One-Shot Triple-Resource Trade-Off in Quantum Channel Coding

We analyze a task in which classical and quantum messages are simultaneously communicated via a noisy quantum channel, assisted with a limited amount of shared entanglement. We derive direct and converse bounds for the one-shot capacity region, represented by the smooth conditional entropies and the error tolerance. The proof is based on the randomized partial decoupling theorem, which is a generalization of the decoupling theorem. The two bounds match in the asymptotic limit of infinitely many uses of a memoryless channel and coincide with the previous result obtained by Hsieh and Wilde. Direct and converse bounds for various communication tasks are obtained as corollaries, both for the one-shot and asymptotic scenarios.


I. INTRODUCTION
One of the major goals of quantum communication theory is to investigate the ultimate capacities of a noisy quantum channel for transmitting classical and quantum information (see, e.g., [1], [2]). In an asymptotic limit of infinitely many uses of a memoryless channel, the classical capacity of a noisy quantum channel was obtained by Holevo [3], Schumacher and Westmoreland [4]; the quantum capacity by Lloyd [5], Shor [6] and Devetak [7]; the entanglement-assisted classical capacity by Bennet et al. [8], [9] and Shor [10]; the entanglementassisted quantum capacity by Devetak et al. [11], [12]; and the capacity of a quantum channel for simultaneously transmitting classical and quantum messages was obtained by Devetak and Shor [13]. One of the most general results in this direction was shown by Hsieh and Wilde [14], which provides a complete characterization of the triple trade-off among the resources of classical communication, quantum communication, and entanglement in quantum channel coding. As proved in [14], the capacity theorems in the special cases mentioned above are obtained from their result as corollaries.
A modern approach addresses the same communication tasks under the assumption that the channel is used only once, which is referred to as the one-shot regime. Along this line, the classical capacity of a noisy quantum channel was first investigated by Mosonyi et al. [15] and was developed by Renes et al. [16] and Wang et al. [17]; the quantum capacity was obtained by Buscemi et al. [18]; the entanglement-assisted classical and quantum capacities by Datta et al. [19]; and the capacity region for simultaneously transmitting classical and quantum information was obtained by Salek et al. [20]. In the asymptotic limit of infinitely many uses of a memoryless channel, all these results (except [15]) are known to coincide with the preexisting results presented above. Numerous studies have been done to develop these approaches further (see e.g. [21]- [29]). However, the one-shot capacity theorem that reduces in the asymptotic limit to the result by Hsieh and Wilde [14] has not been obtained so far.
In this paper, we derive a one-shot generalization of the three-dimensional rate region obtained in [14]. Namely, we consider a task in which classical and quantum messages are transmitted via a noisy quantum channel with the assistance of a limited amount of shared entanglement, all in the one-shot scenario. The main result is that we derive inner and outer bonds for the one-shot rate region, which are represented in terms of the smooth conditional entropy of the channel. To our knowledge, this is the first time that trade-off relations that apply simultaneously to all the three resources of classical communication, quantum communication, and shared entanglement has been obtained in the one-shot channel coding scenario. In the asymptotic limit of infinitely many uses of the channel, the direct and converse bounds match and coincide with the achievable rate region obtained in [14]. As proved in [14], the aforementioned capacity theorems in the special cases, such as the Holevo-Schumacher-Westmoreland theorem [3], [4] and the Lloyd-Shor-Devetak theorem [5]- [7], are obtained from their result as corollaries. Hence, our one-shot result also recovers the capacity theorems in the asymptotic limit in those special cases. Besides, the direct and converse bounds for various communication tasks in the oneshot scenario readily follow from our result. It should be noted, however, that in special cases, our result is not necessarily as strong as those in the previous literature. To compare our results quantitatively with those in the previous literature is left as a future work.
The proof is based on the notion of randomized partial decoupling [30], which is a generalization of decoupling [31]- [36]. Here, we consider a scenario in which a bipartite quantum state on system AR is subject to a unitary operation on A, followed by the action of a linear completely positive (CP) map. Unlike the usual setting of decoupling, we assume that the subsystem is decomposed into a direct-sum form, and the unitary is chosen at random from the set of unitaries that are block-diagonal under the decomposition. In [30], we proved the randomized partial decoupling theorem, which shows that the distance between the final state and the averaged one is bounded in terms of smooth conditional entropies of the initial state and the channel. The previous results of the oneshot decoupling theorem [35] and the dequantization theorem [21] are obtained from this result as corollaries, up to changes in the smoothing and error parameters (see Section III D in arXiv:2004.12593v2 [quant-ph] 11 Feb 2023 [30] for the details). The current paper applies the randomized partial decoupling theorem to the channel coding scenario for proving the one-shot capacity theorems. This paper is organized as follows. In Section II, we introduce notations and definitions that will be used throughout this paper. In Section III, we present formulations of the problem and state the main results. In Section IV, we summarize the statement of the randomized partial decoupling theorem. The proofs of the main theorems are provided in Section V and VI. In Section VII, we apply the one-shot results to the asymptotic scenario of infinitely many uses of the channel and obtain a three-dimensional achievable rate region that is equivalent to the one obtained by Hsieh and Wilde. Proof of the asymptotic result is provided in Section VIII. In Section IX, we apply our results to special cases where one or two of the three resources are assumed to be zero and compare the results to those in the previous literature. Conclusions are presented in Section X.

II. PRELIMINARIES
We summarize notations and definitions that will be used throughout this paper.

A. Notations
We denote the set of linear operators and that of Hermitian operators on a Hilbert space H by L(H) and Her(H), respectively. For positive semidefinite operators, density operators and sub-normalized density operators, we use the following notations, respectively: A Hilbert space associated with a quantum system A is denoted by H A , and its dimension is denoted by d A . A system composed of two subsystems A and B is denoted by AB. When M and N are linear operators on H A and H B , respectively, we denote M ⊗ N as M A ⊗ N B for clarity. In the case of pure states, we often abbreviate |ψ A ⊗ |φ B as |ψ A |φ B . For X AB ∈ L(H AB ), X A represents Tr B [X AB ].
We denote |ψ ψ| simply by ψ. The maximally entangled state between A and A , where H A ∼ = H A , is defined by with respect to a fixed orthonormal basis {|α } d A α=1 . The identity operator is denoted by I. We denote (M A ⊗ I B )|ψ AB as M A |ψ AB and (M A ⊗ I B )ρ AB (M A ⊗ I B ) † as M A ρ AB M A † . When T is a supermap from L(H A ) to L(H B ), we denote it by T A→B . When A = B, we use T A for short. We also denote (T A→B ⊗ id C )(ρ AC ) by T A→B (ρ AC ). When a supermap is given by a conjugation of a unitary U A or a linear operator W A→B , we denote it by its calligraphic font as U A (X A ) := (U A )X A (U A ) † and W A→B (X A ) := (W A→B )X A (W A→B ) † . In that case, the adjoint map of W A→B is defined by W †B→A (·) := (W A→B ) † (·)(W A→B ).
For any linear CP map T A→B , there exist a finite dimensional quantum system E and a linear operator Γ A→BE T such that T A→B (·) = Tr E [Γ T (·)Γ † T ]. The operator Γ T is called the Stinespring dilation of T A→B [37], and the linear CP map defined by Tr B [Γ T (·)Γ † T ] is called the complementary map of T A→B . With a slight abuse of notation, we denote the complementary map by T A→E .

B. Norms and Distances
For a linear operator X, the trace norm is defined as The trace distance between two unnormalized states ρ, ρ ∈ P(H) is defined by 1 2 ρ − ρ 1 . For subnormalized states ρ, ρ ∈ S ≤ (H), the generalized fidelity and the purified distance are defined bȳ respectively [38]. The trace distance and the purified distance are related as for any ρ, ρ ∈ S ≤ (H). The epsilon ball of a subnormalized state ρ ∈ S ≤ (H) is defined by

C. One-shot entropies
For any subnormalized state ρ ∈ S ≤ (H AB ) and normalized state ς ∈ S = (H B ), define The conditional min-and max-entropies (see e.g. [39]) are defined by respectively. The smooth conditional min-and max-entropies are defined by for a smoothing parameter ≥ 0. The properties of the smooth conditional entropies used in this paper are summarized in Appendix B.

