# Entanglement-breaking superchannels

Senrui Chen<EMAIL_ADDRESS>Department of Electronic Engineering, Tsinghua
University, Beijing 100084, China  Center for Quantum Information, Institute
for Interdisciplinary Information Sciences, Tsinghua University, Beijing
100084, China  Eric Chitambar<EMAIL_ADDRESS>Department of Electrical
and Computer Engineering, Coordinated Science Laboratory, University of
Illinois at Urbana-Champaign, Urbana, IL 61801

###### Abstract

In this paper we initiate the study of entanglement-breaking (EB)
superchannels. These are processes that always yield separable maps when
acting on one side of a bipartite completely positive (CP) map. EB
superchannels are a generalization of the well-known EB channels. We give
several equivalent characterizations of EB supermaps and superchannels. Unlike
its channel counterpart, we find that not every EB superchannel can be
implemented as a measure-and-prepare superchannel. We also demonstrate that
many EB 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 EB
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
$k$-EB maps, we construct examples of $(p,q)$-EB superchannels that are not
fully entanglement breaking. Partial results on the characterization of
$(p,q)$-EB superchannels are also provided.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Notations
    2. 2.2 Separable and Entanglement-Breaking Maps
    3. 2.3 Supermaps and Superchannels
  3. 3 Characterization and Realizations of EBSCs
    1. 3.1 Characterization of EBSCs
    2. 3.2 EBSCs realized by EB pre/post-processing maps
    3. 3.3 EBSCs realized by measure-and-prepare superchannels
  4. 4 Superactivation of EBSC
  5. 5 Image of EBSC
    1. 5.1 CPTP-complete Image of EBSC
    2. 5.2 CP-complete Image of EBSC
    3. 5.3 Discussion on CP Image
  6. 6 Alternative definitions of entanglement breaking
    1. 6.1 k-Entanglement Breaking Channel
    2. 6.2 Interplay between $k$-EBC and $(k,\text{complete})$-EBSC
  7. 7 Conclusion and Discussion

##  1 Introduction

Suppose that Alice and Rachel have access to some bipartite quantum channel
$\mathcal{E}^{A_{0}R_{0}\to A_{1}R_{1}}$. 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
$\rho^{A_{0}A_{E}}$, with $A_{E}$ being her memory register, and Rachel does
likewise with the state $\omega^{R_{0}R_{E}}$. Sending systems $A_{0}$ and
$R_{0}$ through the channel leads to the state

$\sigma^{A_{E}A_{1}:R_{E}R_{1}}=\mathrm{id}^{A_{E}R_{E}}\otimes\mathcal{E}^{A_{0}R_{0}\to
A_{1}R_{1}}\left(\rho^{A_{0}A_{E}}\otimes\omega^{B_{0}B_{E}}\right),$

which they hope is entangled. It is known that such a procedure can be used to
generate entanglement if and only if $\mathcal{E}^{A_{0}R_{0}\to A_{1}R_{1}}$
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.

Figure 1: Alice and Rachel can use the their bipartite channel $\mathcal{E}$
with local quantum memories to generate entanglement in
$\sigma^{A_{E}A_{1}:R_{E}R_{1}}$ if and only if $\mathcal{E}$ is not a
separable channel.

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 pre-processing 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
superchannels are not able to transform a separable channel into a non-
separable one. Consequently, if $\mathcal{E}^{A_{0}R_{0}\to A_{1}R_{1}}$ 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 exist 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.

Figure 2: A local superchannel transforms $\mathcal{E}^{R_{0}A_{0}\to
R_{1}A_{1}}$ into $\mathcal{F}^{R_{0}B_{0}\to R_{1}B_{1}}$. An entanglement-
breaking superchannel outputs a separable channel for every initial channel.

Entanglement-breaking superchannels (EBSCs) generalize the class of
entanglement-breaking channels (EBCs), a well-studied object within quantum
information theory [3]. A channel $\mathcal{N}^{A_{1}\to B_{1}}$ is called
entanglement breaking (EB) if $\mathrm{id}^{R_{1}}\otimes\mathcal{N}^{A_{1}\to
B_{1}}(\rho^{R_{1}A_{1}})$ is separable for every $\rho^{R_{1}A_{1}}$ 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 one-
dimensional input. An EBC $\mathcal{N}^{A_{1}\to B_{1}}$ can then be seen as
an EBSC that locally transforms quantum channels with trivial input (see Fig.
3).

Figure 3: A state $\rho^{R_{1}A_{1}}$ represents a special type of bipartite
channel, and a standard EBC can be seen as an EBSC post-processing map.

A central theorem is that every EBC can be realized by a measure-and-prepare
protocol [3]. That is, $\mathcal{N}^{A_{1}\to B_{1}}$ is EB if and only if
there exists a measurement described by a positive operator-valued measure
(POVM) $\\{F^{A_{1}}_{k}\\}_{k}$ along with a family of states
$\\{\omega_{k}^{B_{1}}\\}_{k}$ such that

$\mathcal{N}^{A_{1}\to
B_{1}}(\rho^{A_{1}})=\sum_{k}\operatorname{Tr}[F_{k}^{A_{1}}\rho^{A_{1}}]\omega_{k}^{B_{1}}$
(1)

for all states $\rho^{A_{1}}$ of system $A_{1}$. The interpretation is that
the EBC $\mathcal{N}^{A_{1}\to B_{1}}$ can be implemented simply by first
measuring system $A_{1}$ with POVM $\\{F^{A_{1}}_{k}\\}_{k}$ and then
preparing the state $\omega_{k}^{B_{1}}$ 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 2, we fix notations and provide
necessary preliminaries. In Section 3, 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 4, 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 5, 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 6, 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. 7.

##  2 Preliminaries

###  2.1 Notations

Throughout this paper we adopt most of the notations used in Ref. [10]. We use
capital Latin letters $A,B,C$, etc. to denote physical systems, and
$\mathcal{H}^{A}$, $\mathcal{H}^{B}$, $\mathcal{H}^{C}$, 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 $A$ will be denoted by $\mathcal{B}(A)$, all Hermitian operators by
$\textrm{Herm}(A)$, all positive operators by $\mathcal{P}(A)$, and all
density matrices by $\mathcal{D}(A)$. The set of all linear maps from
$\mathcal{B}(A)$ to $\mathcal{B}(B)$ will be denoted as $\mathcal{L}(A\to B)$,
among which all completely-positive maps (CP) and all completely-positive and
trace-preserving maps (CPTP) are denoted as $\textrm{CP}(A\to B)$ and
$\textrm{CPTP}(A\to B)$, respectively. A CPTP map is also called a quantum
channel.

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
$\mathcal{L}(A):=\mathcal{L}(A_{0}\to A_{1})$,
$\textrm{CP}(A):=\textrm{CP}(A_{0}\to A_{1})$, etc. A linear map from
$\mathcal{L}(A)$ to $\mathcal{L}(B)$ is called a supermap, and the set of all
such supermaps will be denoted by $\mathbb{L}(A\to B)$. The action of a
supermap will be written as a square bracket, like $\Phi[\mathcal{E}]$,
whereas the action of quantum channel will usually denoted by round brackets,
like $\mathcal{E}(\rho)$.

We use $\tilde{A}$ to denote a system with the same dimension of $A$, and
$\phi_{+}^{\tilde{A}A}=\sum_{i,j=1}^{d_{A}}|ii\rangle\langle jj|^{\tilde{A}A}$
is the unnormalized maximally entangled state on space $\tilde{A}A$. For any
linear map $\mathcal{E}\in\mathcal{L}(A_{0}\to A_{1})$, its Choi matrix is
defined as [12, 13]

$\displaystyle
J_{\mathcal{E}}^{A_{0}A_{1}}:=\mathrm{id}^{A_{0}}\otimes\mathcal{E}^{\tilde{A}_{0}\to
A_{1}}(\phi_{+}^{A_{0}\tilde{A}_{0}}),$ (2)

which establishes an isomorphism between $\mathcal{L}(A_{0}\otimes A_{1})$ and
$\mathcal{L}(A_{0}\to A_{1})$, known as the Choi-Jamiolkowski isomorphism. The
inverse of Eq. (2) is

$\mathcal{E}^{A_{0}\to
A_{1}}(\rho^{A_{0}})=\operatorname{Tr}_{A_{0}}\left(\left((\rho^{A_{0}})^{T}\otimes
I^{A_{1}}\right)J_{\mathcal{E}}^{A_{0}A_{1}}\right).$ (3)

Furthermore, $\mathcal{E}^{A_{0}\to A_{1}}$ is a CP map iff
$J_{\mathcal{E}}^{A_{0}A_{1}}\geq 0$ (which means it is positive
semidefinite), and $\mathcal{E}^{A_{0}\to A_{1}}$ is TP iff
$\operatorname{Tr}_{A_{1}}({J_{\mathcal{E}}^{A_{0}A_{1}}})=I^{A_{0}}$.

Throughout this paper, we denote the identity operator on state space
$\mathcal{H}$ as $I$, the identity channel as $\mathrm{id}$ (i.e.
$\mathrm{id}(\rho)=\rho$), and the identity superchannel as $\mathbb{1}$ (i.e.
$\mathbb{1}[\mathcal{E}]=\mathcal{E}$). Hence, when we write
$\mathbb{1}\otimes\Theta[\mathcal{E}]$, it should be unambiguously understood
as $(\mathbb{1}\otimes\Theta)[\mathcal{E}]$ since $\mathbb{1}$ is a supermap,
and should not be confused with $\mathrm{id}\otimes\Theta[\mathcal{E}]$.

###  2.2 Separable and Entanglement-Breaking Maps

We next review the meaning of separable and entanglement-breaking maps.

######  Definition 2.1.

[14, 15] A CP map $\mathcal{E}\in\mathrm{CP}(A_{0}R_{0}\to A_{1}R_{1})$ is
called $A_{0}A_{1}:R_{0}R_{1}$ separable if it can be written as
$\mathcal{E}=\sum_{k}\Phi_{k}\otimes\Psi_{k}$ for some
$\Phi_{k}\in\mathrm{CP}(A_{0}\to A_{1})$ and $\Psi_{k}\in\mathrm{CP}(R_{0}\to
R_{1})$.

It is not difficult to see that $\mathcal{E}^{A_{0}R_{0}\to A_{1}R_{1}}$ is
separable iff its Choi matrix

$J_{\mathcal{E}}^{A_{0}R_{0}A_{1}R_{1}}:=\mathrm{id}^{A_{0}R_{0}}\otimes\mathcal{E}^{\tilde{A}_{0}\tilde{R}_{0}\to
A_{1}R_{1}}(\phi_{+}^{A_{0}\tilde{A}_{0}}\otimes\phi_{+}^{R_{0}\tilde{R}_{0}})$
(4)

is $A_{0}A_{1}:R_{0}R_{1}$ separable [1]. This means that we can write
$J_{\mathcal{E}}^{A_{0}A_{1}:R_{0}R_{1}}=\sum_{k}M_{k}\otimes N_{k}$ for some
$M_{k}\in\mathcal{P}(A_{0}A_{1})$ and $N_{k}\in\mathcal{P}(R_{0}R_{1})$. As
alluded to in the introduction, separability of $\mathcal{E}^{A_{0}R_{0}\to
A_{1}R_{1}}$ 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 using the identity
$\mathcal{E}(\rho)=\operatorname{Tr}_{A_{0}R_{0}}[(\rho^{A_{0}R_{0}})^{T}\otimes
I^{A_{1}R_{1}}\leavevmode\nobreak\ J_{\mathcal{E}}^{A_{0}R_{0}A_{1}R_{1}}]$.

A close cousin to the separable maps are those that are entanglement breaking.

######  Definition 2.2.

[3] A CP map $\mathcal{N}\in\mathrm{CP}(A_{1}\to B_{1})$ is called
entanglement breaking (EB) if
$\mathrm{id}^{R_{1}}\mathbin{\mathop{\otimes}}\mathcal{N}^{A_{1}\to
B_{1}}(\rho^{R_{1}A_{1}})$ is separable for any $\rho^{R_{1}A_{1}}$ and any
system $R_{1}$. An EB map $\mathcal{N}$ 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 2.1.

For a CP map $\mathcal{N}^{A_{1}\to B_{1}}$, the following are equivalent.

  1. (A)

$\mathcal{N}^{A_{1}\to B_{1}}$ is EB.

  2. (B)

Its Choi matrix $J_{\mathcal{N}}^{A_{1}B_{1}}$ is $A_{1}:B_{1}$ separable.

  3. (C)

$\mathcal{N}^{A_{1}\to
B_{1}}(\rho)=\sum_{k}\operatorname{Tr}(F^{A_{1}}_{k}\rho^{A_{1}})\omega_{k}^{B_{1}}$,
for $F^{A_{1}}_{k}\geq 0$ and $\omega_{k}\in\mathcal{D}(B_{1})$.

  4. (D)

For any system $R$ and any bipartite channel $\mathcal{E}^{R_{0}B_{1}\to
R_{1}A_{1}}$, the composition $\mathcal{F}^{R_{0}A_{1}\to
R_{1}B_{1}}=\mathcal{N}^{A_{1}\to B_{1}}\circ\mathcal{E}^{R_{0}B_{1}\to
R_{1}A_{1}}\circ\mathcal{N}^{A_{1}\to B_{1}}$ is a separable map.

###### Proof.

Items (A)–(C) are standard results found in Ref. [3]. From the form of Eq.
(1), it is easy to see that $J_{\mathcal{F}}^{R_{0}A_{1}R_{1}B_{1}}$ is
$R_{0}R_{1}:A_{1}B_{1}$ separable whenever $\mathcal{N}$ is EB, and so (A)
$\Rightarrow$ (D). Conversely, if (D) holds for all bipartite channels
$\mathcal{E}^{R_{0}B_{1}\to R_{1}A_{1}}$, then by considering the discard-and-
prepare channel
$\mathcal{E}(X^{R_{0}B_{1}})=\operatorname{Tr}[X]\frac{1}{d_{A_{1}}}\phi_{+}^{R_{1}A_{1}}$
for $R_{1}\cong A_{1}$, we see that separability of $\mathcal{F}$ implies that
$\mathcal{N}$ is EB; hence (D) $\Rightarrow$ (A). ∎

Remark: In Section 3.3, we will see that the channel
$\mathcal{F}^{R_{0}A_{1}\to R_{1}B_{1}}$ constructed in (D) of this
proposition is the output of a conditional prepare-and-measure superchannel
$\mathcal{E}\mapsto\mathcal{N}\circ\mathcal{E}\circ\mathcal{N}$.

