Entanglement-Breaking Superchannels

In this paper we initiate the study of entanglement-breaking superchannels. These are processes that always yield separable maps when acting on one side of a bipartite completely positive (CP) map. Entanglement-breaking superchannels are a generalization of the well-known entanglement-breaking channels. Unlike its channel counterpart, we find that not every entanglement-breaking superchannel can be implemented as a measure-and-prepare superchannel. We also demonstrate that many entanglement-breaking superchannels can be superactivated, in the sense that they can output non-separable channels when wired in series. We then introduce the notions of CPTP- and CP-complete images of a superchannel, which capture deterministic and probabilistic channel convertibility, respectively. This allows us to characterize the power of entanglement-breaking superchannels for generating CP maps in different scenarios, and it reveals some fundamental differences between channels and superchannels. Finally, we relax the definition of separable channels to include (p,q)-non-entangling channels, which are bipartite channels that cannot generate entanglement using p- and q-dimensional ancillary systems. By introducing and investigating k-entanglement breaking maps, we construct examples of (p,q)-entanglement-breaking superchannels that are not fully entanglement breaking. Partial results on the characterization of (p,q)-entanglement-breaking superchannels are also provided.


I. INTRODUCTION
Suppose that Alice and Rachel have access to some bipartite quantum channel E A 0 R 0 →A 1 R 1 . They are interested in using this channel to generate entangled states across their spatially separated laboratories. As shown in Fig. 1, the most general method for doing so would involve using local quantum memories. Alice prepares a locally entangled state ρ A 0 A E , with A E being her memory register, and Rachel does likewise with the state ω R 0 R E . Sending systems A 0 and R 0 through the channel leads to the state which they hope is entangled. It is known that such a procedure can be used to generate entanglement if and only if E A 0 R 0 →A 1 R 1 does not have the form of a so-called separable channel [1]. Hence for Alice and Rachel's goal of obtaining bipartite entangled states, separable channels are completely useless.
Frustrated with the situation, Alice naively wonders if manipulating her part of the channel could improve their prospects of obtaining entanglement. Any physical procedure she attempts can be described as in Fig. 2; it involves her first applying some preprocessing map that couples her input system A 0 to the memory register A E , and then applying a post-processing map to system A E and her channel output A 1 [2]. Such a process is known as a superchannel, and specifically here it is a local superchannel since it is being implemented only in Alice's laboratory. Unfortunately for Alice, local superchan- An entanglement-breaking superchannel will output a separable channel for every initial channel. nels are not able to transform a separable channel into a non-separable one. Consequently, if E A 0 R 0 →A 1 R 1 is useless for entanglement generation before Alice's manipulation, it will be useless after. On the other hand, it is quite possible that a local superchannel converts a non-separable channel into a separable one. This begs the question of whether there exists certain local superchannels for Alice that convert every bipartite channel into a separable channel. We refer to such processes as entanglement-breaking superchannels since they completely eliminate any channel's ability to distribute entanglement, and they are the focus of this paper.
A channel N A 1 →B 1 is called entanglement breaking (EB) if id R 1 ⊗ N A 1 →B 1 (ρ R 1 A 1 ) is separable for every ρ R 1 A 1 and all systems R 1 . That an EBC is a special case of an EBSC comes from the fact that every quantum state can be regarded as a quantum channel with a onedimensional input. An EBC N A 1 →B 1 can then be seen as an EBSC that locally transforms quantum channels with trivial input (see Fig. 3

).
A central theorem is that every EBC can be realized by a measure-and-prepare protocol. That is, N A 1 →B 1 is EB if and only if there exists a measurement described by positive operator-valued measure (POVM) {F A 1 k } k along with a family of states {ω B 1 k } k such that for all states ρ A 1 of system A 1 . The interpretation is that the EBC N A 1 →B 1 can be imple- Figure 3. A state ρ R 1 A 1 represents a special type of bipartite channel, and a standard EBC can be seen as an EBSC post-processing map.
mented simply by first measuring system A 1 with POVM {F A 1 k } k and then preparing the state ω B 1 k contingent on outcome k. A chief question of interest in this paper is whether there exists a similar type of implementation for EBSCs. We find that obvious forms of EBSC implementation fail to capture the entire class of EBSCs, including the generalization of measure-and-prepare channels to superchannels.
The study of EBSCs falls within the broader research program of understanding dynamical quantum resources. A quantum resource theory (QRT) describes a generic framework for isolating some particular feature of a quantum system, like entanglement or coherence, and analyzing how that feature, or "resource", behaves under a restricted set of operations [4]. Most attention has been previously devoted to studying static resources, that is features that arise in particular states of a quantum system. However, recently, significant progress has been made in developing the theory of dynamical quantum resources, which refer to certain properties of quantum channels that are of interest for quantum information processing [5][6][7][8][9]. For example, in the QRT of entanglement for bipartite channels [10,11], a channel's ability to distribute entanglement is a resource, and when separable processing is taken as free, an EBSC can be interpreted as a one-sided resource-erasing map.
The paper is organized as follows. In Section II we fix notations and provide necessary preliminaries. In Section III we formally define entanglement-breaking superchannels and give several equivalent characterizations. We also characterize a subset of EBSCs which allows for intuitive pre/post-processing realizations. In Section IV we show how two copies of an EBSC can be combined to no longer be an EBSC, thereby demonstrating a type of superactivation. In Section V we introduce the notion of CPTP and CP-complete image and consider what is the largest set of CPTP (CP) maps that can be generated through the action of EBSC, as well as two important subsets of EBSC. In Section VI, we generalize EBSCs to superchannels that output k-non-entangling maps, and we connect these superchannels to the family of k-entanglement-breaking maps, the latter being a type of map that we introduce and thoroughly discuss. We summarize our results and conclude with some discussion in Sec. VII.

