Coherent information of a quantum channel or its complement is generically positive Satvik Singh1, 2, ∗ and Nilanjana Datta2, y 1Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Punjab, India. 2Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom (Dated: July 2, 2021) The task of determining whether a given quantum channel has positive capacity to transmit quantum information is a fundamental open problem in quantum information theory. In general, the coherent information needs to be computed for an unbounded number of copies of a channel in order to detect a positive value of its quantum capacity. However, in this Letter, we show that the coherent information of a single copy of a randomly selected channel is positive almost surely if the channel’s output space is larger than its environment. Hence, in this case, a single copy of the channel typically suffices to determine positivity of its quantum capacity. Put differently, channels with zero coherent information have measure zero in the subset of channels for which the output space is larger than the environment. On the other hand, if the environment is larger than the channel’s output space, identical results hold for the channel’s complement. I. INTRODUCTION tarily evolved composite system then yields the output of the channel, whereas discarding the quantum system According to quantum mechanics, the most general yields the output of a complementary channel. Hence, physical transformation that a quantum system can the leakage of information by the channel to the environ- undergo is described by a quantum channel. Con- ment is modelled by its complement. Unsurprisingly, this sequently, quantum channels serve as quantum ana- leakage crucially affects the channel’s capacity to trans- logues of classical communication channels and are hence mit quantum information. ubiquitous in quantum information-processing protocols. Prime examples of noisy communication channels act- Computing the transmission capacities of a channel – ing on finite-dimensional (or discrete) quantum systems which quantify the fundamental limits on reliable com- (e.g. spin−1=2 electronic qubits) include depolarizing- munication through it – constitute a central problem in [1], amplitude damping- [2], dephasing-[3] and erasure quantum information theory. Unlike a classical channel, channels [4]; see also [5,6]. In contrast, quantum a quantum channel can be used to transmit either classi- communication using continuous variable quantum sys- cal [22, 23] or quantum information [24, 25]. The rate tems (e.g. quantized radiation modes of the electromag- of this communication might be enhanced by the use netic field) is modelled via channels acting on infinite- of auxiliary resources (e.g. shared entanglement between dimensional Hilbert spaces [7–9]. Depending on the type the sender and the receiver [26, 27]) and might also de- of physical medium (e.g. atomic, optical, etc.) used pend on the nature of the states being used as inputs for encoding quantum information, numerous impressive over multiple uses of the channel (i.e. product states or schemes for controlled experimental implementations of entangled states [25]). Moreover, the information to be important quantum channels (both in finite and infinite transmitted might be private [28]. These considerations dimensions) have been reported [10–19]; see the reviews lead to different notions of capacities of a quantum chan- in [20, 21] as well. nel, in contrast to the classical setting where the capacity Mathematically, a quantum channel is a completely of a classical channel is uniquely defined. In this paper, arXiv:2107.00392v1 [quant-ph] 1 Jul 2021 positive and trace-preserving linear map defined between we focus on the quantum capacity of a quantum channel, spaces of operators describing states of quantum systems. which quantifies the maximum rate at which quantum Stinespring’s dilation theorem asserts that the action of information can be transmitted coherently and reliably a channel can be represented as a unitary evolution on through it, in the absence of any auxiliary resource. By an enlarged system consisting of the quantum system on the seminal works of Lloyd [24], Shor [29], and Deve- which the channel acts and its environment; without loss tak [28], we know that the quantum capacity Q(Φ) of a of generality, the latter can initially be assumed to be in a quantum channel Φ admits a regularized formula involv- fixed pure state. Discarding the environment (by tracing ing an optimization of an entropic quantity over infinitely over its associated Hilbert space) from the resulting uni- many successive and independent uses of the channel: (1) ⊗n ∗ [email protected] Q (Φ ) y Q(Φ) = lim ; (1) [email protected] n!1 n 2 where den and Winter [37, 38] to disprove the multiplicativity