Coherent Information of a Quantum Channel Or Its Complement Is Generically Positive

Coherent information of a quantum channel or its complement is generically positive

Satvik Singh1, 2, ∗ and Nilanjana Datta2, † 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 ) † Q(Φ) = lim , (1) [email protected] n→∞ 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 additivity 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 → ∞ 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 ∈ 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). Since all channels Φ that are currently known tum capacity. By exploiting this result, a plethora of to have additive coherent information (e.g. degradable important examples of quantum channels were shown to channels) have Q(Φc) = 0 (i.e. they are more capable), have positive capacities. the above result indicates that the occurrence of channels In this Letter, we apply perturbative techniques to ran- with additive coherent information is very rare. dom quantum channels to obtain an astonishing result: typically (or almost surely), the coherent information of a randomly selected channel (or that of its complement) is guaranteed to be positive, depending on whether the II. PREREQUISITES channel’s output space is larger than its environment (Theorem III.3). Random quantum channels have proved We denote the algebra of all d×d complex matrices by to be very useful in establishing generic properties of Md and the identity matrix in Md by 1d. The convex quantum channels. For instance, they were used by Hay- compact set of all quantum states in Md is denoted by 3

Sd := {ρ ∈ Md : ρ ≥ 0 and Tr ρ = 1}. III. MAIN RESULTS

A linear map Φ: Md → Mdout is called a quantum channel if it is completely positive and trace preserving Let us begin by recalling the main ideas that were (CPTP). We collect all quantum channels Φ: Md → employed in [36] to obtain a simple sufficient condition C Mdout in the convex compact set d,dout . All quantum to detect positive quantum capacities of quantum chan- channels in this paper are assumed to be defined on a nels. Since Q(Φ) ≥ Q(1)(Φ) for any channel Φ, the nat- non-trivial input space (d > 1). The Choi matrix of a ural first step in showing that Q(Φ) > 0 would be to (1) given channel Φ: Md → Mdout is defined as: show that Q (Φ) > 0. To do so, we begin with an arbitrary pure state |ψihψ|, for which it is evident that (1) J(Φ) := (Φ ⊗ id) |ΩihΩ| , where (4) Ic(|ψihψ| ; Φ) = 0 =⇒ Q (Φ) ≥ 0, since both Φ(|ψihψ|) and Φc(|ψihψ|) have identical non-zero eigenvalues (for Pd−1 d d |Ωi := i=0 |ii ⊗ |ii ∈ C ⊗ C is a maximally entangled any Φc ∈ CΦ), see [3, Theorem 3]. Now, the idea is to unnormalized state, and id : Md → Md is the identity slightly perturb the pure input state with a mixed state σ map [46, 47]. and check if the coherent information becomes positive.

We say that two quantum channels Φ: Md → Mdout To this end, let ∈ [0, 1] and deﬁne and Φc : Md → Md are complementary to each other env ρ() = (1 − ) |ψihψ| + σ, if there exists an isometry V : Cd → Cdout ⊗ Cdenv such that for all X ∈ Md, so that

