# Consequences of preserving reversibility in quantum superchannels

Wataru Yokojima  Department of Physics, Graduate School of Science, The
University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan  Marco
Túlio Quintino  Department of Physics, Graduate School of Science, The
University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan  Akihito
Soeda  Department of Physics, Graduate School of Science, The University of
Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan  Mio Murao  Department of
Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1,
Bunkyo-ku, Tokyo 113-0033, Japan

###### Abstract

Similarly to quantum states, quantum operations can also be transformed by
means of quantum superchannels, also known as process matrices. Quantum
superchannels with multiple slots are deterministic transformations which take
causally independent quantum operations as inputs and are enforced to respect
the laws of quantum mechanics but the use of input operations may lack a
definite causal order. While causally ordered superchannels admit a
characterization in terms of quantum circuits, a similar characterization of
general superchannels in terms of standard quantum objects with a clearer
physical interpretation has been missing. In this paper we provide a
mathematical characterization for pure superchannels with two slots (also
known as bipartite pure processes), which are superchannels preserving the
reversibility of quantum operations. We show that the reversibility preserving
condition restricts all pure superchannels with two slots to be either a
causally ordered quantum circuit only consisting of unitary operations or a
coherent superposition of two pure causally ordered circuits. The latter may
be seen as a generalization of the quantum switch, allowing a physical
interpretation for pure two-slot superchannels. An immediate corollary is that
purifiable bipartite processes cannot violate device-independent causal
inequalities.

###### Contents

  1. 1 Introduction
  2. 2 Non-technical overview of concepts and main results
    1. 2.1 Causally ordered superchannels
    2. 2.2 General superchannels with two slots
  3. 3 Review on quantum superchannel
    1. 3.1 Choi-Jamiołkowski isomorphism and linear supermaps
    2. 3.2 Quantum superchannel
    3. 3.3 Quantum comb
  4. 4 Main results and their implications
    1. 4.1 Main result 1: Pure comb
    2. 4.2 Main result 2: Pure superchannel
    3. 4.3 Implications of the main results
    4. 4.4 Properties of pure superchannels
  5. 5 Conclusions
  6. A Relevant properties of composite Hilbert space
    1. A.1 Notations for linear algebra
    2. A.2 Useful properties
    3. A.3 Notation of multiple-ket
    4. A.4 Function from subspaces to a subspace
    5. A.5 Reduced subspace
    6. A.6 Unitary operators representing pure superchannels
    7. A.7 Quantum combs with two slots
  7. B Proof of Thm. 3
  8. C The proof of (1) $\implies$ (2) in Thm. 5
  9. D The proof of (2) $\implies$ (3) in Thm. 5
    1. D.1 Decomposing ${{}^{A_{I}B_{I}}\\!{{\mathcal{V}}_{\alpha\beta}}}$
    2. D.2 Decomposing ${\mathcal{V}}_{\alpha\beta}$
    3. D.3 Decomposing $P$
    4. D.4 Proof of Prop. 6
    5. D.5 Decomposing $F$
    6. D.6 Proof of Prop. 11
    7. D.7 Proof of Prop. 12
    8. D.8 Causality

##  1 Introduction

Understanding physical transformations between quantum systems is one of the
central pillars of quantum mechanics. In quantum information theory,
transformations between states are viewed as devices which take a quantum
state as inputs and outputs. The properties of such devices, mathematically
formalized by quantum channels (deterministic transformations) and quantum
instruments (probabilistic transformations), are a research field on its own
and play a fundamental role in quantum information processing [1, 2, 3].
Similarly to states, quantum operations and the devices implementing the
operations may also be subjected to transformations. In these higher-order
operations, quantum operations play the role of inputs [4, 5]. Deterministic
higher-order operations are named quantum superchannels or also as process
matrices [6, 7]. They are the most general deterministic transformations
between multiple causally independent operations allowed by quantum mechanics.

The quantum circuit formalism provides a concrete way to transform quantum
operations. For instance, one can prepare a quantum circuit which has a few
empty slots where operations may be “plugged” to construct a new operation as
output [8]. Recently, it has been noted that the standard quantum circuit
formalism imposes restrictions on how to transform operations. In particular,
the standard quantum circuit formalism has a strict notion of causal order
between input operations which recent research suggests to be an ad hoc
hypothesis [9, 6] . Namely, quantum theory admits superchannels which make use
of of input operations in an indefinite causal order.

A seminal and illustrative example of a superchannel with indefinite causal
order is the quantum switch [9]. The quantum switch is a superchannel with two
slots that maps a pair of unitary operators $U_{A}$ and $U_{B}$ into the
unitary operator which is a coherent superposition of operator $U_{A}$ before
$U_{B}$ with the operator $U_{B}$ before $U_{A}$. The set of superchannels
with indefinite causal order is in principle compatible with quantum mechanics
and is not only restricted to elements equivalent to the quantum switch. This
set includes superchannels which allow device-independent indefinite causal
order certification [6] and cannot be decomposed by simple coherent
superposition of ordered circuits. Hence, a possible physical interpretation
for an arbitrary superchannels is even less obvious than the quantum switch.

In addition to the fundamental interest, quantum superchannels with indefinite
causal order have shown to be a valuable resource for information processing
tasks. The quantum switch has found applications in several tasks such as
discriminating channels with memory [10], reducing complexity of quantum
computing [12], exponential reduction of certain communication costs [13],
improving classical and quantum communication [14, 15], and semi-device-
independent certification of indefinite causal order [16]. The quantum switch
has also found limitations in the tasks of device-independent indefinite
causal order certification [6, 17] and transforming unitary operations [18,
19] where only non-switch indefinite quantum superchannels display an
advantage.

References [20, 21, 22] have exploited quantum interferometry to obtain
experimental implementations of the quantum switch, and although there is an
ongoing debate on what would be a “fair implementation” of the quantum switch,
the physical implementation/interpretation and the mathematical structure of
the quantum switch are simpler than general superchannels with indefinite
causal order [23, 7]. For instance, there is no simple classification or
universal procedure to analyze general superchannels.

Reversibility is a key concept in several physical theories. Motivated by the
role of reversibility in physical theories, Ref. [7] introduced the definition
of pure superchannels as superchannels preserving the reversibility of input
operations. More formally, a superchannel is said to be pure if it transforms
causally independent unitary channels into a unitary channel. The authors
proposed a purification postulate stating that superchannels which cannot be
purified do not have a fair physical implementation. Reference [23] shows that
pure two-slot superchannels have a realization on suitably defined “time-
delocalized” subsystems and Ref. [24] proves an equivalence between pure
processes and multilinear maps that admit a description in terms of consistent
causal reference frames.

In this paper we analyze the restrictions imposed by the reversibility
preserving property of superchannels and present a simple decomposition for
this class of processes. We show that pure quantum superchannels with two
slots can be divided in two cases: 1) causally ordered superchannels that can
be realized by standard quantum circuits only involving unitary operations; 2)
coherent superposition of two pure causally ordered superchannels which may be
seem as a generalization of the quantum switch. Our characterization provides
a potential interpretation for every pure superchannel with two slots.

This paper is structured as follows. In Sec. 2 we present a non-technical
overview of key concepts on quantum superchannels and state our main results.
In Sec. 3 we review mathematical definitions of quantum superchannels required
for the formal statement of our results. In Sec. 4 we present the formal
statement of our main results and a few direct implications. We conclude in
Sec. 5. In A we present relevant properties for decomposing linear spaces. B
presents a proof of Thm. 3. C and D present a proof of Thm. 5.

##  2 Non-technical overview of concepts and main results

In this Section we exploit the quantum circuit model to illustrate fundamental
concepts and state our main results. In quantum circuits, wires describe
quantum systems and boxes describe quantum channels which are general
deterministic quantum operations111We stress that, unless explicitly
specified, channels represented by white boxes should not be assumed to be
unitary. In Figure 3 we present a connection between general quantum channels
and unitary channels. (Figure 1). Wires take quantum states through boxes from
left to right.

Figure 1: Circuit representation of quantum channels.

We represent quantum states by putting the semi-circular objects from the left
and discarding systems by putting the inverted semi-circular objects from the
right (Figure 2).

Figure 2: Circuit representation of quantum states and the process of
discarding a quantum system.

An important result in characterizing quantum operation is given by the
Stinespring dilation [25], a theorem which proves that every quantum channel
can be realized by a unitary channel with an auxiliary state and possibly
discarding a part of the quantum system (see Figure 3).

