One-Shot Hybrid State Redistribution

We consider state redistribution of a"hybrid"information source that has both classical and quantum components. The sender transmits classical and quantum information at the same time to the receiver, in the presence of classical and quantum side information both at the sender and at the decoder. The available resources are shared entanglement, and noiseless classical and quantum communication channels. We derive one-shot direct and converse bounds for these three resources, represented in terms of the smooth conditional entropies of the source state. Various coding theorems for two-party source coding problems are systematically obtained by reduction from our results, including the ones that have not been addressed in previous literature.


Introduction
Quantum state redistribution is a task in which the sender aims at transmitting quantum states to the receiver, in the presence of quantum side information both at the sender and at the receiver. The costs of quantum communication and entanglement required for state redistribution have been analyzed in [35,13,36] for the asymptotic scenario of infinitely many copies and vanishingly small error, and in [7,10,2] for the one-shot scenario. Various coding theorems for Eyuri Wakakuwa: e.wakakuwa@gmail.com Yoshifumi Nakata: nakata@qi.t.u-tokyo.ac.jp Min-Hsiu Hsieh: min-hsiu.hsieh@foxconn.com two-party quantum source coding problems are obtained by reduction from these results as special cases, such as the Schumacher compression [24], quantum state merging [17] and the fullyquantum Slepian-Wolf [1,9]. However, some of the well-known coding theorems cannot be obtained from those results, such as the (fullyclassical) Slepian-Wolf (see e.g. [8]) and the classical data compression with quantum side information [11]. This is because the results in [35,13,36,7] only cover the fully quantum scenario, in which the information to be transmitted and the available resources are both quantum.
In this paper, we generalize the one-shot state redistribution theorem in [7] to a "hybrid" situation. That is, we consider the task of state redistribution in which the information to be transmitted and the side information at the parties have both classical and quantum components. Not only quantum communication and shared entanglement, but also classical communication is available as a resource. Our goal is to derive trade-off relations among the costs of the three resources required for achieving the task within a small error. The main result is that we provide the direct and the converse bounds for the rate triplet to be achievable, in terms of the smooth conditional entropies of the source state and the error tolerance. For most of the special cases that have been analyzed in the previous literatures, the two bounds match in the asymptotic limit of infinitely many copies and vanishingly small error, providing the full characterization of the achievable rate region. Our result can be viewed as a one-shot generalization of the classically- Figure 1: The task of state redistribution for the classical-quantum hybrid source is depicted. The black dots and the circles represent classical and quantum parts of the information source, respectively. The wavy line represents the entanglement resource.
Coding theorems for most of the redistributiontype protocols, not only for quantum or classical information source but also for hybrid one, in one-shot scenario are systematically obtained from our result by reduction. In this sense, our result completes the one-shot capacity theorems of the redistribution-type protocols in a standard setting. As examples, we show that the coding theorems for the fully quantum state redistribution, the fully quantum Slepian-Wolf, quantum state splitting, quantum state merging, classical data compression with quantum side information, quantum data compression with classical side information and the fully classical Slepian-Wolf and quantum state redistribution with classical side information only at the decoder [3] can be recovered. The last one would further lead to the family of quantum protocols in the presence of classical side information only at the decoder, along the same line as the one without classical side information [1,12]. In addition, our result also covers some redistribution-type protocols that have not been addressed in the previous literatures.
We note that the cost of resources in the hybrid redistribution-type protocols cannot be fully analyzed by simply plugging the hybrid source and the hybrid channel into the fully quantum setting. This is because interconversion of classical and quantum communication channels requires the use of entanglement resource, which is not allowed e.g. in the fully classical scenario.
This paper is organized as follows. In Section 2, we introduce notations and definitions that will be used throughout this paper. In Section 3, we provide the formulation of the problem and present the main results. The results are applied in Section 4 to special cases, and compared with the results in the previous literatures. The proofs of the direct part and the converse part are provided in Section 5 and 6, respectively. Conclusions are given in Section 7. The properties of the smooth entropies used in the proofs are summarized in Appendix A.

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

Notations
We denote the set of linear operators on a Hilbert space H by L(H). For normalized 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 abbreviate |ψ A ⊗|φ B as |ψ A |φ B . We denote |ψ ψ| simply by ψ.
For ρ AB ∈ L(H AB ), ρ A represents Tr B [ρ AB ]. 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 E is a supermap from L(H A ) to L(H B ), we denote it by E A→B . When A = B, we use E A for short. We also denote (E A→B ⊗ id C )(ρ AC ) by E 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 especially denote it by its calligraphic font such 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 ) † .
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 maximally mixed state on A is defined by For any linear CP map T A→B , there exists a finite dimensional quantum system E and a linear operator W A→BE The operator W T is called a Stinespring dilation of T A→B [25], and the linear CP map defined by Tr B [W T (·)W † T ] is called a complementary map of T A→B . With a slight abuse of notation, we denote the complementary map by T A→E .