A. Notations
Throughout this paper we adopt most of the notation used by Gour in Ref. [10]. We use capital Latin letters A, B, C, etc. to denote physical systems, and H A , H B , H C , etc.
to denote their corresponding Hilbert space. Sometimes we also use capital letters to denote Hilbert spaces for simplicity. The collection of all bounded operators on system Since we are considering here dynamical resources, we will always assume that a system A has an associated input and an output system, denoted by A 0 , A 1 , respectively. Therefore, we can use the shorthand notation L(A) : is called a supermap, and the set of all such supermaps will be denoted by L(A → B). The action of a supermap will be written as a square bracket, like Φ[E ], whereas the action of quantum channel will usually denoted by round brackets, like E (ρ).
Finally, we useÃ to denote a system with the same dimension of A, and φÃ A + = ∑ d A i,j=1 |ii jj|Ã A is the unnormalized maximally entangled state on spaceÃA. If E A 0 →A 1 is a quantum channel, then its Choi matrix is the operator defined by (2)

B. Separable and Entanglement-Breaking Maps
We next review the meaning of separable and entanglement-breaking maps. .
. This means that we can write . As alluded to in the introduction, separability of E A 0 R 0 →A 1 R 1 means that it cannot be used for distributing entanglement between Alice and Rachel, and this fact can be most easily established by examining its Choi matrix and

].
A close cousin to the separable maps are those that are entanglement breaking.
is separable for any ρ R 1 A 1 and any system R 1 . An EB map N is called an entanglement breaking channel (EBC) if it is also trace-preserving.
The following provides different characterizations of EB maps and clarifies the relationship between EB and separable maps.
Proposition 1. For a CP map N A 1 →B 1 , the following are equivalent.
(C) N A 1 →B 1 has an implementation given by Eq. (1) for ∑ k F k ≤ I, with equality holding when (D) For any system R and any bipartite channel Proof. Items (A)-(C) are standard results found in Ref. [3]. From the form of Eq. (1), it is easy to see that . Conversely, if (D) holds for all bipartite channels E R 0 B 1 →R 1 A 1 , then by considering the discard-and-prepare channel E (

Remark:
In Section III C, we will see that the channel F R 0 A 1 →R 1 B 1 constructed in (D) of this proposition is the output of a conditional prepare-and-measure superchannel E → N • E • N .

C. Superchannels
We next review the basic structure of superchannels. The following definitions and theorems can be found in [12].
where J A E a 0 a 1 and J B Θ[E a 0 a 1 ] are Choi matrix of E a 0 a 1 ∈ L(A) and Θ[E a 0 a 1 ] ∈ L(B) respectively, and {E a 0 a 1 } a 0 ,a 1 is a complete orthogonal basis of L(A) whose action in the computational basis is given by Alternatively, J AB Θ equals the Choi matirx of (1 The action of Φ AÃ + can be expressed as and hence it can be viewed as a maximally entangled map, in analog with φ + . But notice that Φ + is not trace-preserving and hence not a quantum channel. Similar to the channel-state duality, there is a one-to-one correspondence between supermaps and their Choi matrix. For Θ ∈ L(A → B) and E ∈ L(A), we have Define ∆ Θ ∈ L(A → B) to be the unique map that satisfies Lemma II.1. [2,12] Let Θ ∈ L(A → B). The following are equivalent.

The Choi matrix
pre , then it has been shown that To summarize, the Choi matrix of the superchannel Θ is equal to the Choi matrix of three different CP maps: , as defined above.

A. Characterization of EBSCs
With the background concepts in place from the previous section, we are now able to introduce the notion of an entanglement-breaking superchannel.
Definition III.1. A superchannel Θ A→B is called entanglement-breaking superchannel ] is a separable map for every E ∈ CP(RA), with R being an arbitrary finite-dimensional system.
Our first result provides several equivalent characterizations of EBSCs, in analogy to Proposition 1 for EBCs. However, despite the similarity between the following theorem and Proposition 1, we will see that there are some fundamental differences between EB-SCs and EBCs. In particular, there is evidently no simple implementation result for EBSCs like statement (C) in Proposition 1.