Figure 3: Stinespring dilation: every quantum channel can be realized by a
unitary channel combined with an auxiliary state and possibly discarding a
part of the quantum system.

In addition to these components, we use colored boxes to represent abstract
superchannels, which are deterministic higher-order quantum operations that
can transform quantum channels, and boxes with color white in figures
represent quantum channels. Strictly speaking, general multiple slots
superchannels may not be valid components of quantum circuits, but for
convenience they are depicted as if so.

###  2.1 Causally ordered superchannels

  * •

A quantum superchannel with one slot is a linear supermap which maps quantum
channels into quantum channels (even if the supermap acts on a subsystem of
the input quantum channels) (Figure 4).

Figure 4:
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}$
is a quantum superchannel with one slot, which maps a quantum channel
$\widetilde{\mathcal{C}}{}$ into a quantum channel
$\widetilde{\mathcal{C}}{}^{\prime}$.

It has been shown that any quantum superchannel with one slot can be realized
by a quantum circuit (Figure 5) [4].

Figure 5: Every quantum superchannel
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}$
with one slot is realized with quantum channels
$\widetilde{\mathcal{E}}{},\widetilde{\mathcal{D}}{}$.

A quantum superchannel with one slot is pure if the superchannel maps unitary
channels into unitary channels (even if the superchannel acts on subsystems of
the input unitary channels) (Figure 6).

Figure 6:
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}_{\mathrm{pure}}$
is a pure superchannel, which maps a unitary channel $U$ into a unitary
channel $U^{\prime}$.

  * •

A quantum comb [8], (see also [26, 27]) is a causally ordered superchannel
with one or more slot(s). A quantum comb with no slots is a quantum channel
and a quantum comb with $N$ slot(s) is a linear supermap which maps every
quantum comb with $N-1$ slot(s) into a quantum channel (Figure 7).

Figure 7:
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}$
is a quantum comb with two slots, which maps a quantum comb
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}$
with a slot into a quantum channel $\widetilde{\mathcal{C}}{}^{\prime}$.

It has been shown that any quantum comb can be realized by a standard causally
ordered quantum circuit (Figure 8) [8].

Figure 8: Every quantum comb can be realized by causally ordered quantum
circuits. This picture shows a two-slot comb
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}$
which can be realized by quantum channels
$\widetilde{\mathcal{E}}{}_{0},\widetilde{\mathcal{E}}{}_{1},$ and
$\widetilde{\mathcal{E}}{}_{2}$, which are not necessarily unitary. When
additional auxiliary states and discarding subsystems are allowed, the
Stinespring dilation (see Figure 3) ensures every quantum circuit can be
realized by unitary channels.

A quantum comb is pure if the quantum comb maps a group of causally
independent unitary channels into unitary channels (even if the comb acts on
subsystems of the input unitary channels).

######  Main result 1.

(Thm. 3) Every pure quantum comb, independently of the number of slots, can be
realized by a causally ordered quantum circuit containing only unitary
channels without any need of auxiliary systems or discarding of subsystems
(Figure 9).

Figure 9: Every pure quantum comb with two slots
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}_{\mathrm{pure}}$
can be realized by a causally ordered quantum circuit with some unitary
channels $U_{0}$, $U_{1}$, and $U_{2}$ and does not require discarding part of
the quantum system. This figure should be contrasted with Figure 8, which
presents a circuit construction for general quantum combs and makes use of
auxiliary systems $P^{\prime}$ and $F^{\prime}$. From this perspective, a
quantum comb is pure if and only if it can be realized by quantum circuits
with unitary channels $U_{0}$, $U_{1}$, and $U_{2}$ without making use of the
auxiliary systems $P^{\prime}$ and $F^{\prime}$.

We emphasize that the dimensions of systems $P$, $A_{I}$, $A_{O}$, $B_{I}$,
$B_{O}$, and $F$ represented in Figure 9 are not necessarily the same. Note
that, since the comb
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}_{\mathrm{pure}}$
is represented by a sequence of unitary operators as in Figure 9, these
dimensions should satisfy certain constraints. For instance, when $\dim
A_{I}=\dim A_{O}(=:d_{A})$ and $\dim B_{I}=\dim B_{O}(=:d_{B})$, we must have
$\dim P=\dim F$. Moreover, $\dim P$ must be a multiple of $d_{A}$ and $\dim F$
must be a multiple of $d_{B}$. In particular, when the dimension of systems
$P$, $A_{I}$, $A_{O}$, $B_{I}$, $B_{O}$, and $F$ are all equivalent, pure
combs can be represented without any additional wires, see Figure 10 and Cor.
5 (1).

Figure 10: When the dimensions of involved systems satisfy $\dim P=\dim
A_{I}=\dim A_{O}=\dim B_{I}=\dim B_{O}=\dim F$, the pure comb
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}_{\mathrm{pure}}$
can be realized by a causally ordered quantum circuit with some unitary
channels $U_{0}$, $U_{1}$, and $U_{2}$ without any additional wires and
without discarding any quantum system.

###  2.2 General superchannels with two slots

  * •

A quantum superchannel with $N$ slots ($N\geq 0$) is a linear supermap which
maps every group of $N$ causally independent quantum channel(s) into a quantum
channel (even if the supermap acts on subsystems of the input channels)
(Figure 11).

Figure 11:
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{\mathcal{W}}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{\mathcal{W}}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{\mathcal{W}}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{\mathcal{W}}}}$}}}}{}$
represents a quantum superchannel with two slots. A two slot superchannel maps
any pair of causally independent input-channels represented by the tensor
product
$\widetilde{\mathcal{\mathcal{A}}}{}\otimes\widetilde{\mathcal{\mathcal{B}}}{}$
into an output-channel $\widetilde{\mathcal{\mathcal{C}}}{}$. The shape of
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}$
in this figure illustrates that
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}$
may have indefinite causal order.

  * •

A quantum switch is a quantum superchannel with two slots whose causal order
is coherently controlled by a control qubit (Figure 12). It is defined to
transform any pair of two causally independent unitary channels represented by
unitary operators $U_{A}$ and $U_{B}$ into a unitary channel represented by
$|0\rangle\langle 0|\otimes U_{B}U_{A}+|1\rangle\langle 1|\otimes U_{A}U_{B}$.

Figure 12: $\mathrm{QSWTICH}$ is a quantum switch, which maps a pair of two
causally independent unitary channels represented by unitary operators $U_{A}$
and $U_{B}$ into a unitary channel represented by $|0\rangle\langle 0|\otimes
U_{B}U_{A}+|1\rangle\langle 1|\otimes U_{A}U_{B}=U_{B}U_{A}\oplus U_{A}U_{B}$.
On the right hand side of this image, $U_{B}U_{A}\oplus U_{A}U_{B}$ is
illustrated as a direct sum of two circuits where $P=P_{0}\oplus P_{1}$ and
$F=F_{0}\oplus F_{1}$.

  * •

A quantum superchannel is pure if the quantum superchannel maps every group of
causally independent unitary channels into unitary channels (even if the
superchannel acts on subsystems of the input unitary channels).

It has been shown that every pure superchannel corresponds to a unitary
operator [7] (Figure 13).

Figure 13:
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}_{\mathrm{pure}}$
is a pure superchannel with two slots, which is represented by a unitary
operator $U_{W}\colon P\otimes A_{O}\otimes B_{O}\to A_{I}\otimes B_{I}\otimes
F$ as $W_{\mathrm{pure}}=\ket{U_{{W}}}\rangle\langle\bra{U_{{W}}}$, where
$W_{\mathrm{pure}}$ and $\ket{U_{W}}\rangle$ are the Choi operator of the
superchannel
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}_{\mathrm{pure}}$
and the Choi vector of the unitary operator $U_{W}$, respectively, defined in
Sec. 3.1?.

A quantum switch is an example of pure superchannels and can be represented by
the unitary operator

$\displaystyle U_{\mathrm{QS}}:=$
$\displaystyle\ket{0}^{F_{c}}\bra{0}^{P_{c}}\otimes\mbox{1}\mbox{l}^{P_{t}\to
A_{I}}\otimes\mbox{1}\mbox{l}^{A_{O}\to
B_{I}}\otimes\mbox{1}\mbox{l}^{B_{O}\to F_{t}}$ (1) $\displaystyle+$
$\displaystyle\ket{1}^{F_{c}}\bra{1}^{P_{c}}\otimes\mbox{1}\mbox{l}^{P_{t}\to
B_{I}}\otimes\mbox{1}\mbox{l}^{B_{O}\to
A_{I}}\otimes\mbox{1}\mbox{l}^{A_{O}\to F_{t}},$

where
$\mbox{1}\mbox{l}^{\mathcal{H}_{I}\to\mathcal{H}_{O}}:\mathcal{H}_{I}\to\mathcal{H}_{O}$
is the identity operator (Figure 14).