Norms and Distances
For a linear operator X, the trace norm is defined as For subnormalized states ρ, σ ∈ S ≤ (H), the trace distance is defined by ρ−σ 1 . The generalized fidelity and the purified distance are defined bȳ and respectively (see Lemma 3 in [29]). The trace distance and the purified distance are related as for any ρ, σ ∈ S ≤ (H). The epsilon ball of a subnormalized state ρ ∈ S ≤ (H) is defined by

One-Shot Entropies
For any subnormalized state ρ ∈ S ≤ (H AB ) and normalized state ς ∈ S = (H B ), define and The conditional min-and max-entropies (see e.g. [26]) are defined by (11) and the smoothed versions thereof are given by for ≥ 0. In the case where B is a trivial (onedimensional) system, we simply denote them as H min (A) ρ and H max (A) ρ , respectively. We define We will refer to (15) as the smooth conditional min mutual information. For τ ∈ S(H A ), we also use the "max entropy" in the version of [23] (see Section 3.1.1 therein). Taking the smoothing into account, it is defined by where the infimum is taken over all projections Π such that Tr[Πτ ] ≥ 1 − . The von Neumann entropies and the quantum mutual information are defined by The properties of the smooth conditional entropies used in this paper are summarized in Appendix A. Figure 2: The task of state redistribution for the classical-quantum hybrid source is depicted in the diagram. The black lines and the dashed lines represent classical and quantum systems, respectively.

Formulation and Results
Consider a classical-quantum source state in the form of Here, {p xyz } x,y,z is a probability distribution, |ψ xyz are pure states, and {|x } x , {|y } y , {|z } z , {|xyz } x,y,z are orthonormal bases. The systems X , Y and Z are assumed to be isomorphic to X, Y and Z, respectively. For the simplicity of notations, we denote AX, BY , CZ, X Y Z and RX Y Z byÂ,B,Ĉ, T andR, respectively. Accordingly, we also denote the source state by ΨÂBĈR s . We consider a task in which the sender trans-mitsĈ to the receiver (see Figure 1 and 2). The sender and the receiver have access to systemŝ A andB, respectively, as side information. The systemR is the reference system that is inaccessible to the sender and the receiver. The available resources for the task are the one-way noiseless classical and quantum channels from the sender to the receiver, and an entangled state shared in advance between the sender and the receiver. We describe the communication resources by a quantum system Q with dimension 2 q and a "classical" system M with dimension 2 c . The entanglement resources shared between the sender and the receiver, before and after the protocol, are given by the maximally entangled states Φ E A E B 2 e+e 0 and with Schmidt rank 2 e+e 0 and 2 e 0 , respectively.
Note that, since M is a classical message, the encoding CPTP map E must be such that for any input state τ , the output state is diagonal in M with respect to a fixed orthonormal basis. Note also that we implicitly assume that c, q, e 0 ≥ 0, while the net entanglement cost e can be negative. Our goal is to obtain necessary and sufficient conditions for a tuple (c, q, e, e 0 ) to be achievable within the error δ for a given source state Ψ s . The direct and converse bounds are given by the following theorems:

Theorem 2 (Direct part.)
A tuple (c, q, e, e 0 ) is achievable within an error and H (ι,κ) * is defined by (14). In the case where d C = 1, a tuple (c, 0, 0, 0) is achievable for Ψ s within the error δ if it holds that  (33) and The supremum in (34) is taken over all CPTP maps F :ÂĈ → AG A M A such that F(τ ) is diagonal in M A with a fixed orthonormal basis for any τ ∈ S(HÂĈ), and where we informally denoted ψ AR xyz ⊗ |xyz xyz| T by ψ AR xyz .
The proofs of Theorem 2 and Theorem 3 will be provided in Section 5 and Section 6, respectively. We also consider an asymptotic scenario of infinitely many copies and vanishingly small error. A rate triplet (c, q, e) is said to be asymptotically achievable if, for any δ > 0 and sufficiently large n ∈ N, there exists e 0 ≥ 0 such that the tuple (nc, nq, ne, ne 0 ) is achievable within the error δ for the one-shot redistribution of the state Ψ ⊗n s . The achievable rate region is defined as the closure of the set of achievable rate triplets. The following theorem provides a characterization of the achievable rate region:

Theorem 4 (Asymptotic limit.)
In the asymptotic limit of infinitely many copies and vanishingly small error, the inner and outer bounds for the achievable rate region are given by and respectively. Here, where ∆ ( ,δ) is defined in Theorem 3.
Theorem 4 immediately follows from the oneshot direct and converse bounds (Theorem 2 and Theorem 3). This is due to the fully-quantum asymptotic equipartition property [28], which implies that the smooth conditional entropies are equal to the von Neumann conditional entropy in the asymptotic limit of infinitely many copies.
That is, for any ρ ∈ S = (H P Q ) and > 0, it holds that A simple calculation using this relation and the chain rule of the conditional entropy implies that the R.H.S.s of (22)- (25) and (29)-(31) coincide with those of (36)-(39) and (40)-(42), respectively, in the asymptotic limit of infinitely many copies.
Due to the existence of the term∆ in Inequality (40), the direct and converse bounds in Theorem 4 do not match in general. In many cases, however, it holds that∆ = 0 and thus the two bounds matches. This is due to the following lemma about the property of ∆ ( ,δ) :

Lemma 5
The quantity ∆ ( ,δ) defined in Theorem 3 is nonnegative, and is equal to zero if there is no classical side information at the decoder (i.e. dimY = dimY = 1) or if there is neither quantum message nor quantum side information at the encoder (i.e. dimA = dimC = 1). The quantity∆ satisfies the same property due to the definition (45).
A proof of Lemma 5 will be provided in Section 6.4. To clarify the general condition under which ∆ = 0 is left as an open problem.

Remark.
The results presented in this section are applicable to the case where the sender and the receiver can make use of the resource of classical shared randomness. To this end, it is only necessary to incorporate the classical shared randomness as a part of classical side information X and Y .

Reduction to Special Cases
In this section, we apply the results presented in Section 3 to special cases of source coding (see Figure 3 in the next page). In principle, the results cover all special cases where some of the components A, B, C, X, Y or Z are assumed to be one-dimensional, and where c, q or e is assumed to be zero.
Among them, we particularly consider the cases with no classical component in the source state and with no side information at the encoder, which have been analyzed in previous literatures. We also consider quantum state redistribution with classical side information at the decoder, which has not been addressed before. We investigate both the one-shot and the asymptotic scenarios. The one-shot direct and converse bounds are obtained from Theorem 2 and Theorem 3, respectively, and the asymptotic rate region is obtained from Theorem 4. The analysis presented below shows that, for the tasks that have been analyzed in previous literatures, the bounds obtained from our results coincide with the ones obtained in the literatures. It should be noted, however, that the coincidence in the oneshot scenario is only up to changes of the types of entropies and the values of the smoothing parameters. All entropies are for the source state Ψ s . We will use Lemma 21 in Appendix A for the calculation of entropies.

No Classical Component in The Source State
First, we consider the case where there is no classical component in the source state. It is described by setting X = Y = Z = ∅. By imposing several additional assumptions, the scenario reduces to different protocols.

Fully Quantum State Redistribution
Our hybrid scenario of state redistribution reduces to the fully quantum scenario, by additionally assuming that c = 0. The one-shot direct part is given by  "SI" and "SI-D" stand for "side information" and "side information at the decoder", respectively. See Table 1 below for the notations.
information source available resources side information at the encoder side information at the decoder information to be transmitted communication shared correlation Table 1 An example of the tuple satisfying the above conditions is The achievability of q and e given by (51) and (52) coincides with the result of [7] (see also [2]). The one-shot converse bound is represented as The condition (54) in the above coincides with Inequality (104) in [7]. The rate region for the asymptotic scenario is obtained from Theorem 4, which yields A simple calculation implies that the above rate region is equal to the one obtained in Ref. [13,35].

Fully Quantum Slepian-Wolf
The fully-quantum Slepian-Wolf protocol is obtained by setting A = ∅, c = 0. The one-shot direct part obtained from Theorem 2 reads An example of the rate triplet (q, e, e 0 ) satisfying the above inequalities is The result is equivalent to the one given by [9] (see Theorem 8 therein), with respect to q and e. Note, however, that our achievability bound requires the use of initial entanglement resource of e + e 0 ebits, whereas the one by [9] does not. The one-shot converse bound is obtained from Theorem 3, which yields From Theorem 4, the two-dimensional achievable rate region for the asymptotic scenario is given by which coincides with the result obtained in [1]. It should be noted that various coding theorems for quantum protocols are obtained from that for the fully quantum Slepian-Wolf protocol, which is referred to as the family of quantum protocols [1,12].

