ON FEEDBACK and the CLASSICAL CAPACITY of a NOISY QUANTUM CHANNEL 3 Operation on the Memory State and Next State to Be Trans- Outcome During the Feedback Protocol

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THE CLASSICAL CAPACITY OF A QUANTUM CHANNEL 1 Given a quantum channel Λ, the Shannon mutual informa- C = lim χ(Λ⊗n) (6) tion between a message memory and the average state after n→∞ n ⊗n i ⊗n ⊗n i transmission is bounded above by the von Neumann mutual with χ(Λ ) = max{pi,ρ } S(Λ ρ) − i piS(Λ ρ ) information between the states. A quantum memory for the the HSW capacity for an alphabet in theP Hilbert space of message states may be constructed, without loss of generality, maximum dimension H⊗n, that is, an alphabet that is a using known orthogonal pure states |miihmi| corresponding product state over uses of blocks of n channels but may be to each message mi. The memory state acts classically, in the entangled across different channel uses within the same block. sense that it may measured without error and can be copied The additivity of the HSW capacity is still an open question, arbitrarily many times by measurement and state preparation. although for certain classes of channels it is known to be The total state consisting of the message memory and the input additive [13]–[15], χ(Λ⊗n)= nχ(Λ). states is then,

i i i III. CLASSICAL FEEDBACK AND QUANTUM CHANNELS ρMQ = pi |m ihm |⊗ ρQ (2) Xi To derive the upper bound on the capacity we use the i i technique of attaching a copy of the message, encoded in with each state ρQ corresponding to the message m , where i mutually orthogonal pure states, to the message states to be the message m is sent with apriori probability pi. Following transmitted. In addition, for each use of the feedback channel transmission of the quantum state through the channel the we add a correlated set of quantum operations, where each resultant combined memory and output state is given by, operation on the first output is correlated to a trace preserving i i i operation on the second state. The proof works by induction, ρMQ′ = pi |m ihm |⊗ Λρ . (3) Q showing that for any single step of a feedback protocol the Xi maximum increase of the mutual information between the Any local measurements on the message memory and output sender and receiver cannot exceed the HSW bound. The max- state must give a distribution of measurement outcomes such imum mutual information generated for a multi-step feedback that the Shannon mutual information is bounded by, I(M : protocol cannot exceed the sum of the mutual information ′ ′ Q ) ≤ S(M : Q ) [6], with the von Neumann mutual gained from each step of the protocol, and hence the maximum information between X and Y defined by S(X : Y ) = rate for feedback codes utilized across non-entangled input ′ S(ωX )+ S(ωY ) − S(ωXY ). Here, I(M : Q ) is the Shannon states or feedback codes for entanglement–breaking channels mutual information between the measurement outcomes on the cannot exceed the classical capacity of the channel without memory state and the measurement outcomes on the output feedback. state of the channel Q′. Expanding the von Neumann mutual i information for the alphabet of states {pi,ρQ}, with average i A. Feedback Across Product Input States density matrix ρQ = i piρQ, obtains the bound [9], In order to prove the result for the most general type of ′ P S(M : Q )= S(ρM )+ S(ΛρQ) − S (1M ⊗ ΛQ)ρMQ protocol for product state inputs, the channels may be assumed i ≤ piS(ρM )+ S(ΛρQ) to be of the form Ω ⊗ Λ, where for multiple use of a single ⊗n ⊗m Xi channel Φ we can assume Ω=Φ and Λ=Φ , with m i i and n arbitrary. − piS (1M ⊗ ΛQ)ρ ⊗ ρ M Q The initial total state is of the form, Xi = S(Λρ ) − p S Λρi (4) i i i i Q i Q ρMQ1Q2 = pi |m ihm |⊗ ρQ1 ⊗ ρQ2 (7) Xi Xi i where the inequality follows from the concavity of the condi- with message m being sent with apriori probability pi. The i i tional entropy S(ρX |ρY ) ≥ i piS(ρX |ρY ), with S(X|Y )= state Q1 is then sent through the first channel to produce the S(ωXY ) − S(ωY ) the conditionalP entropy of X given Y [10]. new state, The last line of (4) follows from the additivity of the entropy i i i i ρ ′ p m m ρ ρ . (8) for product states S(ωX ⊗ ωY )= S(ωX )+ S(ωY ). MQ1Q2 = i| ih |⊗ Ω Q1 ⊗ Q2 The Holevo–Schumacher–Westmoreland (HSW) theorem Xi states that a rate equal to the maximum over all such alphabets This is then followed by the feedback operation, which without is asymptotically attainable with vanishing probability of error loss of generality may be represented by classically corre-