###  2.3 Supermaps and Superchannels

We next review the basic structure of superchannels. The following definitions
and theorems can be found in [16].

######  Definition 2.3.

[16, 2] A supermap $\Theta\in\mathbb{L}(A\to B)$ is called a superchannel if
both of the following are satisfied:

  1. (A)

$\Theta$ is completely CP preserving:
$\mathds{1}^{R}\otimes\Theta[\mathcal{E}^{RA}]$ is completely positive for any
CP map $\mathcal{E}$ and arbitrarily large system $R$.

  2. (B)

$\Theta$ is TP preserving: $\Theta[\mathcal{E}]$ is trace preserving for every
TP map $\mathcal{E}$.

We call $\Theta$ a CP supermap if condition (A) is satisfied.

The notion of superchannel defined here is equivalent to the deterministic
supermap defined in [2], and the term CP supermap here is equivalent to
probabilistic supermap or simply supermap in [2]. In contrast to [2], here we
use supermap to refer to any linear map in $\mathbb{L}(A\to B)$.

The Choi matrix of a supermap $\Theta^{A\to B}$ is defined as

$\mathcal{J}_{\Theta}^{AB}=\sum_{a_{0},a_{1}}J^{A}_{\mathcal{E}_{a_{0}a_{1}}}\otimes
J^{B}_{\Theta[\mathcal{E}_{a_{0}a_{1}}]},$ (5)

where $J^{A}_{\mathcal{E}_{a_{0}a_{1}}}$ and
$J^{B}_{\Theta[\mathcal{E}_{a_{0}a_{1}}]}$ are the Choi matrices of
$\mathcal{E}_{a_{0}a_{1}}\in\mathcal{L}(A)$ and
$\Theta[\mathcal{E}_{a_{0}a_{1}}]\in\mathcal{L}(B)$ respectively, and
$\left\\{\mathcal{E}_{a_{0}a_{1}}\right\\}_{a_{0},a_{1}}$ is a complete
orthogonal basis of $\mathcal{L}(A)$ whose action in the computational basis
is given by

$\mathcal{E}_{ijkl}(\rho^{A_{0}})=\mbox{$\left\langle
i\right|$}\rho^{A_{0}}\mbox{$\left|j\right\rangle$}\mbox{$\left|k\right\rangle$}\mbox{$\left\langle
l\right|$}^{A_{1}}.$ (6)

Alternatively, $\mathcal{J}_{\Theta}^{AB}$ equals the Choi matrix of
${(\mathds{1}^{A}\otimes\Theta^{\tilde{A}\to B})[\Phi_{+}^{A\tilde{A}}]}$,
where

$\Phi_{+}^{A\tilde{A}}\equiv\sum_{a_{0},a_{1}}\mathcal{E}^{A}_{a_{0}a_{1}}\otimes\mathcal{E}^{\tilde{A}}_{a_{0}a_{1}}.$
(7)

The action of $\Phi_{+}^{A\tilde{A}}$ can be expressed as

$\Phi_{+}^{A\tilde{A}}(\rho^{A_{0}\tilde{A_{0}}})=\operatorname{Tr}(\rho\phi_{+}^{A_{0}\tilde{A}_{0}})\phi_{+}^{A_{1}\tilde{A}_{1}},$
(8)

and hence it can be viewed as a maximally entangled map, in analogy with
$\phi_{+}$. Note that $\Phi_{+}$ is not trace-preserving and hence not a
quantum channel.

From the Choi-Jamiolkowski duality, there is a one-to-one correspondence
between supermaps and their Choi matrix. Hence, the linear spaces
$\mathbb{L}(A\to B)$, $\mathcal{L}(A_{0}\otimes A_{1}\to B_{0}\otimes B_{1})$,
$\mathcal{L}(A_{0}\otimes B_{0}\to A_{1}\otimes B_{1})$, and
$\mathcal{L}(A_{1}\otimes B_{0}\to A_{0}\otimes B_{1})$ are all isomorphic. We
have already seen that ${(\mathds{1}^{A}\otimes\Theta^{\tilde{A}\to
B})[\Phi_{+}^{A\tilde{A}}]}$ has the same Choi matrix as $\Theta$. Further
define $\Delta_{\Theta}\in\mathcal{L}(A\to B)$ to be the unique map that
satisfies $J_{\Delta_{\Theta}}=\mathcal{J}^{AB}_{\Theta}$, and
$\Gamma_{\Theta}\in\mathcal{L}(A_{1}B_{0}\to A_{0}B_{1})$ to be the unique map
that satisfies $J_{\Gamma_{\Theta}}=\mathcal{J}^{AB}_{\Theta}$ . The
properties of $\Theta^{A\to B}$ are then directly related to these three maps.
Specifically, $\Theta$ is a CP supermap iff
$(\mathds{1}\otimes\Theta)[\Phi_{+}^{A\tilde{A}}]$, $\Gamma^{A_{1}B_{0}\to
A_{0}B_{1}}_{\Theta}$ and $\Delta^{A\to B}_{\Theta}$ are CP. The condition
when $\Theta$ becomes a superchannel is a little bit more involved and is
given by the following lemma.

######  Lemma 2.1.

[2, 16] Let $\Theta\in\mathbb{L}(A\to B)$. The following are equivalent.

  1. (A)

$\Theta$ is a superchannel.

  2. (B)

The Choi matrix $\mathcal{J}^{AB}_{\Theta}\geq 0$ has marginals

$\mathcal{J}^{A_{1}B_{0}}_{\Theta}=I^{A_{1}B_{0}};\quad\mathcal{J}_{\Theta}^{AB_{0}}=\mathcal{J}^{A_{0}B_{0}}\otimes
u^{A_{1}}$ (9)

where $u^{A_{1}}\equiv\frac{1}{d_{A_{1}}}I^{A_{1}}$ is the normalized
maximally mixed state on $A_{1}$.

  3. (C)

There exists a Hilbert space $\mathcal{H}_{E}$ with $d_{E}\leq
d_{A_{0}}d_{B_{0}}$, and two CPTP maps $\Gamma_{\mathrm{pre}}^{B_{0}\to
A_{0}E}$ and $\Gamma_{\mathrm{post}}^{A_{1}E\to B_{1}}$ such that for all
linear maps $\mathcal{E}^{A}$,

$\Theta[\mathcal{E}^{A}]=\Gamma_{\mathrm{post}}^{A_{1}E\to
B_{1}}\circ\left(\mathcal{E}^{A}\otimes\mathrm{id}^{E}\right)\circ\Gamma_{\mathrm{pre}}^{B_{0}\to
A_{0}E}$ (10)

For a superchannel $\Theta^{A\to B}$, one can verify that the Choi matrix of
$\Gamma_{\mathrm{post}}^{A_{1}E\to B_{1}}\circ\Gamma_{\mathrm{pre}}^{B_{0}\to
A_{0}E}$ is $J^{AB}_{\Theta}$, and hence $\Gamma_{\Theta}^{A_{1}B_{0}\to
A_{0}B_{1}}=\Gamma_{\mathrm{post}}^{A_{1}E\to
B_{1}}\circ\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}E}$. On the other hand, by the
Choi-Jamiolkowski duality we have

$J_{\Theta[\mathcal{E}]}^{B}=\operatorname{Tr}_{A}\left(\mathcal{J}_{\Theta}^{AB}\left((J_{\mathcal{E}}^{A})^{\mathrm{T}}\otimes
I^{B}\right)\right).$ (11)

Therefore, the CP map $\Delta_{\Theta}^{A\to B}$ has a property that it
transforms the Choi matrix of a channel to another Choi matrix of a channel,
i.e. $\Delta_{\Theta}(J^{A}_{\mathcal{E}})=J^{B}_{\Theta[\mathcal{E}]}$.

Pictorially, the calculation of the Choi matrix of a superchannel
$\Theta^{A\to B}$ is shown in Fig 4.

Figure 4: Circuit diagram for calculating the Choi matrix
$\mathcal{J}_{\Theta}^{AB}$ of a superchannel $\Theta^{A\to B}$.

##  3 Characterization and Realizations of EBSCs

###  3.1 Characterization of EBSCs

With the background concepts in place from the previous section, we are now
able to introduce the notion of entanglement-breaking supermap and
entanglement-breaking superchannel.

######  Definition 3.1.

A CP supermap $\Theta^{A\to B}$ is called an entanglement-breaking supermap if
$\mathds{1}^{R}\otimes\Theta[\mathcal{E}^{RA}]$ is a separable map for every
$\mathcal{E}\in\textrm{CP}(RA)$, with $R$ being an arbitrary finite-
dimensional system. If $\Theta^{A\to B}$ is furthermore a superchannel, then
it is called an entanglement-breaking superchannel (EBSC).

In this work we primarily focus on EBSCs which is a more physical object.
However, in order to better understand the properties of EBSCs, in this and
the next subsections, we will also look at EB supermaps. Our first result
provides several equivalent characterizations of EB supermaps and EBSCs, in
analogy to Proposition 2.1 for EBCs.

######  Theorem 1.

For a CP supermap $\Theta^{A\to B}$, the following are equivalent.

  1. (A)

$\Theta^{A\to B}$ is an EB supermap.

  2. (B)

$\mathds{1}^{A}\otimes\Theta^{\tilde{A}\to B}[\Phi_{+}^{A\tilde{A}}]$ is
separable for
$\Phi_{+}^{A\tilde{A}}(\rho)=\operatorname{Tr}(\rho\phi_{+}^{A_{0}\tilde{A}_{0}})\phi_{+}^{A_{1}\tilde{A}_{1}}$.

  3. (C)

$\mathcal{J}_{\Theta}^{AB}$ is separable with respect to $A:B$.

  4. (D)

$\Theta$ can be realized with pre/post-processing CP maps
$\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}E}$ and
$\Gamma_{\mathrm{post}}^{A_{1}E\to B_{1}}$ such that
$\Gamma_{\Theta}^{A_{1}B_{0}\to A_{0}B_{1}}:=\Gamma_{\mathrm{post}}^{A_{1}E\to
B_{1}}\circ\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}E}$ is a separable map (with
respect to $A_{1}A_{0}:B_{0}B_{1}$).

  5. (E)

$\Delta_{\Theta}\in\mathcal{L}(A\to B)$ defined as the unique map with Choi
matrix $J_{\Theta}^{AB}$ is an entanglement breaking map.

If $\Theta$ is furthermore a superchannel, then the CP conditions in (D) can
be strengthened to CPTP.

Sketch of the proof of Theorem 1: $(B),\leavevmode\nobreak\
(D),\leavevmode\nobreak\ (E)$ correspond to three different CP maps that have
the same Choi matrix as $\Theta$, as discussed in Sec. 2, so it is easy to
show they are equivalent to $(C)$. $(A)\Rightarrow(B)$ is by definition of
EBSCs. $(C)\Rightarrow(A)$ can be shown by considering the Choi matrix of
$\mathds{1}\otimes\Theta[\mathcal{E}]$ for any bipartite CP map $\mathcal{E}$.

###### Proof.

$(A)\Rightarrow(B)$ is by Def. 3.1 of EBSCs.
$\mathds{1}^{A}\otimes\Theta^{\tilde{A}\to B}[\Phi_{+}^{A\tilde{A}}]$ is
separable if and only if its Choi matrix
$J_{\mathds{1}\otimes\Theta[\Phi_{+}^{A\tilde{A}}]}$ is separable with respect
to $A:B$. Since
$\mathcal{J}^{AB}_{\Theta}=J_{\mathds{1}\otimes\Theta[\Phi_{+}^{A\tilde{A}}]}$
by definition, we see that $(B)\Leftrightarrow(C)$.

We now prove $(C)\Rightarrow(A)$. For any bipartite map
$\mathcal{E}^{RA}\in\mathrm{CP}(R_{0}A_{0}\to R_{1}A_{1})$, according to Eq.
(11), we have

$\displaystyle J^{\tilde{R}B}_{\mathds{1}\otimes\Theta[\mathcal{E}]}$
$\displaystyle=\operatorname{Tr}_{RA}\left(\mathcal{J}_{\mathds{1}\otimes\Theta}^{RA\tilde{R}B}\left((J^{RA}_{\mathcal{E}})^{\mathrm{T}}\otimes
I^{\tilde{R}B}\right)\right)$ (12)
$\displaystyle=\operatorname{Tr}_{RA}\left(\left(\phi_{+}^{R_{0}\tilde{R}_{0}}\otimes\phi_{+}^{R_{1}\tilde{R}_{1}}\otimes\mathcal{J}_{\Theta}^{AB}\right)\left((J^{RA}_{\mathcal{E}})^{\mathrm{T}}\otimes
I^{\tilde{R}B}\right)\right)$ (13)
$\displaystyle=\operatorname{Tr}_{A}\left(\mathcal{J}^{AB}_{\Theta}\left((J^{\tilde{R}A}_{\mathcal{E}})^{\Gamma_{A}}\otimes
I^{B}\right)\right),$ (14)

where the superscript $\Gamma_{A}$ denotes partial transpose on system A.
Since system $R$ is actually unchanged, let $\tilde{R}=R$ and rewrite this
equation as

$J^{RB}_{\mathds{1}\otimes\Theta[\mathcal{E}]}=\operatorname{Tr}_{A}\left(\mathcal{J}^{AB}_{\Theta}\left(J^{RA}_{\mathcal{E}}\right)^{\Gamma_{A}}\right),$
(15)

where we omit the identity operator. Now $\mathcal{J}_{\Theta}^{AB}$ is
separable, which means it can be written as

$\mathcal{J}_{\Theta}^{AB}=\sum_{k=1}^{K}M^{A}_{k}\otimes N^{B}_{k}$ (16)

for some $M_{k}^{A}\in\mathcal{P}(A)$, $N_{k}^{B}\in\mathcal{P}(B)$, and
positive integer $K$. Substituting into Eq. (15), we get

$J^{RB}_{\mathds{1}\otimes\Theta[\mathcal{E}]}=\sum_{k}\operatorname{Tr}_{A}\left(M^{A}_{k}\left(J^{RA}_{\mathcal{E}}\right)^{\Gamma_{A}}\right)\otimes
N^{B},$ (17)

which means $J^{RB}_{\mathds{1}\otimes\Theta[\mathcal{E}]}$ is separable, and
hence $\mathds{1}^{R}\otimes\Theta^{A\to B}[\mathcal{E}^{RA}]$ is separable.
Since $\mathcal{E}^{RA}$ is an arbitrary bipartite CP map, we conclude that
$\Theta$ is an EBSC, which completes the proof of $(C)\Rightarrow(A)$.

