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Mathematically, such a map is called a completely positive trace preserving map, but I’ll just call it a quantum (1) P (N ) ≥ P (N ) := max I(X; B)σ − I(X; E)σ, (2) channel. There is an alternate representation of a quantum px,ρx channel in terms of Kraus operators, which lets us write the † † † where σ = P px|xihx|X ⊗ U(ρx ⊗ |0ih0|E)U and U is the channel’s action as N (ρ)= Pk AkρAk with Pk AkAk = I. x More details on the basics of quantum states and channels unitary extension of the channel N . The expression on the can be found in [5]. right is very similar to the capacity formula found in a classical broadcast setting [4]. III. CODING THEOREMS FOR QUANTUM CHANNELS Just as for the classical capacity of a quantum channel, it I’ll now discuss four fundamental coding theorems of can also be shown that the private classical capacity satisfies 1 (1) ⊗n quantum communications: coding for classical information P (N ) = limn→∞ n P (N ). transmission, private classical transmission, quantum trans- mission, and entanglement assisted classical transmission. I C. Quantum Transmission won’t go into the proofs of these coding theorems but, roughly If our sender has a quantum system whose state she would speaking, they all amount to randomly coding according to like to transmit coherently to a receiver, we enter the realm of some distribution at the input of the channel. Notions like the quantum capacity. We’d like to understand the best rate, typical sequences and conditionally typical sequences, suitably this time in two-level quantum systems (or qubits) per channel generalized to the quantum domain, are key ingredients. use, at which we can coherently transmit quantum information. A. Classical Transmission This requires that an arbitrary quantum state, when encoded and transmitted using a noisy channel, can be recovered by Suppose a sender is interested in using a noisy quantum the receiver. As a result the quantum capacity characterizes channel to send classical information to a receiver. The max- the fundamental limits of quantum error correction. imum rate at which this is possible, measured in bits per A lower bound for the quantum capacity, usually called channel use and with the probability of a transmission error Q(N ), was found by Lloyd [11], Shor [12], and Devetak [9]. vanishing in the limit of many channel uses, is defined as They found that the coherent information is an achievable rate: the classical capacity of a quantum channel, which we’ll call C(N ). Q(N ) ≥ Q(1)(N ) := max (H(B) − H(E)) , (3) Holevo [6] and Schumacher and Westmoreland [7] found a ρ lower bound for C(N ): where the entropies are evaluated on the reduced states of † σBE = Uρ⊗|0ih0|EU . The coherent information can also be C(N ) ≥ χ(N ), (1) expressed as a maximization of −H(A|B)= H(B)−H(AB), evaluated on a state of the form I ⊗N (|φihφ| ′ ), but I won’t where the Holevo Information [8] is defined as χ(N ) = AA use this fact here. max x x I(X; B) , where σ = P p |xihx| ⊗ N (ρ ). The p ,ρ σ x x X x Finally, just like with the classical and private capacities, we mutual information is defined as I(X; B)= H(X)+H(B)− can get a characterization of the quantum capacity as H(XB), with the von Neuman entropy, H(ρ)= − Tr ρ log ρ, Q(N )= 1 (1) ⊗n . replacing the usual Shannon entropy. limn→ n Q (N ) In addition to showing that C(N ) ≥ χ(N ), it’s also D. Classical Transmission with Entanglement Assistance possible to give a characterization of the classical capacity 1 ⊗n At this point let’s go back to trying to understand the as C(N ) = limn→∞ n χ(N ) [6], [7]. Of course this limit is not something that can actually be computed. transmission of classical information with a quantum channel. However, besides letting our sender and receiver use a noisy B. Private Classical Transmission quantum channel, we’ll also give them access to arbitrary Now suppose we have access to a noisy quantum channel, shared quantum states (and specifically, they’ll use entangled and we would again like to transmit classical information from states). On their own, such quantum states are useless for sender to receiver. This time, however, we’re interested in communication due to the locality of quantum mechanics. making sure that only the receiver learns what message we However, the shared quantum states may in principle be useful sent. Specifically, we have to make sure that no information when used together with a noisy channel. In fact, not only will about the transmitted message gets leaked to the environment the shared quantum correlations be useful for assisting many of the channel. This situation is qualitatively similar to the quantum channels, it turns out that they will lead to a dramatic wire-tap channel considered by Wyner [3] and Csiszar-Korner simplification of the theory. [4] and, in fact, the solution looks similar in many ways. Define the entanglement assisted classical capacity to be the The resulting private classical capacity, roughly the best maximum rate at which classical communication is possible rate that we can achieve with both high fidelity for the receiver with low error probability when sender and receiver use a and no information leaked to the environment, is usually called noisy channel together with arbitrary shared quantum states. P (N ). Devetak [9] and Cai-Winter-Yeung [10] showed that Bennett, Shor, Smolin, and Thapliyal [13] showed that this the private classical capacity satisfies capacity is given by the formula. This leaves us without an effectively computable