of the maximal p-norm of a channel for all p > 1. Hast- (1) Q (Φ) := max Ic(ρ; Φ) and (2) ρ ings also employed random channels to disprove that the minimum output entropy of a channel is additive [39], I (ρ; Φ) := S[Φ(ρ)] − S[Φ (ρ)]: c c (3) hence disproving the famous set of globally equivalent ad- ditivity conjectures [40] that had been the focus of much In the above definition, Φc denotes a channel which is research for over a decade. Random channels have also complementary to Φ, and S(ρ) := − Tr(ρ log ρ) is the 2 found applications in various other fields which include, von Neumann entropy of the quantum state ρ. The for example, the study of information scrambling and quantity Q(1)(Φ) is called the coherent information of Φ, chaos in open quantum systems [41, 42], and exploring which trivially bounds the quantum capacity from be- holographic dualities in theories of quantum gravity [43]. low: Q(Φ) ≥ Q(1)(Φ). However, explicit computation See also [44] and references therein. of the quantum capacity is usually intractable because Our result amounts to saying that typically, whenever of two reasons. Firstly, Eq. (2) is an instance of a non- the dimension of the output space of a channel Φ is larger concave optimization problem which allows for the ex- than that of its environment, only a single copy of the istence of local maxima that are not global. Secondly, channel suffices to detect its ability to transmit quantum the coherent information is often strictly superadditive information, in the sense that the regularized formula for [25, 30] (i.e. Q(1)(Φ⊗n) > nQ(1)(Φ)), which implies that the quantum capacity (Eq. (1)) attains a positive value the n ! 1 regularization in Eq. (1) is necessary. at just the n = 1 level: Q(1)(Φ) > 0. This is in stark An important class of channels for which the coherent contrast to the general picture, where it is known that an information is additive (i.e. Q(1)(Φ⊗n) = nQ(1)(Φ)) and unbounded number of uses of a channel may be required hence equal to the quantum capacity (Q(Φ) = Q(1)(Φ)) is to detect its quantum capacity [45], i.e., for every n 2 N, that of degradable channels [31]. These channels strictly there exist examples of channels Φ for which Q(1)(Φ⊗n) = lie within the set of the so-called more capable channels 0 yet Q(Φ) > 0. In light of this result, it is surprising how [32], whose defining property is that their complemen- a simple dimensional inequality between the output and tary channels have zero quantum capacity. Furthermore, environment spaces of a channel drastically simplifies the there exist pairs of quantum channels (say Φ1 and Φ2), problem of quantum capacity detection in almost all the each of which have zero quantum capacity, but which cases: both the hurdles of performing regularization in can be used together to transmit quantum information, Eq. (1) and solving a non-concave optimization problem i.e. Q(Φ1 ⊗ Φ2) > 0. This startling effect (known as in Eq. (2) are eliminated in one stroke! superactivation [33]) is a purely quantum phenomenon An equivalent formulation of Theorem III.3 provides because classically, if two channels have zero capacity, a useful structural insight into the set of quantum chan- the joint channel has zero capacity as well. nels with zero coherent information, which strictly con- The above discussion sheds light on the significance tains the set of channels with zero quantum capacity. We of the set of zero capacity quantum channels. However, show that within the subset of those channels for which even after years of stringent efforts, this set is very poorly the dimension of the output space is larger than that understood. At the heart of it, the problem lies in de- of the environment, channels Φ with Q(1)(Φ) = 0 con- termining if a given quantum channel can be used to tribute no volume (in a well-defined measure-theoretic reliably transmit quantum information in the absence of (1) sense). Similarly, channels Φ with Q (Φc) = 0 con- any other resource. Until recently, there was no known tribute no volume to the subset of those channels for systematic procedure to solve this problem, except for which the environment dimension is larger than that of two special classes of channels, namely, PPT and anti- the output space. In particular, if we consider the set degradable channels [34]. Significant progress, however, of all channels defined between a pair of fixed input and has recently been made on this issue [35, 36]. In [36], we output spaces as a subset of some Euclidean space Rn, employed some basic techniques from analytic perturba- (1) channels Φ with Q (Φc) = 0 reside on the boundary, tion theory of Hermitian matrices to develop a sufficient thus having zero (Lebesgue) volume (Theorems III.4 and condition for a quantum channel to have positive quan- III.5).