† † Φ(X) = Trenv(VXV ), Φc(X) = Trout(VXV ). (5) Φ[ρ()] = (1 − )Φ(|ψihψ|) + Φ(σ), Φc[ρ()] = (1 − )Φc(|ψihψ|) + Φc(σ). If CΦ denotes the set of all quantum channels that are complementary to Φ, then the minimal environment and At = 0, let us focus on the λ = 0 eigenvalue output dimensions of Φ are defined, respectively, as: of the unperturbed outputs Φ(|ψihψ|) and Φc(|ψihψ|) with multiplicities κ = dim ker Φ(|ψihψ|) and κc = ∗ d (Φ) := min{denv : ∃Φc ∈ CΦ, Φc : Md → Md }, dim ker Φc(|ψihψ|), respectively. When > 0, Φ[ρ()] and env env κ ∗ ∗ Φc[ρ()] have exactly κ and κc eigenvalues {λj()}j=1 and dout(Φ) := denv(Φc), (6) c κc {λk()}k=1, respectively, which converge to zero as → 0 and admit convergent power series expansions in a neigh- where Φc ∈ CΦ is complementary to Φ. It is easy to ∗ borhood of = 0 [48, Chapter 1]: see that the definition of dout(Φ) does not depend on the Φ ∈ CΦ 2 c c c 2 choice of (see [36, Remark 2.1] or Lemma II.1 λj() = 0 + λj1 + λj2 . . . λk() = 0 + λk1 + λk2 ... below). The following Lemma provides a simple way to compute the minimal output and environment dimen- By feeding the above analytic expressions for the eigen- sions of any given channel. values into the formula for coherent information I() := Ic(ρ(); Φ) (Eq. (3)), it can be shown that Lemma II.1. [36, Lemma 2.2] For a channel Φ: Md →   Mdout and some complementary channel Φc ∈ CΦ, X X 1 I0() = λ − λc log + K(),  j1 k1 2 ∗ j:λ 6=0 k:λc 6=0 denv(Φ) = rank J(Φ) = rank Φc(1d) j1 k1 ∗ dout(Φ) = rank J(Φc) = rank Φ(1d). where K() is bounded in . Notice that the only unbounded contribution to the derivative comes from the Φ: M → M c Given a quantum channel d dout , we have non-zero first order correction terms λj1 and λj1, which already seen that its quantum capacity Q(Φ) admits the are known to be equal to the non-zero eigenvalues of c c c regularized expression given in Eq. (1). We define the KψΦ(σ)Kψ and KψΦ(σ)Kψ, where Kψ and Kψ are the complementary coherent information and the comple- orthogonal projections onto the unperturbed eigenspaces mentary quantum capacity of Φ, respectively, as ker Φ(|ψihψ|) and ker Φc(|ψihψ|), respectively; see [49,

(1) (1) Section 3.7]. Hence, the expression in the parenthesis Q (Φ) := Q (Φc) and Qc(Φ) := Q(Φc), (7) c c above is Tr(KψΦ(σ)) − Tr(KψΦc(σ)). If this is positive, it is easy to ﬁnd a small enough interval (0, δ) in which Φ ∈ C Φ 0 where c Φ is complementary to . It can be easily I () > 0, implying that I() is also positive in the same shown that the above mentioned capacity expressions do interval (recall that I(0) = 0). We summarize this result Φ ∈ C not depend on the choice of c Φ. Moreover, in all succinctly in the following theorem. quantum capacity computations, it can be assumed with- Theorem III.1. Φ Φ out loss of generality that the channel Φ: Md → Mdout [36, Theorem 3.1] Let and c be and its complement Φc : Md → Mdenv are minimally complementary channels. Then, ∗ ∗ d = d (Φ) d = d (Φ) (1) deﬁned, i.e. out out and env env , see •Q(Φ) ≥ Q (Φ) > 0 if ∃ |ψihψ| , σ such that [36, Remark 2.4]. In the above terminology, a channel c Φ: Md → Mdout is called more capable if Qc(Φ) = 0. Tr(KψΦ(σ)) > Tr(KψΦc(σ)). 4