This has established the equivalence
$(A)\Leftrightarrow(B)\Leftrightarrow(C)$. As for the equivalence of
$(C),\leavevmode\nobreak\ (D),\leavevmode\nobreak\ (E)$, we know that the Choi
matrix of $\Gamma_{\Theta}^{A_{1}B_{0}\to A_{0}B_{1}}$ and
$\Delta_{\Theta}^{A\to B}$ are both $\mathcal{J}^{AB}_{\Theta}$, as shown in
the last section. Therefore, the separability of $\mathcal{J}^{AB}_{\Theta}$
with respect to $A:B$ is equivalent to the separability of
$\Gamma_{\Theta}^{A_{1}B_{0}\to A_{0}B_{1}}$ with respect to
$A_{1}A_{0}:B_{0}B_{1}$, which is also equivalent to the fact that
$\Delta^{A\to B}$ is entanglement breaking.

The final remark on the case $\Theta$ being a superchannel is by Lemma 2.1
that the pre/post-processing map of $\Theta$ can always be chosen to be CPTP
(with $\Gamma_{\Theta}^{A_{1}B_{0}\to A_{0}B_{1}}$ unchanged). ∎

Figure 5: The map $\Gamma_{\Theta}^{A_{1}B_{0}\to
A_{0}B_{1}}=\Gamma_{\mathrm{post}}^{A_{1}E\to
B_{1}}\circ\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}E}$. For any EB supermap
$\Theta^{A\to B}$, Theorem 1 (D) requires this map to be
$A_{0}A_{1}:B_{0}B_{1}$ separable.

###  3.2 EBSCs realized by EB pre/post-processing maps

Theorem 1 (D) provides a structural requirement for the pre/post-processing
maps of an EBSC. However, the separability on $\Gamma_{\Theta}$ only
characterizes the concatenation of the pre/post-processing maps, as depicted
in Fig. 5. It remains unclear what constraints are placed on the pre/post-
processing maps individually in order for the resulting superchannel to be
entanglement-breaking. Here, we obtain such a condition when the physical
constraint of being trace-preserving is relaxed for the pre/post-processing
maps. Namely, we show that the pre/post-processing maps can be expressed as
partly entanglement-breaking maps, as defined below. While this provides some
mathematical insight into the structure of EBSCs, it is not a statement about
physical implementation due to the non-TP property of the maps. When the
pre/post-processing maps are required to valid quantum channels, it is unclear
whether the pre/post-processing maps can always be taken as party-EB maps.
Instead, we show that entanglement needs to be carried by the side channel $E$
that connects the pre and post-processing channels for the implementation of
certain EBSCs.

We begin by generalizing the definition of entanglement breaking for CP maps
that have bipartite input or output.

######  Definition 3.2.

A CP map $\mathcal{E}^{B_{0}\to A_{0}E}$ is called partly entanglement
breaking for output $A_{0}$ if $\mathrm{id}^{R}\otimes\mathcal{E}^{B_{0}\to
A_{0}E}(\rho^{RB_{0}})$ is separable with respect to $A_{0}:RE$ for any
$\rho\in\mathcal{D}(RB_{0})$ and any system $R$ with finite dimension.

######  Definition 3.3.

A CP map $\mathcal{E}^{EA_{1}\to B_{1}}$ is called partly entanglement
breaking for input $A_{1}$ if $\mathrm{id}^{RS}\otimes\mathcal{E}^{EA_{1}\to
B}(\sigma^{RE}\otimes\omega^{SA_{1}})$ is separable with respect to $S:RB_{1}$
for any $\sigma^{RE}\in\mathcal{D}(RE)$ and
$\omega^{SA_{1}}\in\mathcal{D}(SA_{1})$ for any system $R$, $S$ with finite
dimension.

The following two lemmas offer alternative characterizations of partly EB
maps, similar to Proposition 2.1 for standard EB maps.

######  Lemma 3.1.

For a CP map $\mathcal{E}^{B_{0}\to A_{0}E}$, the following are equivalent:

  1. (A)

$\mathcal{E}^{B_{0}\to A_{0}E}$ is partly EB for output system ${A}_{0}$.

  2. (B)

$\mathrm{id}^{B_{0}}\otimes\mathcal{E}^{\tilde{B}_{0}\to
A_{0}E}(\phi_{+}^{B_{0}\tilde{B}_{0}})$ is separable with respect to
$A_{0}:B_{0}E$.

  3. (C)

$\mathcal{E}^{B_{0}\to A_{0}E}$ can be written as

$\mathcal{E}^{B_{0}\to A_{0}E}(\rho^{B_{0}})=\sum_{k}\mathcal{E}_{k}^{B_{0}\to
E}(\rho^{B_{0}})\otimes\sigma^{A_{0}}_{k},$ (18)

for some $\mathcal{E}_{k}^{B_{0}\to E}\in\textrm{CP}(B_{0}\to E)$ and
$\sigma^{A_{0}}_{k}\in\mathcal{D}(B)$ independent of the input.

If $\mathcal{E}$ is furthermore CPTP, then $\\{\mathcal{E}^{B_{0}\to
E}_{k}\\}_{k}$ in (C) is a quantum instrument, which means
$\sum_{k}\mathcal{E}_{k}$ is CPTP.

###### Proof.

$(A)\Rightarrow(B)$ is by definition and $(C)\Rightarrow(A)$ is trivial. It
remains to prove $(B)\Rightarrow(C)$. Note $(B)$ says that the Choi matrix
$J^{B_{0}A_{0}E}_{\mathcal{E}}$ is separable with respect to $A_{0}:B_{0}E$.
Hence we can write

$J^{B_{0}A_{0}E}_{\mathcal{E}}=\sum_{k}M^{B_{0}E}_{k}\otimes\sigma^{A_{0}}_{k}$
(19)

for positive operators $M^{B_{0}E}_{k}$, $\sigma^{A_{0}}_{k}$, and without
loss of generality, $\operatorname{Tr}(\sigma^{A_{0}}_{k})=1$ for all $k$.
Taking $\mathcal{E}_{k}^{B_{0}\to E}$ to be the unique CP map whose Choi
matrix equals $M_{K}^{B_{0}E}$ completes the proof. When $\mathcal{E}$ is
furthermore CPTP, we have
$I^{B_{0}}=J^{B_{0}}_{\mathcal{E}}=\sum_{k}M^{B_{0}}_{k}$. Hence, each
$M^{B_{0}}_{k}$ is the Choi matrix for a CP map $\mathcal{E}_{k}^{B_{0}\to E}$
and their sum is trace-preserving. ∎

Similar conclusions hold for partly EB maps with a bipartite input.

######  Lemma 3.2.

For a quantum maps $\mathcal{E}^{EA_{1}\to B_{1}}$, the following are
equivalent.

  1. (A)

$\mathcal{E}^{EA_{1}\to B_{1}}$ is partly EB for input system $A_{1}$.

  2. (B)

$\mathrm{id}^{A_{1}E}\otimes\mathcal{E}^{\tilde{E}\tilde{A}_{1}\to
B_{1}}(\phi_{+}^{A_{1}\tilde{A}_{1}}\otimes\phi_{+}^{E\tilde{E}})$ is
separable with respect to $A_{1}:EB_{1}$.

  3. (C)

$\mathcal{E}^{EA_{1}\to B_{1}}$ can be written as

$\mathcal{E}^{EA_{1}\to B_{1}}(\rho^{EA_{1}})=\sum_{k}\mathcal{E}_{k}^{E\to
B_{1}}\left(\operatorname{Tr}_{A_{1}}(F^{A_{1}}_{k}\rho^{EA_{1}})\right).$
(20)

for some $\mathcal{E}_{k}^{E\to B_{1}}\in CP(E\to B_{1})$ and
$\\{F^{A_{1}}_{k}\\}\in\mathcal{P}(A_{1})$.

If $\mathcal{E}$ is furthermore CPTP, then $\\{\mathcal{E}_{k}^{E\to
B_{1}}\\}_{k}$ in (C) forms a quantum instrument and $\\{F^{A_{1}}_{k}\\}_{k}$
forms a POVM.

###### Proof.

The proof is very similar to the previous lemma. We only need to show
$(B)\Rightarrow(C)$. For any $\rho\in\mathcal{D}(EA_{1})$,

$\displaystyle\mathcal{E}^{EA_{1}\to B_{1}}(\rho^{EA_{1}})$
$\displaystyle=\operatorname{Tr}_{EA_{1}}(J_{\mathcal{E}}^{EA_{1}B_{1}}(\rho^{EA_{1}})^{\mathrm{T}})$
(21)
$\displaystyle=\sum_{k}\operatorname{Tr}_{EA_{1}}\left((F_{k}^{A_{1}})^{\mathrm{T}}\otimes
N_{k}^{EB_{1}}(\rho^{EA_{1}})^{\mathrm{T}}\right).$ (22)

Taking $\mathcal{E}_{k}^{E\to B_{1}}$ to be the unique CP map whose Choi
matrix equals $N_{k}^{EB_{1}}$ completes the proof. When $\mathcal{E}$ is
furthermore CPTP, we have that
$I^{EA_{1}}=J_{\mathcal{E}}^{EA_{1}}=\sum_{k}(F_{k}^{A_{1}})^{\mathrm{T}}\otimes
N_{k}^{E}$. Hence $\\{F_{k}\\}_{k}$ defines a POVM, and
$\\{N_{k}^{EB_{1}}\\}_{k}$ are Choi matrices for an instrument. ∎

Note that partly-EB maps are equivalent to bipartite separable maps with one
trivial input/output system. That is, $\mathcal{E}^{B_{0}\to A_{0}E}$ is
partly EB for output $A_{0}$ iff it is a separable map w.r.t. $A_{0}:B_{0}E$.
Similarly, $\mathcal{E}^{EA_{1}\to B_{1}}$ is partly EB for input $A_{1}$ iff
$\mathcal{E}^{EA_{1}\to B_{1}}$ is separable w.r.t. $A_{1}:EB_{1}$. This can
be seen from item (B) in Lemmas 3.1 and 3.2. One may also notice from Eq. (18)
and Eq. (20) that partly EB channels take a “measure-and-prepare” form. But we
must be careful here. Although Eq. (18) can indeed be interpreted as preparing
a state conditioned on the measurement outcome, Eq. (20) is like preparing a
CPTNI map conditioned on the measurement outcome, which in general cannot be
done physically. Even when $\mathcal{E}^{EA_{1}\to B_{1}}$ is trace-
preserving, the individual $\mathcal{E}_{k}^{E\to B_{1}}$ need not be
proportional to a trace-preserving map. This is the core reason why EBSCs
differ from prepare-and-measure superchannels as will be discussed later.

To see the relevance of partly EB channels to our problem, let us split system
$E$ in definitions 3.3 and 3.2 into two parts: $E=E_{A}E_{B}$ (see Fig. 6). In
this case, $\mathcal{E}^{B_{0}\to A_{0}E_{A}E_{B}}$ is partly EB for output
$A_{0}E_{A}$ iff it is a separable map w.r.t. $A_{0}E_{A}:B_{0}E_{B}$.
Similarly, $\mathcal{E}^{E_{A}E_{B}A_{1}\to B_{1}}$ is partly EB for input
$A_{1}E_{A}$ iff $\mathcal{E}^{EA_{1}\to B_{1}}$ is separable w.r.t.
$A_{1}E_{A}:B_{1}E_{B}$. Then it is not difficult to show that every EB
supermap can be realized with partly-EB pre/post-processing maps of this form.

######  Theorem 2.

For a CP supermap $\Theta\in\mathbb{L}(A\to B)$, the following are equivalent

  1. (A)

$\Theta^{A\to B}$ is an EB supermap.

  2. (B)

There exists a CPTP map $\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}E_{A}E_{B}}$
that is partly EB for the output system $A_{0}E_{A}$, and a CP map
$\Gamma_{\mathrm{post}}^{A_{1}E_{A}E_{B}\to B_{1}}$ that is partly EB for the
input system $A_{1}E_{A}$, such that

$\Theta^{A\to B}[\mathcal{E}^{A}]=\Gamma_{\mathrm{post}}^{A_{1}E_{A}E_{B}\to
B_{1}}\circ\left(\mathcal{E}^{A}\otimes\mathrm{id}^{E_{A}E_{B}}\right)\circ\Gamma_{\mathrm{pre}}^{B_{0}\to
A_{0}E_{A}E_{B}}$ (23)

Figure 6: A general way to implement an EB supermap.
$\Gamma_{\mathrm{pre}}^{B_{0}\to E_{A}E_{B}A_{0}}$ is partly EB for the output
system $E_{A}A_{0}$ and $\Gamma_{\mathrm{post}}^{A_{1}E_{A}E_{B}\to B}$ is
partly-EB for the input system $E_{A}A_{1}$.

###### Proof.

$(B)\Rightarrow(A)$: As is obvious from Fig. 6, a CP supermap which can be
realized as in (B) must be an EB supermap.

$(A)\Rightarrow(B)$: By Theorem 1, the Choi matrix of any EB supermap can be
written as

$\mathcal{J}_{\Theta}^{AB}=\sum_{j}M_{j}^{A}\otimes N_{j}^{B},$ (24)

for $M_{j}^{A}\in\mathcal{P}(A),\leavevmode\nobreak\
N_{j}^{B}\in\mathcal{P}(B)$. Let $E_{A}\cong A_{0},\leavevmode\nobreak\
E_{B}\cong B_{0}$. Define the following pre/post-processing map

$\displaystyle\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}E_{A}E_{B}}(\rho)$
$\displaystyle=\mathrm{id}^{B_{0}\to
E_{B}}(\rho)\otimes\frac{1}{d_{A_{0}}}\phi_{+}^{A_{0}E_{A}},$ (25)
$\displaystyle\Gamma_{\mathrm{post}}^{A_{1}E_{A}E_{B}\to B_{1}}(\rho)$
$\displaystyle=d_{A_{0}}\sum_{j}\mathcal{N}_{j}^{E_{B}\to
B_{1}}\left(\operatorname{Tr}_{E_{A}A_{1}}\left((M_{j}^{E_{A}A_{1}})^{T}\rho\right)\right),$
(26)

where $\mathcal{N}_{j}^{B_{0}\to B_{1}}$ is defined to be the unique CP map
whose Choi matrix equals to $N_{j}^{B}$. Obviously these two maps are CPTP and
CP respectively.