D. Choi-Jamiolkowski representation
Let T A→B be a linear supermap from L(H A ) to L(H B ) and let Φ AA be the maximally entangled state between A and A . A linear operator J(T A→B ) ∈ L(H AB ) defined by J(T A→B ) := T A →B (Φ AA ) is called the Choi-Jamiołkowski representation of T [40], [41]. The representation is an isomorphism. For an operator X AB ∈ L(H AB ), the inverse map is given by where A T denotes the transposition of A with respect to the Schmidt basis of Φ AA . When T is completely positive, J(T A→B ) is an unnormalized state on AB and is called the Choi-Jamiołkowski state of T .

E. Haar measure
For a unitary group of finite degree, there exists the unique left-and right-unitarily invariant probability measure, known as the Haar measure. We denote it by H. The Haar measure satisfies the property that, for any unitary U and a set of When a unitary U is chosen uniformly at random with respect to the Haar measure, we denote it by U ∼ H.

III. ONE-SHOT CAPACITY THEOREMS
Consider a noisy quantum channel N A→B . Suppose that the sender, Alice, transmits c-bits of classical and q-qubits of quantum messages simultaneously to the receiver, Bob, through a noisy quantum channel assisted by e-ebits of shared entanglement (see Figure 1). We assume that Bob initially has no side information about the messages. Our goal is to obtain the conditions for this task to be achievable within error tolerance δ, in terms of c, q and e.
To be more precise, let M q be a 2 q -dimensional quantum system that represents the quantum message, and R q be a reference system with the same dimension that are inaccessible to Alice and Bob. The quantum message is described by the maximally entangled state Φ MqRq 2 q with Schmidt rank 2 q . With F A and F B being 2 e -dimensional quantum registers, the entanglement resource is given by the maximally entan- with Schmidt rank 2 e . A communication protocol is represented by a set of encoding CPTP maps {E . Let m andm be the original classical message and the decoded one, respectively. The probability that the decoded message iŝ m = k when the original message is m = j is (17) When the decoded message ism = k and the original message is m = j, the state of the quantum part after the decoding is given bŷ The average probability of error in decoding the classical message is calculated to be and that for the quantum message is Both the classical and quantum errors shall be small, i.e., we want δ C and δ Q to be below a certain threshold δ > 0.
A rigorous definition of a code is given as follows. Here, we treat the classical and quantum parts of the message as one large quantum system, because it is more convenient in our analysis.
Definition 1 Consider the setting described above. Let M c and R c be 2 c -dimensional quantum system with a fixed orthonormal basis {|j } 2 c j=1 . We denote M c M q by M and R c R q by R for brevity. Let Φ M R 2 c ,2 q be a source state defined by A pair of an encoding CPTP map E M F A →A and a decoding CPTP map D BF B →M is called a (c, q, e, δ) code for the channel N if it holds that and where C is the completely dephasing operation on M c with respect to the basis {|j } 2 c j=1 . Note that the condition (22) is equivalent to the condition that δ C , δ Q ≤ δ, up to a constant multiplication. The correspondence between the encoding and decoding operations is given by and It should be noted that the capacity theorems obtained in terms of the average probability of error, as in (19), (20) and (22), are translated into those based on the worst-case error, up to halving of the message length. See, for example, Corollary 1 in [18] and Theorem 11 in [19] for the quantum part and Lemma 1 in [16] for the classical part. The latter is known as the expurgation trick (see e.g. [1]). Fig. 1. One-shot channel coding is depicted. The normal arrows represent quantum systems, and the dotted arrows represent the classical part of the source state.
The protocol for communication over the channel N constructed in terms of randomized partial decoupling is depicted. The encoding operation E ρ,s,U is composed of (i) a linear isometry P M F A →S that embeds the message system M and Alice's share of the entanglement resource F A to a larger system S, (ii) the permutation s and the unitary U that appear in randomized partial decoupling, and (iii) a linear CPTP map Eρ that is obtained from ρ SA by the Choi-Jamiolkowski correspondence. The explicit form of the decoder D is left open, because we only prove the existence of a proper decoder D in the proof of the direct part.

A. Channel Capacity with Limited Entanglement
First, we consider the situation in which the amount of the resource of shared entanglement is limited.
Definition 2 A triplet (c, q, e) is said to be achievable within the error δ for the channel N A→B if there exists a (c, q, e, δ) code for N A→B .
The direct part is represented by the following theorem. The proof is based on the direct part of randomized partial decoupling (Theorem 8), and will be provided in Section V. A protocol that achieves the direct bound is depicted in Figure  2.
Theorem 3 Let S r be a finite dimensional quantum system and let S c be a quantum system with a fixed orthonormal basis {|j } d Sc j=1 such that d Sc ≥ 2. We denote S c S r by S. Consider a state in the form of is a set of normalized states on S r A such that ρ S is the full-rank maximally mixed state on S. For any such S c , S r , ρ SA , any δ 1 , δ 2 > 0 and ≥ 0, a triplet (c, q, e) is achievable within the error for the channel N A→B if d Sc ≥ 2 c and the following three inequalities hold: The same statement holds in the cases of (c = 0, d Sc = 1) and (q = e = 0, d Sr = 1). In the former case, the condition (32) is removed and δ 1 in (30) is assumed to be zero. In the latter, the condition (33) is removed and δ 2 in (30) is considered to be zero.
The converse part is stated by the following theorem, which will be proved in Section VI based on the converse part for randomized partial decoupling (Theorem 9).
Theorem 4 Suppose that a triplet (c, q, e) is achievable within the error δ for the channel N A→B . Then, there exist a quantum system S satisfying d S ≤ 2 c+q+e and a state ρ SA such that the following conditions hold. First, S is composed of finite dimensional quantum systems S c and S r , where S c is equipped with an orthonormal basis {|j } d Sc j=1 . Second, the state ρ SA is in the form of where {ρ j } d Sc j=1 is a set of normalized states on S r A and ρ S is the full-rank maximally mixed state on S. Third, for any ι ∈ (0, 1], it holds that q + e ≤ log d Sr , The smoothing parameters λ and λ are given by and x := 2 8 √ δ.

B. Channel Capacity with Free Entanglement
Second, we consider the situation in which the resource of shared entanglement is freely available. Following [19], we assume that the entanglement resource is given in the form of the maximally entangled state.
Definition 5 A pair (c, q) is said to be achievable within the error δ for the channel N A→B with the assistance of entanglement if there exists e ≥ 0 such that a triplet (c, q, e) is achievable within the error δ for N A→B .
The direct and converse bounds for the scenario of free entanglement immediately follow from the direct bound and the converse bound for the case of limited entanglement, i.e., from Theorem 3 and Theorem 4.

Corollary 6
For any ∈ [0, 1/2) and δ ∈ (0, 1 − 2 ], a pair (c, q) is achievable within the error for the channel N A→B with the assistance of entanglement, if there exist a quantum system S and a state ρ SA such that ρ S is the full-rank maximally mixed state on S and the following inequality holds: Corollary 7 Suppose that a pair (c, q) is achievable within the error δ for the channel N A→B with the assistance of entanglement. Then, there exist a quantum system S and a state ρ SA such that ρ S is the full-rank maximally mixed state on S and for any ι ∈ (0, 1], it holds that The smoothing parameter λ is given by (38).
Proof of Corollaries: Corollary 7 immediately follows from Inequalities (35) and (36) in Theorem 4. To prove Corollary 6 from Theorem 3, suppose that there exists a state ρ SA that satisfies the conditions in Corollary 6. Let S c be a system such that d S c ≥ 2 c . Define S r := S, S := S c S r and consider a state Due to the property of the smooth max entropy for product states (Lemma 24), we have It follows from (44) and (41) that Thus, there exists e ∈ R such that We may assume that e ≥ 0, since q ≤ log d S r . This is because the dimension bound for the smooth max entropy (see Lemma 25 in Appendix B) imply The second inequality in (47) leads to due to Inequality (45) and the relation d S c /(d S c − 1) ≤ 2, and to since c ≥ 0. Combining (48), (49) and the first inequality in (47) with Theorem 3, we complete the proof.