(1) •Qc(Φ) ≥ Qc (Φ) > 0 if ∃ |ψihψ| , σ such that and we say that Φ is distributed according to the Stine- µdenv Φ ∼ µdenv c spring probability measure d,dout (denoted d,dout ). Tr(KψΦ(σ)) < Tr(KψΦc(σ)). Now, it can be shown that a doutdenv × d random Haar Now, for any channel Φ and an arbitrary pure state isometry V contains at least one column vector |ψi ∈ dout denv |ψihψ|, it can be shown that C ⊗ C with full Schmidt rank = min{dout, denv} d ∗ ∗ almost surely [54, Theorem S3], so that Φ ∼ µ env rank Φ(|ψihψ|) ≤ min{d (Φ), d (Φ)}, d,dout out env sends at least one pure state to a maximal rank out- see [3, Theorem 3] and Lemma II.1. Hence, if we consider put state almost surely. Moreover, it is known that a pair of minimally defined complementary channels Φ: µdenv the measures d,dout are supported within the subset of M → M Φ : M → M d < d ∗ ∗ d dout and c d denv (with out env, channels with dout(Φ) = min{dout, ddenv} and denv(Φ) = say) and assume that there exists a state |ψihψ| with min{d , dd } Φ ∼ µdenv env out , i.e. d,dout has the stated mini- rank Φ(|ψihψ|) = min{dout, denv} = dout, then it is clear c mal output/environment dimensions almost surely, see that Kψ = 0 but Kψ 6= 0, so that [55, Section 3]. Therefore, if denv 6= dout and Φ ∼ 0 = Tr(K Φ(1 )) < Tr(Kc Φ (1 )) =⇒ Q (Φ) > 0. µdenv ψ d ψ c d c d,dout , Corollary III.2 can be readily applied to infer that Φ or its complement has positive coherent infor- A similar argument can be applied when denv < dout. mation almost surely. The necessary measure-theoretic We thus arrive at the following crucial corollary. background needed to rigorously formulate the above dis- Corollary III.2. [35, Section IV] [36, Corollary 3.2] cussion and the full proof of the main result stated below Let Φ and Φc be complementary channels such that is included in the Supplemental Material [54]. there exists a pure state |ψihψ| with rank Φ(|ψihψ|) = ∗ ∗ Theorem III.3. Φ: Md → Md min{dout(Φ), denv(Φ)}. Then, Let out be a random ∗ ∗ (1) quantum channel distributed according to the Stinespring • dout(Φ) > denv(Φ) =⇒ Q(Φ) ≥ Q (Φ) > 0. µdenv probability measure d,dout . Then, ∗ ∗ (1) • dout(Φ) < denv(Φ) =⇒ Qc(Φ) ≥ Qc (Φ) > 0. (1) • Q(Φ) ≥ Q (Φ) > 0 almost surely if dout > denv. Once the minimal output and environment dimen- (1) sions of a given channel are computed with the help • Qc(Φ) ≥ Qc (Φ) > 0 almost surely if dout < denv. of Lemma II.1, the difficult step in applying the above corollary is to ascertain the existence of a pure state Let us now discuss an important special case of the that gets mapped to a maximal rank output state [50]. above theorem in some detail. For simplicity, let us con- This problem can be linked to the problem of determin- sider the set Cd := Cd,d consisting of quantum channels ing whether a given matrix subspace contains a full rank with input and output dimensions both equal to d. Then, 2 2 matrix, which is known to be hard [51, 52]. It is easy for every Φ ∈ Cd, its Choi matrix J(Φ) is a d × d Her- 4 to find examples of channels that map every pure state mitian matrix defined by d real parameters. Complete to a rank deficient output state (e.g. the Werner-Holevo positivity of Φ forces J(Φ) to be positive semi-definite channel [53]). However, it turns out that the desired pure and the trace-preserving property Tr1[J(Φ)] = 1d en- 2 state exists for almost all channels, in the sense that a forces d additional constraints on J(Φ). Hence, the Choi channel which is selected randomly (according to the dis- isomorphism J identifies Cd with a convex compact set D 4 2 tribution defined below) maps at least one pure state to a J(Cd) ⊂ R with D = d − d , thus allowing us to define maximal rank output state almost surely (i.e. with prob- the uniform Lebesgue measure (denoted µLeb) on Cd by ability one; see [54, Remark S6]). normalizing