Entropy Bound for the Classical Capacity of a Quantum Channel Assisted by Classical Feedback Dawei Ding∗, Yihui Queky, Peter W. Shorz, and Mark M. Wildex ∗Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA, [email protected] yInformation Systems Laboratory, Stanford University, Stanford, California 94305, USA, [email protected] zCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA, [email protected] xHearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA, [email protected] Abstract—We prove that the classical capacity of an arbitrary strong converse statement in [6], [7]. Bowen et al. proved quantum channel assisted by a free classical feedback channel is that the capacity of an entanglement-breaking channel for bounded from above by the maximum average output entropy sending classical messages is not increased by a free classical of the quantum channel. As a consequence of this bound, we conclude that a classical feedback channel does not improve feedback channel [8], and this result was strengthened to a the classical capacity of a quantum erasure channel, and by strong converse statement in [9]. Ref. [10] discussed several taking into account energy constraints, we conclude the same inequalities relating the classical capacity assisted by classical for a pure-loss bosonic channel. The method for establishing the feedback to other capacities. At the same time, it is known that aforementioned entropy bound involves identifying an informa- in general there can be an arbitrarily large gap between the tion measure having two key properties: 1) it does not increase under a one-way local operations and classical communication unassisted classical capacity and the classical capacity assisted channel from the receiver to the sender and 2) a quantum channel by classical feedback [11]. from sender to receiver cannot increase the information measure Our aim here is to go beyond [8] to establish an upper bound by more than the maximum output entropy of the channel. This on the classical capacity of an arbitrary, not just entanglement- information measure can be understood as the sum of two terms, breaking, quantum channel assisted by a classical feedback with one corresponding to classical correlation and the other to entanglement. channel. Due to the fact that a quantum feedback channel is a stronger resource than a classical feedback channel, an I. INTRODUCTION immediate consequence of Bowen’s result [2] is that the A famous result of Shannon is that a free feedback channel entanglement-assisted capacity is an upper bound on the does not increase the capacity of a classical channel for classical capacity assisted by classical feedback. However, communication [1]. That is, the feedback-assisted capacity is since a quantum channel can, in general, establish quantum equal to the channel’s mutual information. Shannon’s result entanglement [12]–[14] and entanglement can increase capac- indicates that the mutual information formula for capacity ity [3]–[5], in such cases it may appear difficult to establish is particularly robust, in the sense that, a priori, one might an upper bound on this capacity other than the entanglement- consider a feedback channel to be a strong resource for assisted capacity. Our main result is that the latter is actually assisting communication. possible: we prove here that the maximum output entropy of With the rise of quantum information theory, several re- a quantum channel is an upper bound on its classical capacity arXiv:1902.02490v2 [quant-ph] 15 Jul 2019 searchers have found variations and generalizations of Shan- assisted by classical feedback. As a generalization of this non’s aforementioned result, in the context of communication result, we find that the maximum average output entropy is over quantum channels. For example, Bowen proved that the an upper bound on the same capacity for a channel that is a capacity of a quantum channel for sending classical messages, probabilistic mixture of other channels. when assisted by a free quantum feedback channel, is equal The approach that we take for establishing the aforemen- to the channel’s entanglement-assisted capacity [2], which tioned bounds is similar in spirit to approaches used to bound is in turn equal to the mutual information of a quantum other assisted capacities or protocols [15]–[18]. We identify channel [3]–[5]. This