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the quantum teleportation phenomenon @xcite has become one of the central aspects in the investigations of quantum entanglement .
its possible applicability for quantum communication and its experimental realization in different physical systems @xcite gave rise to a wide spread interest in several related problems , including possible alternative schemes @xcite , discussion and resolution of experimental and theoretical limitations @xcite and the connection to nonlocality and inseparability @xcite .
the relation of nonlocality and inseparability of mixed states has already been a challenging problem before the advent of quantum teleportation .
werner @xcite introduced inseparable mixed states which do not violate bell s inequalities .
several measures have been defined to quantify the amount of entanglement contained in mixed states @xcite , partly inspired by entanglement distillation protocols @xcite . whether it is possible to employ
a bipartite mixed state for quantum teleportation has turned out to be an additional nontrivial question in this field @xcite .
recently , bose and vedral @xcite have investigated the direct relation between the mixedness of a state and its usefulness for teleportation .
they applied the von neumann entropy @xmath0 as a measure for mixedness , and find that there is a threshold for this quantity above which the bipartite state @xmath1 becomes useless for teleportation .
in order to separate the quantum and classical aspects of communication , n. and b. gisin @xcite and independently steiner @xcite have presented a local hidden variable ( lhv ) model which , supplied by classical communication , is capable of reproducing quantum correlations of two spin 1/2 particles in singlet state .
brassard et al .
@xcite quantify the amount of classical communication required to be supplemented to simulate quantum correlations for given numbers of qubits .
more recently , cerf et al .
@xcite have presented a protocol in an lhv model for teleporting a quantum state via classical communication . on the other hand , m. .
zukowski @xcite has investigated the connection of local realism and the nonclassical part quantum teleportation process , and concluded , that no local hidden variable model can reproduce the quantum prediction .
the effect of mixedness of a bipartite state applied for teleportation is obviously an important aspect in understanding the classical  quantum limit .
the classical limit of the original protocol of bennett has not yet been investigated .
quantum teleportation , the basic primitive of quantum communication networks consists of the transmission of a quantum state via a combination of classical channels and quantum correlations . in lack of either the classical channel or the entangled resource ,
only noise is obtained .
but what is the classical analogue of bennett s scheme ? to answer this question , we investigate a set of bipartite states interpolating between an ideal entangled state and a possible classical limit .
this approach provides a transition from the quantum teleportation to its classical counterpart .    in this paper
we examine , what happens to a teleportation scheme , when the density matrix elements of the entangled state used as a correlated pair , which are off - diagonal on product basis , are reduced .
this approach , turning the entangled state into a classical correlation , obviously offers a possible way of obtaining a classical limit of the teleportation process .
the paper is organized as follows : in section [ sec : telep ] the class of bipartite states in argument is described , and bennett s protocol of quantum teleportation of a qubit is summarized in a consistent density matrix formalism .
the latter can be regarded as a special case of the treatment of e. g. @xcite or @xcite . by replacing the ideal epr pair with the states investigated ,
we obtain our main result . starting from this ,
two examples are studied in detail : in section [ sec : class ] the purely classical limit is introduced , and in section [ sec : stat ] cases between the ideal quantum teleportation and the classical limit are analyzed by examining a gedankenexperiment .
section [ sec : concl ] summarizes the results .
we restrict ourselves to two - level systems , and use the terminology of spin-@xmath2 particles .
the results can be applied to any representation of qubits , e. g. for the polarization of single photon states , for which quantum teleportation is experimentally feasible @xcite .
the spin @xmath3-component eigenstates of the particles are denoted by @xmath4 and @xmath5 .
density matrices of the bipartite systems , if not otherwise stated , are represented on the natural product state basis @xmath6 .
we will also use the notation @xmath7 for the four bell - states .    let us examine bennett s scheme of quantum teleportation @xcite of a qubit : the sender , alice has particle 1 of @xmath2 spin in the state @xmath8 and wants to teleport it to bob .
the upper indices of density matrices ( and other operators ) refer to the number assigned to the particles .
they share particles 2 and 3 in a state @xmath9 as an entangled resource , and there is also a classical communication channel between them .