the standard Lebesgue measure on the am- There are different ways of sampling random channels bient Euclidean space. It turns out that the earlier de- 2 from the set of all channels with given input and output fined Stinespring probability measure on Cd for denv = d d2 dimensions (see e.g. [55] for a detailed review). A natural is equal to the Lebesgue measure, i.e. µd,d = µLeb [55, way to obtain a random channel stems from the consid- Proposition 3]. Thus, one special case of Theorem III.3 eration of the Stinespring isometry of a channel. First, can be restated as follows. we fix the input and output dimensions to be d > 1 Theorem III.4. Φ: M → M and dout, respectively, and hence focus on channels in Let d d be a random the set Cd,d . Then, for every environment dimension quantum channel distributed uniformly according to the out µ denv satisfying d ≤ doutdenv, we select a random isometry Lebesgue measure Leb. Then, V : Cd → Cdout ⊗Cdenv according to the Haar measure on Q (Φ) ≥ Q(1)(Φ) > 0 . the set of all such isometries [54, Eq. (S4)]. This is equiva- c c almost surely lent to a uniformly random selection of a d−dimensional Notice that the above result amounts to saying that subspace from the collection of all d−dimensional sub- quantum channels with zero complementary coherent in- spaces (also called the Grassmanian space) of the com- formation contribute no Lebesgue volume to the set of posite output-environment space Cdout ⊗ Cdenv . The ran- all channels. Perhaps a geometric interpretation would dom channel Φ: Md → Md is then defined as out provide a more insightful description of this result. Since † D Φ(X) = Trenv(VXV ), Jd := J(Cd) is a compact convex subset of R (with 5 the standard Euclidean metric), it makes sense to talk [45]. However, whenever the output space of Φ is larger about the usual topological interior int(Jd) and bound- than its environment, we have shown that only a single ary ∂Jd of Jd, so that the following disjoint splitting copy of Φ typically suffices to detect its ability to reliably (1) occurs: Jd = int(Jd) ∪ ∂Jd. Now, it is easy to show transmit quantum information, i.e., Q (Φ) > 0 almost 2 that J(Φ) ∈ int Jd ⇐⇒ rank J(Φ) = d [56, Propo- surely. In other words, the set of all quantum channels (1) sition 15]. Interestingly, for channels Φ ∈ Cd with with zero coherent information Q (Φ) = 0 contribute ∗ denv(Φ) = rank J(Φ) > d(d − 1), a pure state which gets no volume to the subset of channels for which the output mapped to a maximal rank output state always exists space is larger than the environment. On the other hand, [50], so that Corollary III.2 can be readily applied to de- if the channel’s output space is smaller than its environ- (1) duce that Qc (Φ) > 0. Hence, it is clear that every chan- ment, we have shown that identical results hold for the (1) channel’s complement. Our work opens up several new nel Φ ∈ Cd with J(Φ) ∈ int Jd is such that Qc (Φ) > 0. avenues of research, some of which we list below: In other words, all channels Φ ∈ Cd with zero complementary coherent information must lie within the boundary • Can the above stated results be strengthened if a ∂J d, which clearly has no Lebesgue volume. We state the finite number n ∈ N (with n > 1) of copies of a above conclusion in the form of a theorem below. given channel are allowed to be used in Eq. (1) for detection of its quantum capacity? Theorem III.5. Quantum channels Φ: Md → Md with zero complementary coherent information – including all • What can be said about the coherent information more capable channels and hence all degradable channels and the quantum capacity of a randomly selected as well – lie within the topological boundary of the ambi- D 4 2 channel when its output space is equal in size or ent convex compact set Cd ⊂ R with D = d − d . smaller than the environment?