IV. RANDOMIZED PARTIAL DECOUPLING
In this section, we briefly review a task that we call randomized partial decoupling [30] and present the direct and converse bounds for it. This is a generalization of the decoupling theorem in the version of [35]. For the details and proofs, see the paper by the same authors [30].
Randomized partial decoupling is a task in which a bipartite quantum state Ψ AR is transformed by a unitary operation on A and then is subject to the action of a linear CP map T A→E . We assume that the Hilbert space H A is decomposed into a directsum form as H A = J j=1 H A j , where each H A j (j = 1, · · · , J) has the same dimension r. Let H Ac be a J-dimensional Hilbert space with a fixed orthonormal basis {|j } J j=1 and H Ar be an r-dimensional Hilbert space. The Hilbert space H A is isomorphic to a tensor product Hilbert space H Ac ⊗ H Ar , i.e., A ∼ = A c A r . In terms of this decomposition, any state Ψ AR is written as where Ψ ArR jk := Π A j Ψ AR Π A k with Π j being the projection onto H A j . By properly choosing orthonormal bases, a maximally entangled state |Φ AA is represented by where A c ∼ = A c and A r ∼ = A r . Consider a random unitary U on A in the form of where U j is independently chosen for each j from the Haar measure H on the unitary group on H Ar . For any state Ψ AR , the averaged state after the action of this random unitary is given by Here, π Ar is the maximally mixed state on H Ar , {p j } J j=1 is a probability distribution defined by p j := Tr[ j| Ac Ψ AR |j Ac ], and Ψ R j is a normalized state on H R defined by Ψ R j := p −1 j Ψ R jj . In the following, we denote E U1,...,U J ∼H simply by E U when there is no ambiguity. Let P be the permutation group on {1, · · · , J} and define a unitary G s for each s ∈ P by The permutation s is chosen at random according to the uniform distribution on P. Our concern is how close the final state T A→E • G A s • U A (Ψ AR ) is, on average over all U , to the averaged final state T A→E • G A s (Ψ AR av ), for typical choices of the permutation s (see Figure 4 as well). Fig. 4. The procedure used in the randomized partial decoupling is depicted. For a given initial state Ψ AR , a random unitary U in the form of (52) and a permutation Gs are first applied to A. Then, the system A is mapped to another one E by T .
For simplicity of analysis, we assume that R ∼ = R c R r , where R c is a quantum system with dimension J. We also assume that Ψ AR is decomposed in the form of where ψ kl ∈ L(H Ar ⊗ H Rr ) for each k and l. Such states are called classically coherent states [21]. The following theorem is the direct part of the randomized partial decoupling theorem.
Theorem 8 (Theorem 3 in [30]) Consider a linear CP map T A→E and a state Ψ AR ∈ S = (H AR ) that is decomposed as (56). Let U and G s be random unitaries defined by (52) and (55), respectively. Define the partial decoupling error ∆ s,U (T , Ψ) by Then, for any , µ ≥ 0, it holds that The terms θ I and θ II are represented by where the exponents H I and H II are given by respectively. Here, C is the completely dephasing operation on A c with respect to the basis {|j } J j=1 and τ is the Choi-Jamiolkowski state of the complementary map T A→C of T A→E , i.e. τ = J(T A→C ).
In [30], we also obtained a converse bound for randomized partial decoupling, which is stated by the following theorem.
Theorem 9 (Theorem 4 in [30]) Consider a linear tracepreserving CP map T A→E and a state Ψ AR ∈ S = (H AR ) that is decomposed as (56). Suppose that, for δ > 0, there exists a normalized state in the form of such that Then, for any υ ∈ [0, 1/2) and ι ∈ (0, 1], it holds that The second terms in the L.H.S.s of (63) and (64) are for a purification |Ψ ABR of Ψ AR and the complementary channel T A→C of T A→E , with C being the completely dephasing channel on A c . The smoothing parameters λ and λ are given by

V. PROOF OF THE DIRECT PART (THEOREM 3)
We prove the direct part of the capacity theorem (Theorem 3) based on the direct part of the randomized partial decoupling theorem (Theorem 8). We follow the idea of Ref. [33] that the problem of finding a good code for a quantum channel is equivalent to the problem of finding a good way to decouple a bipartite state constructed from the channel.
Fix an arbitrary triplet (c, q, e), δ > 0, system S ≡ S c S r and a state ρ SA that satisfy the conditions in Theorem 3. We prove achievability of the triplet (c, q, e) within the error δ along the following lines: First, we construct a state Ψ ρ from the state ρ SA , the channel N A→B and a state obtained by "purifying" and "extending" the source state and the resource state. Second, we prove that, if a CP map achieves randomized partial decoupling of the state Ψ ρ for a particular choice of s and U in high precision, there exist an encoder E ρ,s,U and a decoder D that accomplish the communication with a small error. Finally, we evaluate the precision of the randomized partial decoupling based on Theorem 8, by which we complete the proof of Theorem 3.

A. Definitions of States and Operations
We embed M c and M q F A into S c and S r , respectively. This is possible because of the conditions dim S c ≥ 2 c and dim S r ≥ 2 q+e in Theorem 3. Similarly, we embed R c and R q F B to larger systemsR c andR r , respectively, such that R c ∼ = S c andR r ∼ = S r . The embeddings are represented by linear isometries P Mc→Sc , P MqF A →Sr , P Rc→Rc and P RqF B →Rr . For the simplicity of notations, we denoteR cRr byR. In total, the systems M F A and RF B are embedded into systems S andR by linear isometries respectively. The adjoint map corresponding to these isometries are given by Define a "purified" source-resource state by Note that where C Mc the completely dephasing operation on M c with respect to the basis {|j } 2 c j=1 . We also introduce an "extended" one by We properly choose the embedding isometries so that for any j = 1, · · · , 2 c . Using the purified source-resource state and the extended one are simply related as 8 Let E S→A ρ and E Sr→A ρj be linear maps defined by the Choi-Jamiolkowski correspondence from ρ SA and ρ SrA j , respectively. That is, . Due to the condition that ρ S is the full-rank maximally mixed state, the two maps are completely positive and trace-preserving. From the decomposition (29), it follows that We denote by V Sr→AE0 ρj the Stinespring dilation of E Sr→A ρj for each j. Introducing a quantum system E c with a fixed orthonormal basis {|j } d Sc It is straightforward from (79) that with C Sc being the completely dephasing operation on S c with respect to the basis {|j } d Sc j=1 . Let W A→BE N be the Stinespring dilation of N A→B . Using the extended source-resource state (74), we define the following pure state, whereĒ ≡ E c EE 0 (see Figure 3): Defining the state it follows from (80) that We trace out system B to obtain the state ΨRĒ ρ :=