Theorem 1.
For a superchannel Θ A→B , the followings are equivalent.
(C) J AB Θ is separable with respect to A : B.
(D) Θ can be realized with pre/post-processing CPTP maps Γ B 0 →A 0 E pre and Γ A 1 E→B 1 post such that (E) ∆ Θ ∈ L(A → B) defined as the unique map with Choi matrix J AB Θ is an entanglement breaking map.
where the superscript Γ A denotes partial transpose on system A. Since system R is actually unchanged, letR = R and rewrite this equation as where we omit the identity operator. Now that J AB Θ is separable, which means it can be written as for some M A k ∈ P (A), N B k ∈ P (B), and positive integer K. Substitute into Eq. (14), we get This has established the equivalence (A) ⇔ (B) ⇔ (C). As for the equivalence of (C), (D), (E), we know that the Choi matrix of Θ , as shown in the last section. Therefore, the separability of J AB Θ with respect to A : B is equivalent to the separability of Γ A 1 B 0 →A 0 B 1 Θ with respect to A 1 A 0 : B 0 B 1 , which is also equivalent to the fact that ∆ A→B is entanglement breaking. This completes the proof of theorem 1.

B. EBSCs realized by EB pre/post-processing channels
Theorem 1 (D) provides a structural requirement for the pre/post-processing maps of an EBSC. However, the separability on Γ Θ only characterizes the concatenation of the pre/post-processing maps, as depicted in Fig. 4. Since the decomposition of a given superchannel into pre/post-processing channels is not unique, it remains unclear what constraints are placed on the pre/post-processing maps individually in order for the resulting superchannel to be entanglement-breaking. For instance, if both Γ pre and Γ post are entanglement-breaking between the channel input/outputs and the memory, then they will generate an EBSC. But can every EBSC be decomposed into EB pre/post-processing channels? Here we explore this question and find that in general, such a realization is not always possible.
We begin by generalizing the definition of entanglement-breaking for channels that have bipartite input or output.
Definition III.2. A quantum channel E B 0 →A 0 E is called partly entanglement breaking for is separable with respect to A 0 : RE for any ρ ∈ D(RB 0 ) and any system R with finite dimension.
is separable with respect to S : RB 1 for any σ RE ∈ D(RE) and ω SA 1 ∈ D(SA 1 ) for any system R, S with finite dimension.
The following lemmas offer alternative characterizations of partly EB channels. Note the similarity to the structure of EBCs stated in Proposition 1.
Lemma III.1. For a quantum channel E B 0 →A 0 E , the following are equivalent: and their sum is trace-preserving. This completes the proof.
A similar set of conclusions hold for partly EB channels with a bipartite input.
Lemma III.2. For a quantum channel E EA 1 →B 1 , the following are equivalent.
(a) E EA 1 →B 1 is partly EB for input system A 1 .
(c) E EA 1 →B 1 can be written as Proof. The proof is very similar to the previous lemma. We only need to show (b) ⇒ (c).
For any ρ ∈ D(EA 1 ), Since E is CPTP, we have that Remark. Note that partly-EB channels are equivalent to bipartite separable channels with one trivial input/output system. That is, This can be seen from item (b) in Lemmas III.1 and III.2. As a special case (with E being trivial), both types of partly EB channels reduce to standard EBCs, with Choi matries that are separable across input and output systems.
Clearly the use of partly EB pre/post-processing channels yields a superchannel that is EB. A natural question is whether all EBSCs can be realized in this manner. To prove this is not the case, we first make a simple observation.

Proposition 2.
Suppose that Θ is a superchannel realized by a partly EB pre-processing map Then for any ρ RB 0 and ω SA 1 , the output state is separable across A 0 : RB 1 S. Similarly, if Θ is realized by a partly EB post-processing map, then this output state is separable across A 0 RB 1 : S.
will also be separable across A 0 : RB 1 S. An analogous argument proves the second statement when the post-processing map is partly EB.
From this proposition, it follows that if Θ A→B is an EBSC that can be realized by a partly EB pre-processing (resp. post-processing) map, then we must have that J AB Θ is both A 0 A 1 : B 0 B 1 separable as well as A 0 : difficult to construct superchannels that fail to have this separability structure.

Theorem 2.
There exists EBSCs that cannot be realized using either a pre-or post-processing channel that partly EB.
Proof. By the previous proposition, it suffices to construct pre/post-processing maps Hence This leads to the Choi matrix Clearly it A : B separable, and yet it is entangled for both parts A 0 and A 1 since projecting onto |00 B 0 B 1 leads to the maximally entangled state 1 4 φ +A 0 A 1 .