• It would be interesting to see if the results of this IV. CONCLUSION Letter can be extended to randomly selected quantum channels in inﬁnite dimensions. The task of detecting whether a given quantum channel Φ has positive quantum capacity generally requires Acknowledgements. The authors would like to thank computation of the coherent information Q(1)(Φ⊗n) of an Ludovico Lami for his helpful feedback on a ﬁrst draft of arbitrarily large number n ∈ N of copies of the channel this paper.

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Supplemental Material: Coherent information of a quantum channel or its complement is generically positive

I. MATHEMATICAL FOUNDATIONS OF RANDOMNESS

For the sake of completeness, we include some basics of measure theory in this appendix. Measure theory deals with the problem of sensibly defining a notion of volume for subsets of a given set Ω. For example, if Ω = R, one defines a function (called the Lebesgue measure) which is a generalization of the usual notion of length for intervals. It is reasonable to demand that shifting a subset by a real number does not change its length and that the length of a countable disjoint union of subsets is just the sum of the individual lengths of the subsets. However, the axiom of choice forbids such a length function to be defined for all subsets of the real line R [57, Section 16], and hence we must restrict the domain of definition of the length function to only a special kind of collection of subsets, called a σ-algebra. We will denote the empty subset of a set Ω by ∅. For subsets A, B ⊆ Ω, A\B := {x ∈ A : x∈ / B} is the set of all elements that are in A but not in B. Definition S1. A collection F of subsets of a set Ω is called a σ-algebra if: • ∅, Ω ∈ F. • A ∈ F =⇒ Ω \ A ∈ F. S • {An}n∈N ⊆ F =⇒ n∈N An ∈ F. A pair (Ω, F) consisting of a set Ω and a σ-algebra F is called a measurable space, and the subsets in F are called measurable. The analogue of the notion of continuous functions between topological spaces is the notion of measurable functions between measurable spaces.

Definition S2. Let (Ω1, F1) and (Ω2, F2) be measurable spaces. A function f :Ω1 → Ω2 is said to be measurable (with respect to F1 and F2) if A ∈ F2 =⇒ preimf (A) := {x ∈ Ω1 : f(x) ∈ A} ∈ F1. For a finite-dimensional complex vector space W , there is a unique way to turn W into a measurable space. We proceed by equipping W with the standard topology τW induced by some norm on W (note that since W is a finite- dimensional complex vector space, it can be equipped with a norm and moreover, all norms on W are equivalent, so that the induced topology is uniquely defined). Now, let \ σ(τW ) := {F : F is a σ-algebra which contains τW }. (S1)

It is easy to see that σ(τW ) is uniquely deﬁned as the minimal σ-algebra which contains τW , i.e. if G is another σ-algebra which contains τW , then σ(τW ) ⊆ G [58, Theorem 5.2.1]. We say that σ(τW ) is the standard Borel sigma algebra on W (generated by the standard topology τW ). Remark S3. More generally, any topological space (Ω, τ) can be converted into a measurable space by deﬁning F = σ(τ) to be the Borel sigma-algebra generated by τ in the sense of Eq. (S1). 8

Remark S4. It is easy to show that if (Ω1, τ1) and (Ω2, τ2) are two topological spaces and f :Ω1 → Ω2 is a continuous map (with respect to the standard topologies τ1 and τ2), then f is also measurable (with respect to the Borel sigma algebras σ(τ1) and σ(τ2)). We now introduce the central definition of a measure. Definition S5. Let (Ω, F) be a measurable space. A set function µ : F 7→ [0, ∞] is said to be a measure on Ω if µ(∅) = 0 and if µ is countably additive, i.e. for all countable pairwise disjoint collections of measurable subsets S P∞ {An}n∈N ⊆ F, µ( n∈N An) = n=1 µ(An). A triple (Ω, F, µ) consisting of a set Ω, a σ-algebra F, and a measure µ : F → [0, ∞], is called a measure space. Given a measure space (Ω1, F1, µ), a measurable space (Ω2, F2), and a measurable function f :Ω1 → Ω2, we can push the measure µ from (Ω1, F1) forward to (Ω2, F2) via f to define the push-forward measure f∗(µ): F2 → [0, ∞] as

