56 . 
   `` purification '' of the given mixed state @xmath56 can be realized by different methods , as it follows from ( [ init ] ) , but this nonuniqueness has not influence in the results . 
   qcm constructed in accordance with relations ( [ ccm1 ] ) , ( [ ccm2 ] ) produces states ( [ out12 ] ) , ( [ out ] ) , which depend on the parameters of the initial state @xmath56 only . 
   but the set of the initial states is changing , one has to use another parametrization for this set . 
   as such parametrization one can take a bloch sphere , see @xcite . 
   namely , density matrix @xmath56 can be described as @xmath57 where @xmath58 in the spherical coordinates we have : @xmath59 @xmath60 and @xmath61 in order to obtain the parameters of the qcm one has to average the value @xmath62 on the bloch sphere . 
   we suppose that all states in the bloch sphere are equiprobable , so averaging is reduced to the integral @xmath63 note , that if eve has a priori information about transmitted quantum information she has to choose corresponding weight multiplier . 
   we are searching in such values @xmath64 which correspond to the minimal value @xmath65 . as a result 
   we obtain @xmath66 . 
   the plot of fidelity for these parameters differs in a small way from the preceding one , so we omit it . 
   we have discussed here a `` partial '' quantum cloning , when only one component of the two - partite pure state is cloning . such cloning can be considered as a variant of the eavesdropping of the quantum channel . 
   the choice of the parameters of the qcm was realized with help of the some natural criterion . 
   namely , we seek in parameters corresponding to the minimum of the integral average of the `` distance '' between the initial state and output state . 
   note , that fidelity of the initial and output states for the most part of the bloch sphere exceeds value 5/6 which corresponds to the universal qcm for the pure states @xcite . 
   this fact has two reasons . 
   firstly , described qcm copies only part of the two - partite state . 
   second , this qcm is state - dependent one , and this non - universality raises the quality of the cloning . 
   moreover , here was discussed the cloning of the mixed qubit states . in order to consider such device we use the `` purification '' of the mixed state and then apply `` partial '' cloning machine . 
   let s emphasize , that `` purification '' of the mixed state is not unique , but the output state produced by our qcm does not depend on this nonuniqueness . 
   parameters of the qcm was sought by minimization of the integral average of the distance between the initial and output states . 
   note , that this averaging differs from the used above because the sets of the states are different . in this case 
   the fidelity of the initial and output states on the real part of the bloch sphere exceeds the value 5/6 too . 
   evidently , that choice of the parameters @xmath67 for the qcm is defined by the strategy of the eavesdropping . 
   w have discussed here only one possible qcm for the mixed states . evidently , that there are many other variants for the qcm , may be , without restrictions like ( [ ccm1 ] ) , ( [ ccm2 ] ) , ( [ con1 ] ) , asymmetric qcm etc . 
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summary: we discuss the `` partial '' quantum cloning of the pure two - partite states , when the `` part '' of initial state related to the one qubit is copied only . 
   the same approach gives the possibility to design the quantum copying machine for the mixed qubit states . 
   * `` partial '' quantum cloning and quantum cloning of the mixed states *    * a.ya.kazakov *   + laboratory of quantum information , + st .- petersburg state university of aerospace instrumentation ,    67 b.morskaya str . , st .- petersburg , 190000 russia    pacs : 03.67 -a , 03.67 dd