Quantum State Splitting
The task in which B = ∅, c = 0 is called quantum state splitting. The one-shot direct part is represented as Note that if a triplet (q, e, e 0 ) is achievable, then (q, e + e 0 , 0) is also achievable. Thus, an example of an achievable rate pair (q, e) is where we have denoted the R.H.S. of (70) by δe 0 . This coincides with Lemma 3.5 in [6], up to an extra term δe 0 . The one-shot converse bound is given by The rate region for the asymptotic scenario yields An example of a rate pair satisfying this condition is This result coincides with Equality (6.1) in [1], under the correspondence |Ψ s

Quantum State Merging
Quantum state merging is a task in which A = ∅, q = 0. The one-shot direct part is given by The achievability of the entanglement cost (80) is equal to the one given by [15] (see Theorem 5.2 therein). The one-shot converse bound is obtained from Theorem 3, which yields The rate region for the asymptotic setting is obtained from Theorem 4 as

e ≥ H(C|B). (85)
This rate region is equivalent to the results in [16,17]. Note, however, that the protocols in [16,17] are more efficient than ours, in that the catalytic use of entanglement resource is not required.

No Side Information At The Encoder
Next, we consider scenarios in which there is no classical or quantum side information at the encoder. This corresponds to the case where A = X = ∅. We consider three scenarios by imposing several additional assumptions.

Classical Data Compression with Quantum Side Information at The Decoder
The task of classical data compression with quantum side information was analyzed in [11]. This is obtained by additionally setting Y = C = ∅, q = e = e 0 = 0. The one-shot direct and converse bounds are given by respectively. This result is equivalent to the one obtained in [21] (see also [27]). In the asymptotic limit, the achievable rate region is given by c ≥ H(Z|B), which coincides with the result by [11].

Quantum Data Compression with Classical Side Information at The Decoder
The task of quantum data compression with classical side information at the decoder was analyzed in [4]. This is obtained by imposing additional assumptions Z = B = ∅, c = 0. In the entanglement "unconsumed" scenario (e = 0), the direct bounds for the one-shot case is given by Note that the entanglement is used only catalytically. Thus, in the asymptotic regime, the achievable quantum communication rate in the entanglement unassisted scenario (e = e 0 = 0) is obtained due to the cancellation lemma (Lemma 4.6 in [14]), which reads In the case where the unlimited amount of entanglement is available, the converse bounds on the quantum communication cost in the one-shot and the asymptotic scenarios read The asymptotic result (90) coincides with Theorem 7 in [4], and (92) is similar to Theorem 5 therein.
It is left open, however, whether the quantity∆ is equal to the function I (n,δ) that appears in Theorem 5 of [4] (see Definition 2 in the literature).

Fully Classical Slepian-Wolf
In the fully classical scenario, the Slepian-Wolf problem is given by B = C = ∅ in addition to X = A = ∅, and q = e = e 0 = 0. The one-shot achievability is given by and the one-shot converse bound reads which are equivalent to the result obtained in [22]. It is easy to show that the well-known achievable rate region c ≥ H(Z|Y ) follows from Theorem 4.

Quantum State Redistribution with Classical Side Information at The Decoder
We consider a scenario in which X = Z = ∅ and c = 0. This scenario can be regarded as a generalization of the fully quantum state redistribution, that incorporates classical side information at the decoder [3]. The one-shot direct bound is represented by wherẽ H (3 /2, /2) I The converse bound is also obtained from Theorem 3. The inner and outer bounds for the achievable rate region in the asymptotic limit is given by and respectively, wherẽ We may also obtain its descendants by further assuming A = 0 or B = 0, which are generalizations of the fully quantum Slepian-Wolf and quantum state splitting. It is expected that various quantum communication protocols with classical side information only at the decoder are obtained by reduction from the above result, similarly to the family of quantum protocols [1,12]. We, however, leave this problem as a future work.

Proof of The Direct Part (Theorem 2)
We prove Theorem 2 based on the following propositions: In the case where d C = 1 and q = e = e 0 = 0, the classical communication rate c is achievable within the error δ if it holds that whereH ( ) and In the case where d C = 1, a tuple (c, 0, 0, 0) is achievable for Ψ s within the error δ if it holds that Proofs of Proposition 6 and Proposition 7 will be given in the following subsections. In Section 5.1, we prove the partial bi-decoupling theorem, which is a generalization of the bi-decoupling theorem [36,7]. Based on this result, we prove Proposition 6 in Section 5.2. We adopt the idea that a protocol for state redistribution can be constructed from sequentially combining protocols for the (fully quantum) reverse Shannon and the (fully quantum) Slepian-Wolf. In Section 5.3, we extend the rate region in Proposition 6 by incorporating teleportation and dense coding, thereby proving Proposition 7. Finally, we prove Theorem 2 from Proposition 7 in Section 5.4.

Partial Bi-Decoupling
The idea of the bi-decoupling theorem was first introduced in [36], and was improved in [7] to fit more into the framework of the one-shot information theory. The approach in [7] is based on the decoupling theorem in [15]. In this subsection, we generalize those results by using the direct part of randomized partial decoupling [33] to incorporate the hybrid communication scenario.

