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Potential Capacities of Quantum Channels 2 WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 1 Potential Capacities of Quantum Channels Andreas Winter and Dong Yang Abstract—We introduce potential capacities of quantum chan- Entangled inputs between different quantum channels open the nels in an operational way and provide upper bounds for these door to all kinds of effects that are impossible in classical quantities, which quantify the ultimate limit of usefulness of a information theory. An extreme phenomenon is superactivation channel for a given task in the best possible context. Unfortunately, except for a few isolated cases, potential capac- [13]; there exist two quantum channels that cannot transmit ities seem to be as hard to compute as their “plain” analogues. quantum information when they are used individually, but can We thus study upper bounds on some potential capacities: For transmit at positive rate when they are used together. the classical capacity, we give an upper bound in terms of the The phenomenon of superactivation, and more broadly of entanglement of formation. To establish a bound for the quantum super-additivity, implies that the capacity of a quantum channel and private capacity, we first “lift” the channel to a Hadamard channel and then prove that the quantum and private capacity does not adequately characterize the channel, since the utility of a Hadamard channel is strongly additive, implying that for of the channel depends on what other contextual channels are these channels, potential and plain capacity are equal. Employing available. So it is natural to ask the following question: What these upper bounds we show that if a channel is noisy, however is the maximum possible capability of a channel to transmit close it is to the noiseless channel, then it cannot be activated information when it is used in combination with any other into the noiseless channel by any other contextual channel; this conclusion holds for all the three capacities. We also discuss the contextual channels? We introduce the potential capacity to so-called environment-assisted quantum capacity, because we are formally capture this notion. able to characterize its “potential” version. Superactivation can also be rephrased in an alternative way, Index Terms—quantum channel, potential capacity, non- that a zero-capacity channel becomes a positive-capacity one additivity, entanglement, Hadamard channel. under the assistance of another zero-capacity side channel. Superactivation of quantum channel capacity shows that en- tangled inputs across different channel uses can provide a I. MOTIVATION dramatic advantage, but more generally entangling different HE central problem in information theory is to find the channel uses can give rise to superadditivity, i.e., an increase T capacity of a noisy channel for transmitting information of the capacity above the sum of the channel capacities when faithfully. Depending on what type of information is to be the two channels are used jointly. Superactivation exhibits one sent, there are several capacities that can be defined for a regime of entanglement advantage, the regime of low capacity. quantum channel, among them the classical capacity [1], [2], Could entanglement help in this sense at the other extreme? the quantum capacity [3], [4], [5] and the private capacity That is “Can a noisy channel, whose quantum capacity is [5], [6]. In contrast to classical information theory, where log d δ, become perfectly noiseless under the assistance ≤ − the capacity is expressed by Shannon’s famous single-letter of a suitable zero-capacity side channel?” Since it is difficult formula, the status of quantum channel capacities is much to characterize all the zero-capacity channels, it seems hard more complicated. The relevant quantities are known to be to answer this question. Encouraged by superactivation, one non-additive [7], [8], [9], [10], which is at the center of interest might guess that a noisy channel could behave like a noiseless in quantum information science, and the best known formula channel by the assistance of a proper zero-capacity side to calculate the capacities involves optimization over growing channel. In this work, we will provide upper bounds on the arXiv:1505.00907v5 [quant-ph] 18 Jan 2016 numbers of channel uses (“regularization”), where we have potential capacities to exclude this possibility. In this sense, to perform an optimization over an infinite number variables, entanglement can help but cannot help too much. making a head-on numerical approach impossible, cf. [11], This paper is structured as follows. In Section II we [12]. This makes it difficult to answer questions related to introduce notation, definitions and state some basic known capacities, even some