One can verify that the Choi matrix of
$\Gamma_{\mathrm{post}}^{A_{1}E_{A}E_{B}\to
B_{1}}\circ\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}E_{A}E_{B}}$ equals to
$\mathcal{J}_{\Theta}^{AB}$, which means that this pair of pre/post-processing
maps indeed implements $\Theta^{A\to B}$. On the other hand, by Lemma 3.1 and
Lemma 3.2, we see these two maps are partly-EB with respect to system
$E_{A}A_{0}$ and $E_{A}A_{1}$, respectively. This completes the proof. ∎

The above theorem provides a structural condition on the pre/post-processing
maps that can be used to construct any EB supermap. However, it fails to
conclude that every EBSC can be realized by a pair of partly-EB pre/post-
processing channels since the construction involves a non-TP map in Eq. (26).
Nevertheless, we conjecture that it is still possible to realize every EBSC by
partly-EB pre/post-processing channels using a different construction than the
one given in Theorem 2. We leave this conjecture as an open question to pursue
elsewhere. One difficulty in tackling this problem is that the splitting of
side channels $E$ into two parts $E=E_{A}E_{B}$ makes the general structure of
partly-EB pre/post-processing channels rather complex. Here we show that an
implementation as in Fig. 6 but with $E_{A}$ removed is not strong enough to
implement all EBSCs. Hence, implementing an EBSC will in general requires
entanglement to be distributed across a memory side channel. To see this, we
first make a simple observation.

######  Proposition 3.1.

Suppose that $\Theta$ is a superchannel realized by a partly EB pre-processing
map $\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}E}$ for $A_{0}$ and any CPTP post-
processing map $\Gamma_{\mathrm{post}}^{EA_{1}\to B_{1}}$. Then for any
$\rho^{RB_{0}}$ and $\omega^{SA_{1}}$, the output state

$\mathrm{id}^{RS}\otimes\Gamma_{\mathrm{post}}^{EA_{1}\to
B_{1}}\circ\Gamma_{\mathrm{pre}}^{B_{0}\to
A_{0}E}\left(\rho^{RB_{0}}\otimes\omega^{SA_{1}}\right)$ (27)

is separable across $A_{0}:RB_{1}S$. Similarly, if $\Theta$ is realized by a
partly EB post-processing map for $A_{1}$, then this output state is separable
across $A_{0}RB_{1}:S$.

###### Proof.

By definition,
$\sigma^{A_{0}:RE}=\mathrm{id}^{R}\otimes\Gamma_{\mathrm{pre}}^{B_{0}\to
A_{0}E}\left(\rho^{RB_{0}}\right)$ is separable across $A_{0}:RE$. Then
$\mathrm{id}^{S}\otimes\Gamma_{\mathrm{post}}^{EA_{1}\to
B_{1}}(\sigma^{A_{0}:RE}\otimes\omega^{SA_{1}})$ 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 $\Theta^{A\to B}$ is an EBSC that
can be realized by a partly EB pre-processing (resp. post-processing) map for
system $A_{0}$ (resp. system $A_{1}$), then we must have that
$\mathcal{J}_{\Theta}^{AB}$ is both $A_{0}A_{1}:B_{0}B_{1}$ separable as well
as $A_{0}:A_{1}B_{0}B_{1}$ (resp. $A_{1}:A_{0}B_{0}B_{1}$) separable. It is
not difficult to construct superchannels that fail to have this separability
structure.

######  Theorem 3.

There exist EBSCs that cannot be realized using either a pre-processing
channel that is partly EB for $A_{0}$ or a post-processing channel that is
partly EB for $A_{1}$. In other words, in Fig. 6, input system $A_{0}$ must be
entangled with some side channel $E_{A}$ and output system $A_{1}$ must couple
with $E_{A}$ nonlocally.

###### Proof.

By the previous proposition, it suffices to construct pre/post-processing maps
$\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}E}$ and
$\Gamma_{\mathrm{post}}^{EA_{1}\to B_{1}}$ such that

$\mathcal{J}_{\Theta}^{AB}=\mathrm{id}^{B_{0}A_{1}}\otimes\Gamma_{\mathrm{post}}^{E\tilde{A}_{1}\to
B_{1}}\circ\Gamma_{\mathrm{pre}}^{\tilde{B}_{0}\to
A_{0}E}\left(\phi^{+A_{1}\tilde{A}_{1}}\otimes\phi^{+B_{0}\tilde{B}_{0}}\right)$

is $A_{0}A_{1}:B_{0}B_{1}$ separable, but neither $A_{0}:A_{1}B_{0}B_{1}$ nor
$A_{1}:A_{0}B_{0}B_{1}$ is separable. Let $d_{B_{0}}=d_{A_{0}}=d_{A_{1}}=2$
and $d_{B_{1}}=d_{E}=3$. Define the isometry
$\Gamma_{\mathrm{pre}}^{\tilde{B}_{0}\to A_{0}E}$ by

$\displaystyle\mbox{$\left|0\right\rangle$}^{\tilde{B}_{0}}$
$\displaystyle\to\tfrac{1}{\sqrt{2}}(\mbox{$\left|00\right\rangle$}+\mbox{$\left|11\right\rangle$})^{A_{0}E},$
$\displaystyle\mbox{$\left|1\right\rangle$}^{\tilde{B}_{0}}$
$\displaystyle\to\mbox{$\left|02\right\rangle$}^{A_{0}E}.$ (28)

Hence $\mathrm{id}^{B_{0}}\otimes\Gamma_{\mathrm{pre}}^{\tilde{B}_{0}\to
A_{0}E}(\phi^{+B_{0}\tilde{B}_{0}})=|\tau\rangle\langle\tau|$, where
$\mbox{$\left|\tau\right\rangle$}=\frac{1}{\sqrt{2}}(\mbox{$\left|00\right\rangle$}+\mbox{$\left|11\right\rangle$})^{A_{0}E}\mbox{$\left|0\right\rangle$}^{B_{0}}+\mbox{$\left|02\right\rangle$}^{A_{0}E}\mbox{$\left|1\right\rangle$}^{B_{0}}$.
The post-processing map $\Gamma_{\mathrm{post}}^{E\tilde{A}_{1}\to B_{1}}$ is
then a POVM with elements

$\displaystyle\Pi_{0}$ $\displaystyle=\frac{1}{2}\phi^{+E\tilde{A}_{1}},$
$\displaystyle\Pi_{1}$ $\displaystyle=(I-|2\rangle\langle 2|)^{E}\otimes
I^{\tilde{A}_{1}}-\frac{1}{2}\phi^{+E\tilde{A}_{1}}$ $\displaystyle\Pi_{2}$
$\displaystyle=|2\rangle\langle 2|^{E}\otimes I^{\tilde{A}_{1}}.$ (29)

This leads to the Choi matrix

$\displaystyle\mathcal{J}_{\Theta}^{AB}=\tfrac{1}{4}$
$\displaystyle\phi^{+A_{0}A_{1}}\otimes|00\rangle\langle
00|^{B_{0}B_{1}}+\tfrac{1}{2}\left(I^{A_{0}A_{1}}-\tfrac{1}{2}\phi^{+A_{0}A_{1}}\right)\otimes|01\rangle\langle
01|^{B_{0}B_{1}}$ $\displaystyle+|0\rangle\langle 0|^{A_{0}}\otimes
I^{A_{1}}\otimes|12\rangle\langle 12|^{B_{0}B_{1}}.$ (30)

Clearly it is $A:B$ separable, and yet it is entangled for both parts $A_{0}$
and $A_{1}$ since projecting onto
$\mbox{$\left|00\right\rangle$}^{B_{0}B_{1}}$ leads to the maximally entangled
state $\tfrac{1}{4}\phi^{+A_{0}A_{1}}$. ∎

###  3.3 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. Recall that a quantum
instrument is a collection of CP maps $\\{\mathcal{E}_{k}\\}_{k}$ whose sum
$\sum_{k}\mathcal{E}_{k}$ is TP. For a generalized measurement on state $\rho$
described by instrument $\\{\mathcal{E}_{k}\\}_{k}$, outcome $k$ occurs with
probability $p_{k}=\operatorname{Tr}[\mathcal{E}_{k}(\rho)]$ and the post-
measurement state is $\mathcal{E}_{k}(\rho)/p_{k}$. In Ref. [17], the authors
introduce the concept of a quantum super-instrument, which is a set of c-CPTNI
(completely CP preserving and trace-non-increasing preserving) supermaps
$\\{\Theta_{x}\\}$ such that $\sum_{x}\Theta_{x}$ is a superchannel. A quantum
super-instrument therefore describes in one sense how a quantum channel can be
measured. It is proven in [17] that every super-instrument can be realized
with a CPTP pre-processing channel and a post-processing quantum instrument,
as shown on the left side of Fig. 7. We are particularly interested in a
special kind of super-instrument with trivial output channel, which is a
generalization of the positive operator-valued measurement (POVM) to quantum
channels, as shown on the right side of Fig. 7. Such object has been
considered in the study of quantum channel discrimination [18, 19] where it is
called a tester or a process-POVM, and in the study of quantum games [20]
where it is called a co-strategy.

Figure 7: (Left) Implementation of a quantum super-instrument. A CPTP map
$\mathcal{E}$ is performed as pre-processing while a quantum instrument
$\Lambda=\sum_{x}\Lambda_{x}\otimes\mbox{$\left|x\right\rangle$}\mbox{$\left\langle
x\right|$}$ encompasses the post-processing. (Right) A POVM of quantum
channels. The pre-processing is a quantum state $\rho$ and the post processing
is a bipartite measurement channel $P=\sum_{x}\operatorname{Tr}_{x}(\cdot
P_{x})\otimes\mbox{$\left|x\right\rangle$}\mbox{$\left\langle x\right|$}$ for
POVM $\\{P_{x}\\}_{x}$.

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. 8 where the channel $\mathcal{F}^{B}_{x}$ is prepared
contingent on the outcome $x$ of the proceeding channel POVM.

######  Definition 3.4.

A superchannel $\Theta^{A\to B}$ is called a measure-and-prepare superchannel
(MPSC) if it can be realized as

$\Theta^{A\to
B}[\mathcal{E}^{A}]=\sum_{x}\operatorname{Tr}\left({\mathrm{id}^{E}\otimes\mathcal{E}^{A}(\rho^{A_{0}E})P_{x}^{A_{1}E}}\right)\mathcal{F}_{x}^{B},$
(31)

where $\rho^{A_{0}E}$ is some quantum state, $\\{P_{x}^{A_{1}E}\\}$ is some
POVM, and $\\{\mathcal{F}_{x}^{B}\\}$ is a collection of CPTP maps.

Figure 8: A measure-and-prepare superchannel (MPSC).

There is a good physical motivation to consider MPSCs from the point of view
of quantum memories. An experimenter might want to transform an input channel
$\mathcal{E}^{A}$ into a new channel that could be used at some later time.
For an MPSC, no quantum memory is needed to accomplish this goal. Input
channel $\mathcal{E}^{A}$ can be processed at time $t_{0}$ with output $x$
stored in classical memory. Then at some later time $t_{1}$, system $B_{0}$
can be directly processed by map $\mathcal{F}_{x}$. Note, for this
interpretation to hold, it is crucial that $\mathcal{F}_{x}$ be trace-
preserving.

In the definition of an MPSC, the choice of pre-processing state
$\rho^{E_{1}A_{0}}$ provides an extra degree of freedom that is not present in
the channel case. Furthermore, the state $\rho^{E_{1}A_{0}}$ 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 3.5.

A superchannel $\Theta^{A\to B}$ is called a controlled measure-and-prepare
superchannel (CMPSC) if it can be realized as

$\Theta^{A\to
B}[\mathcal{E}^{A}]=\sum_{xy}\operatorname{Tr}\left({\mathrm{id}^{E_{1}}\otimes\mathcal{E}^{A}(\rho_{y}^{A_{0}E_{1}}){P_{x}^{A_{1}E_{1}}}}\right)\mathcal{F}_{x}^{E_{2}\to
B_{1}}\circ\Lambda_{y}^{B_{0}\to E},$ (32)

where $\\{\Lambda_{y}\\}_{y}$ is a quantum instrument, $\rho_{y}^{A_{0}E}$ is
some quantum state, $\\{P_{x}^{A_{1}E}\\}_{x}$ is a POVM, and
$\\{\mathcal{F}_{x}\\}_{x}$ is a family of CPTP maps.

The realization of a CMPSC is shown in Fig. 9.

Figure 9: A controlled measure-and-prepare superchannel (CMPSC).

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.

Figure 10: When $A_{0}$ is one-dimensional, a CMPSC reduces to a 2-way LOCC
map on product states. Since there are separable maps that cannot be
implemented by 2-way LOCC, there exist EBSCs that are not CMPSCs.

######  Theorem 4.

There exist EBSCs that are not CMPSC.

###### Proof.

The construction uses the fact that there exist channels
$\mathcal{E}^{E_{2}A_{1}\to B_{1}}$ that are partly EB for $A_{1}$, yet they
are not implementable by local operations and classical communication (LOCC).
From the remark following Lemma 3.1, a partly EB channel for input $A_{1}$ is
an $A_{1}:E_{2}B_{1}$ separable channel. Let $\mathcal{E}^{E_{2}A_{1}\to
B_{1}}$ be any such channel whose action on a collection of product states
$\\{\mbox{$\left|\alpha_{i}\right\rangle$}^{A_{1}}\otimes\mbox{$\left|\beta_{i}\right\rangle$}^{E_{2}}\\}$
cannot be simulated by LOCC. Such maps arise, for instance, when considering
orthogonal product bases that can be distinguished by separable operations but
not LOCC, a phenomenon famously known as nonlocality without entanglement
[21]. Another example is the so-called double-trine ensemble [22], given by
$\mbox{$\left|\alpha_{i}\right\rangle$}=\mbox{$\left|\beta_{i}\right\rangle$}=U^{i}\mbox{$\left|0\right\rangle$}$
for $i=0,1,2$, where $U=\text{exp}(-i\tfrac{\pi}{3}\sigma_{y})$. While an
optimal minimum-error discrimination measurement on
$\\{\mbox{$\left|\alpha_{i}\right\rangle$}\otimes\mbox{$\left|\beta_{i}\right\rangle$}\\}_{i=0,1,2}$
can be achieved by a separable map, it cannot be implemented by LOCC [23]. Let
$\mathcal{E}^{E_{2}A_{1}\to B_{1}}$ be any such non-LOCC channel whose action
is

$\mathcal{E}^{E_{2}A_{1}\to
B_{1}}:\mbox{$\left|\alpha_{i}\right\rangle$}\mbox{$\left|\beta_{i}\right\rangle$}\mapsto\rho_{i}$

for all $i$. We can regard this as a post-processing map for a superchannel
that converts each state $\left|\alpha_{i}\right\rangle$ into a QC channel
having action $\mbox{$\left|\beta_{i}\right\rangle$}\to\rho_{i}$. Since
$\mathcal{E}^{E_{2}A_{1}\to B_{1}}$ 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. 10, which
is not possible by construction. Therefore the superchannel is not CMPSC. ∎

##  4 Superactivation of EBSC

Entanglement-breaking channels have the property that they are closed under
tensor product. That is, if $\mathcal{E}^{A_{0}\to A_{1}}$ and
$\mathcal{N}^{R_{0}\to R_{1}}$ are both EB channels, then so is their tensor
product, $[\mathcal{E}\otimes\mathcal{N}]^{A_{0}R_{0}\to A_{1}R_{1}}$. In this
section we consider whether an analogous result holds for 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. 11. 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, as shown in Fig. 12. While the full theory
of sequential processing can be described using the formalism of quantum combs
[24, 25] or quantum strategies [20, 26, 27], here we do not need to invoke
these to demonstrate the generic superactivation of EBSC.