Direct Part of Partial Decoupling
We first present the direct part of randomized partial decoupling (Theorem 3 in [33]). Let ΨĈŜ be a subnormalized state in the form of Here, Z and Z are J-dimensional quantum system with a fixed orthonormal basis {|j } J j=1 ,Ĉ ≡ ZC,Ŝ ≡ Z S and ψ jk ∈ L(H C ⊗ H S ) for each j and k. Note that the positive-semidefiniteness of ΨĈŜ implies ψ jj ≥ 0 for all j and the subnormalization condition implies J j=1 Tr[ψ jj ] ≤ 1. Consider a random unitary U onĈ in the form of where U j ∼ H j for each j, and H j is the Haar measure on the unitary group on H C . The averaged state obtained after the action of the randomĈ unitary U is given by where p j := Tr[ψ jj ] and ψ j := p −1 j ψ jj . Consider also the permutation group P on [1, · · · , J], and define a unitary G σ for any σ ∈ P by We assume that the permutation σ is chosen at random according to the uniform distribution on P.
Suppose that the state ΨĈŜ is transformed by unitaries U and G σ , and then is subject to the action of a quantum channel (linear CP map) TĈ →E (see Figure 4). The final state is represented as We consider how close the final state is, on average over all U , to the averaged final state TĈ →E • G Z σ (ΨĈŜ av ), for typical choices of the permutation σ. The following theorem is the direct part of the randomized partial decoupling theorem, which provides an upper bound on the average distance between TĈ →E • G Z σ • UĈ(ΨĈŜ) and TĈ →E • G Z σ (ΨĈŜ av ). Although the original version in [33] is applicable to any J ≥ 1, in this paper we assume that J ≥ 2.

Lemma 8 (Corollary of Theorem 3 in [33])
Consider a subnormalized state ΨĈŜ ∈ S ≤ (HĈŜ) that is decomposed as (121). Let TĈ →E be a linear trace non-increasing CP map with the complementary channel TĈ →F . Let U and G σ be random unitaries given by (122) and (125), Figure 5: The situation of partial decoupling under partial trace is depicted.
respectively, and fix arbitrary , µ ≥ 0. It holds that Here, C is the completely dephasing operation on Z with respect to the basis {|j } J j=1 , and τ is the Choi-Jamiolkowski state of TĈ →F defined by τĈ F := TĈ →F (ΦĈĈ ). The state ΦĈĈ is the maximally entangled state in the form of

Partial Decoupling under Partial Trace
We apply Lemma 8 to a particular case where the channel T is the partial trace (see Figure 5).

Lemma 9
Consider the same setting as in Lemma 8, and suppose that Z = Z L Z R , C = C L C R . We assume that Z L and Z R are equipped with fixed orthonormal bases and {|z R } J R z R =1 , respectively, thus J = J L J R and the orthonormal basis of Z is given by log then it holds that The same statement also holds in the case of d C = 1, in which case the condition (132) can be removed.
Proof: We apply Lemma 8 by the correspon- Here, τ is the Choi-Jamiolkowski state of the complementary channel of TĈ →Z L C L , and is given by Using the additivity of the max conditional entropy (Lemma 15 in Appendix A), the entropies are calculated to be Thus, Inequalities (134) and (135) are equivalent to and that (d Z − 1)/d Z ≥ 1/2, the above two inequalities follow from (131) and (132), respectively.
Thus, the proof in the case of d C ≥ 2 is done.
The proof for the case of d C = 1 proceeds along the same line.

Partial Bi-Decoupling Theorem
Based on Lemma 9, we introduce a generalization of the "bi-decoupling theorem" [36,7] that played a crucial role in the proof of the direct part of one-shot fully quantum state redistribution. We consider the case where systems C and S are composed of three subsystems. The following lemma provides a sufficient condition under which a single pair of σ and U simultaneously achieves partial decoupling of a state, from the viewpoint of two different choices of subsystems (see Figure 6 in the next page).
Lemma 10 (Partial bi-decoupling.) Consider the same setting as in Lemma 8, assume log log there exist σ and U such that The same statement also holds if d C = 1, in which case the conditions (142) and (144) can be removed.
Proof: Suppose that d C ≥ 2 and the inequalities (141)-(144) are satisfied. We apply Lemma 9 under the correspondence C R = C α C 3 , S = S α S 3 and C L = Cᾱ, where α = 1, 2 andᾱ = 2, 1 for each. It follows that Markov's inequality implies that there exist σ and U that satisfy both (145) and (146), which completes the proof in the case of d C ≥ 2. The proof in the case of d C = 1 proceeds along the same line.

Proof of Proposition 6
To prove Proposition 6, we follow the lines of the proof of the direct part of the fully quantum state redistribution protocol in [36]. The key idea is that a protocol for state redistribution can be constructed from sequentially combining a protocol for the fully quantum reverse Shannon and that for the fully quantum Slepian-Wolf. We generalize this idea to the "hybrid" scenario (see Figure 9 in page 34). We only consider the case where d C ≥ 2. The proof for the case of d C = 1 is obtained along the same line.

