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information encoded in qubits can be used for reliable quantum communication or efficient quantum computing @xcite . this information is encoded in a quantum state @xmath4 which in the case of a qubit can be parameterized as |(,)= |0 + e^i |1 ; [ 1 ] where @xmath5 and @xmath6 are basis vectors of the 2-dimensional space of the qubit and @xmath7 ; @xmath8 . qubits are very fragile , that is the state of a qubit can easily be changed by the influence of the environment or a random error . one ( very inefficient ) way to protect the quantum information encoded in a qubit is to measure it . with the help of an optimal measurement one can estimate the state of a qubit , with an average fidelity equal to 2/3 ( see below ) . in this way a quantum information is transformed into a classical information which can be stored , copied , and processed according the laws of classical physics with arbitrarily high precision . however , in order to utilize the full potential of quantum information processing we have to keep the information in states of quantum systems , but then we are forced to face the problem of decoherence . recently it has been proposed that quantum information and quantum information processing can be stabilized via symmetrization @xcite . in particular , the qubit in an unknown state is entangled with a set of @xmath1 ( ancilla ) qubits in a specific reference state ( let us say @xmath5 ) so the symmetric state @xmath9 of @xmath10 qubits , |(|,0  ,0+|0,,  ,0+  + |0,0 ,  ) , [ 2 ] is generated . if we introduce a notation for completely symmetric states @xmath11 of @xmath10 qubits with @xmath12 of them being in the state @xmath6 and @xmath13 of them in the state @xmath5 , then the state ( [ 2 ] ) can be expressed in the simple form |(,|)= |n;0 + e^i| |n;1[3 ] where the parameters @xmath14 and @xmath15 are specified by the relations =  ; [ 4 ] and @xmath16 , while @xmath17 . we see that symmetric @xmath10 qubit state @xmath18 is isomorphic to a single qubit state . but in this case the information is spread among @xmath10 entangled qubits - the original quantum information is `` diluted '' . each of the qubits of the @xmath10-qubit state ( [ 3 ] ) is in the state @xmath19 . we define the average fidelity between the single state @xmath20 and the original qubit @xmath21 as = d(;)|_j(,| ) |(;)[5 ] where @xmath22 is the invariant measure on the state space of the original qubit ( i.e. we assume no _ prior _ knowledge about the pure state @xmath21 ) . for this fidelity we find the expression _ 0 = . [ 5a ] we see that for @xmath23 the fidelity @xmath24 is equal to unity ( as it should , because in this case @xmath25 ) while in the limit @xmath26 we find @xmath27 . in fact in this limit density operators of individual qubits are approximately equal to @xmath28 . in other words , individually the qubits of the symmetric state @xmath18 in the large @xmath10 limit do not carry any information about the original single - qubit state @xmath29 . so how can we extract the information from the @xmath10-qubit symmetric state ( [ 3 ] ) ? the ideal possibility would be to have have a perfect _ universal _ disentangler which would perform a unitary transformation    type of disentangling transformation @xcite .    while the perfect transformation is impossible , there are a number of things we can do to concentrate the information from the @xmath10-qubit state @xmath18 back into a single qubit . in principle , we have the following possibilities : * i ) * we can either optimally measure the @xmath10 qubit state and based on the information obtained prepare a single - qubit state . * ii ) * we can design a quantum disentangler which would perform a transformation as close as possible to the ideal disentangling ( [ 7 ] ) . in this quantum scenario we have several options - the process of disentanglement can be input - state dependent . this means that states ( [ 3 ] ) for some values of the parameters @xmath14 and @xmath30 will be disentangled better than for other values of these parameters . alternatively , we can construct a quantum device which disentangles all the state with the same fidelity . * iii ) * finally , we propose a probabilistic disentangler , such that when a specific projective measurement over an ancilla is performed at the output , the desired single - qubit state is generated . the probability of the outcome of the measurement in this case is state - dependent . in what follows we shall investigate all these possibilities .    