Figure 14: The quantum switch
$\mathrm{QSWTICH}=\ket{U_{\mathrm{QS}}}\rangle\langle\bra{U_{\mathrm{QS}}}$ is
represented by the unitary operator $U_{\mathrm{QS}}$ defined in Eq. (1).

######  Main result 2.

(Thm. 4) Any pure superchannel with two slots is represented by a direct sum
of pure quantum combs with different causal orders (Figure 15).

Figure 15: A pure superchannel
$W_{\mathrm{pure}}=\ket{U_{W}}\rangle\langle\bra{U_{W}}$ with two slots is
represented by a unitary operator $U_{W}=U_{\mathrm{comb}}^{A\prec B}\oplus
V_{\mathrm{comb}}^{B\prec A}$ with $U_{\mathrm{comb}}^{A\prec
B}=U_{2}U_{1}U_{0}$ (identity matrices on auxiliary systems are abbreviated)
representing a quantum comb of $A\prec B$ and $V_{\mathrm{comb}}^{B\prec
A}=V_{2}V_{1}V_{0}$ representing a quantum comb of $B\prec A$ where
$P=P^{A\prec B}\oplus P^{B\prec A}$ and $F=F^{A\prec B}\oplus F^{B\prec A}$.

##  3 Review on quantum superchannel

In this section, we review a mathematical formalism required to present our
main results. In this paper we consider only finite dimensional complex
Hilbert spaces, isomorphic to $\mathbb{C}^{d}$. Notations in the rest of this
paper are set as follows. We sometimes denote a Hilbert space
$\mathcal{H}_{X}$ just by its index $X$ ($X=P,A_{I},A_{O},B_{I},B_{O},F$).
When we refer to an operator, map, and supermap, we mean a linear operator, a
linear map, and a linear supermap, respectively. $\mathcal{L}(\mathcal{H})$
denotes the set of operators on a Hilbert space $\mathcal{H}$ and
$[\mathcal{L}({\mathcal{H}_{I}})\to\mathcal{L}({\mathcal{H}_{O}})]$ denotes
the set of maps from $\mathcal{L}({\mathcal{H}_{I}})$ to
$\mathcal{L}({\mathcal{H}_{O}})$. We often add superscripts on vectors,
operators, maps, and supermaps such as

  * •

a vector $\ket{\phi}^{\mathcal{H}}\in\mathcal{H}$,

  * •

an operator $\rho^{\mathcal{H}}\in\mathcal{L}(\mathcal{H})$,

  * •

a map
$\widetilde{\mathcal{E}}{}^{\mathcal{H}_{I}\to\mathcal{H}_{O}}\colon\mathcal{L}(\mathcal{H_{I}})\to\mathcal{L}(\mathcal{H_{O}})$,
and

  * •

a supermap
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}{}^{(A_{I}\to
A_{O})\to(P\to
F)}\colon[\mathcal{L}({A_{I}})\to\mathcal{L}({A_{O}})]\to[\mathcal{L}({P})\to\mathcal{L}({F})]$,

to specify their attributions. The domain and the range of a unitary operator
are allowed to differ as long as their dimensions are equal.

###  3.1 Choi-Jamiołkowski isomorphism and linear supermaps

We use the Choi-Jamiołkowski (CJ) isomorphism [28, 29] to represent maps,
which act on operators, as operators themselves on a larger Hilbert space. The
Choi operator of a map
$\widetilde{\mathcal{E}}{}\colon\mathcal{L}(\mathcal{H}_{I})\to\mathcal{L}(\mathcal{H}_{O})$
is defined as

$E:=\sum_{ii^{\prime}}|i\rangle\langle
i^{\prime}|\otimes\widetilde{\mathcal{E}}{}(|i\rangle\langle
i^{\prime}|)\in\mathcal{L}(\mathcal{H}_{I}\otimes\mathcal{H}_{O}),$ (2)

where $\\{\ket{i}\\}_{i}$ is the computational basis od $\mathcal{H}_{I}$. The
action of $\widetilde{\mathcal{E}}{}$ on $\rho\in\mathcal{L}(\mathcal{H}_{I})$
can be obtained via the relation

$\widetilde{\mathcal{E}}{}(\rho)=\rho*E,$ (3)

where $*$ is an operation called the link product [8] defined as follows:

######  Definition 1.

(Link product) Let $\mathcal{H}_{0}$, $\mathcal{H}_{1}$, $\mathcal{H}_{2}$ be
different Hilbert spaces. The link product $E_{01}*F_{12}$ between two
operators $E_{01}\in\mathcal{L}(\mathcal{H}_{0}\otimes\mathcal{H}_{1})$ and
$F_{12}\in\mathcal{L}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})$ is

$E_{01}*F_{12}:=\mathrm{Tr}_{1}[(E_{01}\otimes\mbox{1}\mbox{l}_{2})(\mbox{1}\mbox{l}_{0}\otimes
F_{12}^{T_{1}})]$ (4)

where $\mathrm{Tr}_{1}$ denotes the partial trace on
$\mathcal{L}(\mathcal{H}_{1})$ and $\bullet^{T_{1}}$ denotes the partial
transposition on $\mathcal{L}(\mathcal{H}_{1})$ in terms of the computational
basis.

We can consider a pure version of the Choi isomorphism. The Choi vector of a
operator $A\colon\mathcal{H}_{I}\to\mathcal{H}_{O}$ is

$\ket{A}\rangle:=\sum_{i}\ket{i}\otimes(A\ket{i})\in\mathcal{H}_{I}\otimes\mathcal{H}_{O}.$
(5)

Then, the Choi operator of the operation
$\widetilde{\mathcal{E}}{}_{A}\colon\mathcal{L}(\mathcal{H}_{I})\to\mathcal{L}(\mathcal{H}_{O})$
defined as $\widetilde{\mathcal{E}}{}_{A}(\rho):=A\rho A^{\dagger}$ is equal
to $\ket{A}\rangle\langle\bra{A}$. The action of $A$ on
$\ket{\phi}\in\mathcal{H}_{I}$ can be obtained from

$A\ket{\phi}=\braket{\phi^{*}}{A}\rangle$ (6)

where $\ket{\phi^{*}}$ denotes the complex conjugation of $\ket{\phi}$ in the
computational basis, that is, if $\ket{\phi}=\sum_{i}\phi_{i}\ket{i}$, then
$\ket{\phi^{*}}=\sum_{i}\phi_{i}^{*}\ket{i}$.

A supermap is a linear transformation from a set of maps to a set of maps.
Similarly to maps, supermaps can also be represented by operators. This
isomorphism between supermaps and operators can be shown by first showing that
supermaps can be represented by maps [6, 9], and then, by using the CJ
isomorphism on this map representation. In this paper, our map representation
of supermaps is slightly different from those used in previous references [6,
9, 30]. Our convention for representing supermaps as maps provides good
insights for pure superchannels, which will be evident after considering Thm.
1 or Figure 13.