Application of The Partial Bi-Decoupling Theorem
Consider the "purified" source state where we denoted X Y Z simply by T . Let C be isomorphic to C 1 C 2 C 3 and Z to Z L Z R . Fix an arbitrary > 0. We apply Lemma 10 under the following correspondense: Note thatR = RX Y Z . It follows that if the dimensions of C 1 and C 2 are sufficiently small (see the next subsection for the details), there exist σ Figure 6: The situation of partial bi-decoupling is depicted. As represented by the rotary, we consider two cases where S 1 C 2 or S 2 C 1 are traced out. and U that satisfy Let |Ψ σ,1 be a purification of Tr Z R C 1 C 3 • G Z σ (ΨĈÂR av ) with D B being the purifying system. Due to Uhlmann's theorem ( [30]; see also e.g. Chapter 9 in [34]), there exist linear isometries We particularly choose C 1 , C 2 , C 3 and Z R so that they satisfy the isomorphism In addition, we introduce systems C , Z , A 1 and B 2 such that We consider the purifying systems to be D A ≡ Z RĈ Â A 1 and D B ≡ Z RĈ B B 2 , whereĈ = C Z .

Explicit Forms of The Purifications
To obtain explicit forms of the purifications Ψ σ,1 and Ψ σ,2 , we define a state Ψ σ by From the definition (20) of the source state Ψ s , (148) of the purified source state Ψ and (157) of the state Ψ σ , it is straightforward to verify that the states are related simply by and Here, Let P Z →Z Z be a linear isometry defined by and C be the completely dephasing operation on T with respect to the basis {|xyz } x,y,z . The state Ψ σ is simply represented as It is convenient to note that Due to (148) and (124), the averaged state in (150) is calculated to be where p z = x,y p xyz . It follows that Thus, a purification Ψ σ,1 of this state is given by where φ 1 is the maximally entangled state of Schmidt rank d C 1 . In the same way, the purification Ψ σ,2 is given by with φ 2 being the maximally entangled state of Schmidt rank d C 2 . Substituting these to (153) and (154), we arrive at Inequality (169) implies that the operation (G Z σ • UĈ) † • V is a reverse Shannon protocol for the state ΨĈÂ (BR) , up to the action of a linear isometry G Z σ • P Z →Z Z by which Ψ σ is obtained from Ψ as (158). Similarly, Inequality (170) implies that the operation W • G Z σ • UĈ is a Slepian-Wolf protocol for the state ΨĈB (ÂR) , up to the action of G Z σ • P Z →Z Z (see Figure 9 in page 34). We combine the two protocols to cancel out (G Z σ • UĈ) † and G Z σ • UĈ. Due to the triangle inequality, it follows from (169) and (170) that

Construction of The Encoding and Decoding Operations
Define a partial isometry Applying the map Tr Z ⊗ C T to Inequality (171), and using (158) and (160), it follows that We construct a protocol for state redistribution as follows: In the first step, the sender performs the following encoding operation: where C Z is the completely dephasing operation on Z with respect to the basis {|z L |z R } z L ,z R . The sender then sends the classical system Z R ∼ = M and the quantum system C 3 ∼ = Q to the receiver, who performs the decoding operation defined by Noting that Tr Z = Tr Z L ⊗ Tr Z R , we obtain from (173) that From (172) and (174), it is straightforward to verify that E(τ ) is diagonal in Z R for any input state τ . Thus, the pair (E, D) is a state redistribution protocol for the state Ψ s within the error 4 √ 12 + 6δ.

Evaluation of Entropies
We analyze conditions on the size of systems C 1 and C 2 , in order that inequalities (150) and (151) are satisfied. We use the partial bi-decoupling theorem (Lemma 10) under the correspondence (149), which reads It follows that inequalities (150) and (151) are satisfied if it holds that log log Using the duality of the smooth conditional entropy (Lemma 12), and noting that ΨÂBĈ = ΨÂBĈ s , the min entropies in the first and the third inequalities are calculated to be Similarly, due to Lemma 23 and Lemma 26 in Appendix A, and noting that C(Ψ) = Ψ s because of (20) and (148) In addition, the isomorphism (155) implies Substituting these relations to (178)-(181), and noting that d q + e ≥ H max (C|BXY Z) Ψs − log δ 2 (195) and q + e + 2e 0 = log d C . Combining these all together, we obtain the set of Ineqs. (107)-(111) as a sufficient condition for the tuple (c, q, e) to be achievable within the error 4 √ 12 + 6δ.

