Reverse Coherent Information Ra´ul Garc´ıa-Patr´on,1 Stefano Pirandola,1 Seth Lloyd,1 and Jeffrey H. Shapiro1 1Research Laboratory of Electronics, MIT, Cambridge, MA 02139 In this letter we define a family of entanglement distribution protocols assisted by feedback clas- sical communication that gives an operational interpretation to reverse coherent information, i.e., the symmetric counterpart of the well known coherent information. This lead to the definition of a new entanglement distribution capacity that exceeds the unassisted capacity for some interesting channels. PACS numbers: 03.67.-a, 03.67.Hk Shannon’s great result was proving that sending in- The channel Λ is called degradable if there exists a formation through a noisy channel can be achieved map that transforms Bob’s output ρ into the en- N M B with vanishing error, in the limit of many uses of the vironment state ρE, i.e., (ρB) = ρE, where ρE = channel [1]. Shannon’s key idea was to add redundancy Tr [ φ φ ] and φ M is the purification of ρ . RB | ih |RBE | iRBE RB to the message in order to compensate for the channel’s Similarly if there is a map such that (ρE) = ρB the noise. He showed that the channel’s communication ca- channel is called antidegradableG and (Λ)G = 0. pacity ( ) between two partners, called Alice and Bob, Having free access to a classical communicationQ chan- is givenC byN the maximal mutual information between Al- nel Alice and Bob can improve the quantum communi- ice’s input a and Bob’s output b = (a), i.e., cation protocol, as opposed to Shannon’s theory where N using feedback gives no improvement [7]. One can define ( ) = max H(a:b) (bits/channel use). (1) three new quantum communication capacities depend- C N a ing on the use of the classical channel: forward classical Quantum information theory [2] is a generalization of communication ( ); feedback classical communication Shannon’s information theory that has attracted huge in- ( ); two-way classicalQ→ communication ( ). In Fig. 1 terest in the last decade, as it allows for new potential weQ← review the relations between these fourQ↔ capacities. applications, such as quantum communication and entan- glement distribution. Quantum communication allows faithful transfer of quantum states through a quantum 1 4 5 noisy channel Λ. The quantum communication capacity (Λ) gives the number of qubits per channel use that Q = Q Q Q Q ĺ ĸ ļ can be reliably transmitted, preserving quantum coher- 1 2 5 2 = = = ence. It was shown in [3] that the coherent information I(Λ,ρ ), a function of Alice’s input ρ on channel Λ, E = E E E A A ĺ ĸ ļ plays a crucial role in the definition of the quantum com- 3 4 munication capacity. The coherent information is I(Λ,ρ )= I( Λ( ψ ψ )) = I(ρ ), (2) A I⊗ | ih |RA RB FIG. 1: (color online) Relations between the quantum com- where ψ is the purification of ρ , is the identity munication and entanglement distribution capacities. We | iRA A I arXiv:0808.0210v2 [quant-ph] 26 Jun 2009 operator and I(ρ ) = S(B) S(RB), where S(X) is first start by two general remarks: (I) Being able to send RB − the von Neumann entropy of ρX . By analogy with Shan- a noiseless qubit is a stronger resource than distributing units of entanglement (e-bits): x x for all x. (II) Increas- non’s theory, one would expect (Λ) to be calculated by E ≥ Q maximizing over a single use ofQ the channel, ing the complexity of the assistance cannot decrease the ca- pacity: ← ↔. The following remarks concern X ≤X ≤ X (1) their corresponding number on the figure. (1) The equality (Λ) = max I(Λ,ρA). (3) Q ρA = = → was shown in [12]. (2) Any entanglement dis- tributionE Q protocolQ with free forward classical communication Unfortunately, the quantum case is more complicated, can be transformed into a quantum communication protocol as (1)(Λ) is known to be non-additive [4]. The correct by appending teleportation to it. (3) Results from combining capacityQ definition [5] is, 1 and 2. (4) Combining 1, 3 and II. (5) It is easy to prove that ← = ↔ for the erasure channel [13, 14]. In [14] it was shownE thatQ the erasure channel satisfies the strict inequality 1 n (Λ) = lim max I(Λ⊗ ,ρA¯). (4) ← < ↔, which gives ← = ←. n ρA¯ Q →∞ n Q Q E 6 Q Only for the restricted class of degradable channels [6], Entanglement is another important resource for quan- is (Λ) known to be additive, i.e., Q(Λ) = Q(1)(Λ). tum information processing. Therefore, the study of the Q 2 entanglement distribution capacity of quantum channels Reverse entanglement distribution.