before proceeding we note that a different type of disentangler has been considered by terno and mor @xcite - @xcite . they considered two different operations . the first would take the state of a bipartite quantum system and transform it into a state that is just the product of the reduced density matrixes of the two subsystems . the second , which is a generalization of the first , would again start with a state of a bipartate quantum system , and map it into a separable state which has the same reduced density matrixes as the original state . they showed that while both of these processes are impossible in general , they can be realized for particular sets of input states . an approximate disentangler of the first type has been considered by bandyopadhyay , et . al .  @xcite . the disentanglers we are considering extract , to some degree of approximation , an unknown state from an entangled state formed from that state and a known state . here we first describe a measurement scenario utilizing a set of specific projection operators . then we present the optimal measurement - based approach to quantum disentanglement and we derive an upper bound on the fidelity of the measurement - based disentangler . we utilize the fact that the @xmath10 qubit system prepared in the state @xmath18 is isomorphic to a single qubit . therefore we first consider a strategy based on a a projective measurement with two projectors @xcite @xmath31 ( @xmath32 ) with |_0(,)&= &     @xmath33 and @xmath34 , where the angles @xmath35 and @xmath36 are chosen randomly if no _ prior _ information about the measured @xmath10-qubit state is available . we can use the result of the measurement to manufacture a a single - qubit state . specifically , if the result of the measurement is positive for @xmath37 then the single qubit is prepared in the state    @xmath38 then the single qubit is prepared in the orthogonal state @xmath39 . for a particular orientation of the measurement apparatus ( i.e. the angles @xmath40 ) this measurement - based scenario gives us a single qubit prepared in the state described by the density operator ^(meas ) ( , |;,)= _ j=0 ^ 1 ||_j|^2 |_j_j| [ 9 ] after we average over all possible orientations of the measurement apparatus we obtain on average a single qubit prepared in the state ^(est ) ( , |)= |(,|)(,|)| + . [ 10 ] to find the average fidelity of this measurement - based disentangling procedure we have to evaluate the mean fidelity @xmath41 , that is the overlap between the state ( [ 10 ] ) and the original input state @xmath42 averaged over all possible orientations of the input qubit : _ 1 = d (,)|^(est ) ( , |)| ( , ) . [ 11 ] taking into account the relation ( [ 4 ] ) we perform the integration in eq.([11 ] ) and we find _ 1 = ( 1+f_n ) [ 12 ] where the function @xmath43 reads f_n= . [ 6 ] for @xmath23 : @xmath44 which is the optimal fidelity of estimation of the state of a single qubit . from fig . [ fig1 ] we see that the fidelity ( [ 12 ] ) is a decreasing function of @xmath10 and in the limit @xmath45 we find @xmath46 , which is equal to the fidelity of a _ random _ guess associated with a binary system such as the two projectors under consideration . in other words , when the original qubit is diluted into an infinite qubit state of the form ( [ 3 ] ) no relevant information can be gained from the measurement . the estimated density operator ( [ 10 ] ) in this case is simply equal to @xmath47 , which is understandable , because as we have shown earlier in this limit the @xmath10-qubit state is approximately in the state @xmath48 , so information about the original is `` almost '' totally lost . we now want to find an upper bound @xmath49 for the average fidelity which can be achieved by a wide class of measurement - based disentanglement procedures . we assume that it is _ a priori _ known that our @xmath10-qubit is prepared in the symmetric state ( [ 2 ] ) with unknown parameters @xmath50 and @xmath51 associated with a single - qubit state ( [ 1 ] ) . the integration measure on the state space of the single qubit is @xmath52 and the corresponding prior probability density distribution on this state space is constant .    