B. Construction of Encoding and Decoding Operations
We consider randomized partial decoupling of the "bipartite" state ΨRĒ ρ by a linear CP map and a permutation s on {1, · · · , d Sc }, define the partial decoupling error ∆ s,U by Here, we have defined where An evaluation of the partial decoupling error ∆ s,U will be given in the next subsection, based on the direct part of the randomized partial decoupling theorem (Theorem 8). We introduce the operator by which (85) is simply represented as Let |Ψ ρ,s M RĒM0 be a purification ofΨ RĒ ρ,s with M M 0 being a purifying system. Due to Uhlmann's theorem ( [42]; see also e.g. Chapter 9 in [1]) and (89), there exists a linear isometryṼ BF B →M M0 such that Note thatṼ BF B →M M0 depends on ρ, s and U in general. Defining and tracing out systemsĒ and M 0 in (90), we obtain To obtain an explicit form of the stateΨ M R ρ,s , we use (84) and (87) to have Combining this with (86), and by using (77), we obtaiñ A purification of this state is given by The first term in the L.H.S. of (94) is calculated as follows. Note that the state |Φ ext SR defined by (74) is the maximally entangled state on SR. Thus, from (88) and (78) (see Figure  5), it holds that 5. Transformation of the protocol for randomized partial decoupling of the state Ψρ into that for communication over the channel N is depicted. We first trace outĒ in Figure 3 and apply these transformations to obtain Figure 2. (i) is due to the fact that Φext is the maximally entangled state between S andR. (ii) is from Equality (78), and (iii) is because E follows from (73).
where we have defined a linear isometry We now construct an encoding operation E ρ,s,U by Substituting this to (94), we arrive at It remains to prove that the encoding operation defined by (98) satisfies the condition (23) (see Figure 5), i.e., From (67), (76) and (96), we have Thus, it holds that with C Sc being the completely dephasing operation on S c with respect to the basis {|j } J j=1 . Combining this with (81), we obtain which implies (102). Substituting this to (101), and by using the relation (73), we finally arrive at The deformations of expressions that we have done in this subsection are depicted in Figure 5. The figure explains the way how the communication protocol depicted in Figure 2 is obtained from the randomized partial decoupling protocol in Figure 3.

C. Evaluation of The Errors
To evaluate the partial decoupling error ∆ s,U defined by (85), we apply the direct part of the randomized partial decoupling theorem (Theorem 8) under the following correspondence: It follows that there exist a unitary UR = d Sc and a permutation s such that the partial decoupling error is bounded as Here, the exponents H I and H II are given by and τR F B is the Choi-Jamiolkowski state of the complementary channel of Tr F B •PR →RF B . Using (72)-(75), it holds that A simple calculation yields Using the duality of the conditional smooth entropies, we have Noting thatR c andR r are isomorphic to S c and S r , respectively, it follows from the definition (84) of Ψ ρ that which leads to Substituting all these equalities to (113) and (114), we obtain We now use the conditions (32) and (33) to have Substituting these inequalities to (112), we finally arrive at and complete the proof of Theorem 3.

VI. PROOF OF THE CONVERSE PART (THEOREM 4)
We prove the converse part of the capacity theorem (Theorem 4) based on the converse part of randomized partial decoupling (Theorem 9). The proof proceeds as follows: First, we construct a state Ψ E from the source state Φ M R 2 c ,2 q , the resource state Φ 2 e , the channel N and an encoding operation E. Second, we prove that the small error condition (22) implies that a partial-trace operation achieves partial decoupling of the state Ψ E . Applying the converse part for randomized partial decoupling, we obtain a set of inequalities represented in terms of the conditional entropies of the state Ψ E . Finally, we evaluate the entropies of the state to complete the proof of Theorem 4.

A. Application of the Converse Bound for Randomized Partial Decoupling
Suppose that a triplet (c, q, e) is achievable within the error δ for the channel N . By definition, there exist an encoding operation E M F A →A and a decoding operation D BF B →M that satisfy the conditions and where V E,j is a linear isometry for each j. We introduce notationsR ≡ RF B ,M ≡ M F A , and define a "purified" source-resource state ΦMR pur (see Figure 6) by , with M 0Ē being a purifying system. Due to the small error condition (128) and Uhlmann's theorem (see also Lemma 17 where |ω j are normalized pure states, and satisfies 6. The definition of the state Ψ E given by (132) and its transformation by partial decoupling are depicted. SD stands for the Stinespring dilation. Note that R ≡ RcRq, M ≡ McMq and that |Φpur Tracing out M M 0 , it follows that From (133), we have Thus, the condition (135) implies that the map id R ⊗ Tr F B : R → R achieves partial decoupling of the state ΨRĒ E (see Figure 6). We apply the converse bound for randomized partial decoupling (Theorem 9) under the following correspondence: Noting that the complementary channel of id R ⊗Tr F B is given by where C is the completely dephasing operation on E c . Substituting υ = 0 to (65) and (66), the smoothing parameters λ and λ are given by (38) and (39), respectively.

B. Evaluation of Entropies
By using the duality of the smooth conditional entropies [38] (see also Lemma 27 and 29 in [30]), the first terms in To calculate the second term in (143), note that (130), (131) and (132) imply Similarly, the second term in (144) is calculated to be Substituting all these equalities into (143) and (144), we arrive at Finally, we calculate the reduced state of Ψ E by using (130) and (132) to obtain ). (157) We relabel R c and R q F B by S c and S r , respectively, in which case we have d S ≤ 2 c+q+e . Noting that ρ RqF B j is the maximally mixed state, we complete the proof of Theorem 4.

VII. APPLICATION TO A MEMORYLESS CHANNEL
In this section, we consider an asymptotic limit of infinitely many uses of a memoryless channel. We consider a scenario of transmitting classical and quantum messages with the assistance of shared entanglement. First, we review a complete characterization of the achievable rate region, which was obtained by Hsieh and Wilde [14]. Second, we present another characterization that is obtained from the direct and converse bounds for the one-shot scenario presented in Section III (Theorem 3 and 4). A complete proof of the latter will be provided in Section VIII. Despite their seeming inconsistency, the two characterizations are equivalent. A detailed proof for this equivalence will be provided in Appendix D. In Section IX, we will apply the achievable rate region obtained in this section to special cases such as the classical capacity, the quantum capacity and the entanglement assisted classical/quantum capacities. We will see that the known results for each case are recovered from our result.
The definition of a code for the channel N in the scenario of many uses of the channel follows from Definition 1 by the correspondence (c, q, e) → (nC, nQ, nE) and N → N ⊗n , where n is the number of the uses of the channel. The threedimensional achievable rate region is defined as follows: Definition 10 A rate triplet (C, Q, E) is achievable if, for any δ > 0, there exists n 0 ∈ N such that for any n ≥ n 0 , there exists an (nC, nQ, nE, δ) code for (N A→B ) ⊗n . The closure of the set of all achievable rate triplets is called the achievable rate region, and is denoted by C(N ).

A. Hsieh-Wilde Theorem
Hsieh and wilde [14] analyzed the coding problem described above and obtained a complete characterization of the achievable rate region as follows: Definition 11 Let N A→B be a quantum channel. Consider finite dimensional quantum systems S c and S r , the former of which is equipped with a fixed orthonormal basis {|j } d Sc j=1 . We denote S c S r by S. Consider a state ρ SA in the form of where {p j , ρ j } d Sc j=1 is an ensemble of quantum states on S r A. Let Λ(N , ρ) ∈ R 3 be the set of all triplets (C, Q, E) that satisfy Here, the union is taken over all S c , S r and ρ that is decomposed as (158), for which we assume that ρ j is a pure state on S r A for each j. The set Λ p (N ⊗n ) is defined for any n ∈ N along the same line. The regularized version of Λ p is defined by Theorem 12 (Theorem 1 in [14]) For any quantum channel N A→B , it holds that C(N ) = Λ ∞ p (N ). Here, the overline represents the closure of the set.
It was proved in [14] that, in the special cases where one or two of C, Q or E vanish, the above theorem recovers the known results cases such as the Holevo-Schumache-Westmoreland theorem for the classical capacity [3], [4], the Lloyd-Shor-Devetak theorem for the quantum capacity [5]- [7], the entanglement-assisted classical [8]- [10] and quantum [11], [12] capacity theorems. Note that, due to the chain rule of the quantum entropies, the R.H.S. of (167) is equal to

B. Alternative Expression
We present an alternative expression for the achievable rate region that is obtained from the one-shot result presented in Section III.