∀ A ∈ F2 : f∗(µ)(A) := µ(preimf (A)). (S2) Remark S6. If (Ω, F, µ) is a measure space such that µ(Ω) = 1, then it is called a probability space. The measurable subsets A ∈ F are then interpreted as events occurring with probability µ(A) under the given probability rule defined by the set function µ : F → [0, 1]. An event A is said to occur almost surely if µ(A) = 1 ⇐⇒ µ(Ω \A) = 0. Remark S7. In this paper, every finite dimensional complex vector space W comes equipped with the standard topology τW and the associated Borel sigma algebra σ(τW ). Similarly, any subset K ⊆ W comes equipped with the standard subset topology τK (inherited from the standard topology on W ) and the associated Borel sigma algebra σ(τK ). Hence, any measure on W (or K) will always be assumed to be defined on the standard Borel sigma algebra σ(τW ) (or σ(τK )).

d1d2 Complex Ginibre distribution. The set Md1×d2 ' C of complex d1 × d2 matrices comes equipped with the standard Lebesgue measure, which we denote by µLeb. The complex Ginibre distribution, µG, on Md1×d2 is a probability measure deﬁned as a product of the standard complex Gaussian distributions (one for each matrix entry):

d d 2 Y1 Y2 e−|zij | dµG(Z) = P (Z)dµLeb(Z), where ∀Z ∈ Md ×d : P (Z) = (S3) 1 2 π i=1 j=1

th and zij denotes the ij entry of Z [59]. Lemma S8. M∗ M The set d1×d2 of all full rank matrices in d1×d2 is Borel measurable. Further, µ (M∗ ) = 1. G d1×d2 P Proof. Assume d1 ≤ d2. Let f : Md1×d2 → R be deﬁned as f(X) = I | det X[I]|, where the sum runs over all index sets I ⊆ {1, 2, . . . , d2} of size |I| = d1 and X[I] is a matrix consisting of columns of X that are labelled by I. Clearly, f M∗ = M \ preim ({0}) is continuous, so that d1×d2 d1×d2 f is Borel measurable.

The second claim amounts to saying that the set {v1, v2, . . . , vd1 } of vectors chosen independently and identically from Cd2 according to the complex Ginibre distribution is linearly independent with probability one, which is a well-known fact.

Haar measures. Let U(d) denote the set of unitary matrices in Md. Being a compact topological group with respect to the standard subset topology and matrix multiplication as the group operation, U(d) can be equipped with 0 a unique left (or right) invariant Haar probability measure, µHaar. For any d ≤ d, we denote the set of isometries d0 d 0 † 0 V : C → C by V(d , d) := {V ∈ Md×d0 : V V = 1d0 }. Then, the continuous map f : U(d) → V(d , d) whose action is to remove the rightmost d − d0 columns from the input unitary matrix, can be used to obtain the push-forward probability measure

V µHaar := f∗(µHaar), (S4)

0 V which is called the Haar measure on V(d , d). There is an equivalent way to deﬁne µHaar, which we will employ in the proof of Theorem S3 below. We start with the complex Ginibre distribution µG on the matrix space Md×d0 . Recall ∗ ∗ that according to Lemma S8, µG(Md×d0 ) = 1, where Md×d0 is the set of all full rank matrices in Md×d0 . ∗ 0 Now, if GS : Md×d0 → V(d , d) denotes the (continuous) Gram Schmidt orthonormalization map (performed on the linearly independent columns of the full rank input matrix), then the push-forward meausre GS∗(µG) turns out V 0 to be the Haar measure µHaar on V(d , d), see [60, Sections 4 and 5], i.e.,

V µHaar = GS∗(µG). (S5) 9

II. RANDOM CHANNELS

Consider the set Cd,dout of all quantum channels Φ: Md → Mdout and for each denv ∈ N such that d ≤ doutdenv, d dout denv let V(d, doutdenv) be the set consisting of all isometries V : C → C ⊗ C . Let us deﬁne a continuous map

Φ: V(d, doutdenv) → Cd,dout in the following fashion:

† ∀ V ∈ V(d, doutdenv), ∀ X ∈ Md :ΦV (X) = Trenv(VXV ). (S6)

Now, by implementing a two step push-forward process via the mappings

M∗ −−→VGS (d, d d ) −→Φ C , doutdenv×d out env d,dout (S7)

we can construct the following probability measure on Cd,dout

µdenv := Φ (µV ) = Φ (GS (µ )). d,dout ∗ Haar ∗ ∗ G (S8)

Hence, for every input dimension d ∈ N and output dimension dout ∈ N, the set Cd,dout of quantum channels can be endowed with a sequence of probability measures deﬁned as in Eq. (S8), one for each positive integer denv ∈ N satisfying d ≤ doutdenv. For other probability measures that can be deﬁned on Cd,dout , see e.g. [55].

III. PROOF OF THE MAIN RESULT

The following lemma establishes the continuity of the quantum capacity and coherent information functions.

(1) (1) Lemma S1. The quantum capacity functions Q, Qc : Cd,dout → R and the coherent information functions Q , Qc :

Cd,dout → R are continuous.

Proof. The diamond norm k · k on the set Cd,dout is deﬁned as (see e.g. [5, Chapter 9]):

kΦk := sup{k(idd ⊗Φ)(X)k1 : X ∈ Md ⊗ Md, kXk1 ≤ 1}, where k · k1 is the usual trace norm on the relevant matrix algebras. In [61], the authors proved the continuity of (1) Q, Q : Cd,dout → R by showing that for all Φ, Ψ ∈ Cd,dout and ∈ [0, 1]:

(1) (1) kΦ − Ψk ≤ =⇒ |Q(Φ) − Q(Ψ)|, |Q (Φ) − Q (Ψ)| ≤ 8 log dout − 4[ log + (1 − ) log(1 − )].

Now, by employing [62, Theorem 1], it is clear√ that if kΦ − Ψk ≤ , then there exist Φc ∈ CΦ and Ψc ∈ CΨ with Φc, Ψc : Md → Mdenv and kΦc − Ψck ≤ c = 2 , implying that

(1) (1) |Qc(Φ) − Qc(Ψ)|, |Qc (Φ) − Qc (Ψ)| ≤ 8c log denv − 4[c log c + (1 − c) log(1 − c)], thus completing the proof.

For ﬁxed input and output dimensions d ∈ N (with d > 1) and dout ∈ N, respectively, we consider the set Cd,dout consisting of all quantum channels Φ: Md → Mdout . Let us collect all channels Φ ∈ Cd,dout with zero quantum capacity and zero complementary quantum capacity, respectively, in the sets

Z := {Φ ∈ C : Q(Φ) = 0} ⊂ {Φ ∈ C : Q(1)(Φ) = 0} := Z(1) d,dout d,dout d,dout d,dout (S9) Zc := {Φ ∈ C : Q (Φ) = 0} ⊂ {Φ ∈ C : Q(1)(Φ) = 0} := Z(1c) d,dout d,dout c d,dout c d,dout (S10) Before proving the main result, let us recall Corollary III.2 from the main text.

Corollary S2. Let Φ and Φc be complementary channels such that there exists a pure state |ψihψ| with ∗ ∗ rank Φ(|ψihψ|) = min{dout(Φ), denv(Φ)}. Then, ∗ ∗ (1) • dout(Φ) > denv(Φ) =⇒ Q(Φ) ≥ Q (Φ) > 0.