our strategy is to measure the input state @xmath9 along the vector @xmath53 [ see eq . ( [ 8 ] ) ] , where the angles @xmath54 and @xmath55 are chosen according to the distribution @xmath56 , which will be left unspecified for the moment . if the answer is positive , we produce the output density matrix @xmath57 , and if it is negative we produce @xmath58 , where @xmath59 with @xmath32 and @xmath60 given by eq . ( [ 8a ] ) . we shall also leave the conditional probabilities , @xmath61 unspecified , as this allows us to consider a wide range of strategies . for a fixed @xmath53 , the probability of the output being @xmath57 is @xmath62 and the probability of it being @xmath58 is @xmath63 . averaging over all vectors , @xmath64 gives us ^(out)(,| ) & = & d^[|_0|    & + & |_1 ||^2_1(^,^ ) ] q(^ , ^ ) . in order to find the average fidelity of the output produced by this procedure , we compute the fidelity for a particular input state and average over the input ensemble @xmath65 where @xmath14 is a function of @xmath50 [ see eq.([4 ] ) ] . this can be expressed as @xmath66 where @xmath67 is a normalized joint probability distribution , and @xmath68 where @xmath69 and the supremum is taken over the variables @xmath70 . we then have that @xmath71 , \ ] ] where the supremum is now taken over @xmath72 and @xmath73 . in order to calculate this upper bound we must find explicit expressions for @xmath74 and @xmath75 . after performing the necessary calculations we find for @xmath49 the expression ^max= . this fidelity for @xmath23 is equal to 2/3 while in the limit @xmath26 is equal to 1/2 . for any other @xmath10 is larger than the fidelity @xmath41 of the measurement given by eq.([12 ] ) as discussed in our previous example . nevertheless , as we will show later it is alway smaller than the fidelity of the universal quantum device . in what follows we show that a quantum disentangler which preserves quantum coherences can distill the information back to a single qubit more efficiently than can the measurement - based method . as we have already said in the introduction quantum mechanics does not allow one to construct a perfect disentangler which would perform transformation ( [ 7 ] ) for an arbitrary ( unknown ) state @xmath42 diluted in the @xmath10 qubit symmetric state ( [ 3 ] ) . nevertheless , we can try to design optimal disentanglers which perform best under given constraints . so let us assume our quantum disentangler , @xmath76 , is a quantum system with a @xmath77-dimensional hilbert space spanned by basis vectors @xmath78 ( @xmath79 ) . the disentangler is always initially prepared in the state @xmath80 , and then it interacts with the @xmath10-qubit system in the state ( [ 3 ] ) . at the output we want to disentangle the @xmath1 ancilla qubits from the original qubit , so we expect to have |(,|)|d_0|n-1;0_k=1^k _ j=0 ^ 1 c_j(,| ) |j|d_k . [ 13 ] as seen from eq.([13 ] ) during the disentanglement process the entanglement between the @xmath1 ancilla qubits and the original qubit is transferred ( swapped ) into the entanglement between the original qubit and the disentangler itself . by tracing over the disentangler we then expect to obtain the best possible disentangled qubit in the state @xmath81 . now we impose several constraints which would specify what we mean by the optimal covariant ( universal ) disentangler : the fidelity between the output of the disentangler and the original state @xmath42 has to be invariant with respect to rotations of the original qubit , so the fidelity has to be input - state independent . this universality of the disentangler would then guarantee that the information from the symmetric state ( [ 3 ] ) is extracted for all states equally well . we are looking for the _ optimal _ disentangler which would disentangle the information with the highest fidelity .    