[11], [12]. Therefore the classical information capacity for a lated operations on the feedback state BQ1 and a combined BOWEN AND NAGARAJAN : ON FEEDBACK AND THE CLASSICAL CAPACITY OF A NOISY QUANTUM CHANNEL 3 operation on the memory state and next state to be trans- outcome during the feedback protocol. This may be seen by mitted AMQ2 , where the copy of the message memory must attaching an initially pure ancilla state |0Aih0A| which after ′ remain invariant under the operation AM . The total state after the new operation on the output state and ancilla BQ1A, gives the feedback operation and transmission of the second state a classical copy of the measurement outcome in the ancilla j ′ j† through the channel is then, state B ρQ1 B ⊗ |jAihjA|. The mutual information between i i i the message memory and the output state combined with the ρ ′ ′ p m m ω ′ (9) MQ1Q2 = i| ih |⊗ 1Q1 ⊗ ΛQ2 Q1Q2 ′ ancilla following the measurement operation BQ1A, must also Xi necessarily be less than the initial mutual information between where, the message memory and the initial output state and ancilla

i (j)k i i i (j)k† state. Because the ancilla is initially in a product state, the ω ′ = TrM A |m ihm |⊗ ρ A Q1Q2 MQ2 Q2 MQ2 first inequality in (15) then follows from the additivity of the Xjk ′ ′ entropies of product states S ρQ1 ⊗ |0Aih0A| = S(ρQ1 ). j i j † Obtaining the required bound on the final term of (13) ⊗ BQ1 ΩρQ1 (BQ1 ) is only slightly more difficult, and we begin by expanding j i j† (ij)k i (ij)k† the terms according to the basic definition in terms of the = BQ1 ΩρQ1 BQ1 ⊗ AQ2 ρQ2 AQ2 . (10) Xjk conditional von Neumann entropies, such that,