Let
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}\colon[\mathcal{L}({A_{I}})\to\mathcal{L}({A_{O}})]\to[\mathcal{L}({P})\to\mathcal{L}({F})]$
be a supermap from the set of maps
$\mathcal{L}({A_{I}})\to\mathcal{L}({A_{O}})$ to the set of maps
$\mathcal{L}({P})\to\mathcal{L}({F})$. Define a map
$\widetilde{\mathcal{S}}{}\colon\mathcal{L}(P\otimes
A_{O})\to\mathcal{L}(A_{I}\otimes F)$ as follows:

$\widetilde{\mathcal{S}}{}^{PA_{O}\to
A_{I}F}(\rho^{P}\otimes\sigma^{A_{O}}):=\\\
\left(\left({\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{I}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{I}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{I}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{I}}}$}}}}{}^{(\mathbb{C}\to
A_{I})\to(\mathbb{C}\to
A_{I})}\otimes{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}^{(A_{I}\to
A_{O})\to(P\to F)}\right)\left(\widetilde{\mathcal{I}}{}^{A_{I}\to
A_{I}}\otimes\tilde{\sigma}^{\mathbb{C}\to
A_{O}}\right)\right)(|0\rangle\langle 0|^{\mathbb{C}}\otimes\rho^{P})$ (7)

for all $\rho^{P}\in\mathcal{L}(P)$ and $\sigma^{A_{O}}\in\mathcal{L}(A_{O})$
where $\ket{0}^{\mathbb{C}}$ is a unit vector in $\mathbb{C}$,
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{I}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{I}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{I}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{I}}}$}}}}{}^{(\mathbb{C}\to
A_{I})\to(\mathbb{C}\to
A_{I})}\colon[\mathcal{L}(\mathbb{C})\to\mathcal{L}(A_{I})]\to[\mathcal{L}(\mathbb{C})\to\mathcal{L}(A_{I})]$
is the identity supermap on the set of maps
$\mathcal{L}(\mathbb{C})\to\mathcal{L}(A_{I})$,
$\widetilde{\mathcal{I}}{}^{A_{I}\to
A_{I}}\colon\mathcal{L}(A_{I})\to\mathcal{L}(A_{I})$ is the identity map on
$\mathcal{L}(A_{I})$ and $\tilde{\sigma}^{\mathbb{C}\to
A_{O}}\colon\mathcal{L}(\mathbb{C})\to\mathcal{L}(A_{O})$ is an isometry which
creates $\sigma^{A_{O}}$ defined as

$\tilde{\sigma}^{\mathbb{C}\to
A_{O}}(\tau^{\mathbb{C}}):=\mathrm{Tr}[\tau^{\mathbb{C}}]\sigma^{A_{O}}\quad\forall\tau^{\mathbb{C}}\in\mathcal{L}(\mathbb{C})$
(8)

(see Figure 16). Note that there is only one density operator on $\mathbb{C}$,
which is $|0\rangle\langle 0|^{\mathbb{C}}$.

Figure 16: A supermap
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}$
can be seen as a map $\widetilde{\mathcal{S}}{}$ defined in Eq. (7). The gray
line represents $\mathbb{C}$, which is a one-dimensional quantum system.

Let $S$ be the Choi operator of $\widetilde{\mathcal{S}}{}$. The action of
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}$
on $\widetilde{\mathcal{E}}{}:\mathcal{L}(A_{I})\to\mathcal{L}(A_{O})$ can be
retrieved via the relation in the CJ isomorphism

$E^{\prime}=S*E$ (9)

where $E$, $E^{\prime}$ are the Choi operators of $\widetilde{\mathcal{E}}{}$,
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}(\widetilde{\mathcal{E}}{})$,
respectively. Now that
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}$,
$\widetilde{\mathcal{S}}{}$ and $S$ are isomorphic, we can define $S$ as the
Choi operator of
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}$.

In Refs. [6, 9], the authors represent a supermap
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}\colon[\mathcal{L}({A_{I}})\to\mathcal{L}({A_{O}})]\to[\mathcal{L}({P})\to\mathcal{L}({F})]$
as a map
$\widetilde{\mathcal{S}}{}_{\mathrm{prev}}:=\mathfrak{C}^{\prime}\circ{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}\circ\mathfrak{C}^{-1}\colon\mathcal{L}(A_{I}\otimes
A_{O})\to\mathcal{L}(P\otimes F)$, where
$\mathfrak{C}\colon[\mathcal{L}({A_{I}})\to\mathcal{L}({A_{O}})]\to\mathcal{L}(A_{I}\otimes
A_{O})$ and
$\mathfrak{C}^{\prime}\colon[\mathcal{L}({P})\to\mathcal{L}({F})]\to\mathcal{L}(P\otimes
F)$ transform maps into their Choi operators and they obtain the Choi operator
of $\widetilde{\mathcal{S}}{}_{\mathrm{prev}}$ as the Choi operator of
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}$.
Although our map representation $\widetilde{\mathcal{S}}{}$ of the supermap
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}$
is different from previous one $\widetilde{\mathcal{S}}{}_{\mathrm{prev}}$,
the Choi operators of $\widetilde{\mathcal{S}}{}_{\mathrm{prev}}$ and
$\widetilde{\mathcal{S}}{}$ coincide. In Ref. [30], there is another
representation of supermap as a map, but the Choi operator of that map also
coincides with ours.

###  3.2 Quantum superchannel

We review the notions related to quantum superchannels. We recall that quantum
channels are completely positive and trace-preserving maps (CPTP maps).
Quantum superchannels with one slot are supermaps which transform quantum
channels into quantum channels even if the supermaps act on a subsystem of the
input quantum channels (see Figure 4).

######  Definition 2.

(Quantum superchannel with one slot) Let $W$ be the Choi operator of a
supermap
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}\colon[A_{I}\to
A_{O}]\to[P\to F]$. We say that
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}$
is a quantum superchannel with one slot if for all auxiliary Hilbert spaces
$A_{I}^{\prime}$, $A_{O}^{\prime}$ and all quantum channels
$\widetilde{\mathcal{E}}{}\colon\mathcal{L}(A_{I}\otimes
A_{I}^{\prime})\to\mathcal{L}(A_{O}\otimes A_{O}^{\prime})$, $G=W\ast E$ is
the Choi operator of a quantum channel
$\widetilde{\mathcal{G}}{}\colon\mathcal{L}(P\otimes
A_{I}^{\prime})\to\mathcal{L}(F\otimes A_{O}^{\prime})$, where $E$ is the Choi
operator of $\widetilde{\mathcal{E}}{}$.

In this paper we also consider superchannels with more than a single slot.
Multiple slot superchannels are supermaps transforming causally independent
quantum channels, which are represented by tensor product of channels, into a
quantum channel222Note that due to linearity, the action of superchannels with
multiple slots is also defined for linear combinations of tensor product of
quantum channels. The set of multipartite quantum channel which can be written
as a linear combination of tensor product of single-party quantum channels is
the set of non-signalling channels [31, 32, 33]. (see Figure 11).

######  Definition 3.

(Quantum superchannel with multiple slots) Let $N$ be a positive integer,
$\mathcal{H}_{m}$ be Hilbert spaces, $m\in\\{0,\cdots,2N+1\\}$, and $W$ be the
Choi operator of a supermap
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}\colon\bigotimes_{n=1}^{N}[\mathcal{L}(\mathcal{H}_{2n-1})\to\mathcal{L}(\mathcal{H}_{2n})]\to[\mathcal{L}(\mathcal{H}_{0})\to\mathcal{L}(\mathcal{H}_{2N+1})]$.
We say that
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}$
is a quantum superchannel with $N$ slots if for all auxiliary Hilbert spaces
$\mathcal{H}_{m}^{\prime}$, $m\in\\{1,\cdots,2N\\}$, and all quantum channels
$\widetilde{\mathcal{E}}{}_{n}\colon\mathcal{L}(\mathcal{H}_{2n-1}\otimes\mathcal{H}_{2n-1}^{\prime})\to\mathcal{L}(\mathcal{H}_{2n}\otimes\mathcal{H}_{2n}^{\prime})$,
$n\in\\{1,\cdots,N\\}$, $G=W\ast(\bigotimes_{n=1}^{N}E_{n})$ is the Choi
operator of a quantum channel
$\widetilde{\mathcal{G}}{}\colon\mathcal{L}(\mathcal{H}_{0}\otimes\bigotimes_{n=1}^{N}\mathcal{H}_{2n-1}^{\prime})\to\mathcal{L}(\mathcal{H}_{2N+1}\otimes\bigotimes_{n=1}^{N}\mathcal{H}_{2n}^{\prime})$,
where $E_{n}$ are the Choi operators of $\widetilde{\mathcal{E}}{}_{n}.$

Quantum superchannels with a single slot are introduced in Ref. [4] as
deterministic quantum supermaps. Quantum superchannels with $N$ slots appeared
in Ref. [9] as quantum supermaps on product channels. The Choi operators of
superchannels with $N$ slot are equal to $N$-partite process matrices [6] with
a global past and global future [17, 7]. $N$-partite process matrices with no
global past and no global future are equivalent to the Choi operators of
quantum superchannels with $N$ slots, where the first input space
$\mathcal{H}_{0}$ and the last output space $\mathcal{H}_{2N+1}$ is the one-
dimensional linear space $\mathbb{C}$. In this paper, we mainly focus on
quantum superchannels with two slots
$[\mathcal{L}(A_{I})\to\mathcal{L}(A_{O})]\otimes[\mathcal{L}(B_{I})\to\mathcal{L}(B_{O})]\to[\mathcal{L}(P)\to\mathcal{L}(F)]$
(see Figure 17). We name slots for maps
$\mathcal{L}(A_{I})\to\mathcal{L}(A_{O})$,
$\mathcal{L}(B_{I})\to\mathcal{L}(B_{O})$ as $A$, $B$, respectively. Slots
like $A$ and $B$ in a quantum superchannel can be identified as parties of
Alice and Bob in a process matrix.