imposing these two conditions we have found the unitary transformation which realizes the _ optimal covariant _ disentangler , i.e. which disentangle the qubit - state @xmath29 from the @xmath10-qubit state @xmath9 in the optimal and the @xmath29-state independent way ( see appendix ) . this disentangler is described by the transformation : |n;0|d_0 & & |n-1;0 ; +    basis vectors of the disentangler . the amplitudes @xmath82 and @xmath83 given by the relation _ n=()^1/2 ; _ n=. [ 15 ]    we can directly verify , that the fidelity @xmath84 is input - state independent and equal to @xmath85 . moreover , it can be shown that the transformation ( [ 15 ] ) is optimal , i.e. among all unitary transformations satisfying the given conditions the transformation ( [ 15 ] ) has the largest fidelity . we see that for @xmath23 the fidelity @xmath86 , which is obvious , because the original qubit has not been entangled with ancilla qubits . we plot @xmath87 in fig . we see , that it is _ always _ larger than the fidelity of the disentanglement via measurement . in the limit @xmath26 even the quantum disentangler gives us a totally random outcome . so in this limit , even optimal quantum entangler on which we impose the universality condition , is not able to extract information from the state ( [ 3 ] ) . this is one of the main results of our paper - the optimal covariant quantum disentangler operates better than if the information is extracted ( disentangled , distilled ) from the symmetrized state ( [ 3 ] ) with the help the of optimal measurement . this is due to the fact that @xmath88 . one can also ask the opposite question , how can we generate out of a qubit in an unknown state @xmath29 the symmetric state of the form ( [ 3 ] ) . it can be shown that within quantum mechanics perfect universal entanglers , which would realize the inverse of the relation ( [ 7 ] ) do not exist . if one wants to create a state ( [ 3 ] ) from a qubit in an unknown state and @xmath1 ancilla qubits in the known state @xmath5 again two scenarios are possible , the measurement - based and quantum scenarios . it is not surprising that the quantum scenario works better . we have found the optimal universal ( covariant with respect to rotations of the input qubit ) quantum entangler given by the transformations : |0|n-1;0|e_0 & & ; +    basis states of the quantum entangler , @xmath89 is its initial state and the parameters @xmath90 and @xmath83 are given by eq.([15 ] ) . one can check that the fidelity between the output of this entangler described by the density operator @xmath91 and the ideally entangled state ( [ 3 ] ) is input - state independent ( i.e. does not depend on the parameters @xmath92 ) and is equal to @xmath93 . this is the best possible universal ( covariant ) entangler . the universal disentangler gives a higher fidelity than does the best measurement - based procedure , but it is not obvious that this is the best that one can do . in the case of quantum cloning , the universal cloners are the ones which maximize the average fidelity @xcite . as we shall see , however , in the case of disentanglers this is no longer the case ; there are state - dependent devices which are better . consider the general disentangler transformation     @xmath94 , are states of the disentangler itself and need not be orthogonal . they must , however , satisfy the constraints imposed by the unitarity of the above transformation . the input state for the device is assumed to be @xmath95 , and the ideal output state , to which the actual output should be compared , is @xmath96 . the output state is calculated by starting with the input state , using the above transformation , and then tracing over the disentangler to obtain an output density matrix , @xmath97 . one then finds the average fidelity for this process , which we shall call @xmath98 , from @xmath99 note that we are assuming a specific ensemble of input states ; the probability of the one - qubit state @xmath100 is assumed to be constant on the bloch sphere . our result for the average fidelity for a state - dependent device depends on our choice of input ensemble , while for a state - independent device the average fidelity is independent of this ensemble . the calculation of the average fidelity is given in the appendix , and will not be given in detail . we find that @xmath101 and @xmath102 . this implies that the final state is just a product of the state of the @xmath10 particles and the entangler state , which means that the entangler states can be dropped from the problem . therefore , the transformation which maximizes the average fidelity is just     a kind of state swapping transformation . the average fidelity itself is given by @xmath103 , where the coefficient @xmath43 is given by eq.