alice has an ideal bell - state detector , and bob can carry out unitary transformations on particle 3 given to him . as a starting point of our investigation , let us suppose , that the state of particles 2 and 3 is described by the following density matrix : @xmath10 the lower indices in the kets indicate the number assigned to the particles .
let @xmath11 be real parameter between @xmath12 and @xmath13 . for @xmath14 , @xmath15 ,
thus in this case alice and bob share an epr pair .
this is the case of ideal quantum teleportation . otherwise , the off - diagonal elements of the density matrix are multiplied by @xmath16 . for @xmath17
the density matrix is diagonal in the product state basis , describing classical statistics only .
this state is the mixture of the product states @xmath18 .
such a state can be generated by a _ classically stochastic _ source emitting particles with opposite spins , with equal probability of sending `` up '' and `` down '' state both for particles 2 and 3 .
the states in equation ( [ eq : entangled ] ) may be rewritten in the bell - basis , we find @xmath19 that is , the state is a mixture of the two @xmath20 bell - states .
the class of states examined here is a subclass of werner states ( which were examined from teleportation s point of view in references  @xcite ) where the two @xmath21 states do not occur .
let us consider now the entire teleportation process .
initially , the state of the whole system of the three particles is the product of the states in eqs .
( [ eq : instate ] ) and ( [ eq : entangled ] ) : @xmath22 alice carries out a bell - state measurement , which is described by one of the operators including projection of subsystems 1 and 2 to a bell - state , @xmath23 where @xmath24 stands for one of the four bell - states , and @xmath25 is the identity operator for the hilbert - space of particle 3 .
the result of the measurement is @xmath26 , corresponding to the @xmath26-th bell - state .
this information is sent to bob via the classical channel . because the detection of each bell - state occurs with probability @xmath27
, the operator is multiplied by @xmath28 in order to preserve the norm of the state obtained .
the state of the system after the measurement is given by applying the operator in equation  ( [ eq : bellmeas ] ) to the state in equation  ( [ eq : in123 ] ) . from this
we obtain the state of bob s particle by tracing out in the other two particles : @xmath29 in the last step , bob has to apply a unitary transformation @xmath30 on state @xmath31 , according to alice s measurement result @xmath26 , which is the identity operator in case alice has detected @xmath32 , and @xmath33 for detecting @xmath34,@xmath35 , and @xmath36 respectively . carrying out the calculations described , we obtain our main result : the state teleported to bob reads @xmath37 it can be seen that _ the reduction of the off - diagonal elements of the density matrix describing the entangled resource is inherited by the teleported state .
_ thus teleportation acts as a phase - damping channel .
as a first example , let us examine the case of `` teleporting '' a classical bit .
assume , that @xmath17 , that is the density matrix describing the entangled resource is diagonal . according to ( [ eq : result ] ) , only the diagonal matrix elements of the density matrix of the input state , i.e. the statistics of the measurement of the spin-@xmath3 component is preserved .
therefore , let us suppose , that the input state in equation  ( [ eq : instate ] ) is already diagonal : @xmath38 . in this section we consider measurement of the @xmath3 components of the spins .    under these circumstances our particles can be exactly identified with classical bits .
the states of these classical bits , denoted by @xmath39 , and @xmath40 , are identical with the basis quantum states @xmath4 and @xmath5 . the diagonal density matrix of the input quantum state describes a classical probability distribution of bit 1 .
this is transferred into bit 3 via a classical communication channel and a _ classical _ correlation .
the process itself can be interpreted in the following way : the source of bits 2 and 3 broadcasts correlated bit - pairs @xmath41 or @xmath42 with equal probability .
this can be regarded as the classical limit of an epr - pair .
bit 2 is obtained by alice , who makes a measurement , which tells whether bit 1 and 2 are the same or different .
one can not speak of superpositions in this classical context , and therefore the two @xmath20 and the two @xmath21 bell - states coincide in this limit : the former two mean simply `` the two bits are different '' ( @xmath20-detection ) , and the latter `` the two bits are the same '' ( @xmath21-detection ) . thus the bell - state measurement degenerates to an `` exclusive or '' operation , resulting in a single bit of information , communicated to bob .