Figure 17:  A quantum superchannel
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}\colon[\mathcal{L}(A_{I}\otimes
B_{I})\to\mathcal{L}(A_{O}\otimes B_{O})]\to[\mathcal{L}(P)\to\mathcal{L}(F)]$
with two slots $A$ and $B$.

Next, we review the concept of pure superchannels, which is first defined in
Ref. [7] as a pure process333Note that Ref. [17] considers a different
definition of pure process, where the authors define the pure process matrix
to be a rank-one process matrix.. We define unitary channels as maps
$\widetilde{\mathcal{U}}{}(\rho):=U\rho U^{\dagger}$ for all
$\rho\in\mathcal{L}(\mathcal{H}_{I})$ with a unitary operator
$U\colon\mathcal{H}_{I}\to\mathcal{H}_{O}$. Pure superchannels are quantum
superchannels, which transform a group of causally independent unitary
channels into unitary channels.

######  Definition 4.

(Pure superchannel) Let $W$ be the Choi operator of a quantum superchannel
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}\colon\bigotimes_{n=1}^{N}[\mathcal{L}(\mathcal{H}_{2n-1})\to\mathcal{L}(\mathcal{H}_{2n})]\to[\mathcal{L}(\mathcal{H}_{0})\to\mathcal{L}(\mathcal{H}_{2N+1})]$
with $N$ slots. We say that $W$ (and
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}$)
is pure if for all auxiliary Hilbert spaces $\mathcal{H}_{m}^{\prime}$,
$m\in\\{1,\cdots,2N\\}$, and unitary operators
$U_{n}:\mathcal{H}_{2n-1}\otimes\mathcal{H}_{2n-1}^{\prime}\to\mathcal{H}_{2n}\otimes\mathcal{H}_{2n}^{\prime}$,
$n\in\\{1,\cdots,N\\}$, there exists a unitary operator
$U_{G}:\mathcal{H}_{0}\otimes\bigotimes_{n=1}^{N}\mathcal{H}_{2n-1}^{\prime}\to\mathcal{H}_{2N+1}\otimes\bigotimes_{n=1}^{N}\mathcal{H}_{2n}^{\prime}$
such that
$\ket{U_{G}}\rangle\langle\bra{U_{G}}=W\ast\bigotimes_{n=1}^{N}\ket{U_{n}}\rangle\langle\bra{U_{n}}$.

Any pure superchannel can be represented as a unitary channel (see Figure 13)
from the following theorem:

######  Theorem 1.

Let $W$ be the Choi operator of a quantum superchannel
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}\colon\bigotimes_{n=1}^{N}[\mathcal{L}(\mathcal{H}_{2n-1})\to\mathcal{L}(\mathcal{H}_{2n})]\to[\mathcal{L}(\mathcal{H}_{0})\to\mathcal{L}(\mathcal{H}_{2N+1})]$.
$W$ is pure if and only if there exists a unitary operator
$U_{W}\colon\bigotimes_{n=0}^{N}\mathcal{H}_{2n}\to\bigotimes_{n=0}^{N}\mathcal{H}_{2n+1}$
such that $W=\ket{U_{W}}\rangle\langle\bra{U_{W}}$.

In this paper, we say that $U_{W}$ represents the superchannel
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{W}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{W}}}$}}}}{}$
in such cases. A proof of this theorem follows the same steps of Theorem 2 in
Ref. [7] which considers the case $N=2$. Note that the unitary operator
representing a quantum superchannel is uniquely determined up to a global
phase factor.

######  Lemma 1.

Let $U_{W}\colon P\otimes A_{O}\to A_{I}\otimes F$ and $U_{A}\colon
A_{I}\otimes A_{I}^{\prime}\to A_{O}\otimes A_{O}^{\prime}$ be operators, not
necessarily unitary. There exists an operator $U_{G}\colon P\otimes
A_{I}^{\prime}\to F\otimes A_{O}^{\prime}$ such that

$\ket{U_{G}}\rangle\langle\bra{U_{G}}=\ket{U_{W}}\rangle\langle\bra{U_{W}}\ast\ket{U_{A}}\rangle\langle\bra{U_{A}}.$
(10)

In addition,

$U_{G}=\mathrm{Tr}_{A_{O}}\left[\left(\mbox{1}\mbox{l}^{{F}}\otimes
U_{A}\right)\left(U_{W}\otimes\mbox{1}\mbox{l}^{A_{I}^{\prime}}\right)\right].$
(11)

A proof of this lemma follows from direct calculations and the fact that a
global phase factor is irrelevant.

From this lemma, $U_{G}$ in Def. 4 can be calculated using $U_{W}$ in Thm. 1:

$U_{G}=\mathrm{Tr}_{\mathcal{H}_{2}\mathcal{H}_{4}\cdots\mathcal{H}_{2N}}\left[\left(\mbox{1}\mbox{l}^{\mathcal{H}_{2N+1}}\otimes\bigotimes_{n=1}^{N}U_{n}\right)\left(U_{W}\otimes\bigotimes_{n=1}^{N}\mbox{1}\mbox{l}^{\mathcal{H}_{2n-1}^{\prime}}\right)\right].$
(12)

###  3.3 Quantum comb

Quantum combs are causally ordered superchannels (see Figure 7), which is
first introduced in Ref. [8] and are mathematically equivalent to quantum
channels with memory [26] and to quantum strategies [27].

######  Definition 5.

(Quantum comb)

We say that
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}{}^{(2)}$
is a quantum comb with one slot if
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}{}^{(2)}$
is a quantum superchannel with one slot. Let $N\geq 2$ be a positive integer
and let $\mathcal{H}_{m}$ be Hilbert spaces, $m\in\\{0,\cdots,2N+1\\}$, and
$R^{(N+1)}$ be the Choi operator of a supermap
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}{}^{(N+1)}\colon\bigotimes_{n=1}^{N}[\mathcal{L}(\mathcal{H}_{2n-1})\to\mathcal{L}(\mathcal{H}_{2n})]\to[\mathcal{L}(\mathcal{H}_{0})\to\mathcal{L}(\mathcal{H}_{2N+1})]$.
We say that
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}{}^{(N+1)}$
is a quantum comb with $N$ slots if for all auxiliary Hilbert spaces
$\mathcal{H}_{1}^{\prime}$ and $\mathcal{H}_{2N}^{\prime}$ and all quantum
combs with $N-1$ slots
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}{}^{(N)}\colon\bigotimes_{n=1}^{N-1}[\mathcal{L}(\mathcal{H}_{2n})\to\mathcal{L}(\mathcal{H}_{2n+1})]\to[\mathcal{L}(\mathcal{H}_{1}\otimes\mathcal{H}_{1}^{\prime})\to\mathcal{L}(\mathcal{H}_{2N}\otimes\mathcal{H}_{2N}^{\prime})]$,
$G=R^{(N+1)}*S^{(N)}$ is the Choi operator of a quantum channel
$\widetilde{\mathcal{G}}{}\colon\mathcal{L}(\mathcal{H}_{0}\otimes\mathcal{H}_{1}^{\prime})\to\mathcal{L}(\mathcal{H}_{2N+1}\otimes\mathcal{H}_{2N}^{\prime})$
where $S^{(N)}$ is the Choi operator of
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{S}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{S}}}$}}}}{}{}^{(N)}$.

Note that the slots of a quantum comb have an order. In the above definition,
the $n$-th slot is a slot whose input space is $\mathcal{H}_{2n-1}$ and whose
output space is $\mathcal{H}_{2n}$ ($n=1\cdots N$). The orders of slots of
quantum combs are suppressed throughout the remaining text unless otherwise
misunderstood.

Similarly to pure superchannels, we can define the pure combs.