([6 ] ) . this average fidelity is larger than the fidelity of the optimal _ universal _ disentangler ( see fig .  [ fig1 ] ) . in this case , the fact that the universality condition forces us to use an additional quantum device , the disentangler , with which the qubit at the output becomes partially entangled , results in a net loss of information . as a result the fidelity of the universal ( covariant ) entangler is smaller . analogously , we find that quantum state - dependent entanglement can also be performed by a kind of state swapping transformation , i.e. |(,)|n;0|0    @xmath104 . nevertheless , when averaged over all values of @xmath105 we find the mean fidelity of this state - dependent entangler to be equal to @xmath43 which on average is larger than the fidelity of the state - independent entangler . let us examine a simple quantum network which takes as an input the @xmath10-qubit state ( [ 3 ] ) . the network is composed of a sequence of @xmath1 c - not gates @xmath106 where @xmath107 is the c - not with @xmath108 being the control bit and @xmath12 being the target bit . this sequence of the c - not gates acts on the two vectors @xmath109 and @xmath110 as [ 20 ] p_n|n;0 & & |n-1;0|0 + p_n |n;1 & & ( |n-1;1+|n-1;0 ) |1from which it follows that the input vector ( [ 3 ] ) is transformed as |(,| ) & & ( |v_+| ( , ) + & & + |v_-|0 ) [ 21 ] where @xmath111 is the normalization constant . in eq.([21 ] ) we have introduced two orthogonal vectors of @xmath1 qubits @xmath112 . measurement on the first @xmath1 qubits is performed in order to determine whether they are in the state @xmath113 or @xmath114 . if the result @xmath115 is obtained , then the @xmath10th qubit is in the desired state @xmath42 . the probability of this outcome is given by p_|v_+= . [ 23 ] this probability is input - state - dependent , and it decreases with @xmath10 .    there is a difference between this probabilistic process and those considered previously , such as probabilistic cloning @xcite . those only work for set of input states which is finite . the process considered above , however , works for a continuous , and hence infinite , set of input states . it , in fact , works for all input states of the type we are considering . therefore , we can conclude that the range of applicability of probabilistic devices depends on the process being considered . we have considered a number of different methods of extracting an unknown state from an entangled state formed from that state and a known state . measuring the state is , as expected , the least effective method . in the case of quantum devices , the universal device was not best one , at least if average fidelity is used as the criterion . probabilistic quantum devices were seen to work very well for this operation in that they can be used for the entire set of input states . this work was supported by the national science foundation under grant phy-9970507 , by the ist project equip under the contract ist-1999 - 11053 and by the crest , research team for interacting career electronics . let us consider the optimal quantum disentangler which acts as close as possible to the ideal transformation ( [ 7 ] ) . the disentangler maps the space spanned by the vectors @xmath116 and @xmath117 , into the space spanned by @xmath118 and @xmath119 . this suggests that we consider a transformation of the following form @xmath120 where @xmath80 is the initial state of the disentangler which is supposed to be the same for all inputs and @xmath121 ( @xmath122 ) are some unnormalized disentangler state - vectors . our task is to determine these vectors . unitarity immediately implies that @xmath123 we shall now use our disentangler transformations ( a.1 ) to calculate the fidelity of the actual output to the ideal output ( [ 7 ] ) the input of the disentangler is given by eq . ( [ 3 ] ) . if we introduce a notation @xmath124 and @xmath125 we can write the result of the transformation ( a.1 ) @xmath126 we now use this expression to find the output density matrix and trace out the disentangler itself . we define the @xmath10-qubit output density matrix to be @xmath127 the output fidelity is given by @xmath128 where @xmath129 is given by eq . ( [ 7 ] ) . if we denote @xmath130 and @xmath131 we can express