Remark:
One interpretation of this result is that there exists EBSCs that must entangle their memory system E with the channel input A 0 for some inputs to B 0 . Likewise, the channel output A 1 must couple with the memory E nonlocally.
One may consider a more general way to apply partly-EB channels as pre/postingprocessing maps. As shown in Fig. 5, one can split the side-channel in half, , and require the pre-processing channel to be partly-EB for output system , the post-processing channel partly-EB for input system This EBSC becomes what we discussed in Thm. 2 if E A is taken to be trivial.

C. EBSCs realized by measure-and-prepare superchannels
On the level of channels, entanglement-breaking is equivalent to measuring and preparing, as described in Eq. (1). A natural question is whether the same holds for EBSCs. To this end, we first need a generalized notion of measurement that also applies to dynamical resources. In Ref. [13], the authors introduce the concept of a quantum super-instrument, which is a set of c-CPTNI (completely CP preserving and trace-nonincreasing preserving) supermaps {Θ x } such that ∑ x Θ x is a superchannel. A quantum super-instrument therefore describes in one sense how a quantum channel can be measured. It is proven in [13] that every super-instrument can be realized with a CPTP preprocessing channel and a post-processing quantum instrument, as shown on the left side of Fig. 6. We are also interested in a special kind of super-instrument with trivial output channel, which we call a POVM of quantum channels, as shown on the right side of Fig. 6.
With the concept of a channel POVM in place, we next combine it with a channel preparation step to obtain a measure-and-prepare superchannel. This is depicted in Fig. 7 where the channel F B x is prepared contingent on the outcome x of the proceeding channel POVM.
Definition III.4. A superchannel Θ A→B is called a measure-and-prepare superchannel (MPSC) if it can be realized as where In short, an MPSC conducts a measurement of the original channel and then prepares a new channel conditioned on the measurement outcome. This seems like a reasonable superchannel analogy for a measure-and-prepare channel. However, notice that in the superchannel case, the choice of pre-processing state ρ E 1 A 0 provides an extra degree of freedom that is not present in the channel case. In the definition of an MPSC, the preprocessing state ρ E 1 A 0 is independent of the input system B 0 . One may also want to apply a quantum instrument on B 0 and generate different pre-processing states conditioned on the measurement outcome of this instrument. We will refer to a superchannel having this type of structure as a controlled measure-and-prepare superchannel, or CMPSC for short.
Its rigorous definition is as follows.
Definition III.5. A superchannel Θ A→B is called a controlled measure-and-prepare superchannel (CMPSC) if it can be realized as where {Λ y } y is a quantum instrument, ρ A 0 E y is some quantum state, {P A 1 E x } x is a POVM, and {F x } x is a family of CPTP maps.
The realization of a CMPSC is shown in Fig. 8. One can easily verify that every CMPSC is an EBSC. It is therefore natural to conjecture that every EBSC can be realized as an MPSC, similar to the case of EBCs. However, we will now show below that this conjecture fails to be true.

Theorem 3.
There exists EBSCs that are not CMPSC.   [14]. Hence, E E 2 A 1 →B 1 is a non-LOCC channel whose action is for all i. We can regard this as a post-processing map for a superchannel that converts states |α i into QC channels |β i → i. Since E E 2 A 1 →B 1 is separable, it is clearly an EBSC. However, with system A 0 being one-dimensional, the possible implementation by a CMPSC reduces to two-way LOCC, as shown in Fig. 9, which is not possible by construction. Therefore the superchannel is not CMPSC.

IV. SUPERACTIVATION OF EBSC
Entanglement-breaking channels have the property that they are closed under tensor product. That is, if E A 0 →A 1 and N R 0 →R 1 are both EB channels, then so is their tensor prod- EBSCs. The problem has an added level of complexity when dealing with superchannels since the dynamic nature of channels allows them to be processed in different ways. On the one hand, two copies of a superchannel can be used for parallel processing, which means that their pre/post-processing occurs simultaneously, as shown in Fig. 10. Alternatively, the two superchannels can implement a processing of channels in series, such that the output of one superchannel can be used as the input for the other. While the full theory of sequential processing can be described using the formalism of quantum combs [15,16], here we do not need to invoke the latter to demonstrate the generic superactivation of EBSC.
Likewise, let E R A 1 be the channel that discards and prepares 1 . Then with the Figure 11. A series implementation of two EBSCs that can yield a non-separable channel. The channels E RA 0 and E R A 1 discard their inputs and prepare maximally entangled states.
wiring shown in Fig. 11, it follows that the resulting channel will be the state However, this is proportional to J with A 1 replaced by R 1 and B 0 replaced by R 1 .
The A 1 B 0 : B 1 entanglement implies that Ω R 1 R 1 B 1 is entangled between Rachel and Bob.
Hence E RA 0 and E R A 1 is transformed into an entangled channel by two copies of the EBSC Θ A→B .