Figure 11: A parallel implementation of two EBSCs is still an EBSC.

######  Theorem 5.

Suppose $\Theta^{A\to B}$ is an EBSC. Then two copies of $\Theta^{A\to B}$
will be an EBSC when used in parallel. However, if
$\mathcal{J}_{\Theta}^{A_{1}B_{0}B_{1}}$ is entangled across
$A_{1}B_{0}:B_{1}$ and $d_{A_{1}}\geq d_{B_{0}}$, then two copies of
$\Theta^{A\to B}$ is no longer an EBSC when used in series.

###### Proof.

When two copies of $\Theta^{A\to B}$ are used in parallel, they become a
single superchannel $\Theta^{AA^{\prime}\to BB^{\prime}}$. Its Choi matrix
satisfies
$\mathcal{J}_{\Theta}^{AA^{\prime}BB^{\prime}}=\mathcal{J}_{\Theta}^{AB}\otimes\mathcal{J}_{\Theta}^{A^{\prime}B^{\prime}}$,
and since two copies of a separable operator is separable, it follows that
$\mathcal{J}_{\Theta}^{AA^{\prime}BB^{\prime}}$ represents an EBSC. On the
other hand, suppose that each copy of $\Theta^{A\to B}$ can be used in series.
Let $d_{R_{0}}=d_{A_{0}}$, $d_{R_{1}}=d_{A_{1}}$, and define the channel
$\mathcal{E}_{0}^{RA}$ that discards its input and prepares the state
$\frac{1}{d_{B_{0}}}\phi_{+}^{R_{1}B^{\prime}_{0}}$. Likewise, let
$\mathcal{E}_{1}^{R^{\prime}A^{\prime}}$ be the channel that discards and
prepares $\frac{1}{d_{A_{1}}}\phi_{+}^{R^{\prime}_{1}A_{1}^{\prime}}$. Then
with the wiring shown in Fig. 12, it follows that the resulting channel will
be the state

$\Omega^{R_{1}R_{1}^{\prime}B_{1}^{\prime}}=d^{-1}_{B_{0}}d^{-1}_{A_{1}}\operatorname{Tr}_{A_{0}^{\prime}}\left(\mathrm{id}^{R^{\prime}_{1}}\otimes\Gamma_{\mathrm{post}}^{EA_{1}^{\prime}\to
B_{1}^{\prime}}\circ\mathrm{id}^{R_{1}}\otimes\Gamma_{\mathrm{pre}}^{B_{0}^{\prime}\to
A_{0}^{\prime}E}\left(\phi_{+}^{R_{1}B_{0}^{\prime}}\otimes\phi_{+}^{R_{1}^{\prime}A_{1}^{\prime}}\right)\right).$
(33)

However, this is proportional to
$\mathcal{J}_{\Theta}^{A^{\prime}_{1}B^{\prime}_{0}B^{\prime}_{1}}$ with
$A_{1}^{\prime}$ replaced by $R_{1}^{\prime}$ and $B_{0}^{\prime}$ replaced by
$R_{1}$. The $A_{1}^{\prime}B^{\prime}_{0}:B^{\prime}_{1}$ entanglement
implies that $\Omega^{R_{1}R_{1}^{\prime}B_{1}^{\prime}}$ is entangled between
Rachel and Bob. Hence $\mathcal{E}_{0}^{RA}$ and
$\mathcal{E}_{1}^{R^{\prime}A^{\prime}}$ is transformed into an entangled
channel by two copies of the EBSC $\Theta^{A\to B}$. ∎

Figure 12: A series implementation of two EBSCs that can yield a non-separable
channel. The channels $\mathcal{E}_{0}^{RA}$ and
$\mathcal{E}_{1}^{R^{\prime}A^{\prime}}$ discard their inputs and prepare
maximally entangled states.

##  5 Image of EBSC

In Section 3 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 channel-generation power as EBSC. For this
purpose, we introduce the notion of _CPTP-complete image_.

######  Definition 5.1.

Given a superchannel $\Theta^{A\to B}$, the CPTP image of $\Theta^{A\to B}$ is
defined as

$\mathrm{Im}_{\Theta}=\left\\{\mathcal{G}^{B}\in\mathrm{CPTP}(B)\leavevmode\nobreak\
|\leavevmode\nobreak\ \mathcal{G}^{B}=\Theta^{A\to
B}[\mathcal{E}^{A}],\leavevmode\nobreak\
\mathcal{E}^{A}\in\mathrm{CPTP}(A)\right\\}.$ (34)

The CPTP-complete image of $\Theta^{A\to B}$ is defined as

$\mathrm{C}_{\Theta}=\bigcup_{R}\mathrm{Im}_{\mathds{1}^{R}\otimes\Theta},$
(35)

which is the union of CPTP images $\mathds{1}^{R}\otimes\Theta^{A\to B}$ for
all systems $R$. The CPTP-complete image of a set of superchannels
$\mathcal{S}$ is defined as
$\mathrm{C}_{\mathcal{S}}=\bigcup_{\Theta\in\mathcal{S}}\mathrm{C}_{\Theta}.$

Similarly, we introduce the _CP-complete image_ as follows.

######  Definition 5.2.

Given a superchannel $\Theta^{A\to B}$, the CP image of $\Theta^{A\to B}$ is
defined as

$\mathrm{Im}^{*}_{\Theta}=\left\\{\mathcal{G}^{B}\in\mathrm{CP}(B)\leavevmode\nobreak\
|\leavevmode\nobreak\ \mathcal{G}^{B}=\Theta^{A\to
B}[\mathcal{E}^{A}],\leavevmode\nobreak\
\mathcal{E}^{A}\in\mathrm{CP}(A)\right\\}.$ (36)

The CP-complete image of $\Theta^{A\to B}$ is defined as

$\mathrm{C}^{*}_{\Theta}=\bigcup_{R}\mathrm{Im}^{*}_{\mathds{1}^{R}\otimes\Theta},$
(37)

which is the union of CP images $\mathds{1}^{R}\otimes\Theta^{A\to B}$ for all
systems $R$. The CP-complete image of a set of superchannels $\mathcal{S}$ is
defined as
$\mathrm{C}^{*}_{\mathcal{S}}=\bigcup_{\Theta\in\mathcal{S}}\mathrm{C}^{*}_{\Theta}.$

Alternatively, we can say that the CPTP image of $\Theta^{A\to B}$ is its
image when the domain is restricted to CPTP maps, and likewise the CP image of
$\Theta^{A\to B}$ is its image when the domain is restricted to CP maps.

With these definitions in place, a superchannel $\Theta^{A\to B}$ is an EBSC
if and only if $\mathrm{C}^{*}_{\Theta}\subseteq\mathrm{SEP}^{*}$. Here we use
SEP∗ to denote the set of separable CP maps while SEP denotes the set of all
separable CPTP maps. An interesting question is whether
$\mathrm{C}_{\mathrm{EBSC}}=\mathrm{SEP}$ and
$\mathrm{C}^{*}_{\mathrm{EBSC}}=\mathrm{SEP}^{*}$ 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 $\mathcal{E}$, the set
$\mathrm{Im}_{\mathcal{E}}$ (resp. $\mathrm{Im}^{*}_{\mathcal{E}}$) is the
image of $\mathcal{E}$ when the domain is restricted to density operators
(resp. positive operators). In this case, it is easy to see that both
$\mathrm{C}_{\mathrm{EBC}}=\mathrm{SEP}$ and
$\mathrm{C}^{*}_{\mathrm{EBC}}=\mathrm{SEP}^{*}$ hold. Indeed for an arbitrary
separable positive operator
$\rho^{RB}=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|^{R}\otimes|\alpha_{i}\rangle\langle\alpha_{i}|^{B}$,
one need only consider the action of the EBC $\mathcal{E}^{A\to B}$ on the
positive operator
$\sigma^{RA}=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|^{R}\otimes|i\rangle\langle
i|^{A}$, where $\mathcal{E}^{A}(X)=\mbox{$\left\langle
i\right|$}X\mbox{$\left|i\right\rangle$}^{A}|\alpha_{i}\rangle\langle\alpha_{i}|^{B}$,
and $\sigma^{RA}$ is a density matrix if and only if $\rho^{RB}$ 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 $\mathrm{C}_{\mathrm{EBSC}}$, we relate
$\mathrm{C}_{\mathrm{MPSC}}$ and $\mathrm{C}_{\mathrm{CMPSC}}$ to LOCC
channels of certain communication rounds. As for the CP-complete image, we
show that $\mathrm{C}^{*}_{\mathrm{EBSC}}=\mathrm{SEP}^{*}$ 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.

###  5.1 CPTP-complete Image of EBSC

As noted above, one of our primary interests is determining whether
$\mathrm{C}_{\mathrm{EBSC}}=\mathrm{SEP}$. 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.

######  Definition 5.3.

[28] A channel $\mathcal{E}\in\mathrm{CPTP}(R_{0}B_{0}\to R_{1}B_{1})$ is
called a ($B\to R\to B$) two-round LOCC channel if it can be written as

$\mathcal{E}^{RB}=\sum_{ij}\Gamma^{R_{0}\to
R_{1}}_{j|i}\otimes(\mathcal{F}^{B_{2}\to B_{1}}_{ij}\circ\Lambda^{B_{0}\to
B_{2}}_{i})$ (38)

for some quantum instrument $\\{\Lambda_{i}^{B_{0}\to B_{2}}\\}_{i}$ and
family of channels $\\{\mathcal{F}^{B_{2}\to B_{1}}_{ij}\\}_{ij}$ for Bob, and
a family of instruments $\\{\Gamma^{R_{0}\to R_{1}}_{j|i}\\}_{i,j}$ for
Rachel. An ($R\to B$) one-way LOCC channel is defined similarly except with
the added condition that all the $\Lambda_{i}$ are trivial; i.e.
$\mathcal{E}^{RB}=\sum_{j}\Gamma_{j}^{R}\otimes\mathcal{F}_{j}^{B}$.

######  Proposition 5.1.

The CPTP-complete images of EBSC, MPSC and CMPSC satisfy the following
relation.

$\mathrm{C}_{\mathrm{MPSC}}=\mathrm{LOCC_{1}}\subsetneq\mathrm{LOCC_{2}}=\mathrm{C}_{\mathrm{CMPSC}}\subsetneq\mathrm{C}_{\mathrm{EBSC}}\subseteq\mathrm{SEP},$
(39)

where $\mathrm{LOCC_{1}}$ denotes the set of all ($R\to B$) one-round LOCC
channels, $\mathrm{LOCC_{2}}$ denotes the set of all ($B\to R\to B$) two-round
LOCC channels, and $\mathrm{SEP}$ denotes the set of all bipartite separable
channels.

Figure 13: A CMPSC superchannel transforms any $\mathcal{E}^{RA}$ into a
channel $\mathcal{F}^{RB}$ that can implemented by a two-round LOCC channel
from Bob to Rachel.

###### Proof.

For the second equality, first notice that if $\Theta$ is CMPSC, then
$\mathds{1}\otimes\Theta[\mathcal{E}]$ is a two-round LOCC channel for every
quantum channel $\mathcal{E}^{RA}$, as can be seen from Fig. 13. This
establishes $\mathrm{C}_{\mathrm{CMPSC}}\subseteq\mathrm{LOCC}_{2}$. On the
other hand, consider an arbitrary ($B\to R\to B$) two-round LOCC channel
$\mathcal{G}$. By definition 5.3, it can be written as

$\mathcal{G}^{RB}=\sum_{ij}\Gamma^{R}_{j|i}\otimes(\mathcal{F}^{B_{2}\to
B_{1}}_{ij}\circ\Lambda^{B_{0}\to B_{2}}_{i})$ (40)

for some CPTP maps $\mathcal{F}^{B}_{ij}$ and deterministic quantum
instruments $\\{\Lambda^{B}_{i}\\}$, $\\{\Gamma^{R}_{j|i}\\}$. Set
$B_{2}=E_{2}$ and construct a superchannel $\Theta$ with the following
realization:

$\displaystyle\Gamma^{B_{0}\to A_{0}\hat{E}_{2}E_{2}}_{\mathrm{pre}}(\cdot)$
$\displaystyle=\sum_{i}\mbox{$\left|i\right\rangle$}\mbox{$\left\langle
i\right|$}^{A_{0}}\otimes|i\rangle\langle
i|^{\hat{E}_{2}}\otimes\Lambda^{B_{0}\to E_{2}}_{i}(\cdot),$ (41)
$\displaystyle\Gamma^{A_{1}\hat{E}_{2}E_{2}\to B_{1}}_{\mathrm{post}}(\cdot)$
$\displaystyle=\sum_{i,j}\mbox{$\left\langle
j\right|$}\cdot\mbox{$\left|j\right\rangle$}^{A_{1}}\mbox{$\left\langle
i\right|$}\cdot\mbox{$\left|i\right\rangle$}^{\hat{E}_{2}}\mathcal{F}_{ij}^{E_{2}\to
B_{1}}(\cdot).$ (42)

Here, system $E_{1}$ is taken to be trivial. It is easy to see that $\Theta$
is a CMPSC by Def. 3.5. Then consider the quantum channel

$\mathcal{E}^{RA}(\cdot)=\sum_{ij}\Gamma_{j|i}^{R}(\cdot)\otimes\mbox{$\left\langle
i\right|$}\cdot\mbox{$\left|i\right\rangle$}^{A_{0}}\otimes\mbox{$\left|j\right\rangle$}\mbox{$\left\langle
j\right|$}^{A_{1}}.$ (43)

One can immediately verify that
$\mathcal{G}^{RB}=\mathds{1}^{R}\otimes\Theta^{A\to B}[\mathcal{E}^{RA}],$
which means $\mathcal{G}^{RB}\in\mathrm{C}_{\mathrm{CMPSC}}$. By the
arbitrariness of $\mathcal{G}$ we have
$\mathrm{LOCC_{2}}\subseteq\mathrm{C_{CMPSC}}$, hence
$\mathrm{LOCC_{2}}=\mathrm{C}_{\mathrm{CMPSC}}$.