Definition 13
Let N A→B be a quantum channel. Consider finite dimensional quantum systems S c and S r . Let S be S c S r , and {|j } d Sc j=1 be a fixed orthonormal basis of S c . Consider a state ρ SA in the form of where is an ensemble of quantum states on S r A. Let Θ(N , ρ) ⊂ R 3 be the set of all triplets (C, Q, E) that satisfy and define Here, the union is taken over all finite dimensional systems S c , S r and all states ρ that is decomposed as (165), for which we assume that ρ S is the full-rank maximally mixed state on S. The set Θ π (N ⊗n ) is defined for any n ∈ N along the same line. The regularized version of Θ π is defined by Theorem 14 For any quantum channel N A→B , it holds that C(N ) = Θ ∞ π (N ). Here, the overline represents the closure of the set.
A proof of Theorem 14 will be provided in Section VIII.
In Section IX, we will apply Theorem 14 to the special cases where one or two of C, Q or E vanish. We prove that the known results for those special cases such as the Holevo-Schumache-Westmoreland theorem for the classical capacity [3], [4], the Lloyd-Shor-Devetak theorem for the quantum capacity [5]- [7] and the entanglement-assisted capacity theorems [8]- [12] are recovered from Theorem 14, in the same way as those are obtained from Theorem 12.

C. Equivalence of The Two Expressions
The region defined by Definition 13 may look different from the one defined by Definition 11. But the two regions are actually the same, i.e., it holds that The point is that we take the union over S c , S r and ρ both in (170) and in (163). Taking the union eliminate the seeming mismatch between the conditions (159) and (166).
To be more precise, we introduce two alternative expressions for the achievable rate region. We define where Θ(N , ρ) ∈ R 3 is given by Definition 13. Here, the union is taken over all S and ρ that is in the form of (165). We do not require the condition that ρ S is the full-rank maximally mixed state on S. We also define where Λ(N , ρ) is defined in Definition 11. The union is taken over all S and ρ that is decomposed as (158), but we do not require that {ρ SrA j } j are pure states. Note that the only difference between Θ(N , ρ) and Λ(N , ρ) is in one of the inequality conditions for (C, Q, E). That is, we require Q+E ≤ H(S r |S c ) ρ for Θ(N , ρ) and C +2Q ≤ I(S : B) N (ρ) for Λ(N , ρ) (see Inequalities (166) and (159)). The two sets are regularized into In appenxid Appendix D, we prove the following proposition that implies (172): Proposition 15 For any quantum channel N , it holds that The first equality in (177) implies that the condition that the reduced state of the state ρ on system S c S r is the completely mixed state does not impose any restriction on the achievable rate region. I.e., the rate region does not change when we drop this condition. In Section IX, we will invoke this fact to show that the known capacity theorems in special cases are obtained from Theorem 14.

D. Remark On The Non-Optimality of Time-Sharing Strategies
Ref. [13] addressed the task of simultaneously transmitting classical and quantum messages through a noisy quantum channel without the assistance of shared entanglement, i.e. the case where E = 0 in Definition 10. A strategy for this task is to use nλ copies of the channel to transmit classical messages and the remaining n(1 − λ) copies for quantum messages (0 ≤ λ ≤ 1), which is often referred to as the time sharing strategy. It achieves the rate pair with C(N ) and Q(N ) being the classical and quantum capacities of the channel. Ref. [13] proved that the time sharing strategy is not optimal in general: For a certain channel N , there exists an achievable rate pair (C, Q) that cannot be represented as (178). In contrast, the time sharing strategy is optimal when the entanglement resource is freely available. This is because t > 0 qubits of quantum communication is converted to 2t bits of classical communication and vice versa, by means of superdense coding and quantum teleportation.
In the case where a limited amount of shared entanglement is available, it is not clear whether or not the time sharing strategy is optimal. This problem may be of independent interest because it would be closely related to the non-additivity of the classical capacity of a quantum channel under a limited amount of entanglement assistance [43]. We leave this problem as an open question.

VIII. PROOF OF THEOREM 14
In this section, we provide a proof for Theorem 14.

A. Proof of The Direct Part
We prove the direct part of Theorem 14, i.e.
Since C(N ) is a closed set, it suffices to prove that, for any n, a rate triplet (C, Q, E) is achievable if (C, Q, E) is an inner point of 1 n Θ π (N ⊗n ). We only consider the case where n = 1. It is straightforward to generalize the proof for n ≥ 2. Fix an arbitrary state ρ that satisfies the condition (165), in addition to the condition that ρ S is the full-rank maximally mixed state on S. Fix an arbitrary triplet (C, Q, E) that is an inner point of Θ (N , ρ). Then, there exists ν > 0 such that Fix arbitrary , δ > 0 and choose sufficiently large m. Due to the fully-quantum asymptotic equipartition property ( [44]: see also Lemma 22 in Appendix B), it holds that Combining this with (180)-(182), and noting that H(S r |S c ) ρ ≤ log dim S r and H(S c ) ρ = log d Sc , we obtain We choose m sufficiently large so that we have −mν ≤ log δ.
Denoting mC, mQ, mE by c, q, e, respectively, it follows that where we have used d m Sc /(d m Sc − 1) ≤ 2. Combining this with (186) and (188), it follows from Theorem 3 that there exists a (c, q, e, δ ) code for the channel N ⊗m , where δ := 2 √ 2δ + √ δ + 4 . For the latter case, we introduce a system S c such that d m Sc d S c ≥ max{2 c , 2}. Denoting S c S m c byŜ c and S c S m r = S c S m byŜ, we define the statê Due to the property of the smooth max entropy for product states (see Lemma 24 in Appendix B), we have

Thus, Inequalities (187) and (188) yield
where we have used dŜ c /(dŜ c − 1) ≤ 2 in the first line. These two inequalities together with (186) imply that there exists a (c, q, e, δ ) code for the channel N ⊗m . Since and δ can be arbitrarily small in both cases, we complete the proof of the direct part (179).

B. Proof of The Converse Part
The converse part of Theorem 14 is given by and is proved as follows. Suppose that a rate triplet (C, Q, E) is an inner point of C(N ). By definition, for any δ > 0 and sufficiently large n, there exist a (nC, nQ, nE, δ) code for the channel N ⊗n . Due to Theorem 4, there exists a quantum system S ≡ S c S r and a state in the form of satisfying d S ≤ 2 n(C+Q+E) and ρ S = π S , such that for any ι ∈ (0, 1], it holds that The entropies are for the state N ⊗n (ρ n ), and the smoothing parameters λ and λ are given by (38) and (39). Note that we have ρ S = N (ρ) S since N acts only on A. From the condition ρ S = π S , we have log dim S r = H(S r |S c ). Using the relation between the smooth max entropy and the von Neumann entropy (Lemma 26 in Appendix B), we also have where η is a function that satisfies lim x→0 η(x) = 0. Combining these inequalities, we arrive at By taking the limit of n → ∞ and ι, δ → 0, we obtain (C, Q, E) ∈ Θ ∞ π (N ) and complete the proof.

IX. COMPARISON TO PREVIOUS RESULTS IN SPECIAL CASES
In this section, we apply the results presented in Section III and Section VII to the special cases where one or two of C, Q and E is/are equal to zero and compare the results with those in the previous literature. That is, we investigate the classical capacity, the quantum capacity, the entanglementassisted classical capacity, the entanglement-assisted quantum capacity and the simultaneous capacity for classical and quantum information without the entanglement assistance, and discuss the relation with the existing results. We consider a quantum channel N with the input system A and the output system B.
For the one-shot scenario, we will invoke Theorem 3, Theorem 4, Corollary 6 and Corollary 7. The smoothing parameters (δ, δ ), λ(δ, ι) and λ (δ, ι) are set to be (δ, δ ) := δ 2 /16 − √ δ /4, (38) and (39), respectively. When we write sup, the supremum is taken over all finite dimensional quantum systems S ≡ S c S r and all classical-quantum states ρ ScSrA between S c and S r such that ρ ScSr is the full-rank maximally mixed state. For each of the special cases, we compare our results with several results in the previous literature. We, however, do not provide quantitative comparisons because all the results, including ours, involve complex optimizations, which makes quantitative analyses intractable. All the one-shot results presented or reviewed in this section shall coincide with the known coding theorems in the asymptotic limit of infinitely many uses of a memoryless channel. For the asymptotic scenario, we invoke Theorem 14 and prove that the known capacity theorems proved in the previous literature are recovered. As mentioned in the discussion below Proposition 15, and as proved in Appendix D-A in detail, the restriction in Definition 13 that the reduced state of ρ on S is the completely mixed state does not impose any restriction on the achievable rate region. Therefore, in the following subsections, we assume that this restriction is removed. It is convenient to note that the R.H.S.s of (166)-(168) are evaluated as (201)