result indicates that the mutual infor- an information measure that has two key properties: 1) it mation of a quantum channel is robust, in a sense similar does not increase under a free operation, which in this case to that mentioned above. The result also indicates that the is a one-way local operations and classical communication best strategy, in the limit of many channel uses, is to use the (1W-LOCC) channel from the receiver to the sender, and 2) quantum feedback channel once in order to establish sufficient a quantum channel from sender to receiver cannot increase shared entanglement between the sender and receiver, and the information measure by more than the maximum output to subsequently employ an entanglement-assisted communi- entropy of the channel. This information measure can be cation protocol [3]–[5]. Bowen’s result was strengthened to a understood as the sum of two terms, with one corresponding to classical correlation and the other to entanglement. W We organize the rest of the paper as follows. Section II provides a formal definition of a protocol for classical com- A’ A’ A’ A’ munication over a quantum channel assisted by classical F A B F A B F A B feedback. Section III discusses explicitly how to purify such N N N Ŵ a protocol, which is an important conceptual step for our B’ B’ B’ analysis. Section IV introduces our key information measure and several important supplementary lemmas regarding it. Fig. 1. A protocol for classical communication over three uses of a quantum Section V then employs this information measure and the sup- channel NA!B , when assisted by a classical feedback channel. plementary lemmas to establish the maximum output entropy bound for classical capacity assisted by classical feedback. We 0 apply this bound to the erasure channel and pure-loss bosonic Bob processes his systems B1B1 with the decoding chan- 1 channel in Section VI. We conclude in Section VII. nel 0 0 and Alice acts with the encoding channel B1B1!F1B2 2 D A0 F !A0 A , resulting in the state II. PROTOCOL FOR CLASSICAL COMMUNICATION OVER A E 1 1 2 2 QUANTUM CHANNEL ASSISTED BY CLASSICAL FEEDBACK (2) 2 1 (1) ! 0 0 ( 0 0 0 0 )(ρ 0 0 ): WA A B A F1!A A2 B1B !F1B WA B B 2 2 2 ≡ E 1 2 ◦ D 1 2 1 1 1 To begin with, let n; M N, let " [0; 1], and let E 0. (5) 2 2 ≥ Let A!B be a quantum channel, and let H be a Hamiltonian This process iterates n 2 more times, resulting in the N − acting on the input system A of A!B. An (n; M; H; E; ") following states: N protocol for classical communication over a quantum channel (i) (i) ρWA0 B B0 Ai!Bi (!WA0 A B0 ); (6) A!B consists of n uses of the quantum channel A!B, i i i ≡ N i i i N N along with the assistance of a classical feedback channel from (i+1) !WA0 A B0 the receiver Bob to the sender Alice, in order for Alice to send i+1 i+1 i+1 ≡ i+1 i (i) one of M messages to Bob such that the error probability is ( 0 0 0 0 )(ρ 0 0 ); (7) A Fi!A Ai+1 BiBi!FiBi+1 WA BiB no larger than ". Furthermore, the average state at the input of E i i+1 ◦ D i i each channel use should have energy no larger than E, when for i 2; : : : ; n 1 . The final decoding (measurement) channel2 fn −resultsg in the following state: taken with respect to the Hamiltonian H. B B0 !W^ D n n In more detail, the protocol consists of an initial classical– n (n) 0 0 0 ρ ^ (TrA 0 )(ρ 0 0 ): (8) quantum state σF B , with F0 classical and B quantum, of W W n BnB !W^ WAnBnBn 0 1 1 ≡ ◦D n the form Figure 1 depicts the above protocol for n = 3. X f0 0 For an (n; M; H; E; ") protocol, the following is satisfied σF0B = p(f0) f0 f0 F0 σB0 : (1) 1 j ih j ⊗ 1 f 1 0 Φ ^ ρ ^ "; (9) 2 W W − W W 1 ≤ It also involves n encoding channels, with each one denoted 1 PM i where ΦW W^ M m=1 m m W m m W^ is by A0 F !A0 A for i 1; : : : ; n , as well as n decoding ≡ j ih j ⊗ j ih j i−1 i−1 i i the maximally classically correlated state. Note that E 2 f g i channels, with each of them denoted by 0 0 for 1 BiB !FiB Φ ρ = Pr W^ = W W D i i+1 2 W W^ W W^ 1 , where here denotes i 1; : : : ; n 1 . Note that all F systems are classical the uniform− random variablef 6 correspondingg to the message 2 f − g because the feedback channel is constrained to be a classical choice and W^ denotes the random variable corresponding channel. So this means that each decoding channel is a quan- ^ to the classical value in the register W of the state ρW W^ .