For the first equality in Eq. (39), we can similarly verify that
$\mathrm{C_{MPSC}}\subseteq\mathrm{LOCC}_{1}$ from Fig. 13 when the classical
channel from Bob to Rachel is removed. For an arbitrary ($R\to B$) one-way
LOCC channel
$\mathcal{G}^{RB}=\sum_{i}\Gamma^{R}_{i}\otimes\mathcal{F}_{i}^{B}$ with CPTP
maps $\mathcal{F}_{i}$ and quantum instrument $\\{\Gamma_{i}\\}_{i}$,
construct a superchannel $\Theta^{A_{1}\to B}$ with trivial $A_{0}$ system as

$\Theta^{A_{1}\to B}[\cdot]=\sum_{i}\mbox{$\left\langle
i\right|$}\cdot\mbox{$\left|i\right\rangle$}^{A_{1}}\otimes\mathcal{F}_{i}^{B},$
(44)

and further define a quantum channel $\mathcal{E}^{R_{0}\to
R_{1}A_{1}}=\sum_{i}\Gamma^{R}_{i}\otimes\mbox{$\left|i\right\rangle$}\mbox{$\left\langle
i\right|$}^{A_{1}}$. Then we have $\mathds{1}^{R}\otimes\Theta^{A_{1}\to
B}[\mathcal{E}]=\mathcal{G}^{RB}$. It is obvious that $\Theta^{A_{1}\to B}$ is
an MPSC, and so $\mathcal{G}\in\mathrm{C_{MPSC}}$. Combining the above results
we get $\mathrm{C_{MPSC}}=\mathrm{LOCC}_{1}$.

As for $\mathrm{C_{CMPSC}}\subsetneq\mathrm{C_{EBSC}}$, recall the EBSC we
constructed to prove Theorem 4. By applying that EBSC at an input noiseless
channel $\mathrm{id}^{R_{0}\to A_{1}}$, we obtain a separable channel that
cannot be implemented by LOCC. This complete the proof of proposition 5.1. ∎

The above proposition precisely characterizes the CPTP-complete image of MPSC
and CMPSC. Also, it tells us $\mathrm{C_{EBSC}}$ contains all ($B\to R\to B$)
two-round LOCC channels and some non-LOCC separable channels. It remains an
open problem whether $\mathrm{C_{EBSC}}=\mathrm{SEP}$. We next turn to the CP-
complete image.

###  5.2 CP-complete Image of EBSC

Interestingly, we can easily prove $\mathrm{C^{*}_{EBSC}}=\mathrm{SEP^{*}}$.
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 5.2.

The CP-complete image of EBSC and CMPSC satisfy

$\mathrm{C^{*}_{CMPSC}}=\mathrm{C^{*}_{EBSC}}=\mathrm{SEP^{*}},$ (45)

where $\mathrm{SEP^{*}}$ is the set of all bipartite separable CP maps.

###### Proof.

By definition, $\mathrm{C^{*}_{EBSC}}\subseteq\mathrm{SEP^{*}}$. We only need
to prove that $\mathrm{C^{*}_{EBSC}}\supseteq\mathrm{SEP^{*}}$. It is enough
to show that $\mathrm{C^{*}_{EBSC}}$ 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 $\mathcal{G}^{RB}$, its Choi matrix can
be written as $J^{RB}_{\mathcal{G}}=\sum_{k=1}^{r}P^{R_{0}R_{1}}_{k}\otimes
Q^{B_{0}B_{1}}_{k}$ such that $P_{k},\leavevmode\nobreak\ Q_{k}$ are positive
operators. Without loss of generality we take $\operatorname{Tr}(P_{k})=1$.
Since $\mathcal{G}^{RB}$ is CPTNI, we have $\sum_{k}P^{R_{0}}_{k}\otimes
Q^{B_{0}}_{k}\leq I^{R_{0}B_{0}}$, and so $\sum_{k}Q^{B_{0}}_{k}\leq
I^{B_{0}}$. Let $F^{B_{0}B_{1}}$ be a positive operator satisfying
$F^{B_{0}}=I^{B_{0}}-\sum_{k}Q^{B_{0}}_{k}$. Then for an $(r+1)$-dimensional
system $A_{0}$ and system $A_{1}$ being one-dimensional, construct a supermap
$\Theta$ whose Choi matrix is as follows.

$\mathcal{J}^{AB}_{\Theta}=\sum_{k=1}^{r}\mbox{$\left|k\right\rangle$}\mbox{$\left\langle
k\right|$}^{A_{0}}\otimes
Q_{k}^{B_{0}B_{1}}+\mbox{$\left|r+1\right\rangle$}\mbox{$\left\langle
r+1\right|$}^{A_{0}}\otimes F^{B_{0}B_{1}}.$ (46)

Since $A_{1}$ is one-dimensional, it holds trivially that
$\mathcal{J}^{AB_{0}}_{\Theta}=\mathcal{J}^{A_{0}B_{0}}_{\Theta}\otimes
u^{A_{1}}$ and $\mathcal{J}^{A_{1}B_{0}}_{\Theta}=I^{A_{1}B_{0}}$, and hence
$\Theta$ is a superchannel. Furthermore, since $\mathcal{J}^{AB}_{\Theta}$ is
separable with respect to $A:B$, it is also an EBSC, by Theorem 1. Define a CP
map $\mathcal{E}$ whose Choi matrix is given by

$J^{RA}_{\mathcal{E}}=\sum_{k=1}^{r}P_{k}^{R}\otimes\mbox{$\left|k\right\rangle$}\mbox{$\left\langle
k\right|$}^{A_{0}}.$ (47)

Note the condition $\sum_{k}P_{k}^{R_{0}}=I$ need not be enforced here since
we allow $\mathcal{E}$ to be non-trace-preserving. Then,

$\displaystyle J^{RB}_{\mathds{1}\otimes\Theta[\mathcal{E}]}$
$\displaystyle=\operatorname{Tr}_{A}\left(\mathcal{J}^{AB}_{\Theta}\left(J^{RA}_{\mathcal{E}}\right)^{\Gamma_{A}}\right)$
(48) $\displaystyle=\sum_{k}P^{R}_{k}\otimes Q^{B}_{k}=J^{RB}_{\mathcal{G}},$
(49)

hence $\mathcal{G}=\mathds{1}\otimes\Theta[\mathcal{E}]$. By the arbitrariness
of $\mathcal{G}$ and the argument before, we have $\mathrm{SEP^{*}\subseteq
C^{*}_{EBSC}}$, hence $\mathrm{C^{*}_{EBSC}=SEP^{*}}$.

Notice that the EBSC $\Theta$ in the above proof is actually a CMPSC. To see
this, take the pre/post-processing map to be

$\displaystyle\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}E}$
$\displaystyle=\sum_{k=1}^{r}\Lambda_{k}^{B_{0}\to
E}\otimes\mbox{$\left|k\right\rangle$}\mbox{$\left\langle
k\right|$}^{A_{0}}+\mathcal{F}^{B_{0}\to
E}\otimes\mbox{$\left|r+1\right\rangle$}\mbox{$\left\langle
r+1\right|$}^{A_{0}},$ (50) $\displaystyle\Gamma_{\mathrm{post}}^{A_{1}E\to
B_{1}}$ $\displaystyle=\mathrm{id}^{E\to
B_{1}}\otimes\operatorname{Tr}_{A_{1}},$ (51)

where $\Lambda_{k}^{B_{0}\to E}$ (resp. $\mathcal{F}^{B_{0}\to E}$) is the
unique map with Choi matrix $Q_{k}^{B_{0}E}$ (resp. $F^{B_{0}E}$). It is easy
to check $\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}E}$ is CPTP. Therefore, we have
$\mathrm{C^{*}_{CMPSC}=C^{*}_{EBSC}=SEP^{*}}$. This complete the proof of
Proposition 5.2. ∎

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
$\mathcal{G}^{RB}=\sum_{k=1}^{r}P^{R}_{k}\otimes Q^{B}_{k}$, Bob first
performs a quantum instrument $\\{\Lambda_{1},...,\Lambda_{r},\mathcal{F}\\}$
and sends the outcome (stored as classical information in system $A_{0}$) to
Rachel. Rachel then acts as follows. If Bob’s action is $\Lambda_{k}$, Rachel
implements the CPTNI map with Choi matrix $P_{k}$ to complete the procedure.
If Bob’s action is $\mathcal{F}$, 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 [29] that every separable map can be implemented by an SLOCC
protocol, which provides a rough explanation for why Proposition 5.2 holds. It
also helps shed light on the physical significance of CP image, as defined in
Definition 5.2. This is related to stochastic quantum processes, which we
discuss further in the next subsection.

###  5.3 Discussion on CP Image

Suppose that, for superchannel $\Theta^{A\to B}$ and CP maps $\mathcal{G}^{B}$
and $\mathcal{E}^{A}$ one has

$\mathcal{G}^{B}=\Theta^{A\to B}[\mathcal{E}^{A}].$ (52)

If both $\mathcal{G}$ and $\mathcal{E}$ are CPTP, the physical interpretation
of this equation is clearly deterministic channel conversion. What if they are
not trace-preserving? Let us consider the case when $\mathcal{G}^{B}$ is CPTNI
and $\mathcal{E}^{A}$ is a general CP map. We can always find a positive
number $p\leq 1$ such that $p\mathcal{E}$ is a CPTNI map, hence we can find
another CPTNI map $\mathcal{T}$ to make up a quantum instrument
$\\{p\mathcal{E},\mathcal{T}\\}^{A}$. By applying $\Theta^{A\to B}$ to this
instrument (see Fig. 14) we get a new instrument
$\\{p\mathcal{G},\Theta[\mathcal{T}]\\}^{B}$. This provides an implementation
of the CPTNI map $\mathcal{G}$ 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.

Figure 14: A quantum instrument $\left\\{\Theta[\Lambda_{i}]\right\\}_{i}^{B}$
resulting from applying superchannel $\Theta^{A\to B}$ at a bipartite quantum
instrument $\left\\{\Lambda_{i}\right\\}_{i}^{A}$.

Comparing Propositions 5.1 and 5.2, which say that
$\mathrm{C_{CMPSC}=LOCC_{2}}$ but $\mathrm{C^{*}_{CMPSC}=SEP^{*}}$, we see
that there are CPTP maps that can be probabilistically implemented by CMPSC
but not deterministically, since there are non-LOCC CPTP maps in
$\textrm{SEP}\subset\textrm{SEP}^{*}$ that are not in $\mathrm{LOCC_{2}}$.
Hence, there are separable channels $\mathcal{G}^{RB}$ and valid superchannels
$\Theta^{A\to B}$ for which
$\mathcal{G}^{RB}=\mathds{1}^{R}\otimes\Theta^{A\to B}[\mathcal{E}^{RA}]$
holds only for some CP non-TP map $\mathcal{E}^{RA}$. 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 more challenging to formalize than QRTs of states (see also
Ref. [17] for a related discussion on this point).

##  6 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 that a bipartite channel
$\mathcal{E}\in\mathrm{CP}(A_{0}R_{0}\to A_{1}R_{1})$ is separable if
$\sigma^{R^{\prime}A^{\prime}R_{1}A_{1}}=\mathrm{id}^{R^{\prime}A^{\prime}}\otimes\mathcal{E}^{A_{0}R_{0}\to
A_{1}R_{1}}(\rho^{R^{\prime}R_{0}}\otimes\omega^{A^{\prime}A_{0}})$ is
$R^{\prime}R_{1}:A^{\prime}A_{1}$ separable for all
$\rho^{R^{\prime}R_{0}}\otimes\omega^{A^{\prime}A_{0}}$ and all systems
$R^{\prime}$ and $A^{\prime}$. In fact, it suffices to consider systems
$R^{\prime}$ and $A^{\prime}$ of dimension $d_{R_{0}}$ and $d_{A_{0}}$,
respectively. A relaxation would be to require that
$\sigma^{R^{\prime}A^{\prime}R_{1}A_{1}}$ is separable only for systems
$R^{\prime}$ and $A^{\prime}$ of smaller dimension.

######  Definition 6.1.

A bipartite channel $\mathcal{E}\in\mathrm{CP}(A_{0}R_{0}\to A_{1}R_{1})$ is
called $(p,q)$-non-entangling if
$\mathrm{id}^{R^{\prime}A^{\prime}}\otimes\mathcal{E}^{A_{0}R_{0}\to
A_{1}R_{1}}(\rho^{R^{\prime}R_{0}}\otimes\omega^{A^{\prime}A_{0}})$ is
$R^{\prime}R_{1}:A^{\prime}A_{1}$ separable for all
$\rho^{R^{\prime}R_{0}}\otimes\omega^{A^{\prime}A_{0}}$ with any systems
$R^{\prime}$ and $A^{\prime}$ of dimension $d_{A^{\prime}}=p$ and
$d_{R^{\prime}}=q$.

A $(k,k)$-non-entangling channel is also called $k$-non-entangling in previous
literature [30]. We call a channel $(p,\text{complete})$-non-entangling if it
is $(p,q)$-non-entangling for every positive integer $q$. A separable channel
is then a $(\text{complete},\text{complete})$-non-entangling channel.

######  Definition 6.2.

A superchannel $\Theta^{A\to B}$ is called a (p,q)-EBSC if
$\mathds{1}^{R}\otimes\Theta^{A\to B}[\mathcal{E}^{RA}]$ is a $(p,q)$-non-
entangling map ($p$ with $B$ and $q$ with $R$) for every
$\mathcal{E}\in\textrm{CP}(RA)$, with $R$ being an arbitrary finite-
dimensional system.

Although the general structure of $(p,q)$-EBSCs could be rather complicated,
we can obtain some decent results when restricting our attention to
superchannels that have an implementation without a side-channel (see Def 6.4
below), and only $(k,\text{complete})$-EBSCs. Before doing that, we first
introduce a class of $k$-entanglement breaking channels in the following
subsection.