For each i the state is an action utilizing only local operations ′ ′ S(M : Q |Q )= S(ρQ′ |ρQ′ ) − S(ρQ′ |ρMQ′ ) and classical communication (LOCC). Any LOCC action on 2 1 2 1 2 1 ≤ S(ρ ′ ) − S(ρ ′ |ρ ′ ) (16) a separable state necessarily results in a separable state, hence Q2 Q2 MQ1 i each ω ′ is separable, and the convex sum of these separa- Q1Q2 where the inequality follows from the fact that conditioning ble states ω ′ is also separable. As each state following the Q1Q2 cannot increase the entropy in the first term [10]. As the con- feedback operation is separable, it may be written as a convex ′ ′ ditional entropy is concave, the second term −S(ρQ2 |ρMQ1 ) sum over product states, is convex, and from the decomposition of each of the states i ij ij i ′ ω ′ as a separable state, the bound ω = qj ω ′ ⊗ ω (11) Q1Q2 Q1Q2 Q1 Q2 Xj i ij ′ ′ ′ −S(ρQ2 |ρMQ1 )= S piqj ρM ⊗ ωQ where qj ≥ 0 and j qj =1. The total state in (9), following 1 Xij both the feedback operationP and the second transmission, may i ij ij then be rewritten in the form, − S piqj ρ ⊗ ω ′ ⊗ Λω M Q1 Q2 i i ij ij Xij ′ ′ ′ ρMQ1Q2 = piqj |m ihm |⊗ ω ⊗ ΛωQ2 . (12) Q1 i ij Xij ≤ piqj S ρM ⊗ ωQ′ 1 The mutual information between the message memory and Xij i ij ij the combined output states may be rephrased in terms of − piqj S ρ ⊗ ω ′ ⊗ Λω M Q1 Q2 the reduced mutual information and the conditional mutual Xij information, ij = − piqj S ΛωQ2 (17) ′ ′ ′ ′ ′ Xij S(M : Q1Q2)= S(M : Q1)+ S(M : Q2|Q1) (13) with the conditional mutual information defined in terms of is obtained. Substituting (17) into (16) then gives, the conditional entropies by, ′ ′ ij S(M : Q2|Q1) ≤ S ΛωQ2 − piqj S ΛωQ2 ′ ′ ′ ′ ′ ′ S(M : Q2|Q1)= S(Q2|Q1) − S(Q2|MQ1). (14) Xij The conditional information for quantum states differs from ≤ χQ2 (18) the classical counterpart, in that, the two terms on the right ′ with ρ =Λω 2 , and the decomposition, hand side of (14) can each be negative, but only for entangled Q2 Q states. The first term on the right hand side of (13) may be ij p q ω = ω 2 . (19) explicitly written as, i j Q2 Q Xij ′ S(M : Q )= S(ρ )+ S(BΩρ 1 ) − S(BΩρ 1 ) 1 M Q MQ The second inequality in (18) follows from the fact that ≤ S(ρM )+ S(ΩρQ1 ) − S(ΩρMQ1 ) the HSW capacity is the maximum over all such ensembles.

≤ χQ1 (15) Hence, the total capacity of the feedback protocol across the channels is bounded above by the separate HSW capacities of where χ is the HSW capacity of the first channel. The first Q1 the channels, inequality follows from the fact that any quantum operation B acting on part of a bipartite state cannot increase the von FB ′ ′ χQ1Q2 ≤ S(M : Q1Q2) ≤ χQ1 + χQ2 . (20) Neumann mutual information [16], and the second inequality follows from the definition of χQ. The first inequality in (15) The same result follows for an arbitrary feedback protocol, incorporates any information gained from the measurement across any product states inputs, by induction. 4 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY

B. Feedback Across Non–entangled Input States the major difference being that in (8) the alphabet states are The upper bound on any feedback protocol in the previous product states rather than separable states after the use of section is easily extended to include the class of input states the channel. It may be noted, however, that for a feedback that are non-entangled, that is, all separable states. This is due protocol to possibly exceed the bound for the non-feedback ′ ′ to the fact that any separable input states will remain separable capacity the average total states, ρMQ1Q2 and ωMQ1Q2 , must after any LOCC feedback operation, and can therefore be remain entangled between the states that have been sent and written in the form of (12). Therefore feedback across non- the states held by the sender, at some step of the protocol. entangled input states cannot increase the maximum asymp- Any channel for which all possible ensembles do not obey totic rate at which classical information may be sent through this property must therefore have CFB = C, which is the a memoryless quantum channel. deﬁning characteristic of any entanglement–breaking channel.