V. IMAGE OF EBSC
In Section III we introduced EBSC and its two subsets MPSC and CMPSC. Here, we investigate the comparative powers of these operational classes. In particular, we consider whether every bipartite separable channel can be generated by applying an EBSC on one part of some bipartite channel, and whether MPSC/CMPSC has the same channelgeneration power as EBSC. For this purpose, we introduce the notion of CPTP-complete image.
Definition V.1. Given a superchannel Θ A→B , the CPTP image of Θ A→B is defined as The CPTP-complete image of Θ A→B is defined as which is the union of CPTP images 1 R ⊗ Θ A→B for all systems R. The CPTP-complete image of a set of superchannels S is defined as C S = Θ∈S C Θ .
Similarly, we introduce the CP-complete image as follows.
Definition V.2. Given a superchannel Θ A→B , the CP image of Θ A→B is defined as The CP-complete image of Θ A→B is defined as which is the union of CP images 1 R ⊗ Θ A→B for all systems R. The CP-complete image of a set of superchannels S is defined as C * S = Θ∈S C * Θ .
Alternatively, we can say that the CPTP image of Θ A→B is its image when the domain is restricted to CPTP maps, and likewise the CP image of Θ A→B is its image when the domain is restricted to CP maps.
With these definitions in place, a superchannel Θ A→B is an EBSC if and only if C * Θ ⊆ SEP * . Here we use SEP * to denote the set of separable CP maps while SEP will denote the set of all separable CPTP maps. An interesting question is whether C EBSC = SEP and C * EBSC = SEP * hold, which will be the main concern of this section. Let us first consider this question for EBCs. Note that our definition of CPTP (CP) complete image also applies for channels since they are a special case of superchannels.
Specifically, for a channel E , the set Im E (resp. Im * E ) is the image of E when the domain is restricted to density operators (resp. positive operators). In this case, it is easy to see that both C EBC = SEP and C * EBC = SEP * hold. Indeed for an arbitrary separable positive operator ρ RB = ∑ i p i |ψ i ψ i | R ⊗ |α i α i | B , one need only consider the action of the EBC E A→B on the positive operator σ RA = ∑ i p i |ψ i ψ i | R ⊗ |i i| A , where E A (X) = i|X|i A |α i α i | B , and σ RA is a density matrix if and only if ρ RB is.
In following part of this section, we set out to study the image of general EBSC. Regarding the CPTP-complete image, while we are unable to precisely characterize C EBSC , we relate C MPSC and C CMPSC to LOCC channels of certain communication rounds. As for the CP-complete image, we show that C * EBSC = SEP * holds exactly. These results reveal some fundamental differences between channels and superchannels of physical significance, and we discuss this further at the end of this section.

A. CPTP-complete Image of EBSC
As noted above, one of our primary interests is determining whether C EBSC = SEP.
We consider here a special subset of the separable channels that can be generated by LOCC. While LOCC is a notoriously difficult class of operations to analyze, here it will be sufficient to just consider one-round and two-round LOCC. A precise definition of these operational classes is as follows.
for some CPTP maps F B ij and deterministic quantum instruments {Λ B i }, {Γ R j|i }. Set B 2 = E 2 and construct a superchannel Θ with the following realization: Here, system E 1 is taken to be trivial. It's easy to see that Θ is a CMPSC by Def. III.5. Then consider the quantum channel One can immediately verify that By the arbitrariness of G we have LOCC 2 ⊆ C CMPSC , hence LOCC 2 = C CMPSC .
For the first equality in Eq. (34), we can similarly verify that C MPSC ⊆ LOCC 1 from Fig. 12 when the classical channel from Bob to Rachel is removed. For an arbitrary Figure 12. A CMPSC superchannel transforms any E RA into a channel F RB that can implemented by a two-round LOCC channel from Bob to Rachel.
and further define a quantum channel It is obvious that Θ A 1 →B is an MPSC, and so G ∈ C MPSC . Combining the above results we get C MPSC = LOCC 1 .
As for C CMPSC C EBSC , recall the EBSC we constructed to prove theorem 3. By applying that EBSC at an input noiseless channel id R 0 →A 1 , we obtain a separable channel that cannot be implemented by LOCC. This complete the proof of proposition 3.
The above proposition precisely characterizes the CPTP-complete image of MPSC and CMPSC. Also, it tells us C EBSC contains all (B → R → B) two-round LOCC channels and some non-LOCC separable channels. It remains an open problem whether C EBSC = SEP.
We next turn to the CP-complete image.

B. CP-complete Image of EBSC
Interestingly, we can easily prove C * EBSC = SEP * . We first present a proof by direct construction, and then later discuss how this result is related to the notion of stochastic LOCC (SLOCC) strategy.