Proof of Proposition 7 from Proposition 6
We prove Proposition 7 based on Proposition 6 by (i) modifying the first inequality (107), and (ii) extending the rate region by incorporating teleportation and dense coding. (107) and (112) We argue that the smooth conditional max entropy in the R.H.S. of Inequality (107) is modified to be H max (C|AXZ) Ψs . Consider a "modified" redistribution protocol as follows: In the beginning of the protocol, the sender prepares a copy of Z, which we denote byZ. The sender then uses XZ as the classical part of the side information, instead of X alone, and apply the protocol presented in Section 5.2.1. The smooth max entropy corresponding to the first term in (107) is then given by (see Lemma 24)

Modification of Inequalities
For the same reason, the term H max (Z|AX) Ψs in the condition (112) is modified to be H max (Z|AXZ) Ψs , which is no greater than zero (see Lemma 21 and Lemma 24). It should be noted that the entropies in the other three inequalities are unchanged by this modification.

Extension of the rate region by Teleportation and Dense Coding
To complete the proof of Theorem 2, we extend the achievable rate region given in Proposition 6 by incorporating teleportation and dense coding. More precisely, we apply the following lemma that follows from teleportation and dense coding (see the next subsection for a proof):

Proof of Theorem 2 from Proposition 7
The achievability for the case of d C = 1 immediately follows from the condition (120) in Proposition 7. Thus, we only consider the case where d C ≥ 2.
Let Π be a projection onto a subspace H C Π ⊆ . Such a projection exists due to the definition of H max given by (16). Consider the "modified" source state defined by From the gentle measurement lemma (see Lemma 32 in Appendix B), it holds that Thus, due to the definitions of the smooth entropies (12) and (13), we have and so forth. Suppose that the tuple (c, q, e, e 0 ) satisfies Inequalities (22)- (25) in Theorem 2. It follows that Thus, due to Proposition 7, the tuple (c, q, e, e 0 ) is achievable within an error 4 √ 12 + 6δ for the state Ψ s,Π . That is, there exists a pair of an encoding CPTP map EÂĈ Π E A →ÂQM F A Π and a decod- Define an encoding map EÂĈ E A →ÂQM F A and a decoding map DB QM E B →BĈF B for the state Ψ s by where ξ 0 is an arbitrary fixed state onÂQM F A , and D = D Π . Note that the system C Π is naturally embedded into C. By the triangle inequality, we have Here, Inequality (231) follows from ) and the monotonicity of the trace distance, and the last line from (220) and (228). Hence, the tuple (c, q, e, e 0 ) is achievable within an error 4 √ 12 + 6δ + √ 2 for the state Ψ s , which completes the proof of Theorem 2.

Proof of The Converse Part (Theorem 3 and Lemma 5)
We prove the one-shot converse bound (Theorem 3). The proof proceeds as follows: First, we construct quantum states that describe the state transformation in a redistribution protocol in a "purified picture". Second, we prove four entropic inequalities that hold for those states. Finally, we prove that the four inequalities imply the three inequalities in Theorem 3, thereby completing the proof of the converse bound. We also analyze the properties of the function ∆ ( ,δ) , and prove Lemma 5.

Construction of States
Let UÂĈ E A →ÂQM F AĜA E and UB QM E B →BĈF BĜB D be the Stinespring dilations of the encoding operation E and the decoding operation D, respectively, i.e., We define the "purified" source state |Ψ by and consider the states The stateΨ is a purification of the state after the encoding operation, and Ψ f is the one after the decoding operation. See Figure 7 for the diagram.
Due to the relation (6) between the trace distance and the purified distance, the condition (21) implies that with C T being the completely dephasing operation on T with respect to the basis {|xyz }.
Due to an extension of Uhlmann's theorem (see Lemma 30 in Appendix B), there exists a pure state |Γ ÂBĈĜ AĜBR , which is represented in the form of Using this state, we define Due to the isometric invariance of the purified distance, it follows from (239) and (236) that Relations among the states defined as above are depicted in Figure 8. Some useful properties of these states are presented in the following, and will be used in the proof of the converse part. Since M is a classical system, we may, without loss of generality, assume that U E and U D are decomposed as Since Z is a classical system, we may further assume that v m are decomposed as where Z is a system isomorphic to Z with the fixed orthonormal basis {|z } z andG A ≡ G A Z . The operators v m,z are linear operators such that

Properties ofΨ and Ψ f
Since |Ψ is defined as (235) by U E that is in the form of (242), it is decomposed into with some probability distribution {q m } m and pure states {|Ψ m } m . Thus, we have where C M is the completely dephasing operation on M with respect to the basis {|m } m . Similarly, due to (244), (235) and (236), the state |Ψ f is decomposed into Both states are ensembles of pure states on ABCRĜ AĜB , classically labelled by xyz on XY Z or T , that are decoupled between ABCR andĜ AĜB . It follows from (250) that Due to (248), (249) and Lemma 31 in Appendix B, we may, without loss of generality, assume that |φ xyz is in the form of and |φ xyz Substituting this to (250), we have Thus, the state C T (Γ) given by is classically coherent in ZZ . Denoting p xyz p m|xyz by p m,xyz , it follows from (250) that with C M A being the completely dephasing operation on M A with respect to the basis {|m } m . It should also be noted that