A. Classical Capacity without Entanglement Assistance
The δ-classical capacity c(N , δ) of a channel N is defined as the supremum of c such that the triplet (c, q = 0, e = 0) is achievable within the error δ for the channel N . Lower and upper bounds on the δ-classical capacity are obtained from Theorem 3 and Theorem 4 by letting q = e = 0. We have where the map C is the completely dephasing operation on S with respect to a fixed orthonormal basis. Dupuis et al. [21] addressed the one-shot classical capacity of a channel N A→B with classical input and quantum output (the classical-quantum channel), namely a channel that satisfies N A→B = N A→B • C A with C being the completely dephasing map on A with respect to a fixed orthonormal basis. Based on the 'dequantizing theorem' (Theorem 3.1 in [21]), they obtained a lower bound that holds for any 0 < δ < 1: where ω is the Choi-Jamiołkowski state of N . Renes et al. [16] addressed the same task and obtained both lower and upper bounds (see Theorem 1 in [16]). The channel is modeled by an input alphabet X and the set of output quantum states {θ x } x∈X . They obtained lower and upper bounds that yield for any δ > 0, where the maximization is taken over all probability distribution P X on alphabet X and the entropies are for the state ϑ = x∈X p x |x x| X ⊗ θ B x . From both results, by taking supremum over all sets of output states of the form {N (ρ x )} x∈X , we obtain one-shot lower bounds that are similar to (203). The upper bounds in (202) and (204) are different in that (202) includes system S r . We shall see below that this seeming mismatch vanishes in the asymptotic scenario.
Wang et al. [17] addressed the same task in terms of the hypothesis testing approach and obtained lower and upper bounds that yield Here, D H is the hypothesis testing relative entropy and ϑ is defined in the same way as above. The hypothesis testing approach to this problem was further improved by Datta et al. [23] and Matthews et al. [24]. In particular, the result in Ref. [23] for the one-shot scenario recovers the strong converse rate [45], [46] when applied to the asymptotic scenario.
In the asymptotic limit, we obtain from (167) and (171)  where the supremum is taken over all finite-dimensional quantum systems S c , S r and all classical-quantum states ρ between S c and S r A. Due to the argument in [1] (see the argument after Theorem 24.2.2 therein), we may, without loss of generality, assume that S r is a trivial (one-dimensional) system. Hence, if C is below the R.H.S. of (206), both the conditions (166) and (168) are satisfied as well. Therefore the inequality (206) is achievable and we recover the Holevo-Schumacher-Westmoreland theorem [3], [4], which yields

B. Quantum Capacity without Entanglement Assistance
The δ-quantum capacity q(N , δ) of a channel N is defined as the supremum of c such that the triplet (c = 0, q, e = 0) is achievable within the error δ for the channel N . Lower and upper bounds on the δ-quantum capacity are obtained from Theorem 3 and Theorem 4 by letting c = e = 0, which yields Datta et al. [22] and Buscemi et al. [18] addressed this task based on the decoupling approach. Ref. [22] defines the one-shot δ-error quantum capacity q (1) δ (N ) in terms of the entanglement transmission fidelity (see Definition 11 therein) and obtained the following lower and upper bounds that holds for any 0 < δ ≤ 1: Here, the maximization is taken over all subspaces M ⊆ H A , |Φ M ∈ M ⊗ H Sr is the maximally entangled state with Schmidt rank dim M, and ε > 0 is chosen so that δ = 2(5(ε+ 4 √ ε − 4ε)) 1/2 + 2 √ ε. A comparison of this result with the one in [18] was discussed in Section 3.3 of [22]. Besides the values of the smoothing parameters, the difference between (208) and (209) is in whether the supremum over the states is limited to the maximally entangled state or not. Tomamichel et al. [26] addressed the same task from the hypothesis-testing approach.
In the asymptotic limit, we have where the supremum is taken over all finite-dimensional quantum systems S c , S r and a classical-quantum state ρ between S c and S r A. Since the von Neumann entropy conditioned by a classical system is equal to the entropy averaged over the values of the classical system, without loss of generality we can assume that S c is a trivial (one-dimensional) system. Thus, we recover the Lloyd-Shor-Devetak theorem [5]- [7], which yields One can also obtain (211) directly from (208).

C. Classical Capacity with Free Entanglement Assistance
The entanglement-assisted δ-classical capacity c e (N , δ) is defined as the supremum of c such that there exists e > 0 and the triplet (c, q = 0, e) is achievable within the error δ for the channel N . Lower and upper bounds on the entanglementassisted δ-classical capacity is obtained from Corollary 6 and Corollary 7 by letting q = 0, which yields Datta et al. [19] investigated this task and obtained lower and upper bounds based on the decoupling approach. Their proof is based on the fact that the classical capacity is twice as large as the quantum capacity if entanglement resource is freely available. They defined the one-shot entanglementassisted classical capacity c (1) ea, (N ) in terms of the worst-case error probability (see Definition 13 in [19]) and obtained lower and upper bounds thereof. They proved that for any 0 < ε < 1 and ε such that ε = 2 4 where φ SA is a purification of ∈ S(H A ). Datta et al. also addressed this task from the hypothesis testing approach and obtained a lower bound in terms of the average error probability [25]. Namely, they proved that for any 0 < δ < 1 and 0 < < δ/2, it holds that where ϑ AA is an arbitrary pure state that can be represented as a superposition of the maximally entangled state in orthogonal subspaces and κ is the probabilistic mixture of the maximally mixed state in those subspaces. Matthews et al. [24] obtained a converse bound from the hypothesis testing approach, which yields where φ is a purification of ρ. These two bounds are more general than ours, in that they do not assume that the resource state is the maximally entangled state. The hypothesis testing approach to the entanglement-assisted quantum capacity has further been developed based on the convex splitting and position based coding [27]- [29].
In the asymptotic limit, the conditions (166)-(169) are equivalent to where The two-dimensional achievable rate region in this scenario was originally proved by Shor [10], which is represented by the condition The regions (216) and (218) coincide with each other by taking the union over all classical-quantum states ρ and between S c and S r A. To show this, observe that the nontrivial vertices of the region (216) are given by In both cases, there exists a state such that C i ≤ I(S c S r : B) N ( ) and E ≥ H(S r |S c ) . For (C 2 , E 2 ), we may choose = |0 0| Sr ⊗ρ ScA . Conversely, let (C * , E * ) be the nontrivial vertex of the region defined by (218), i.e., let C * and E * be equal to the R.H.S.s of (218). It is straightforward to verify that (C * , E * ) satisfies the condition (216). Thus, by taking the union over all ρ in (216) and by regularizing it, we obtain the two-dimensional achievable rate region proved by Shor [10]. As proved in [10], the single-letter formula for the entanglement-assisted classical capacity by Bennet et al. [8], [9] is obtained from (218). An alternative proof is obtained from Theorem 14 as follows. One can see if a pair (C, E) satisfies the conditions (166)-(168) (which is equivalent to (216) when Q = 0), it holds that Conversely, if the above inequality holds, we have where the supremum is taken over all finite-dimensional quantum systems S c , S r and a classical-quantum state ρ between S c and S r A n . Since the von Neumann entropy conditioned by a classical system is equal to the entropy averaged over the values of the classical system, without loss of generality we can assume that S c is a trivial (one-dimensional) system. Thus, we have It is known that the mutual information of quantum channels is additive, and that it is sufficient to take the supremum over all pure states (see e.g. Section 12.4 in [1]). Thus, we arrive at the entanglement-assisted classical capacity theorem by Bennet et al. [8], [9], which yields