Proposition 4. The CP-complete image of EBSC and CMPSC satisfy
where SEP * is the set of all bipartite separable CP maps.
Proof. By definition, C * EBSC ⊆ SEP * . We only need to prove that C * EBSC ⊇ SEP * . It is enough to show that C * EBSC contains all CPTNI separable maps, because every CP map can be normalized to be CPTNI by dividing a positive factor, and we can always multiply this factor to the input CP map.
For any bipartite separable CPTNI map G RB , its Choi matrix can be written as such that P k , Q k are positive operators. Without loss of generality we take Tr(P k ) = 1. Since G RB is CPTNI, we have ∑ k P R 0 k ⊗ Q B 0 k ≤ I R 0 B 0 , and so ∑ k Q B 0 k ≤ I B 0 . Let F B 0 B 1 be a positive operator satisfying Then for an (r + 1)dimensional system A 0 and system A 1 being one-dimensional, construct a supermap Θ whose Choi matrix is as follows.
Since A 1 is one-dimensional, it holds trivially that J AB 0 and hence Θ is a superchannel. Furthermore, since J AB Θ is separable with respect to A : B, it is also an EBSC, by theorem 1. Define a CP map E whose Choi matrix is given by Note the condition ∑ k P R 0 k = I need not be enforced here since we allow E to be nontrace-preserving. Then, . By the arbitrariness of G and the argument before, we have Notice that the EBSC Θ in the above proof is actually a CMPSC. To see this, take the pre/post-processing map to be where Λ B 0 →E k (resp. F B 0 →E ) is the unique map with Choi matrix Q B 0 E k (resp. F B 0 E ). It is easy to check Γ B 0 →A 0 E pre is CPTP. Therefore, we have C * CMPSC = C * EBSC = SEP * . This complete the proof of Proposition 4.
It is interesting to consider the physical interpretation of the superchannel and input CP used in the previous proof. In order to implement the CPTNI map G RB = ∑ r k=1 P R k ⊗ Q B k , Bob first performs a quantum instrument {Λ 1 , ..., Λ r , F } and sends the outcome (stored as classical information in system A 0 ) to Rachel. Rachel then acts as follows. If Bob's action is Λ k , Rachel implements the CPTNI with Choi matrix P k to complete the procedure. If Bob's action is F , the protocol aborts. Of course, it may only be possible for Rachel to implement the CPTNI map P k with some nonzero probability. In this case, Rachel needs an extra round of classical communication to let Bob know whether her implementation is successful. Bob's final CPTP map would then be the identity in the case that she succeeds, and some other fixed "failure" channel in the case that she does not.
The above procedure describes a general stochastic LOCC (SLOCC) protocol. It is shown in [18] that every separable map can be implemented by an SLOCC protocol, which provides a rough explanation for why Proposition 4 holds. It also helps shed light on the physical significance of CP image, as defined in Definition V.2. This is related to stochastic quantum processes, which we discuss further in the next subsection.

C. Discussion on CP Image
Suppose that, for superchannel Θ A→B and CP maps G B and E A one has If both G and E are CPTP, the physical interpretation of this equation is clearly deterministic channel conversion. What if they are not trace-preserving? Let's consider the case when G B is CPTNI and E A is a general CP map. We can always find a positive number p ≤ 1 such that pE is a CPTNI map, hence we can find another CPTNI map T to make up a quantum instrument {pE , T } A . By applying Θ A→B to this instrument (see Fig.13) we get a new instrument {pG, Θ[T ]} B . This provides an implementation of the CPTNI map G with success probability p. Therefore, the physical interpretation of the above equation is probabilistic channel conversion. Hence the CPTP (resp. CP) image describes the channels (resp. CP maps) that can be deterministically (resp. probabilistically) generated by a superchannel.
Comparing Propositions 3 and 4, which say that C CMPSC = LOCC 2 but C * CMPSC = SEP * , we see there are CPTP maps that can be probabilistically implemented by CMPSC but not deterministically, since there are non-LOCC CPTP maps in SEP ⊂ SEP * that are not in LOCC 2 . Hence, there are separable channels G RB and valid superchannels Θ A→B for which G RB = 1 R ⊗ Θ A→B [E RA ] holds only for some CP non-TP map E RA . Such a phenomenon does not exist at the level of channels and states because we cannot transform an unnormalized state into a normalized one by a quantum channel.
This result shows that the structure of unnormalized maps is much more complex than the structure of unnormalized states, which is one of the reasons why QRTs of channels are often harder than QRTs of states (see also Ref. [13] for a related discussion on this point).