this fidelity as @xmath132   \nonumber \\ & + & \alpha^{\ast}\beta           \langle d_{3}|d_{4}\rangle ) \nonumber \\   & + & ( \alpha^{\ast})^{2}\beta^{2}\sqrt{n}\langle d_{2}| d_{3}\rangle   + \alpha^{2}(\beta^{\ast})^{2}\sqrt{n } \langle d_{3}|d_{2}\rangle \}. \label{a.5 } \eqnum{a.5}\end{aligned}\ ] ]        demanding that the fidelity be independent of phases of @xmath133 and @xmath134 we find that @xmath135 assuming these conditions to be satisfied the fidelity becomes @xmath136 \ } . \nonumber\end{aligned}\ ] ] in order for this to be independent of @xmath133 and @xmath134 , the term in brackets must be proportional to @xmath137 comparing eqs . ( a.7 ) and ( a.8 ) we find that @xmath138 combining these requirements with those imposed by unitarity we conclude that @xmath139 and @xmath140 . this means that in order to maximize @xmath141 , we must maximize @xmath142 . our first step in accomplishing this is to note that by combining the results of eqs . ( a.9 ) and ( a.10 ) we have that @xmath143 where @xmath144 and @xmath145 . solving for @xmath142 we find that @xmath146 which , assuming @xmath147 , is greatest when @xmath148 . this implies that @xmath149 and that @xmath150 imposing now the conditions on inner products we find that @xmath151    we can summarize our results in the following way . let @xmath152 be a set of three orthonormal vectors and define two parameters @xmath82 and @xmath83 given by eq . ( [ 15 ] ) we then have that @xmath153 and the universal _ optimal _ disentangler transformation is given explicitly by eq . ( [ 14 ] ) .      in order to find the optimal input - state dependent disentangler we find the explicit form of the transformation ( a.1 ) such that the _ averaged _ fidelity @xmath154 ( with @xmath141 given by eq . ( a.5 ) ) is maximized . here , as usually , the integration measure is @xmath155 . therefore after the integral over the phase @xmath51 is performed we can write the average fidelity as @xmath156\ } \nonumber\end{aligned}\ ] ] with [ a.18 ] _ 1&=&_0^ ^4 + _ 2 & = & _ 0^ ^4 + _ 3&=&_0^ ^2 ^2 after the integration over the parameter @xmath50 we find [ a.19 ] _ 1&= & + _ 2&= & + _ 3&= &    from the unitarity of the disentangling transformation it follows that @xmath157 and @xmath158 . when we introduce the notation @xmath159 where @xmath160 , and @xmath161 ; @xmath162 we can rewrite the average fidelity ( a.17 ) as & = & [ _ 1 n ( _ 1-_3 ) + _ 4(_2-_3 ) + & + & 2_3 u + _ 3(1+n ) ] . [ a.21 ] taking into account that @xmath163 and @xmath164 we easily find that the maximum of the mean fidelity ( a.21 ) is achieved for @xmath165 and @xmath166 . in this case we rewrite ( a.21 ) as = [ _ 1 n + _ 2 + 2 _ 3 ] . [ a.22 ] when we substitute into eq . ( a.22 ) the explicit expression for the parameters @xmath167 given by eq .  ( a.19 ) we find that the mean fidelity is equal to the function @xmath43 given by eq . ( [ 6 ] ) . this exactly is equal to the mean fidelity of the input - state disentanglement performed via the state swapping transformation described by eq . ( [ 18a ] ) . in fact , from our conditions @xmath166 it directly follows that @xmath168 while @xmath169 . in addition , from @xmath165 it follows that @xmath102 , so that the optimal state - dependent disentangling transformation is indeed equal to eq . ( [ 18a ] ) , which we wanted to prove .
it is not possible to disentangle a qubit in an _ unknown _ state @xmath0 from a set of @xmath1 ancilla qubits prepared in a specific reference state @xmath2 . 
 that is , it is not possible to _ perfectly _ perform the transformation @xmath3 . 
 the question is then how well we can do ? 
 we consider a number of different methods of extracting an unknown state from an entangled state formed from that qubit and a set of ancilla qubits in an known state . 
 measuring the whole system is , as expected , the least effective method . 
 we present various quantum `` devices '' which disentangle the unknown qubit from the set of ancilla qubits . 
 in particular , we present the _ optimal universal _ disentangler which disentangles the unknown qubit with the fidelity which does not depend on the state of the qubit , and a probabilistic disentangler which performs the perfect disentangling transformation , but with a probability less than one .    * 
 pacs number : 03.67.-a , 03.65.bz *        2