bob has to carry out the proper transformation to regain the `` teleported '' bit .
there is no phase of the probability amplitudes for classical probability , thus the transformations in equation  ( [ eq : u ] ) degenerate to a conditional not operation : in case of `` @xmath20 detection '' , bob has obtained bit 3 in the proper state , while in case of `` @xmath21 detection '' he has to invert bit 3 .
finally bit 3 is left with the original value of bit 1 .
since the values of bits 1 or 2 are irrelevant from the `` bell - state measurement '' , it is not necessary for alice to be aware of the actual value of bit 1 to be `` teleported '' .
therefore the method works for `` teleporting '' an unknown classical bit as well , similarly to the quantum protocol .    the classical protocol described here is well known as the one - time - pad in classical cryptography @xcite .
this is , in some sense , the classical protocol most similar to quantum teleportation .
it is the `` teleportation '' of the ( optionally unknown ) state of a classical bit via a classical communication channel and a classical correlation .
the classical correlation provides noise in the lack of the classical communication channel , while the classical channel itself is useless for reconstruction of the result without the correlation .
note that instead of measuring the state of bit 1 and simply communicating its value , only a comparison with a reference has been made .
having described the classical analogue , we may now examine @xmath43 , which interpolates between the classical and the quantum case .
we calculate the consequence of ( [ eq : result ] ) to the result of a teleportation experiment . for simplicity ,
let us suppose , that we want to teleport the state @xmath44 , having @xmath3-component of @xmath45 , rotated by a given angle @xmath46 around the @xmath47 axis of the coordinate system .
the operator of this rotation ( @xmath48 ) is @xmath49 @xmath50 being the first pauli - matrix , and thus we have the state @xmath51 to be teleported . according to ( [ eq : result ] ) , we obtain @xmath52 as result of the teleportation process . in order to verify the teleportation , one may measure the spin along an axis obtained by rotating the @xmath3 axis with the angle @xmath53 around the x axis .
the probability of finding this component of the spin @xmath45 is in this case is @xmath54 for @xmath14 we obtain the familiar cosine - type result valid for ideal teleportation , @xmath55 which is equal to @xmath13 for @xmath56 , meaning perfect teleportation to any direction . for a given input state @xmath57 , the probability for finding the output in the input state after the teleportation is the fidelity of the teleportation of the input state .
this fidelity is @xmath58.\ ] ] in figure  [ fig : fig1 ] we have plotted this function .
it is equal to 1 in the case of ideal teleportation .
the basis states @xmath59 and @xmath60 are always properly teleported , as we have seen in the classical case .
the minimum of the fidelity for any @xmath16 occurs for the equal superposition of the two basis states ( @xmath61 ) .
the minimum value for @xmath17 is @xmath2 , expressing that the result of the measurement can be either @xmath62 or @xmath63 with equal probability , thus this state is not at all teleported .
increasing the purity of the mixed state increases the domain of the angles in which teleportation can be regarded as reliable .
we have studied the role of the off - diagonal density matrix elements of the entangled pair in the teleportation of a qubit . by setting these to zero ,
we have obtained the classical one - time pad cypher , as a classical limit of quantum teleportation .
we have demonstrated the result on the behavior of statistics and fidelity of a gedankenexperiment .
we have found , that the fidelity of the teleportation is most sensitive to the loss of the quantum entanglement in the epr pair for a state at a right angle to the quantization direction .
this work was supported by the research fund of hungary ( otka ) under contracts nos .
t023777 , t034484 and f032346 . t. k. acknowledges the support of the hungarian academy of sciences ( bolyai jnos kutatsi sztnd ' ij ) .
the role of the off - diagonal density matrix elements of the entangled pair is investigated in quantum teleportation of a qubit . the dependence between them and the off - diagonal elements of the teleported density matrix 
 is shown to be linear . in this way 
 the ideal quantum teleportation is related to an entirely classical communication protocol : the one - time pad cypher . 
 the latter can be regarded as the classical counterpart of bennett s quantum teleportation scheme . 
 the quantum - to - classical transition is demonstrated on the statistics of a gedankenexperiment . + pacs nubmers : 03.67.hk , 03.65.bz