- A big practical (distributed e-bits per use of the channel) is of crucial im- disadvantage of the previous protocol is that Alice has portance. As for quantum communication, we can also to wait until Bob sends the message bi before apply- b1...bi define four types of assisted (unassisted) capacities for ing i and subsequently sending qubit Ai+1, which entanglement distribution: , , , . As shown greatlyA decreases the transmission rate. A way of avoid- in Fig. 1, all the entanglement{E E→ distributionE← E↔} capacities ing this problem is to simplify the protocol to a single are equivalent to their quantum communication counter- round of classical feedback after Alice has sent all her parts, except for (Λ). qubits A1A2...An through the quantum channel Λ, see Entanglement distributionE← assisted by feedback classi- Fig. 3. We call this familly of simplified protocols reverse cal communication.- The entanglement distribution pro- tocol assisted by classical feedback communication, as A B described in [8], goes as follows. Alice starts preparing 1 ȁ 1 A2 B2 a bipartite entangled state ΨR A1,A2,...,An , where R is a R | Ab ȁ B group of qubits entangled with the qubits Ai sent, one by Ȍ one, through the channel Λ. The first round of the pro- An Bn tocol, see Fig. 2, consists of three steps: i) Alice sends ȁ qubit A through the quantum channel Λ; ii) Bob ap- 1 b plies an incomplete quantum measurement 1 over his received qubit B and communicates the classicalB out- 1 Classical Communication come b1 to Alice. iii) Alice, conditioned on the classical Alice Bob b1 message b1, applies a global quantum operation 1 over the joint system of R and the remaining n A1 qubits − A2A3...An. The next n 1 rounds are a slight modifica- FIG. 3: (color online) A simplification of the general entan- − b1...bi−1 tion of the first one: First, Bob’s measurement glement distribution protocol assisted by classical feedback Bi acts on all his received qubits B1B2...Bi, conditioned on (Fig.2) limits the protocol to a last single round of process- ing. After Alice has sent all her qubits (A1A2...An) through his previous measurement outcomes b1...bi 1. Second, b1...bi − the quantum channel Λ, Bob applies a collective incomplete Alice’s operation i , acts on all her remaining qubits A measurement among all the qubits B1B2...Bn and commu- RAi+1...An, conditioned on all previous classical commu- nicates the classicalB outcome b to Alice. Finally, conditioned nication messages. By properly choosing Alice’s opera- on the message b, Alice applies the quantum operation b on system R. A A B 1 1 entanglement distribution protocols, by analogy with the ȁ B1 quantum key distribution scenario [9]. Before the single R Ȍ ȁ post-processing round Alice and Bob’s shared state is n ρR B1,B2,...,Bn = Λ⊗ (ΨR A1,A2,...,An ). (5) ȁ | I⊗ | By properly choosing Alice’s and Bob’s operations both b1 A1 b1 partners extract n ⊳(Λ) e-bits, where ⊳(Λ) is the reverse entanglement≈ distributionE capacity, satisfyingE the Alice Classical Bob inequality ⊳(Λ) (Λ). Communication RemarkE that, in≤E the← particular case where Alice’s in- puts are independent and identically distributed, i.e., n ρR A1,A2,...,An = ρ⊗ , the post-processing of the re- | R A FIG. 2: (color online) The first round of the entanglement verse entanglement distribution| protocol is the dynam- distribution protocol assisted by classical feedback consists of ical equivalent of an entanglement distillation protocol three steps: i) Alice sends qubit A1 through the quantum n channel Λ; ii) Bob applies an incomplete quantum measure- over the static resource ρR⊗ B [10]. | ment 1 over his received qubit B1 and communicates the Reverse coherent information capacity.- In what fol- B outcome b1 to Alice; iii) Alice applies a global quantum oper- lows we consider a subset of the reverse entanglement ation b1 over the joint system of R and the remaining n 1 A1 − distribution protocols with a strikingly simple capacity qubits Ai. The next rounds are straightforward extensions of that lower bounds ⊳(Λ). By exchanging the roles of Al- the first one. ice and Bob in theE family of static distillation protocol assisted by one-way classical communication defined in tions and Bob’s incomplete measurements both partners [10], we obtain a new family of static distillation proto- extract n (Λ) units of entanglement (e-bits) at the cols with rate end of the≈ protocol.E← Unfortunately, the calculation of (Λ) is extremely challenging in full generality. IR(ρRB)= S(R) S(RB). (6) E← − 3 By analogy with the quantum key distribution scenario to the class ρ = diag(1 p,p) without loss of general- A − [9], we call the quantity IR(ρRB ) the reverse coherent ity (see appendix). For a given input population p, the information.