VI. ALTERNATIVE DEFINITIONS OF ENTANGLEMENT BREAKING
In the final section, we examine certain relaxations to the definition of entanglement breaking. To this end, we first consider relaxations to separable channels. Recall

and all systems
R and A . In fact, it suffices to consider systems R and A of dimension d R 0 and d A 0 , respectively. A relaxation would be to require that σ R A R 1 A 1 is separable only for systems R and A of smaller dimension. A (k, k)-non-entangling channel is also called k-non-entangling in previous literature [19].
We call a channel (p, complete)-non-entangling if it's (p, q)-non-entangling for every positive integer q. A separable channel is then a (complete, complete)-non-entangling channel.
non-entangling map (p with B and q with R) for every E ∈ CP(RA), with R being an arbitrary finite-dimensional system.
As the general structure of (p, q)-EBSCs is rather complicated, we will only consider superchannels that have an implementation without using a side-channel, and only (k, complete)-EBSCs. Before doing that, we first introduce k-entanglement breaking channels in the following subsection.

A. k-Entanglement Breaking Channel
For simplicity, in the following we only consider channels with the same input and output dimension, d ≥ 2. As noted above, a k-EBC is completely-EBC whenever k ≥ d.
For 1 ≤ k < d, the set of k-EBC is clearly a subset of (k+1)-EBC, and the set of all 1-EBCs trivially equals CPTP(B 0 → B 1 ). To summarize, we have In Theorem 5 below, we show that each of the above inclusion is strict. That is, there exists k-EBC which is not (k+1)-EBC for all 1 ≤ k < d.
Note that our definition of k-EBC differs from the type of channels studied in Ref. [20].
That paper defines a k entanglement-breaking channel Λ B 0 →B 1 to be one that satisfies SN(id R ⊗ Λ(ρ)) ≤ k for any ρ ∈ D(R ⊗ B 0 ) with arbitrarily large system R, where SN denotes the Schmidt number. Despite the mismatch in terminology, we will still use our definition of a k-EBC since the name accurately conveys a correspondence with k-nonentangling maps, as we will clarify later.
The main result of this subsection is the following theorem, which shows that there exists a non-trivial structure of k-EBC.

Theorem 5. For all integers k and d with
The proof of this theorem is inspired by Theorem 20 in [19], which establishes the existence of k-non-entangling map that is not (k + 1)-non-entangling. Interestingly, theorem 20 in [19] turns out to be a corollary of Theorem 5 presented here.
Now we set out to prove Theorem 5. In the following, we assume H R = C k and for some operator X ∈ L(C d → C k ). Let X = ∑ r i=1 λ i |α i β i | denote a singular value decomposition of X, and define X −1 := ∑ r i=1 λ −1 i |β i α i | (with λ i > 0 for i = 1, · · · r). Then X −1 X = P = ∑ r i=1 |β i β i | is a projector of rank r ≤ k. Then since local operators preserve separability, we have that is separable if and only if is separable. We summarize this observation in the following lemma.
is separable for all projectors P with dimension no larger than k.
Now we introduce the Werner states [21] which will play an important role in our construction of k-EBC. The Werner states are a family of states on C d ⊗ C d which takes the following form, where F d = ∑ ij |i j| ⊗ |j i| is the swap operator (or the partial transpose of φ + d ). A crucial observation is that the partial trace of a Werner state is 1 d I d , which means it is a valid Choi matrix of some CPTP quantum channel (with some normalization). Specifically, consider the following map, This map is clearly completely positive, and it is also trace-preserving since where the second equation is from our observation on the partial trace of Werner states.
Since ρ W d (β) is the Choi matrix of Λ β up to normalization, we can apply Lemma VI.1 by studying the separability properties of ρ W d (β). The following lemma is modified from Lemma 19 in [19].
Lemma VI.2. Let k and d be integers such that 1 ≤ k < d and let −(d is separable for all projectors P with dimension no greater than k if and only if β ≤ (d − k)/k.
Proof. (Proof of lemma VI.2) We first require P to be a k-dimensional projector in L(B 0 → B 1 ). By direct calculation, we have for some normalized pure state |ψ = (P * ⊗ 1 If β > (d − k)/k, then β+1 d k > 1 and so is obviously not positive. A non-positive partial transpose implies that ρ proj RB 1 is entangled. On the other hand, if β ≤ (d − k)/k, we apply Theorem 1 from [22] which states that the is the Frobenius norm. Here, we have since ψ is a normalized pure state. We therefore conclude that ρ proj RB 1 is separable. This establishes that ρ proj RB 1 is separable for all k-dimensional projectors P if and only if β ≤ (d − k)/k. Since any projector of dimension strictly less than k can be implemented by first performing a k-dimensional projector and then projecting into a smaller subspace, it immediately follows that ρ

B. Interplay between k-EBC and (k, complete)-EBSC
In this subsection, we discuss the interplay between generalized EBC and generalized EBSC. For simplicity, the systems A, B, we consider are both required to have ddimensional input and output systems. We will also restrict attention to the special class of superchannels that allow for a realization without the side-channel E.
Definition VI.4. A superchannel Θ is said to be without side-channel, if it can be realized as for some CPTP maps Γ pre and Γ post . Proof. This simply follows from Thm. 1. (D) and the definition of EB channels.
The following proposition discusses the relation between k-EBC and (k, complete)-EBSC, for any positive integer k ≤ d.