######  Definition 6.

(Pure comb)

A pure comb is a quantum comb which is a pure superchannel.

Quantum combs are special cases of quantum superchannels. Any quantum
superchannel with one slot is a quantum comb with one slot [4]. Suppose a
quantum comb with two slots has the first slot $A$ and the second slot $B$,
then we call it a quantum comb of444We stress that $A\prec B$ does not
necessarily imply that $A$ signals to $B$ but that $B$ is forbidden to signal
to $A$. $A\prec B$ (read as “$A$ to $B$”). Similarly, we can consider the
notion of a quantum comb of $B\prec A$. If a quantum superchannel can be
regarded as a quantum comb of $A\prec B$ and as a quantum comb of $B\prec A$,
it is parallel and we call it a quantum comb of $A\parallel B$ (read as “$A$
parallel to $B$”).

A necessary and sufficient condition for a supermap to be a quantum comb can
be obtained from the following theorem [34] and any quantum comb is equivalent
to a causally ordered quantum circuit [8] (see Figure 7):

######  Theorem 2.

Let $N$ be a positive integer and let $\mathcal{H}_{m}$ be Hilbert spaces,
$m\in\\{0,\cdots,2N+1\\}$, and $R^{(N+1)}$ be the Choi operator of a supermap
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}{}^{(N+1)}\colon\bigotimes_{n=1}^{N}[\mathcal{L}(\mathcal{H}_{2n-1})\to\mathcal{L}(\mathcal{H}_{2n})]\to[\mathcal{L}(\mathcal{H}_{0})\to\mathcal{L}(\mathcal{H}_{2N+1})]$.
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}{}^{(N+1)}$
is a quantum comb with $N$ slots if and only if $R^{(N+1)}\geq 0$ and for each
$n\in\\{0,\cdots,N\\}$,

$\mathrm{Tr}_{\mathcal{H}_{2n+1}\cdots\mathcal{H}_{2N-1}\mathcal{H}_{2N+1}}R^{(N+1)}=\mbox{1}\mbox{l}^{\mathcal{H}_{2n}\cdots\mathcal{H}_{2N-2}\mathcal{H}_{2N}}\otimes
R^{(n)}$ (13)

holds where $R^{(n)}$ ($n\geq 2$) is the Choi operator of a quantum comb with
$n-1$ slots, $R^{(1)}$ is the Choi operator of a quantum channel (“quantum
comb with $0$ slot”) and $R^{(0)}=1$ (“quantum comb with $-1$ slot”).

##  4 Main results and their implications

We are now ready to precisely state our main results and their implications.

###  4.1 Main result 1: Pure comb

Every pure comb with $N$ slots can be realized by $(N+1)$ unitary channels and
does not require discarding any system (see Figure 9).

######  Theorem 3.

Let
$U^{(N+1)}:\bigotimes_{n=0}^{N}\mathcal{H}_{2n}\to\bigotimes_{n=0}^{N}\mathcal{H}_{2n+1}$
be a unitary operator representing a pure quantum comb
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{{\mathcal{R}}}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{{\mathcal{R}}}$}}}}{}{}^{(N+1)}\colon\bigotimes_{n=1}^{N}[\mathcal{L}(\mathcal{H}_{2n-1})\to\mathcal{L}(\mathcal{H}_{2n})]\to[\mathcal{L}(\mathcal{H}_{0})\to\mathcal{L}(\mathcal{H}_{2N+1})]$
with $N$ slots. There exist integers $\\{k_{n}\\}_{n=0}^{N+1}$ such that
$k_{0}=k_{N+1}=1$ and

$d_{2n}k_{n}=d_{2n+1}k_{n+1},\quad n\in\\{0,\cdots,N\\},$ (14)

where $d_{m}:=\dim\mathcal{H}_{m}$. $U^{(N+1)}$ can be decomposed into

$U^{(N+1)}=U_{N}\cdots U_{1}U_{0}$ (15)

by using some unitary operators
$U_{n}:\mathcal{H}_{2n}\otimes\mathcal{A}_{n}\to\mathcal{H}_{2n+1}\otimes\mathcal{A}_{n+1}$
($n=0\cdots N+1$) for a Hilbert spaces $\mathcal{A}_{n}$ ($n=0\cdots N+1$)
such that $\dim\mathcal{A}_{n}=k_{n}$. (In Eq. (15), it is implicitly assumed
that $U_{n}$ trivially acts on all the spaces but
$\mathcal{H}_{2n}\otimes\mathcal{A}_{n}$.)

###### Proof.

See B. ∎

In Ref. [11] the authors prove that every quantum comb with $N$ slots can be
realized by $(N+1)$ isometries followed by a partial trace which represents
discarding part of the quantum system. Theorem 3 shows that a quantum comb is
pure if and only if all these isometries can be replaced with unitaries and no
quantum system is discarded. We refer to Figure 8 for a pictorial illustration
of general combs and Figure 9 for a pictorial illustration of pure combs.

###  4.2 Main result 2: Pure superchannel

Before our second main statement, we introduce direct sum of pure combs as a
generalization of the quantum switch (see Figure 12). We observe that the
quantum switch is represented by the following unitary operator (see Figure 14
and Eq. (1)):

$\displaystyle U_{\mathrm{QS}}=$
$\displaystyle\ket{0}^{F_{c}}\bra{0}^{P_{c}}\otimes\mbox{1}\mbox{l}^{P_{t}\to
A_{I}}\otimes\mbox{1}\mbox{l}^{A_{O}\to
B_{I}}\otimes\mbox{1}\mbox{l}^{B_{O}\to
F_{t}}+\ket{1}^{F_{c}}\bra{1}^{P_{c}}\otimes\mbox{1}\mbox{l}^{P_{t}\to
B_{I}}\otimes\mbox{1}\mbox{l}^{B_{O}\to
A_{I}}\otimes\mbox{1}\mbox{l}^{A_{O}\to F_{t}}$ (16) $\displaystyle\cong$
$\displaystyle\mbox{1}\mbox{l}^{P^{A\prec B}\to
A_{I}}\otimes\mbox{1}\mbox{l}^{A_{O}\to
B_{I}}\otimes\mbox{1}\mbox{l}^{B_{O}\to F^{A\prec
B}}\oplus\mbox{1}\mbox{l}^{P^{B\prec A}\to
B_{I}}\otimes\mbox{1}\mbox{l}^{B_{O}\to
A_{I}}\otimes\mbox{1}\mbox{l}^{A_{O}\to F^{B\prec A}},$

where555$\oplus$ denotes the orthogonal direct sum of linear subspaces. See
A.1. $X=X_{c}\otimes X_{t}=X^{A\prec B}\oplus X^{B\prec A}$, $\dim X_{c}=2$,
$\dim X_{t}=\dim X^{A\prec B}=\dim X^{B\prec A}=\dim A_{I}=\dim A_{O}=\dim
B_{I}=\dim B_{O}$ ($X=P,F$). Clearly, $\mbox{1}\mbox{l}^{P^{A\prec B}\to
A_{I}}\otimes\mbox{1}\mbox{l}^{A_{O}\to
B_{I}}\otimes\mbox{1}\mbox{l}^{B_{O}\to F^{A\prec B}}$ is a pure quantum comb
of $A\prec B$ and $\mbox{1}\mbox{l}^{P^{B\prec A}\to
B_{I}}\otimes\mbox{1}\mbox{l}^{B_{O}\to
A_{I}}\otimes\mbox{1}\mbox{l}^{A_{O}\to F^{B\prec A}}$ is a pure quantum comb
of $B\prec A$. The Choi vector form of this operator is

$\ket{U_{\mathrm{QS}}}\rangle=\ket{0}^{P_{c}}\ket{\mbox{1}\mbox{l}}\rangle^{P_{t}A_{I}}\ket{\mbox{1}\mbox{l}}\rangle^{A_{O}B_{I}}\ket{\mbox{1}\mbox{l}}\rangle^{B_{O}F_{t}}\ket{0}^{F_{c}}+\ket{1}^{P_{c}}\ket{\mbox{1}\mbox{l}}\rangle^{P_{t}B_{I}}\ket{\mbox{1}\mbox{l}}\rangle^{B_{O}A_{I}}\ket{\mbox{1}\mbox{l}}\rangle^{A_{O}F_{t}}\ket{1}^{F_{c}}.$
(17)

Now we define the direct sum of pure combs.

