By ρ q e q ρ q tr e bracketleftbig i r u qe ρ rq e

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by ρ Q = E Q ( ρ Q ) = Tr E bracketleftBig ( I R U QE )( ρ RQ ⊗ | E )( E | ) parenleftBig I R U QE parenrightBigbracketrightBig . (1) As discussed in Ref.[28], the amount of information that is ex- changed between the system Q and the environment E during the interaction is measured by the von Neumann entropy S e . Since the environment is initially in a pure state, the entropy exchange is S e = S ( ρ E ) . The coherent quantum informa- tion , I c as introduced in Ref.[27], is given by I c ( R ) Q ) = S ( ρ Q ) S e = S ( ρ Q ) S ( ρ RQ ) . (2) In this paper, we will use the notation I c = I c ( R ) Q ) = I c ( A ) B ) , as the subsystem R is with Alice and the subsys- tem Q is with Bob, after Q passes through a noisy chan- nel. The coherent information captures how much entan- glement can be retained between Alice and Bob when Alice sends one-half of an entangled pair through a noisy channel. The notion of coherent information plays an important role in quantum data processing and quantum error correction. It is an intrinsic quantity and satisfies the following properties [29]: (i) the absolute value of the coherent information obeys | I c ( R ) Q ) | ≤ log dim H R , (ii) under quantum operation it can never increase, i.e., it satisfies the data processing inequal- ity I c ( R ) Q ) I c ( R ) Q ) as ρ RQ = I R ⊗E Q ( ρ RQ ) , and (iii) S ( ρ Q ) I c ( R ) Q ) . Note that iff S ( ρ Q ) = I c ( R ) Q ) , then perfect quantum error-correction is possible. Schumacher and Westmoreland [26] have shown that the “optimal guaranteed privacy” of the communication channel between Alice and Bob, as depicted in Fig. 1 , is lower bounded by the coherent information I c ( A ) B ) . Next, imagine a situation where Alice wants to send classi- cal information to Bob and she prepares a quantum system Q by encoding information in one of the “signal states”, ρ Q k with a priori probability p k such that the average state ρ Q is given by ρ Q = summationdisplay k p k ρ Q k . (3)
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3 She sends the state over a noisy quantum channel and Bob receives the k th signal state as ρ Q k = E Q ( ρ Q k ) . Since the superoperator is linear, the average state received by Bob is given by ρ Q = summationdisplay k p k E Q ( ρ Q k ) = E Q ( ρ Q ) . (4) The amount of classical information H Bob that can be con- veyed from Alice to Bob is governed by the Holevo quantity χ Q , where χ Q = S ( ρ Q ) summationdisplay k p k S ( ρ Q k ) . (5) The evolution superoperator E Q that represents the effect of the eavesdropper can be represented as a unitary U QE evolu- tion of a larger quantum system that includes an environment E . The evolution can be shown to be ρ RQ ⊗ | E )( E | U QE −−−→ ρ RQ E . (6) The amount of classical information H Eve that is available to Eve is governed by the Holevo quantity χ E , where χ E = S ( ρ E ) summationdisplay k p k S ( ρ E k ) . (7) The quantum “privacy” of a channel between Alice and Bob is defined as P AB = H Bob H Eve . (8) Classically, any positive difference between ( H Bob H Eve )
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