Proposition 5.
For a superchannel Θ A→B without side-channel as in Eq. 62, the following are equivalent.
2. Γ B 0 →A 0 pre is a k-EBC, and Γ A 1 →B 1 post is a completely-EBC.
As a result, the realization of a (k, complete)-EBSC without side-channel is shown in  Proof. 1 ⇒ 2: Choose system R to have d R 0 = d R 1 = d, Let D be a d-dimensional system, K be a k-dimensional system. Consider the CP map Φ RA + (·) = Tr((·)φ R 0 A 0 + )φ R 1 A 1 + , the maximally entangled state φ R 0 D + , and an arbitrary quantum state ρ ∈ D(B 0 K). We have Since Θ is (k, complete)-EBSC, the above state must be separable with respect to KB 1 : DR 1 , which means both Γ B 0 →D pre (ρ B 0 K ) and Γ A 1 →B 1 post (φ R 1 A 1 + ) are separable. The former implies Γ pre is k-EBC since ρ is an arbitrary state on B 0 K. The latter implies Γ post is (completely) EBC by Proposition. 1 [3]. This complete the proof of 1 ⇒ 2.
2 ⇒ 1: Consider the circuit in Fig. 14. For any separable input ρ B 0 K ⊗ ϕ R 0 D , it's easy to see the output state is K : B 1 : R 1 D separable, which means Θ A→B is indeed (k,d)-

EBSC.
Since the hierarchy of k-EBC is non-trivial, there is also a non-trivial hierarchy of (k, complete)-EBSC, for 1 ≤ k ≤ d.

VII. CONCLUSION AND DISCUSSION
In this paper, we introduce and thoroughly study the notion of entanglement-breaking superchannels (EBSCs). These are objects that generalize and extend the standard notion of entanglement-breaking channels (EBCs) to "higher-order" quantum maps. On the one hand, EBSCs allow for relatively simple characterization via the Choi matrix, just like its channel counterpart. On the other hand, they can also exhibit some interesting properties, which make them much more complex than EBCs in many aspects. Firstly, we show that not all EBSCs can be decomposed into partly-EB pre/post-processing channels. Secondly, we show that EBSC is more general than measure-and-prepare superchannels (MPSCs), and even control-measure-and-prepare superchannels (CMPSCs), while for EBC these three classes coincide. Finally, we illustrate a super-activation phenomenon of EBSCs.
We further investigate which quantum channels can be generated using EBSCs, as well as the smaller classes of MPSCs and CMPSCs. We show that the CPTP-complete image of MPSC/CMPSC equals one/two-round LOCC maps, respectively. Although we are not able to precisely characterize the CPTP-complete image of EBSC, we show that its CPcomplete image equals the collection of all separable maps. We argue that the notion of CP image captures some fundamental difference between channels and superchannels, and we hope these results might inspire new lines of investigation into probabilistic channel conversion.
In the final section of this paper, we establish a relationship between k-EBC, k-nonentangling channel and (k, complete)-EBSC without side-channel. By generalizing the method of [19], we show that all these three objects have a non-trivial hierarchy for 1 ≤ k ≤ d. We remark that other alternative definitions of EBSC are also possible. One can require the output of EBSC to be not only separable, but also LOCC, or even LOSR (local operations and shared randomness). In this sense, our definition of MPSC (CMPSC) is just an example of one-round (two-round) LOCC-EBSC, but it remains unclear whether every one-round (two-round) LOCC-EBSC can be realized this way. In other words, whether MPSC ⊆ LOCC 1 -EBSC and CMPSC ⊆ LOCC 2 -EBSC are strict inclusions needs further investigation.
Our work provides a useful tool for the dynamical resource theory of quantum entanglement. Many results in entanglement theory based on EBCs can possibly be generalized to the dynamical resource theory with EBSC. For example, inspired by the resource theory of quantum memory where EBCs are a free resource [23], one can consider the ability to faithfully store a quantum operation, perhaps call it a "super-memory", where EBSC may serve as free resource. We leave this for future work. Also, since we have characterized the Choi matrix of an EBSC, it is straightforward to calculate a robustness-type quantity with respect to it, similar to what has been done in [24]. We anticipate there being other applications of EBSC within the quantum the study of dynamical quantum resource theories.
There are some problems left open in our work. The first is whether the deterministic image of EBSC equals the set of all separable channels, namely whether C EBSC = SEP or not. Currently we only know that all two-round LOCC and some non-LOCC separable channels are in C EBSC . The second is whether every EBSC can be realized as in Fig. 5.
Answering these questions will help us better understand the intricate structure of EBSCs.