D. Quantum Capacity with Free Entanglement Assistance
The entanglement-assisted δ-quantum capacity q e (N , δ) is defined as the supremum of q such that there exists e > 0 and the triplet (c = 0, q, e) is achievable within the error δ for the channel N . Lower and upper bounds on the entanglementassisted δ-quantum capacity is obtained from Corollary 6 and Corollary 7 by letting c = 0, which yields A similar result was obtained in Datta et al. [19] based on the decoupling approach. They define the one-shot entanglement-assisted quantum capacity q (1) ea, in terms of the entanglement transmission fidelity (see Definition 6 and 7 therein) and obtained lower and upper bounds thereof. They proved that for any 0 < ε < 1 and ε such that ε = 2 2 √ 27ε + 27ε , it holds that where φ SA is a purification of ∈ S(H A ). They also proved that similar bounds hold for the entanglement-assisted quantum capacity defined in terms of the minimum output fidelity (see Theorem 11 in [19]).
In the asymptotic limit, one can see that if there exists (Q, E) satisfying the conditions (166) and (168), it holds that Conversely, if both of the above inequality hold, there exists ∆ ≥ 0 such that where the second line follows from the first condition in (228). For any m ∈ N, let S m be an 2 m∆ -dimensional quantum system, π m be the maximally mixed state thereon, and m ≡ π m ⊗ ρ. Denoting S m S m r ≡Ŝ r , we obtain Taking the limit of m to infinity, we observe that the pair (Q, E) belongs to the region defined by Definition 13 for C = 0. Thus, we obtain the two-dimensional rate region obtained by Devetak et al. [12] (Theorem 7.7 in [12]). I.e., the region obtained from the set of all pairs (Q, E) satisfying by taking the union over all pure states |φ AA and by regularizing it. Note that the region defined in [12] is convex because of regularization. Hence, the conditioning by a classical system S c does not change the rate region.

E. Simultaneous Capacity of Classical and Quantum Information without Entanglement Assistance
We consider the one-shot capacity region for simultaneously transmitting classical and quantum messages through the channel without entanglement assistance. The δ-simultaneous capacity region of a channel N is defined as the set of all pairs (c, q) ∈ R 2 ≥ such that the triplet (c, q, e = 0) is achievable within the error δ for the channel N . We denote the achievable rate region by Γ δ (N ), and assume that δ ∈ (0, 2]. For an arbitrary system S ≡ S c S r and an arbitrary state ρ SA that is diagonal in S c with respect to a fixed orthonormal basis, let Γ in δ,δ , (N , ρ) be the set of all pairs (c, q) ∈ R 2 ≥ that satisfy and let Γ out δ,ι (N , ρ) be the one that satisfy It follows from Theorem 3 and Theorem 4 that Here, the union over δ and the intersection over ι are taken in the intervals δ ∈ (0, δ 4 /16] and ι ∈ (0, 1], respectively. The union over ρ is taken over all classical-quantum states ρ between S c and S r A such that ρ S is the full-rank maximally mixed. The same communication task has been analyzed by Salek et al. [20] for the one-shot scenario based on the hypothesistesting approach. They first proved the inner and outer bounds on the two-dimensional capacity region for simultaneously transmitting public and private messages (see Theorem 2 and 3 therein). Then they apply the argument by Devetak et al. [7], which states that the private classical capacity of a quantum channel is equal to the quantum capacity. The outer bound is given as the union, over all states ρ, of the set of rate pairs (c, q) satisfying Here, I H and I max are the hypothesis testing mutual information and the max mutual information, and δ and δ are the error parameters for the classical message and the quantum message, respectively. The mutual informations are for the state of the form = x,y p(x, y)|x x| ⊗ |y y| ⊗ N A→B,E (ρ x,y ), where N A→E is the complementary channel of N A→B . The inner bound is given in a similar form, with the additional terms on the R.H.S.s of the two inequality that depends only on the smoothing parameters and the errors. One may think that our bound (236) is not consistent with those obtained (237). The condition (237) defines a rectangle, while (235) defines a trapezoid. Note, however, that the actual bounds are obtained by taking the union over states ρ in both cases, which resolves this seeming mismatch.
In the asymptotic limit, we obtain from the conditions (166)-(168) that which is equivalent to the conditions obtained by Hsieh and Wilde in [14] (see Theorem 5). Originally, Devetak and Shor [13] addressed this problem and obtained the achievable rate region which is characterized by the following conditions: As discussed in [14], by taking the union over all state ρ and by regularizing it, the above regions coincide with each other.

X. CONCLUSION
In this paper, we analyzed the task of simultaneously transmitting classical and quantum messages through a noisy quantum channel assisted by a limited amount of shared entanglement in the one-shot scenario. We have derived direct and converse bounds for the one-shot achievable rate region. To our knowledge, this is the first time that trade-off relations that apply simultaneously to all the three resources of classical communication, quantum communication, and shared entanglement has been obtained in the one-shot channel coding scenario. We then applied the one-shot result to the asymptotic scenario of infinitely many uses of a memoryless channel. We obtained a complete characterization of the triple resource trade-offs, which coincides with the prior result by Hsieh and Wilde [14]. Numerical calculations of the achievable rate region for specific qubit channels are given in [47]. In [48], we analyzed quantum state redistribution for a classical-quantum hybrid source in the one-shot scenario in terms of randomized partial decoupling. To investigate relations between these two tasks is left as future work.
To investigate quantum channel coding problems in the one-shot scenario, an approach from hypothesis testing (see e.g. [15], [17], [20], [23]- [26], [49]) is complementary to the one based on decoupling. The hypothesis-testing approach has an advantage over the decoupling approach in that it recovers the optimal asymptotic results not only in the i.i.d. setting but also in the non-i.i.d. (information spectrum) setting. This approach has been widely used for quantum channel coding problems e.g. to investigate the strong converse bounds and the second-order asymptotics. Along this line, the concepts of convex splitting [50] and position-based coding [28] have recently proved useful in various settings of one-shot quantum channel coding [20], [27]- [29], [51]- [58]. One future direction is to apply the hypothesis-testing approach to the problem addressed in this paper and obtain triple-resource trade-off bounds that are asymptotically tight in the non-i.i.d. setting.

APPENDIX A TECHNICAL LEMMAS
We introduce some technical lemmas that are used in the main text.
Lemma 16 Consider two states ρ and σ in the form of where {|k } k is an orthonormal basis of H X and H Y , {p(k |k)} K k =1 is a conditional probability distribution, and ρ k and σ kk are normalized states on A for each k and k . Suppose that we have for δ > 0. Then, it holds that Proof: Using the properties of the trace distance (see e.g. Section 9.1 in [1]), we have where Inequality (246) follows due to the triangle inequality.
Lemma 17 Consider a state ρ on XA and a pure state |φ on XY AB that take the forms of where {p k } k is a probability distribution, k is a state on A and |ϕ k is a pure state on AB for each k. Suppose that Then, there exists a purification |Ψ XY AB of ρ that takes the form of and satisfies |Ψ Ψ| − |φ φ| 1 ≤ 2 √ δ.
Proof: Any purification |Ψ of ρ is represented as where |ψ k is a purification of k for each k. It follows that where we have defined The maximization in the fourth line is taken over all purifications |ψ k AB of A k . We consider a state Due to (256), it holds that where F is the fidelity defined by F (σ, τ ) := √ σ √ τ 1 . The maximization is taken over all purifications of ρ XA and the first equality follows from Uhlmann's theorem [42]. By using the relation between the trace distance and the fidelity (see e.g. Section 9.2.3 in [2]), we obtain Combining these all together, we complete the proof.