CE(N ) = max I(A; B)σ , (4) characterization of the capacities, but in each case we also find φAA′ that the natural guess for capacity was overly pessimistic— where the quantum mutual information is evaluated on the by using more complicated coding strategies we can transmit state σ = I ⊗N (φAA′ ). Note that the only difference between more information. Eq. (4) and the Holevo information of Eq. (1) is that the Although χ, P (1), and Q(1) are not additive in general, Holevo information is the maximum mutual information we it is possible to show additivity for some special classes of channels. For example, C(N ) = χ(N ) for all entanglement can generate using a state of the form φXA = Px px|xihx|⊗ 2 ρx, whereas in Eq. (4) the state φAA′ is unrestricted. breaking channels [16], depolarizing channels [17], and unital The truly remarkable thing about Eq. (4) is that we have qubit channels. The quantum capacity is equal to Q(1)(N ) for an equality—the formula is single-letter. There is no need to both degradable channels [18] and PPT channels [19], [20] take a limit over many channels uses, so this formula gives although for the latter it is always zero. Finally, the private a complete characterization of the channel’s capability for capacity is P (1)(N ) for degradable channels [21]. classical transmission given free access to entanglement. The The additivity questions considered so far concern the need 1 ⊗n appearance of a single letter formula here, and the absence for regularization (as taking limn→∞ n f(N ) is called) of single-letter formulas for the three other capacities we’ve in our characterization of capacities. Some entropic function discussed is closely related to the question of additivity of f(N ) is shown to be an achievable rate, and its regularization information theoretic quantities, which I’ll consider in the next equal to the capacity, so if we can show f is additive we section. get a tractable capacity formula. Note that this concerns the additivity of f on two copies of the same channel. However, IV. ADDITIVITY QUESTIONS once we regularize f (consider, say, the formula Q(N ) = 1 (1) ⊗n A real function, , on the set of channels is called additive limn→∞ n Q (N )), the resulting capacity always satisfies f ⊗n if f(N ⊗M) = f(N )+ f(M). Many important questions Q(N )= nQ(N ). Capacities are always additive on parallel in quantum information theory boil down to asking whether a uses of the same channel. certain function is additive. Very often, we have some function The need for regularization is a mathematical question about that it’s fairly easy to see satisfies f(N⊗M) ≥ f(N )+f(M), the existence of a simple formula for capacity, but there but we would like to show equality. is a distinct, more operational, question about the additivity For example, the single-letter formula for the entanglement of capacities themselves. Since capacities are additive on assisted classical capacity is shown in three steps. Let g(N )= products of the same channel, this is a question about how different noisy channels interact and enhance each others’ maxφ ′ I(A; B)σ with σAB = I ⊗N (φAA′ ). The first step is AA capabilities3. This sort of additivity thus tells us whether to show that CE (N ) ≥ g(N ) by a random coding argument. 1 ⊗n the communication potential of a channel depends on the The next step is to show that CE (N ) ≤ limn→∞ n g(N ), basically by using the continuity properties of von Neuman context in which it is used or is independent of what other entropy together with the requirement for asymptotically near- channels are available with it. The quantum capacity is very perfect transmission. Finally, one shows (in this case, using strongly nonadditive: there are pairs of channels N and M strong subadditivity of entropy) that g is additive, from which with Q(N ) = Q(M)=0 but Q(N ⊗M) > 0 [22]. The 1 ⊗n private capacity displays nonadditivity that is almost as strong, we get CE(N ) ≤ limn→∞ g(N ) = g(N ), which gives n with P (N )=0 and P (M) ≤ 2 but P (N ⊗M) ≥ 1/8 log d CE(N )= g(N ). Conversely, often where we would have liked to show addi- for sufficiently large dimensional channels [23], [24], [25]. It is tivity it turns out not to be true. One