###  6.1 k-Entanglement Breaking Channel

######  Definition 6.3.

[31] A quantum channel $\Lambda\in\mathrm{CPTP}(B_{0}\to B_{1})$ is called a
k-entanglement breaking channel ($k$-EBC) if
$\mathrm{id}^{R}\mathbin{\mathop{\otimes}}\Lambda(\rho)$ is separable for any
$\rho\in\mathcal{D}(R\otimes B_{0})$, where $R$ is any $k$-dimensional Hilbert
space.

This class of $k$-EBC was first introduced in [31] to study the condition when
concatenated maps become entanglement breaking. Note that, this definition
differs from the $k$-partially entanglement-breaking channel studied in Ref.
[32], which is a channel $\Lambda^{B_{0}\to B_{1}}$ that satisfies
$\mathrm{SN}(\mathrm{id}^{R}\mathbin{\mathop{\otimes}}\Lambda(\rho))\leq k$
for any $\rho\in\mathcal{D}(R\mathbin{\mathop{\otimes}}B_{0})$ with
arbitrarily large system $R$, where $\mathrm{SN}$ denotes the Schmidt number.

For simplicity, in the following we only consider channels with the same input
and output dimension, $d\geq 2$. A $k$-EBC is a completely-EBC whenever $k\geq
d$ [3]. For $1\leq k<d$, the set of $k$-EBC is clearly a subset of
($k$+1)-EBC, and the set of all $1$-EBCs trivially equals
$\mathrm{CPTP}(B_{0}\to B_{1})$. To summarize, we have

$\text{completely-
EBC}=\text{$d$-EBC}\subseteq\text{($d$-$1$)-EBC}\subseteq...\subseteq\text{$1$-EBC}=\text{CPTP}.$
(53)

In Theorem 6 below, we show that each of the above inclusions is strict. That
is, there exists a $k$-EBC that is a not ($k$+1)-EBC for all $1\leq k<d$.

######  Theorem 6.

For all integers $k$ and $d$ with $1\leq k<d$, there exists a $k$-EBC
$\Lambda\in\mathrm{CPTP}(\mathbb{C}^{d}\to\mathbb{C}^{d})$ that is not
$(k+1)$-EBC.

The proof of this theorem is inspired by Theorem 20 in [30], which establishes
the existence of $k$-non-entangling map that is not $(k+1)$-non-entangling.
Interestingly, Theorem 20 in [30] turns out to be a corollary of Theorem 6
presented here.

######  Corollary 6.1.

[30] For all integers $k$ and $d$ with $1\leq k<d$, there exists a $k$-non-
entangling map
$\Lambda:\mathcal{L}(\mathbb{C}^{d}\mathbin{\mathop{\otimes}}\mathbb{C}^{d})\to\mathcal{L}(\mathbb{C}^{d}\mathbin{\mathop{\otimes}}\mathbb{C}^{d})$
that is not $(k+1)$-non-entangling.

###### Proof.

The channel $\mathrm{SWAP}\circ(\Lambda\mathbin{\mathop{\otimes}}\Lambda)$
with $k$-EBC but not ($k$+1)-EBC $\Lambda$ is by definition $k$-non-entangling
but not ($k$+1)-non-entangling map, for all $1\leq k<d$. ∎

Now we set out to prove Theorem 6. In the following, we assume
$\mathcal{H}_{R}=\mathbb{C}^{k}$ and
$\mathcal{H}_{B_{0}}=\mathcal{H}_{B_{1}}=\mathbb{C}^{d}$ and
$\Lambda\in\mathrm{CPTP}(B_{0}\to B_{1})$. First notice that, in order to
prove a channel $\Lambda$ is $k$-EBC, it suffices to show
$\mathrm{id}_{R}\mathbin{\mathop{\otimes}}\Lambda(\varphi)$ is separable for
all pure states
$\mbox{$\left|\varphi\right\rangle$}\in\mathcal{H}_{R}\mathbin{\mathop{\otimes}}\mathcal{H}_{B_{0}}$,
since we can then apply convexity to cover mixed states. Any such pure state
can be written as

$\mbox{$\left|\varphi\right\rangle$}=X\mathbin{\mathop{\otimes}}\mathrm{id}_{d}\mbox{$\left|\phi_{d}^{+}\right\rangle$}$
(54)

for some operator $X\in\mathcal{L}(\mathbb{C}^{d}\to\mathbb{C}^{k})$. Let
$X=\sum_{i=1}^{r}\lambda_{i}|\alpha_{i}\rangle\langle\beta_{i}|$ denote a
singular value decomposition of $X$, and define
$X^{-1}:=\sum_{i=1}^{r}\lambda_{i}^{-1}|\beta_{i}\rangle\langle\alpha_{i}|$
(with $\lambda_{i}>0$ for $i=1,\cdots r$). Then
$X^{-1}X=P=\sum_{i=1}^{r}|\beta_{i}\rangle\langle\beta_{i}|$ is a projector of
rank $r\leq k$. Then since local operators preserve separability, we have that

$\mathrm{id}\otimes\Lambda[|\phi\rangle\langle\phi|]=\mathrm{id}\otimes\Lambda[(X\otimes\mathbb{I}_{d})\phi^{+}_{d}(X\otimes{I}_{d})^{\dagger}]$

is separable if and only if

$(X^{-1}\otimes\mathbb{I})\mathrm{id}\otimes\Lambda[(X\otimes\mathbb{I}_{d})\phi^{+}_{d}(X\otimes
I_{d})^{\dagger}](X^{-1}\otimes
I)^{\dagger}=(P\mathbin{\mathop{\otimes}}\mathrm{id}_{d})J_{\Lambda}(P^{\dagger}\mathbin{\mathop{\otimes}}\mathrm{id}_{d})$
(55)

is separable. We summarize this observation in the following lemma.

######  Lemma 6.1.

A channel $\Lambda\in\mathrm{CPTP}(B_{0}\to B_{1})$ is $k$-EBC if and only if
the operator

$(P\mathbin{\mathop{\otimes}}I_{d})J_{\Lambda}(P^{\dagger}\mathbin{\mathop{\otimes}}I_{d})$
(56)

is separable for all projectors $P$ with dimension no larger than $k$.

Now we introduce the Werner states [33] which will play an important role in
our construction of k-EBC. The Werner states are a family of states on
$\mathbb{C}^{d}\mathbin{\mathop{\otimes}}\mathbb{C}^{d}$ which take the
following form,

$\rho^{\mathcal{W}}_{d}(\beta)=\cfrac{1}{d^{2}-(\beta+1)}\left(I_{d^{2}}-\frac{\beta+1}{d}F_{d}\right),$
(57)

for $-(d+1)\leq\beta\leq d-1$, where
$F_{d}=\sum_{ij}\mbox{$\left|i\right\rangle$}\mbox{$\left\langle
j\right|$}\mathbin{\mathop{\otimes}}\mbox{$\left|j\right\rangle$}\mbox{$\left\langle
i\right|$}$ is the swap operator (or the partial transpose of $\phi_{d}^{+}$).
A crucial observation is that the partial trace of a Werner state is
$\frac{1}{d}I_{d}$, which means it is a valid Choi matrix of some CPTP quantum
channel (with some normalization). Specifically, consider the following map,

$\Lambda_{\beta}(\rho_{B_{0}}):=d\operatorname{Tr}_{B_{0}}[\rho^{\mathcal{W}}_{d}(\beta)(\rho^{\mathrm{T}}_{B_{0}}\mathbin{\mathop{\otimes}}I_{B_{1}})].$
(58)

This map is clearly completely positive, and it is also trace-preserving since

$\displaystyle\operatorname{Tr}\left(\Lambda_{\beta}(\rho_{B_{0}})\right)$
$\displaystyle=\operatorname{Tr}[d\operatorname{Tr}_{B_{1}}\left(\rho^{\mathcal{W}}_{d}(\beta)\right)\rho_{B_{0}}]$
(59) $\displaystyle=\operatorname{Tr}(I_{d}\ \rho_{B_{0}})$
$\displaystyle=\operatorname{Tr}(\rho_{B_{0}}),$

where the second equation is from our observation on the partial trace of
Werner states. We conclude that $\Lambda_{\beta}\in\mathrm{CPTP}(B_{0}\to
B_{1})$.

Since $\rho_{d}^{\mathcal{W}}(\beta)$ is the Choi matrix of $\Lambda_{\beta}$
up to normalization, we can apply Lemma 6.1 by studying the separability
properties of $\rho_{d}^{\mathcal{W}}(\beta)$. The following lemma is modified
from Lemma 19 in [30].

######  Lemma 6.2.

Let $k$ and $d$ be integers such that $1\leq k<d$ and let $-(d+1)\leq\beta\leq
d-1$. The operator

$\displaystyle\rho^{\mathrm{proj}}_{RB_{1}}:=(P\mathbin{\mathop{\otimes}}I_{d})\rho_{d}^{\mathcal{W}}(\beta)(P\mathbin{\mathop{\otimes}}I_{d})$

is separable for all projectors P with dimension no greater than $k$ if and
only if $\beta\leq(d-k)/k$.

###### Proof.

(Proof of lemma 6.2) We first require $P$ to be a $k$-dimensional projector in
$\mathcal{L}(B_{0}\to B_{1})$. By direct calculation, we have

$\displaystyle\rho^{\mathrm{proj}}_{RB_{1}}$ $\displaystyle\propto
I_{kd}-\frac{\beta+1}{d}\left((P\otimes I_{d})F_{d}(P^{\dagger}\otimes
I_{d})\right)$ (60) $\displaystyle=I_{kd}-\frac{\beta+1}{d}\left((P^{*}\otimes
I_{d}){\phi}_{d}^{+}(P^{\mathrm{T}}\otimes I_{d})\right)^{\Gamma}$ (61)
$\displaystyle=I_{kd}-\frac{\beta+1}{d}c\ \psi^{\Gamma},$ (62)

for some normalized pure state
$\mbox{$\left|\psi\right\rangle$}=(P^{*}\otimes\mathrm{id}_{d})\mbox{$\left|{\phi}_{d}^{+}\right\rangle$}/\sqrt{c}$,
with normalizing factor

$\displaystyle c$ $\displaystyle=\operatorname{Tr}\left((P^{*}\otimes
I_{d}){\phi}_{d}^{+}(P^{\mathrm{T}}\mathbin{\mathop{\otimes}}I_{d})\right)$
(63)
$\displaystyle=\operatorname{Tr}\left(\overline{P}P^{\mathrm{T}}\right)=\operatorname{Tr}\left(PP^{\dagger}\right)=k.$
(64)

If $\beta>(d-k)/k$, then $\frac{\beta+1}{d}k>1$ and so

$\left(\rho^{\mathrm{proj}}_{RB_{1}}\right)^{\Gamma}=I_{kd}-\frac{\beta+1}{d}k\
\psi$ (65)

is obviously not positive. A non-positive partial transpose implies that
$\rho^{\mathrm{proj}}_{RB_{1}}$ is entangled. On the other hand, if
$\beta\leq(d-k)/k$, we apply Theorem 1 from [34] which states that the
operator $\mathrm{id}+A$ is separable for all Hermitian $A$ with
$\|A\|_{2}\leq 1$, where $\|A\|_{2}=\sqrt{\operatorname{Tr}A^{\dagger}A}$ is
the Frobenius norm. Here, we have

$\left\|\frac{\beta+1}{d}k\
\psi^{\Gamma}\right\|_{2}\leq\|\psi^{\Gamma}\|_{2}=\|\psi\|_{2}=1,$ (66)

since $\psi$ is a normalized pure state. We therefore conclude that
$\rho^{\mathrm{proj}}_{RB_{1}}$ is separable. This establishes that
$\rho^{\mathrm{proj}}_{RB_{1}}$ is separable for all $k$-dimensional
projectors $P$ if and only if $\beta\leq(d-k)/k$. 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 $\rho^{\mathrm{proj}}_{RB_{1}}$ is separable for all
projectors $P$ of dimension no greater than $k$ if and only if
$\beta\leq(d-k)/k$. This completes the proof of Lemma 6.2. ∎

As a direct result of Lemma 6.1 and Lemma 6.2, the channel $\Lambda_{\beta}$
constructed in Eq. (58) is $k$-EBC if and only if $\beta\leq(d-k)/k$, for
$1\leq k<d$. This completes the proof of Theorem 6.

Finally, we note that, a special case of Theorem 6 for $2$-EBCs is also
established in [31, Corollary III.4] using the Holevo-Werner maps similar as
in the above proof.

###  6.2 Interplay between $k$-EBC and $(k,\text{complete})$-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 $d$-dimensional 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 6.4.

A superchannel $\Theta$ is said to be without side-channel, if it can be
realized as

$\Theta^{A\to B}[\mathcal{E}^{A}]=\Gamma_{\mathrm{post}}^{A_{1}\to
B_{1}}\circ\mathcal{E}^{A}\circ\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}}$ (67)

for some CPTP maps $\Gamma_{\mathrm{pre}}$ and $\Gamma_{\mathrm{post}}$.

######  Corollary 6.2.

If an EBSC $\Theta^{A\to B}$ has a realization without side-channel as in Eq.
(67), then both $\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}}$ and
$\Gamma_{\mathrm{post}}^{A_{1}\to B_{1}}$ must be entanglement breaking
channels.

###### Proof.

This simply follows from Thm. 1. (D) and the definition of EB channels. ∎

The following proposition discusses the relation between $k$-EBC and
$(k,\text{complete})$-EBSC, for any positive integer $k\leq d$.

######  Proposition 6.1.

For a superchannel $\Theta^{A\to B}$ without side-channel as in Eq. (67), the
following are equivalent.

  1. (A)

$\Theta^{A\to B}$ is a $(k,\text{complete})$-EBSC.

  2. (B)

$\Gamma_{\mathrm{pre}}^{B_{0}\to A_{0}}$ is a k-EBC, and
$\Gamma_{\mathrm{post}}^{A_{1}\to B_{1}}$ is a completely-EBC.

As a result, the realization of a $(k,\text{complete})$-EBSC without side-
channel is shown in Fig. 15.

Figure 15: Realization of a $(k,\text{complete})$-EBSC without side-channel
(in the blue dash box).