Lemma 25
For any ρ AB ∈ S = (H AB ) and ≥ 0, it holds that Proof: Letρ AB ∈ B (ρ) be such that H max (A|B) ρ = H max (A|B)ρ. Due to Lemma 20, it holds that Using the triangle inequality for the trace distance, we have where the last inequality follows from the relation between the trace distance and the purified distance (7). Substituting this to (274), we complete the proof.
Lemma 26 For any 0 ≤ < 1 and any state ρ ∈ S = (H AB ), it holds that where η is a function that satisfies lim x→0 η(x) = 0 and is independent of the dimensions of the systems.
For any state ρ on system AB, it holds that Proof: The first inequality follows from the isometric invariance of the conditional quantum entropy and its monotonicity under the completely dephasing operation (see e.g. Corollary 11.9.4 in [1]). Note that the map E is represented as A Stinespring dilation of the map E is given by E = Tr X • V. It holds that where the first line follows from the isometric invariance of the conditional quantum entropy, the second line from the chain rule, the third line due to H(X|B) V(ρ) ≥ 0 and H(X |ABX) V(ρ) ≥ − log dim X, and the last line from V(ρ) AB = E(ρ).

Lemma 28
Let {p x , ρ x } x∈X be an ensemble of states on system A, and suppose that there exists a POVM {M x } x∈X such that Then, for the state it holds that where η is a function that satisfies lim →0 η( ) = 0 and is independent of the dimensions of the systems.
Proof: LetX be a quantum system with a fixed orthonormal basis {|x } x∈X , and define a CPTP map M : A →X by M(·) := Tr[M x (·)]|x x|X . It follows that where {p x |x } x ∈X is a conditional probability distribution defined by p x |x = Tr[M x ρ x ]. Thus, due to the monotonicity of the conditional quantum entropy, we have With a slight abuse of notation, let (X,X) be a pair of random variables that takes values in X × X according to a joint probability distribution P ≡ {p(x,x)}, where p(x,x) := p x p x |x . Due to the condition (286), it holds that Thus, due to Fano's inequality (see e.g. Theorem 2.10.1 in [61]), it follows that Combining this with (291), we complete the proof.

APPENDIX C METHOD OF TYPES AND TYPE SUBSPACES
In this section, we briefly review the definitions and properties of types and type subspaces. For the details, see e.g. Section 13.7 and 14.3 in [1]. The properties of type subspaces presented in this section will be used in Appendix D to prove the equivalence between the achievable rate region for the asymptotic limit given by Theorem 14 in the main text and the one obtained in Ref. [14] Let X be a finite alphabet and n ∈ N. A probability distribution {t(x)} x∈X is called a type of length n if nt(x) ∈ N for all x ∈ X . Let x n ≡ x 1 · · · x n be a sequence of variables of length n such that x i ∈ X for each i. The sequence x n is said to be of type t if for all x ∈ X , where N (x|x n ) is the number of the symbol x ∈ X that appears in the sequence x n . The type class corresponding to the type t of length n, which we denote by T n t , is the set of all sequences whose type is t. Let T(X , n) be the set of all types of length n on the alphabet X . It holds that The size of T(X , n) is bounded above by a polynomial function of n as |T(X , n)| ≤ (n + 1) |X | .
By definition, any two sequences in the same type class are transformed with each other by permuting the elements. That is, for any t ∈ T(X , n) and x n , x n ∈ T n t , there exists a permutation s such that x i = x s(i) for any 1 ≤ i ≤ n.
Let H be a Hilbert space with a fixed orthonormal basis {|x } x∈X . For any n ∈ N, the type subspace corresponding to the type t of length n is defined by where |x n = |x 1 · · · |x n , and the type projector is defined by It follows from (295) that Π n t Π n t = 0 (t = t ), I = t∈T(X ,n) Consider a state ρ ∈ S = (H) and suppose that the eigen decomposition of ρ is given by ρ = x∈X p x |x x|. By definition, it holds that where The projectors {Π n t } t∈T(X ,n) are called the type projectors corresponding to the state ρ ⊗n .
We define the set of conditional type projectors {Π ν|j } ν on H ⊗n by By definition, it holds that (307) Note that the number of conditional type projectors for the sequences j is equal to that for j if the two sequences belong to the same type class.

APPENDIX D PROOF OF PROPOSITION 15
In this appendix, we prove Proposition 15 in Section VII. For the simplicity of notations, we denote d Sc by J. .
By using (310) and (312), the state (313) is rewritten into For the later convenience, we introduce a map E j : S m r → Y S m r by for each j, which leads to A useful relation which follows from the properties of the conditional type projectors is that for any j and j in the same type set, there exits a permutation s such that and (318) . The points of intersection of the three lines and the axes are denoted by u 1 , u 2 , u 3 , t 2 , t 2 and t 3 . At C = 0, both u 1 and u 2 are on the right than u 3 . Thus u 3 and t 3 are vertices of the region that yield P + 1 and P 2 , respectively. The two points u 1 and u 2 approaches to the origin as C increases. At C = I(Sc : B), the point u 2 coincides u 3 , in which case the points u 2 = u 3 and t 2 = t 3 are the vertices P + 3 and P 4 . In C = H(Sc) − H(S|B), the point u 2 meets the origin, and yields P 5 . Finally, the point u 1 coincides the origin at C = I(S : B), in which case the point of u 2 = t 2 yields P 6 .  161), respectively, and the crossing points are denoted by u 1 , u 2 , u 3 , t 2 and t 3 . At C = 0, the point u 2 is below u 3 , in which case u 3 and t 3 are vertices of the region that yield P − 1 and P 2 , respectively. The point u 2 approaches u 3 as C increases, and coincides it at C = I(Sc : B). In this case, the points u 2 = u 3 and t 2 = t 3 correspond to the vertices P − 3 and P 4 . At C = I(S : B), the point u 1 coincides the origin, where the point u 2 = t 2 yields P 6 .
3) Proof of convΘ ∞ π (N ) ⊆ Θ ∞ π (N ): We prove that the convex hull of Θ ∞ π (N ) is a subset of Θ ∞ π (N ). Fix arbitrary λ 1 , λ 2 > 0 such that λ 1 + λ 2 = 1, and suppose that (C (i) , Q (i) , E (i) ) ∈ Θ ∞ π (N ) for i = 1, 2. By definition, for any sufficiently large n, there exist quantum systems S (i) ≡ S (i) c S (i) r and a quantum state ρ i on S (i) C nλi such that ρ S (i) i is the maximally mixed state, and it holds that (353) It is straightforward to verify that the state is diagonal on S c with respect to a fixed basis, and thatρ S is the full-rank maximally mixed state on S. Due to the additivity of the conditional quantum entropy, we have and so forth. Define (C n , Q n , E n ) ≡ i=1,2 n i (C (i) , Q (i) , E (i) ).
1) Vertices of Λ(N , ρ): Consider the following points in R 3 , where all entropies and mutual informations are for the state N (ρ): The vertices of Λ(N , ρ) are P 0 , P + 1 , P 2 , P + 3 , P 4 , P 5 , P 6 in the case of −H(S r |BS c ) N (ρ) > 0 and P − 1 , P 2 , P − 3 , P 4 , P 6 when −H(S r |BS c ) N (ρ) < 0 (Figures 7 and 8: see also Section VI in [14]). Note that, by the chain rule of the mutual information, it holds that By a simple calculation, it is straightforward to verify that all of the above points except P 2 belong to Θ(N , ρ), and consequently to Θ ∞ (N ).
2) Proof of P 2 ∈ Θ ∞ (N ): Consider the point P 2 represented by the coordinate (C 2 , Q 2 , E 2 ), where Fix arbitrary , δ > 0 and choose sufficiently large n. Due to the data compression theorem for classical information source with quantum side information (Theorem 1 in [62]), there exist a countable set Y n,δ satisfying |Y n,δ | ≤ 2 n(H(Sc|B) N (ρ) +δ) , We introduce a |Y n,δ |-dimensional quantum system Y and define a state We denote the system S n c S n r byŜ r . It is straightforward to verify that ρ Y S n A n n = (ρ SA ) ⊗n .
where η is a function that satisfies lim →0 η( ) = 0 and is independent of the dimensions of the systems. From (369), Combining these relations with (365)-(367), we arrive at Since these relations hold for any small , δ > 0 and sufficiently large n, we obtain P 2 ∈ Θ ∞ (N ).