simple qualitative ones, such as whether, facts. In particular, we review the regularized formulas of given a quantum channel, it is useful to transmit quantum three capacities (classical, quantum, and private capacity), and information. Non-additivity in quantum Shannon theory is the results about additivity of degradable channels, further- due to entanglement, which has no classical counterpart. more the entanglement-assisted and the environment-assisted Employing entangled inputs for the channels, it is possible capacities. In Section III, we introduce the notion of potential to transmit more information than just using product inputs. capacity and in Section IV evaluate it or give upper bounds for it, and prove that an imperfect channel cannot be activated Andreas Winter is with ICREA and F´ısica Te`orica: Informaci´o i into a perfect one. Finally we end with a summary and open Fen`omens Qu`antics Universitat Aut`onoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain. Email: [email protected]. questions in Section V. Dong Yang is with F´ısica Te`orica: Informaci´o i Fen`omens Qu`antics Universitat Aut`onoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain II. NOTATION AND PRELIMINARIES and Laboratory for Quantum Information, China Jiliang University, Hangzhou, We assume that all Hilbert spaces, denoted , are finite Zhejiang 310018, China. Email: [email protected]. H Manuscript date 16 January 2016. dimensional. Recall that a quantum state ρ is a linear operator WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 2 BE A B on satisfying ρ 0 and Tr ρ =1. A quantum channel is a where ρ = Uρ U †, U is the isometry of , and S(ρ ) completelyH positive≥ and trace preserving (CPTP) linear map, S(ρE)= S(ρB) S(ρRB) is known as coherentN information− from an input system A to output system B (we shall generally [16], [17], [18], [19].− use the same names for the underlying Hilbert spaces). From The private capacity of the quantum channel is given by the Stinespring dilation theorem [14], we know that for a [5], [6] N 1 (1) n (channel , there always exist an isometry U : A ֒ B E P ( ) = lim P ( ⊗ ), (7 N → ⊗ n n for some environment space E, i.e. U †U = 11, such that N →∞ N (1) (ρ) = TrE UρU †. The complementary channel of , which with P ( ) defined as weN denote c, is the channel that maps fromN the input N N P (1)( )= max (I(T : B) I(T : E)), system A to the environment system E, obtained by taking N pt,ρt − { } the partial trace over system B rather than the environment: TBE T A c with respect to ρ = pt t t Uρt U †. (ρ) = TrB UρU †. Since the Stinespring dilation is unique t | ih | ⊗ upN to a change of basis of environment E, c is well- X defined up to unitary operations on E. A quantumN channel Now we recall the definition of a degradable channel and has another representation known as the Kraus representation: its properties on quantum and private capacities. (ρ) = i KiρKi†, where Ki are called Kraus operators N ABC Definition 1 A channel is called degradable [20] if it satisfying i Ki†Ki = 11. Given a multipartite state ρ , we N c A P ABC can simulate its complementary channel , i.e. there is a write ρ = TrBC ρ for the corresponding reduced state. N c P A degrading CPTP map such that = . The von Neumann entropy is defined as S(A)ρ = S(ρ ) = D D ◦ N N A A Tr ρ log ρ . The conditional von Neumann entropy of A Lemma 2 (Devetak/Shor [20]) If and are degradable − AB B given B is defined as S(A B) = S(ρ ) S(ρ ), the channels, then their single-letter quantumN capacityM is additive: | A B− AB mutual information I(A : B) = S(ρ )+ S(ρ ) S(ρ ), Q(1)( )= Q(1)( )+ Q(1)( ). and the conditional mutual information as I(A −: B C) = N ⊗M N M S(ρAC )+ S(ρBC ) S(ρABC ) S(ρC ). When there| is no − − Lemma 3 (Smith [21]) If a quantum channel is a degrad- ambiguity as to which state is being referred, we simply write able channel, then its quantum capacity is equalN to its private A S(A)= S(ρ ). capacity, and both are given by the single-letter coherent We review the regularization formulas of the three principal information: Q( )= P ( )= Q(1)( )= P (1)( ). capacities: the classical, quantum, and private capacity. N N N N The classical capacity of a quantum channel is the rate at We furthermore recall two other capacities: The which one can reliably send classical information through a entanglement-assisted classical capacity of [22], which is quantum channel, and is given by [1], [2], the capacity for transmitting classical informationN through the 1 n channel with the help of unlimited prior entanglement shared C( ) = lim χ ⊗ , (1) n n N →∞ N between the sender and the receiver and which is given by with the Holevo capacity χ( ) defined as the simple and beautiful formula N CE( ) = max I(R : B), (8) χ( )= max S pi (φi) piS (φi) .
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