C. Entanglement–breaking Channels IV. CONCLUSION An entanglement–breaking quantum channel is one which The use of classical feedback for the transmission of clas- cannot transmit entanglement. If part of any bipartite entangled sical information through a memoryless quantum channel has state is transmitted through the channel, then the bipartite state been shown to give no increase in the capacity of the channel following transmission is necessarily separable. Explicitly, for when the feedback is used across non-entangled input states. Additionally, it has been shown that feedback cannot increase any initial state |φRQi, the output ρRQ′ given by, the classical capacity of entanglement–breaking channels. The ρRQ′ = 1R ⊗ ΛQ |φRQihφRQ| (21) question of whether or not feedback can increase the capacity is always separable. The classical capacity for entanglement– of memoryless quantum channels when used across entangled breaking channels has previously been shown to be additive, input states remains open. and hence the classical information capacity for such channels is simply C = χ(Λ) [14]. REFERENCES The proof that feedback cannot increase the classical capac- [1] C. E. Shannon, “A mathematical theory of communication,” Bell Sys. ity of an entanglement–breaking channel is straightforward. Tech. J., vol. 27, pp. 379–423, 623–56, 1948. [2] T. M. Cover and J. A. Thomas, Elements of Information Theory. New After the ﬁrst state Q1 is sent through an entanglement– York: Wiley, 1991. breaking channel, the total state is necessarily separable and [3] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, may be written in the form, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A, vol. 54, pp. 3824–3851, 1996. ′ k k [4] H. Barnum, E. Knill, and M. A. Nielsen, “On quantum ﬁdelities and ρMQ Q2 = πk ρ ′ ⊗ ρ (22) 1 MQ1 Q2 channel capacities,” IEEE Trans. Inform. Theory, vol. 46, pp. 1317– Xk 1329, 2000. for which the feedback operation will result in a new separable [5] G. Bowen, “Quantum feedback channels,” quant-ph/0209076. state, [6] C. Adami and N. J. Cerf, “von Neumann capacity of noisy quantum k k channels,” Phys. Rev. A, vol. 56, pp. 3470–3483, 1997. ω ′ η ω ′ ω . (23) MQ1Q2 = k MQ1 ⊗ Q2 [7] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, Xk “Entanglement-assisted classical capacity of noisy quantum channels,” Phys. Rev. Lett., vol. 83, pp. 3081–3084, 1999. Following transmission of the second state, the mutual infor- [8] ——, “Entanglement-assisted capacity of a quantum channel and the ′ ′ mation between MQ1 and Q2 is bound by, reverse Shannon theorem,” IEEE Trans. Inform. Theory, vol. 48, pp. 2637–2655, 2002. ′ ′ ′ ′ ′ [9] A. S. Holevo, “Bounds for the quantity of information transmitted by S(MQ1 : Q2)= S(M)+ S(Q2) − S(MQ1Q2) a quantum communication channel,” Probl. Peredachi Inf., vol. 9, pp. ′ k S Q η S ω ′ 3–11, 1973. ≤ ( 2)+ k ( MQ1 ) Xk [10] N. J. Cerf and C. Adami, “Quantum extension of conditional probability,” Phys. Rev. A, vol. 60, pp. 893–897, 1999. k k η S ω ′ ω [11] A. S. Holevo, “The capacity of the quantum channel with general signal − k ( MQ1 ⊗ Λ Q2 ) Xk states,” IEEE Trans. Inform. Theory, vol. 44, pp. 269–273, 1998. [12] B. Schumacher and M. D. Westmoreland, “Sending classical information k = S(ΛωQ2 ) − ηkS(ΛωQ2 ) via noisy quantum channels,” Phys. Rev. A, vol. 56, pp. 131–138, 1997. Xk [13] C. King, “Additivity for unital qubit channels,” J. Math. Phys., vol. 43, pp. 4641–4653, 2002. ≤ χQ2 (24) [14] P. W. Shor, “Additivity of the classical capacity of entanglement- k breaking quantum channels,” J. Math. Phys., vol. 43, pp. 4334–4340, with ωQ2 = k ηk ωQ2 . The total mutual information over 2002. the two channelsP is therefore bound by, [15] C. King, “The capacity of the quantum depolarizing channel,” IEEE Trans. Inform. Theory, vol. 49, pp. 221–229, 2003. ′ ′ ′ ′ ′ S(M : Q1Q2)= S(M : Q1)+ S(M : Q2|Q1) [16] G. Lindblad, “Completely positive maps and entropy inequalities,” ′ ′ ′ Commun. Math. Phys., vol. 40, pp. 147–151, 1975. ≤ S(M : Q1)+ S(MQ1 : Q2)

≤ χQ1 + χQ2 (25) and consequently feedback cannot increase the classical information capacity of entanglement–breaking channels. The derivation from (23) to (25) is essentially a less detailed, but otherwise almost identical, version of (8) to (20), with