∗ ∗ (1) • dout(Φ) < denv(Φ) =⇒ Qc(Φ) ≥ Qc (Φ) > 0. We are now ready to prove our main result, which was stated as Theorem III.3 in the main text. 10

Theorem S3. Z(1) , Z(1c) ⊆ C The sets d,dout d,dout d,dout as deﬁned above are Borel measurable. Moreover, d < d : µdenv (Z(1) ) = 0 ⇐⇒ µdenv (C \Z(1) ) = 1 • for all env out d,dout d,dout d,dout d,dout d,dout .

d > d : µdenv (Z(1c) ) = 0 ⇐⇒ µdenv (C \Z(1c) ) = 1 • for all env out d,dout d,dout d,dout d,dout d,dout . Q(1), Q(1) : C → R Z(1) = Proof. Clearly, since both the coherent information functions c d,dout are continuous, the sets d,dout (1c) preimQ(1) ({0}) and Zd,d = preim (1) ({0}) are Borel measurable. Now, in order to prove the remaining assertions, out QC let us deﬁne the following subsets of quantum channels (for denv ∈ N): n o Adenv := Φ ∈ C : max rank Φ(|ψihψ|) = min{d , d } d,dout d,dout out env |ψihψ|∈Sd n o Bdenv := Φ ∈ C : d∗ (Φ) = min{d , dd } d∗ (Φ) = min{d , dd } . d,dout d,dout out out env and env env out From Corollary S2, we can immediately deduce that

d < d =⇒ Adenv ∩ Bdenv ⊆ C \Z(1) env out d,dout d,dout d,dout d,dout d > d =⇒ Adenv ∩ Bdenv ⊆ C \Z(1c) . env out d,dout d,dout d,dout d,dout µdenv (Adenv ∩ Bdenv ) = 1 d Hence, in order to prove the required claims, it suﬃces to show that d,dout d,dout d,dout for all env such that d ≤ d d µdenv Bdenv µdenv (Bdenv ) = 1 out env. Now, it has been shown in [55, Section III] that d,dout is supported on d,dout , i.e. d,dout d,dout . Adenv r ∈ N So we only need to worry about the set d,dout . Let us deﬁne a continuous map for :

fr : Cd,dout → R Φ 7→ max detr Φ(|ψihψ|). |ψihψ|∈Sd P Here, for X ∈ Md, we define detr X = I | det X[I]|, where the sum runs over all index sets I ⊆ {1, 2, . . . , d} of size |I| = r and X[I] is the |I| × |I| principle submatrix consisting of entries Xij with i, j ∈ I. It is easy to see that rank X < r ⇐⇒ detr X = 0. Hence, we can choose r = min{dout, denv}, so that Adenv = preim ({0}) \ preim ({0}) d,dout fr+1 fr is Borel measurable. µdenv Now, from the definition of d,dout (Eq. (S8)), we have µdenv (Adenv ) = µV (preim (Adenv )) = µ preim (preim (Adenv )) , d,dout ed,dout Haar Φ ed,dout G GS Φ ed,dout Φ: V(d, d d ) → C GS : M∗ → V(d, d d ) where out env d,dout is defined as in Eq. (S6), doutdenv×d out env is the Gram Schmidt Adenv = C \Adenv orthonormalization map, and ed,dout d,dout d,dout . Clearly,

denv preimΦ(Aed,d ) ⊆ (S × ... × S) ∩ V(d, doutdenv), out | {z } d times

dout denv where S ⊆ C ⊗ C is the set of all unit vectors with Schmidt rank < min{dout, denv}. Further,

doutdenv doutdenv ∗ preimGS((S × ... × S) ∩ V(d, doutdenv)) ⊆ (Se × C × ... × C ) ∩ Md d ×d, | {z } out env d−1 times

dout denv where Se ⊆ C ⊗C is the set of all vectors with Schmidt rank < min{dout, denv}, so that Se can be easily identiﬁed M \M∗ M with the set dout×denv dout×denv of rank deﬁcient matrices in dout×denv . From Lemma S8, we can infer that µ (S × Cdoutdenv × ... × Cdoutdenv ) ∩ M∗ = 0 =⇒ µdenv (Adenv ) = 0, G e doutdenv×d d,dout ed,dout

µdenv (Adenv ) = 1 so that d,dout d,dout , and we obtain the desired conclusion: µdenv (Adenv ∩ Bdenv ) = 1. d,dout d,dout d,dout