######  Definition 7.

(Direct sum of pure combs) Let $U^{A\prec B}:P^{A\prec B}\otimes A_{O}\otimes
B_{O}\to A_{I}\otimes B_{I}\otimes F^{A\prec B}$ be a unitary operator
representing a quantum comb of $A\prec B$ and $U^{B\prec A}:P^{B\prec
A}\otimes A_{O}\otimes B_{O}\to A_{I}\otimes B_{I}\otimes F^{B\prec A}$ be a
unitary operator representing a quantum comb of $B\prec A$. A supermap
represented by a unitary operator $U\colon P\otimes A_{O}\otimes B_{O}\to
A_{I}\otimes B_{I}\otimes F$ defined as

$U=U^{A\prec B}\oplus U^{B\prec A}$ (18)

is a direct sum of pure combs with two slots $A$ and $B$, where $P:=P^{A\prec
B}\oplus P^{B\prec A}$ and $F:=F^{A\prec B}\oplus F^{B\prec A}$.

This definition can also be written by in terms of Choi vectors. Equation (18)
is equivalent to $\ket{U}\rangle=\ket{U^{A\prec B}}\rangle+\ket{U^{B\prec
A}}\rangle$. Note that $\ket{U^{A\prec B}}\rangle\in P^{A\prec B}\otimes
A_{O}\otimes B_{O}\otimes A_{I}\otimes B_{I}\otimes F^{A\prec B}$ and
$\ket{U^{B\prec A}}\rangle\in P^{B\prec A}\otimes A_{O}\otimes B_{O}\otimes
A_{I}\otimes B_{I}\otimes F^{B\prec A}$ although the sum of the two Choi
vectors is taken in $P\otimes A_{O}\otimes B_{O}\otimes A_{I}\otimes
B_{I}\otimes F$, which is valid because $P^{A\prec B}\otimes A_{O}\otimes
B_{O}\otimes A_{I}\otimes B_{I}\otimes F^{A\prec B}$ and $P^{B\prec A}\otimes
A_{O}\otimes B_{O}\otimes A_{I}\otimes B_{I}\otimes F^{B\prec A}$ are
subspaces of $P\otimes A_{O}\otimes B_{O}\otimes A_{I}\otimes B_{I}\otimes F$.
Moreover, $P=P^{A\prec B}\oplus P^{B\prec A}$ and $F=P^{A\prec B}\oplus
F^{B\prec A}$ are satisfied. If not satisfied, then $U$ is not unitary. The
quantum switch (Eq. (16)) is an example of a direct sum of pure combs such
that

$U^{A\prec B}=\ket{0}^{F_{c}}\bra{0}^{P_{c}}\otimes\mbox{1}\mbox{l}^{P_{t}\to
A_{I}}\otimes\mbox{1}\mbox{l}^{A_{O}\to
B_{I}}\otimes\mbox{1}\mbox{l}^{B_{O}\to F_{t}},$ (19) $U^{B\prec
A}=\ket{1}^{F_{c}}\bra{1}^{P_{c}}\otimes\mbox{1}\mbox{l}^{P_{t}\to
B_{I}}\otimes\mbox{1}\mbox{l}^{B_{O}\to
A_{I}}\otimes\mbox{1}\mbox{l}^{A_{O}\to F_{t}}$ (20)

where

$P^{A\prec B}=\text{SPAN}[\ket{0}^{P_{c}}]\otimes P_{t},$ (21) $P^{B\prec
A}=\text{SPAN}[\ket{1}^{P_{c}}]\otimes P_{t},$ (22) $F^{A\prec
B}=\text{SPAN}[\ket{0}^{F_{c}}]\otimes F_{t},$ (23) $F^{B\prec
A}=\text{SPAN}[\ket{1}^{F_{c}}]\otimes F_{t},$ (24)

and $\text{SPAN}[\ket{\psi}]$ is the one-dimensional subspace spanned by
$\ket{\psi}$, e.g.,

$\text{SPAN}[\ket{0}^{P_{c}}]:=\\{c\ket{0}^{P_{c}}|c\in\mathbb{C}\\}.$ (25)

Our second main result is that a pure superchannel with two slots is a direct
sum of pure combs.

######  Theorem 4.

All pure superchannels with two slots are equivalent to direct sums of pure
combs.

More formally, we prove the following theorem.

######  Theorem 5.

Let $U:P\otimes A_{O}\otimes B_{O}\to A_{I}\otimes B_{I}\otimes F$ be a
unitary operator. The following conditions are equivalent:

(1)

$U$ represents a pure quantum superchannel with two slots $A$ and $B$.

(2)

For all $\ket{\pi}^{P},\,\ket{{\pi}^{\prime}}^{P}\in P$,
$\ket{\alpha}^{A_{O}},\,\ket{\alpha^{\prime}}^{A_{O}},\,\ket{\alpha^{\perp}}^{A_{O}}\in
A_{O}$ such that $\ket{\alpha}^{A_{O}}\perp\ket{\alpha^{\perp}}^{A_{O}}$,
$\ket{\beta}^{B_{O}},\,\ket{\beta^{\prime}}^{B_{O}},\,\ket{\beta^{\perp}}^{B_{O}}\in
B_{O}$ such that $\ket{\beta}^{B_{O}}\perp\ket{\beta^{\perp}}^{B_{O}}$,
$\ket{\Gamma}\rangle^{A_{I}B_{I}},\,\ket{{\Gamma}^{\prime}}\rangle^{A_{I}B_{I}}\in
A_{I}\otimes B_{I}$, $\ket{\phi}^{A_{I}},\,\ket{{\phi}^{\prime}}^{A_{I}}\in
A_{I}$ and $\ket{\psi}^{B_{I}},\,\ket{{\psi}^{\prime}}^{B_{I}}\in B_{I}$,

$\left({}^{A_{I}B_{I}}\langle\bra{\Gamma}U\left(\ket{\pi}^{P}\ket{\alpha}^{A_{O}}\ket{\beta}^{B_{O}}\right),{}^{A_{I}B_{I}}\langle\bra{{\Gamma}^{\prime}}U\left(\ket{{\pi}^{\prime}}^{P}\ket{\alpha^{\perp}}^{A_{O}}\ket{\beta^{\perp}}^{B_{O}}\right)\right)=0,$
(a)
$\left({}^{B_{I}}\bra{\psi}U\left(\ket{\pi}^{P}\ket{\alpha}^{A_{O}}\ket{\beta}^{B_{O}}\right),{}^{B_{I}}\bra{{\psi}^{\prime}}U\left(\ket{{\pi}^{\prime}}^{P}\ket{\alpha^{\prime}}^{A_{O}}\ket{\beta^{\perp}}^{B_{O}}\right)\right)=0,$
(b)
$\left({}^{A_{I}}\bra{\phi}U\left(\ket{\pi}^{P}\ket{\alpha}^{A_{O}}\ket{\beta}^{B_{O}}\right),{}^{A_{I}}\bra{{\phi}^{\prime}}U\left(\ket{{\pi}^{\prime}}^{P}\ket{\alpha^{\perp}}^{A_{O}}\ket{\beta^{\prime}}^{B_{O}}\right)\right)=0.$
(c)

(3)

$U$ represents a direct sum of pure combs with two slots $A$ and $B$.

The implication (1) $\implies$ (2) is proved from straightforward calculations
(C). The condition (2) can be simply rephrased in terms of the reduced
subspace, introduced in A.5. After rephrasing, we provide the proof of (2)
$\implies$ (3) in D. The last implication (3) $\implies$ (1) is clear since if
we insert two arbitrary unitary operators $U_{A}:A_{I}\otimes
A_{I}^{\prime}\to A_{O}\otimes A_{O}^{\prime}$, $U_{B}:B_{I}\otimes
B_{I}^{\prime}\to B_{O}\otimes B_{O}^{\prime}$, in the two slots $A$ and $B$,
the output is also a unitary operation. This can be rigorously verified via
Lem. 7.

###  4.3 Implications of the main results

Recall that the Choi operators of quantum superchannels with two slots
$[[\mathcal{L}(A_{I})\to\mathcal{L}(A_{O})]\otimes[\mathcal{L}(B_{I})\to\mathcal{L}(B_{O})]]\to[\mathcal{L}(P)\to\mathcal{L}(F)]$
are equivalent to bipartite process matrices [6] with two parties $A$ and $B$,
global past $P$ and global future $F$. Thus, from Thm. 4, we have three
implications about a bipartite process matrix, in particular, about a
purifiable process matrix defined in Ref. [7].