of the earliest examples unknown whether the classical capacity of a quantum channel of this was the finding of [14] that the coherent information is additive in this sense. is not additive on several copies of a very noisy qubit depo- V. OUTLOOK larizing channel1. Specifically, they showed that for a range (1) ⊗5 (1) There has been an enormous amount of progress on under- of depolarizing parameter p, Q (Np ) > 5Q (Np), thus standing the various communication capacities of a quantum (1) dashing hopes of proving Q(N ) = Q (N ). More recently, channel in the past decade or so. From 1997-2003, the basic (1) it was shown in [15] that the private information, P , is tools necessary to understand random coding in quantum not additive on several copies of a channel closely related communication were developed and, more recently, much to the Bennett-Brassard-84 quantum cryptography protocol more streamlined approaches have been found [27]. In the [2]. Finally, Hastings recently showed that there are channels past couple of years several fundamental questions about the such that χ(N ⊗ N ) > 2χ(N ). In each of these cases— the nonadditivity of Q(1), P (1), and χ—there was a natural 2Entanglement breaking channels are so noisy that they don’t allow sender guess for a simple formula giving the associated capacity, and receiver to establish any entanglement. Alternatively, an entanglement breaking channel has a set of rank one Kraus operators. but the nonadditivity implied that the capacity did not equal 3Note that it is possible to have a nonadditive f that, when regularized, gives an additive capacity. Similarly, one could have a simple formula for a 1The qubit depolarizing channel is the most natural quantum analogue of nonadditive capacity. So, one type of (non)additivity does not generally imply I a binary symmetric channel. It acts as Np(ρ) = (1 − p)ρ + p 2 . the other. Information \ Additivity Capacity Single-letter nonadditivity of χ, there remains essentially a single class of Classical ? No [26] nonconstructive examples [26] (though, this class has been Private No [23], [24], [25] No [15] better understood recently [33], [34], [35]). There are a few Quantum No [22] No [14] examples of nonadditivity for Q(1), P (1), Q, and P [14], Entanglement Yes [13] Yes [13] [15], [22], [24], [25], but it is still not entirely clear when Assisted nonadditivity should be expected (though see [36] for some Fig. 1. Status of additivity questions for quantum capacities. The different ideas). For the quantum and private capacity, essentially the rows correspond to using the channel for different kinds of information only additivity results we have concern degradable channels, transmission. The right column indicates whether regularization is necessary with PPT channels also understood for the quantum capacity. in our characterization of the associated capacity. The left column indicates whether the capacity itself is additive. Note that the right column concerns the It would be fantastic to find the quantum capacity of some additivity of entropic quantities when evaluated on multiple copies of the same channels that are not related to the degradable channels. channel, while the left concerns the behavior of a capacity when evaluated on two copies of different channels. There is a qualitative similarity between quantum channels and classical broadcast channels [37]—in point-to-point quan- tum problems, we always implicitly have a second receiver in the form of the environment. So, when we look at the private additivity of capacities have been resolved, with the general classical transmission with a quantum channel, the achievable trend that most things are not additive. rates we find are closely related to the classical results on At first glance this nonadditivity seems like bad news. broadcasting privacy [4]. Furthermore, most of the channels The need for regularization means that we don’t have simple whose quantum capacity we can compute are degradable, a capacity formulas available, and the nonadditivity of capacities notion that comes directly from the classical idea of a degraded implies that the capacity of a channel might not even be the broadcast channel [38], [37]. Can this correspondence be relevant measure of its communication capability. However, developed into an explicit mapping between (perhaps a subset in both cases there is a positive side—regularization means of) quantum channels and classical broadcast channels, with that the capacities we’re interested in are generally higher the capacities of the former derived from the capacity region than expected, and we can use communication strategies of the latter? like entangled signal states and structured error correction I’m grateful to Charlie Bennett for comments an an earlier codes to enhance our communication abilities. 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