###### Proof.

$(A)\Rightarrow(B)$: 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
$\Phi_{+}^{RA}(\cdot)=\operatorname{Tr}((\cdot)\phi_{+}^{R_{0}A_{0}})\phi_{+}^{R_{1}A_{1}}$,
the maximally entangled state $\phi_{+}^{R_{0}D}$, and an arbitrary quantum
state $\rho\in\mathcal{D}(B_{0}K)$. We have

$\displaystyle\left(\mathrm{id}^{KD}\otimes(\mathds{1}^{R}\otimes\Theta^{A\to
B}[\Phi_{+}^{RA}])\right)(\rho^{B_{0}K}\otimes\phi_{+}^{R_{0}D})$ (68)
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\Gamma_{\mathrm{pre}}^{B_{0}\to
D}(\rho^{B_{0}K})\otimes\Gamma_{\mathrm{post}}^{A_{1}\to
B_{1}}(\phi_{+}^{R_{1}A_{1}}).$ (69)

Since $\Theta$ is $(k,\text{complete})$-EBSC, the above state must be
separable with respect to $KB_{1}:DR_{1}$, which means both
$\Gamma_{\mathrm{pre}}^{B_{0}\to D}(\rho^{B_{0}K})$ and
$\Gamma_{\mathrm{post}}^{A_{1}\to B_{1}}(\phi_{+}^{R_{1}A_{1}})$ are
separable. The former implies $\Gamma_{\mathrm{pre}}$ is $k$-EBC since $\rho$
is an arbitrary state on $B_{0}K$. The latter implies $\Gamma_{\mathrm{post}}$
is (completely) EBC by Proposition. 2.1 [3]. This complete the proof of
$(A)\Rightarrow(B)$.

$(B)\Rightarrow(A)$: Consider the circuit in Fig. 15. For any separable input
$\rho^{B_{0}K}\otimes\varphi^{R_{0}D}$, it is easy to see the output state is
$K:B_{1}:R_{1}D$ separable, which means $\Theta^{A\to B}$ is indeed
(k,d)-EBSC. ∎

Since the hierarchy of $k$-EBC is non-trivial, there is also a non-trivial
hierarchy of $(k,\text{complete})$-EBSC, for $1\leq k\leq d$.

As a final comment, the series of study about when composed maps become
entanglement breaking [35, 36, 37, 38, 39, 31] may also be generalized to
supermaps. As an example, the Schmidt number iteration theorem [31] can be
directly applied here to get the following corollary.

######  Corollary 6.3.

If $\Theta^{A\to A}$ is a (k,complete)-EBSC without side-channel and $M$ is an
integer no less than $\left\lceil\frac{d-1}{k-1}\right\rceil$, then an $M$
times concatenation of $\Theta$ as

$\Theta^{M}\equiv\underbrace{\Theta\circ\Theta\circ\dots\circ\Theta}_{M\leavevmode\nobreak\
times}$

is an EBSC.

###### Proof.

The Schmidt number iteration theorem [31] states that a
$\left\lceil\frac{d-1}{k-1}\right\rceil$ times concatenation of a $k$-EBC
$\mathcal{E}^{A}$ becomes completely-EB. Combining it with Proposition 6.1
gives this corollary. ∎

We leave the question when general composed superchannels become EBSC as an
interesting future research direction.

##  7 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 (and EB supermaps) 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 all entanglement-breaking supermaps can be decomposed
into partly-EB pre/post-processing CP maps. While left open the question
whether every EBSC can be implemented with partly-EB pre/post-processing
channels, we show that a generic implementation must allow entangling the
input system to the memory system of a superchannel. Secondly, we show that
EBSC is more general than measure-and-prepare superchannels (MPSCs), and even
controlled-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 CP-complete 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, $k$-non-entangling channel and $(k,\text{complete})$-EBSC without
side-channel. By generalizing the method of [30], we show that all these three
objects have a non-trivial hierarchy for $1\leq k\leq d$. 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
$\mathrm{MPSC}\subseteq\mathrm{LOCC_{1}\text{-}EBSC}$ and
$\mathrm{CMPSC}\subseteq\mathrm{LOCC_{2}\text{-}EBSC}$ 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 [40], 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 [41]. We
anticipate there being other applications of EBSC within 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 $\mathrm{C_{EBSC}=SEP}$ or not. Currently we only know that all two-
round LOCC and some non-LOCC separable channels are in $\mathrm{C_{EBSC}}$.
The second is whether every EBSC can be realized as in Fig. 6 with the
pre/post-processing maps being CPTP. Answering these questions will help us
better understand the intricate structure of EBSCs.

We thank the anonymous reviewers for their helpful suggestions to improve this
paper. SC acknowledges the support from the Tsinghua Scholarship for
Undergraduate Oversea Studies. EC is supported by the National Science
Foundation (NSF) Award No. 1914440.

## References

  * Cirac et al. [2001] J. I. Cirac, W. Dür, B. Kraus, and M. Lewenstein.  Entangling operations and their implementation using a small amount of entanglement.  _Phys. Rev. Lett._ , 86:544–547, Jan 2001.  doi: 10.1103/PhysRevLett.86.544.
  * Chiribella et al. [2008a] G. Chiribella, G. M. D’Ariano, and P. Perinotti.  Transforming quantum operations: Quantum supermaps.  _EPL (Europhysics Letters)_ , 83(3):30004, jul 2008a.  doi: 10.1209/0295-5075/83/30004.
  * Horodecki et al. [2003] Michael Horodecki, Peter W. Shor, and Mary Beth Ruskai.  Entanglement breaking channels.  _Reviews in Mathematical Physics_ , 15(06):629–641, 2003.  doi: 10.1142/S0129055X03001709.
  * Chitambar and Gour [2019] Eric Chitambar and Gilad Gour.  Quantum resource theories.  _Rev. Mod. Phys._ , 91:025001, Apr 2019.  doi: 10.1103/RevModPhys.91.025001.
  * Theurer et al. [2019] Thomas Theurer, Dario Egloff, Lijian Zhang, and Martin B. Plenio.  Quantifying operations with an application to coherence.  _Phys. Rev. Lett._ , 122:190405, May 2019.  doi: 10.1103/PhysRevLett.122.190405.
  * Berk et al. [2019] Graeme D Berk, Andrew JP Garner, Benjamin Yadin, Kavan Modi, and Felix A Pollock.  Resource theories of multi-time processes: A window into quantum non-markovianity, 2019.  arXiv:1907.07003.
  * Liu and Yuan [2020] Yunchao Liu and Xiao Yuan.  Operational resource theory of quantum channels.  _Physical Review Research_ , 2(1):012035, 2020\.  doi: 10.1103/PhysRevResearch.2.012035.
  * Liu and Winter [2019] Zi-Wen Liu and Andreas Winter.  Resource theories of quantum channels and the universal role of resource erasure.  2019\.  arXiv:1904.04201.
  * Gour and Winter [2019] Gilad Gour and Andreas Winter.  How to quantify a dynamical quantum resource.  _Phys. Rev. Lett._ , 123:150401, Oct 2019.  doi: 10.1103/PhysRevLett.123.150401.
  * Gour and Scandolo [2019] Gilad Gour and Carlo Maria Scandolo.  The entanglement of a bipartite channel.  2019\.  arXiv:1907.02552.
  * Bäuml et al. [2019] Stefan Bäuml, Siddhartha Das, Xin Wang, and Mark M. Wilde.  Resource theory of entanglement for bipartite quantum channels.  2019\.  arXiv:1907.04181.
  * Jamiołkowski [1972] Andrzej Jamiołkowski.  Linear transformations which preserve trace and positive semidefiniteness of operators.  _Reports on Mathematical Physics_ , 3(4):275–278, 1972.  doi: 10.1016/0034-4877(72)90011-0.
  * Choi [1975] Man-Duen Choi.  Completely positive linear maps on complex matrices.  _Linear algebra and its applications_ , 10(3):285–290, 1975.  doi: 10.1016/0024-3795(75)90075-0.
  * Vedral et al. [1997] Vlatko Vedral, Martin B Plenio, Michael A Rippin, and Peter L Knight.  Quantifying entanglement.  _Physical Review Letters_ , 78(12):2275, 1997\.  doi: 10.1103/PhysRevLett.78.2275.
  * Barnum et al. [1998] Howard Barnum, M. A. Nielsen, and Benjamin Schumacher.  Information transmission through a noisy quantum channel.  _Phys. Rev. A_ , 57:4153–4175, Jun 1998.  doi: 10.1103/PhysRevA.57.4153.
  * Gour [2019] Gilad Gour.  Comparison of quantum channels by superchannels.  _IEEE Transactions on Information Theory_ , 65(9):5880–5904, 2019.  doi: 10.1109/TIT.2019.2907989.
  * Burniston et al. [2020] John Burniston, Michael Grabowecky, Carlo Maria Scandolo, Giulio Chiribella, and Gilad Gour.  Necessary and sufficient conditions on measurements of quantum channels.  _Proceedings of the Royal Society A_ , 476(2236):20190832, 2020.  doi: 10.1098/rspa.2019.0832.
  * Chiribella et al. [2008b] Giulio Chiribella, Giacomo M D’Ariano, and Paolo Perinotti.  Memory effects in quantum channel discrimination.  _Physical review letters_ , 101(18):180501, 2008b.  doi: 10.1103/PhysRevLett.101.180501.
  * Ziman [2008] Mário Ziman.  Process positive-operator-valued measure: A mathematical framework for the description of process tomography experiments.  _Phys. Rev. A_ , 77:062112, Jun 2008.  doi: 10.1103/PhysRevA.77.062112.
  * Gutoski and Watrous [2007] Gus Gutoski and John Watrous.  Toward a general theory of quantum games.  In _Proceedings of the thirty-ninth annual ACM symposium on Theory of computing_ , pages 565–574, 2007.  doi: 10.1145/1250790.1250873.
  * Bennett et al. [1999] Charles H. Bennett, David P. DiVincenzo, Christopher A. Fuchs, Tal Mor, Eric Rains, Peter W. Shor, John A. Smolin, and William K. Wootters.  Quantum nonlocality without entanglement.  _Phys. Rev. A_ , 59:1070–1091, Feb 1999.  doi: 10.1103/PhysRevA.59.1070.
  * Peres and Wootters [1991] Asher Peres and William K. Wootters.  Optimal detection of quantum information.  _Phys. Rev. Lett._ , 66:1119–1122, Mar 1991.  doi: 10.1103/PhysRevLett.66.1119.
  * Chitambar and Hsieh [2013] Eric Chitambar and Min-Hsiu Hsieh.  Revisiting the optimal detection of quantum information.  _Phys. Rev. A_ , 88:020302, Aug 2013.  doi: 10.1103/PhysRevA.88.020302.
  * Chiribella et al. [2008c] G. Chiribella, G. M. D’Ariano, and P. Perinotti.  Quantum circuit architecture.  _Phys. Rev. Lett._ , 101:060401, Aug 2008c.  doi: 10.1103/PhysRevLett.101.060401.
  * Chiribella et al. [2009] Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti.  Theoretical framework for quantum networks.  _Phys. Rev. A_ , 80:022339, Aug 2009.  doi: 10.1103/PhysRevA.80.022339.
  * Gutoski [2012] Gus Gutoski.  On a measure of distance for quantum strategies.  _Journal of Mathematical Physics_ , 53(3):032202, 2012.  doi: 10.1063/1.3693621.
  * Gutoski [2009] Gus Gutoski.  _Quantum strategies and local operations_.  PhD thesis, University of Waterloo, 2009.  Available at arXiv.org e-Print quant-ph/1003.0038.
  * Chitambar et al. [2014] Eric Chitambar, Debbie Leung, Laura Mančinska, Maris Ozols, and Andreas Winter.  Everything you always wanted to know about LOCC (but were afraid to ask).  _Communications in Mathematical Physics_ , 328(1):303–326, May 2014.  doi: 10.1007/s00220-014-1953-9.
  * Dür et al. [2000] W. Dür, G. Vidal, and J. I. Cirac.  Three qubits can be entangled in two inequivalent ways.  _Phys. Rev. A_ , 62:062314, Nov 2000.  doi: 10.1103/PhysRevA.62.062314.
  * Chitambar et al. [2020] Eric Chitambar, Julio I de Vicente, Mark W Girard, and Gilad Gour.  Entanglement manipulation beyond local operations and classical communication.  _Journal of Mathematical Physics_ , 61(4):042201, 2020.  doi: 10.1063/1.5124109.
  * Christandl et al. [2019] Matthias Christandl, Alexander Müller-Hermes, and Michael M Wolf.  When do composed maps become entanglement breaking?  In _Annales Henri Poincaré_ , volume 20, pages 2295–2322. Springer, 2019.  doi: 10.1007/s00023-019-00774-7.
  * Chruściński and Kossakowski [2006] Dariusz Chruściński and Andrzej Kossakowski.  On partially entanglement breaking channels.  _Open Systems & Information Dynamics_, 13(1):17–26, Mar 2006.  ISSN 1573-1324.  doi: 10.1007/s11080-006-7264-7.
  * Werner [1989] Reinhard F. Werner.  Quantum states with einstein-podolsky-rosen correlations admitting a hidden-variable model.  _Phys. Rev. A_ , 40:4277–4281, Oct 1989.  doi: 10.1103/PhysRevA.40.4277.
  * Gurvits and Barnum [2002] Leonid Gurvits and Howard Barnum.  Largest separable balls around the maximally mixed bipartite quantum state.  _Phys. Rev. A_ , 66:062311, Dec 2002.  doi: 10.1103/PhysRevA.66.062311.
  * Christandl [2012] M Christandl.  PPT square conjecture.  In _Banff International Research Station workshop: Operator structures in quantum information theory_ , 2012.
  * Lami and Giovannetti [2015] Ludovico Lami and Vittorio Giovannetti.  Entanglement–breaking indices.  _Journal of Mathematical Physics_ , 56(9):092201, 2015.  doi: 10.1063/1.4931482.
  * Lami and Giovannetti [2016] Ludovico Lami and Vittorio Giovannetti.  Entanglement-saving channels.  _Journal of Mathematical Physics_ , 57(3):032201, 2016.  doi: 10.1063/1.4942495.
  * Rahaman et al. [2018] Mizanur Rahaman, Samuel Jaques, and Vern I Paulsen.  Eventually entanglement breaking maps.  _Journal of Mathematical Physics_ , 59(6):062201, 2018.  doi: 10.1063/1.5024385.
  * Kennedy et al. [2018] Matthew Kennedy, Nicholas A. Manor, and Vern I. Paulsen.  Composition of ppt maps.  _Quantum Info. Comput._ , 18(5-6):472–480, 2018\.
  * Rosset et al. [2018] Denis Rosset, Francesco Buscemi, and Yeong-Cherng Liang.  Resource theory of quantum memories and their faithful verification with minimal assumptions.  _Phys. Rev. X_ , 8:021033, May 2018.  doi: 10.1103/PhysRevX.8.021033.
  * Yuan et al. [2019] Xiao Yuan, Yunchao Liu, Qi Zhao, Bartosz Regula, Jayne Thompson, and Mile Gu.  Universal and operational benchmarking of quantum memories, 2019.  arXiv:1907.02521.