Inequalities for Proving Theorem 3
As an intermediate goal for the proof of Theorem 3, we prove that the following four inequalities hold for the states Ψ s and Γ defined by (20) and (238), respectively: where f (x) := − log (1 − √ 1 − x 2 ). The proof of these inequalities will be given in the following subsections. We will extensively use the properties of the smooth conditional entropies, which are summarized in Appendix A.

Proof of Inequality (259)
We have Here, (280) is from the fact that Γ is a pure state onÂBĈRĜ AĜB as (238), which is transformed by C T to an ensemble of classically-labelled pure states, to which Lemma 27 is applicable; (281) from the dimension bound (Lemma 19); (282) from the fact thatΓ is obtained from Γ ⊗ Φ 2 e 0 by an isometry as (240) under which the smooth conditional entropy is invariant (Lemma 14); (283) from the chain rule (359); and (284) from the dimension bound (Lemma 18).
The first term in (284) is further calculated to be The second term in (279) is bounded as Here, (293) follows fromĜ A ≡ G A M A Z and the fact that C T (Γ) is classically coherent in XX and in ZZ because of (255); (294) from the chain rule (360); and (295) from Γ AXY Z = Ψ AXY Z s and the fact that the system A in the conditioning part is decoupled from G A M A when conditioned by XY Z as (251) in addition to Lemma 25. Combining these all together, we arrive at = H min (BĈ|T ) C T (Γ) + e 0 = H min (BĈ|T ) Ψs + e 0 (332) where (335) This completes the proof of Inequality (261).

Proof of Theorem 3 from Inequalities (258)-(261)
Since Γ is diagonal in M A XY Z as (257), and due to the properties of the smooth conditional entropies for classical-quantum states (Lemma 25), we have Thus, Inequalities (260) and (261)

Conclusion
In this paper, we investigated the state redistribution of classical and quantum hybrid sources in the one-shot scenario. We analyzed the costs of classical communication, quantum communication and entanglement. We obtained the direct bound and the converse bound for those costs in terms of smooth conditional entropies. In most of the cases that have been analyzed in the previous literatures, the two bounds coincide in the asymptotic limit of infinitely many copies and vanishingly small error. Various coding theorems for two-party source coding tasks are systematically obtained by reduction from our results, including the ones that have not been analyzed in the previous literatures.
To investigate the protocol that are covered by our result, but have not been addressed in the previous literature, in detail is left as a future work. Another direction is to explore the family of quantum communication protocols in the presence of classical side information only at the decoder. It would also be beneficial to analyze the relation between our results and the one-shot bounds for entanglement-assisted communication of classical and quantum messages via a noisy quantum channel [32].
In addition, if Thus, without loss of generality, we may assume that bothρ AK and ς are diagonal in {|k } k . That is, we may assume thatρ AK and ς are in the form of Suppose that the Schmidt decomposition of |ψ k is given by and V := k v k ⊗ |k k| K . It is straightforward to verify that ρ BK = V ρ AK V † . Thus, due to the monotonicity of the purified distance under trace non-increasing CP maps (Lemma 7 in [29]), it holds that Applying V to the both sides in condition (397), it follows that Noting that I B ≥ (v † k v k ) B , this implies that Thus, we arrive at H min (A|K) ρ ≤ H min (B|K) ρ .
By exchanging the roles of A and B, we also obtain the converse inequality. This completes the proof of Equality (394).

B Properties of The Purified Distance
We summarize the properties of the purified distance, used in Appendix A to prove the properties of the smooth conditional entropies.

Lemma 29
For any normalized state ρ on system A and any normalized pure state |φ on system AB, the purified distance satisfies It follows that In addition, the states |Ψ p and |Γ * p are obtained by a linear isometry P K→KK := k |k K |k K k| from |Ψ and |Γ as Thus, due to the property of the purified distance (Lemma 29 and Lemma 28), it follows that P |Γ Γ |, |Ψ Ψ| = P |Γ * p Γ * p |, |Ψ p Ψ p |, = P Γ KAB , C K •Tr CD (|Ψ Ψ|) , which completes the proof. follows from the fact that the state |Ψ σ is obtained from |Ψ by applying P and G σ , due to (158). In (iii), we trace out Z ≡ Z L Z R and apply the completely dephasing operation C to X Y Z . See Inequalities (173) and (176) that are obtained from (171). Note that the source state Ψ s is obtained from |Ψ and |Ψ σ as (160).