A process matrix with a global past and global future is pure if it is a pure
superchannel. A process matrix $W$ with a global past and global future is
purifiable if it can be obtained from a pure process matrix
$W^{\prime}_{\mathrm{pure}}$ with global past $P$ and global future $F$ by
inserting a fixed state $\ket{0}^{P_{\mathrm{sub}}}$ in a subsystem
$P_{\mathrm{sub}}$ of $P$ and tracing out a subsystem $F_{\mathrm{sub}}$ of
$F$, that is, $W=W^{\prime}_{\mathrm{pure}}*(|0\rangle\langle
0|^{P_{\mathrm{sub}}}\otimes\mbox{1}\mbox{l}^{F_{\mathrm{sub}}})$. These
notions of a pure process matrix and a purifiable process matrix are
introduced based on the principle of “reversibility of the transformations
between quantum states” in Ref. [7] and the authors propose a postulate that a
process matrix admits a “fair” physical realization if and only if it is
purifiable.

The first implication of our main result concerns causally separable [6]
purifiable process matrices without a global future. A process matrix with a
global past and global future is causally ordered if it is a quantum comb. In
the bipartite case, a causally separable process matrix defined in Ref. [6] is
a probabilistic mixture of causally ordered process matrices of two different
causal orders $A\prec B$ and $B\prec A$. Now, we present the first
implication.

######  Corollary 1.

All purifiable bipartite processes without a global future are causally
separable.

###### Proof.

From Thm. 4, we see that all bipartite pure processes can be written as a
direct sum of pure combs. Note that when the partial trace on the total global
future space of a direct sum of pure comb is taken, the resulting operator is
a convex combination of causally ordered processes. ∎

The next implication is about device-independent certifications of indefinite
causal order, concept that we now present a brief introduction. One method to
certify that an uncharacterized superchannel does not respect any definite
causal order is to design an experiment where one prepares a set of well
characterized quantum instruments, “plug” these instruments and analyze the
statistical data collected after many rounds of this experiment. This approach
where the quantum instruments are totally characterized is referred as device-
dependent and it is the scenario considered in previous experiments involving
superchannels [20, 21, 22]. All superchannels with indefinite causal order can
be certified in a device-dependent way by means of a causal witness [17],
which is an inequality is respected by all causally separably superchannels
but may be violated by indefinite causally order ones.

In a device-independent approach, the goal is to certify indefinite causal
order of a superchannel even without characterizing the instruments involved
in the experiment. That is, the data analysis of the experiment is conducted
without assuming any particular internal structure or knowledge on instruments
performed at each slot. Moreover, with due experimental hypotheses, device-
independent indefinite causal order certification can identify indefinite
causal order without assuming all the laws of quantum theory [6]. The device-
independent approach then allows us to state strictly stronger claims about
Nature and experiments than the device-dependent approach. Reference [6] shows
there exists two-slot superchannels (or equivalently, bipartite process
matrices) allowing device-independent certification of indefinite causal
order. Also, this device independent certification can always be made by a
(device-independent) causal inequality [35], which is the device-independent
analogue of the device-dependent (causal witness). Although all indefinite
causally ordered superchannels violate a device-independent causal witness,
not all indefinite causally ordered superchannels can violate a device-
independent causal inequality [17, 36]. Processes which do not violate any
device-independent causal inequality are said to be device-independent causal.

In Ref. [37], the authors define an extensibly causal process as a device-
independent causal process which remains device-independent causal even when
the parties have access to additional arbitrary entangled input state. The set
of extensibly causal process is known to be strictly smaller than the set of
device-independent causal ones [36]. Reference [37] shows that the quantum
switch is extensibly causal. Here we show that every purifiable bipartite
process (potentially with a global past and global future) is extensibly
causal. In Ref. [7] the authors claim “In fact, we haven’t found a
purification for any bipartite process that was able to violate a causal
inequality.”, and here we prove a strictly stronger statement by showing that
all purifiable process are extensibly causal.

######  Corollary 2.

All purifiable bipartite process matrices (including the ones with a non-
trivial global past and global future) are extensibly causal.

###### Proof.

First note that, if a bipartite process $W$ is purifiable,
$W^{\prime}=W\otimes\rho_{A_{I}^{\prime}B_{I}^{\prime}}$ is also purifiable
for any quantum state $\rho_{A_{I}^{\prime}B_{I}^{\prime}}$. To see this,
first note that the quantum state $\rho_{A_{I}^{\prime}B_{I}^{\prime}}$ is a
process where the dimensions $A_{O}$ and $B_{O}$ are set to one. Now let
$W_{P}$ be a purification of $W$ and $\rho_{P}$ be a process purification of
the process represented by the state $\rho_{A_{I}^{\prime}B_{I}^{\prime}}$.
Straightforward calculation shows that $W_{P}\otimes\rho_{P}$ is a valid
purification for $W\otimes\rho_{A_{I}^{\prime}B_{I}^{\prime}}$. Now, by Thm.
4, $W^{\prime}$ is a direct sum of pure processes. Hence, when the partial
trace over the total global future space is taken, $W^{\prime}$ becomes
causally separable. Theorem 4 of Ref. [17] implies that $W^{\prime}$ is
device-independent causal, ensuring that $W$ is extensibly causal. ∎

The results obtained here also allow to tackle the semi-device-independent
indefinite causal order certification scenario [16]. The semi-device-
independent scenario is as an intermediate step between the device-independent
and the device-independent case. While the device-independent scenario does
not assume the instruments involved are characterized and the device-dependent
scenario assume all instruments involved are characterized, the semi-device-
independent scenario assumes the instruments performed in some slots (or
equivalently, by some parties) are characterized and the instruments of some
slots are uncharacterized. In a semi-device-independent scenario we analyze
the given statistical data under more hypothesis than a device-independent one
but we have less hypothesis than a device-dependent one. Reference [16] shows
that, although the quantum switch is extensibly causal in a device-independent
scenario, its indefinite causally order property can be certified in a semi-
device independent way by characterizing some, but not all, of the involved
instruments. When contrasted with the device-dependent case, this semi-device-
independent approach allows one to certify the same physical property with
less hypothesis on the experiment.

Here we show that bipartite purifiable processes are extensibly causal in a
semi-device-independent scenario where the instruments performed by Alice and
Bob are not characterized but the measurement performed in the global future
may be characterized. Note that this claim is stronger than Cor. 2, which
states that all purifiable process are (device-independent) extensibly causal.

######  Corollary 3.

All purifiable bipartite process matrices are semi-device-independent
extensibly causal in a scenario where the instruments performed by Alice and
Bob are uncharacterized but the measurement performed in the global future may
be characterized.

###### Proof.

The proof for the semi-device-independent case for the scenario where
measurements performed in the future part are trusted follows the same steps
used when proving Cor. 2. Since the partial trace over the total global future
of $W^{\prime}$ leads to a causally separable process, Theorem 7 of Ref. [16]
ensures that $W$ is extensibly causal when the instruments of Alice and Bob
are untrusted. ∎

###  4.4 Properties of pure superchannels

In this subsection, we analyze general properties of pure superchannels with
two slots and discuss a few examples. We start by presenting necessary
conditions on the dimensions of quantum systems for a two-slot superchannel to
be pure.

######  Corollary 4.

Let $U:P\otimes A_{O}\otimes B_{O}\to A_{I}\otimes B_{I}\otimes F$ be the
unitary operator representing a direct sum of pure combs. Suppose $\dim
A_{I}=\dim A_{O}=:d_{A}$, $\dim B_{I}=\dim B_{O}=:d_{B}$ and $\dim P=\dim
F=:D$. Then, both $d_{A}$ and $d_{B}$ have to be a divisor of $D$.

This corollary can be proved from Thm. 3. Namely, if $d_{A}=d_{B}=d$, then
$D=kd$ should hold for some integer $k\geq 1$ due to Thm. 3. Next, we look at
simple examples of direct sums of pure combs satisfying the dimensional
conditions $\dim A_{I}=\dim A_{O}=\dim B_{I}=\dim B_{O}$ and $\dim P=\dim F$.

######  Corollary 5.

Let $U:P\otimes A_{O}\otimes B_{O}\to A_{I}\otimes B_{I}\otimes F$ be the
unitary operator representing a direct sum of pure combs and suppose $\dim
A_{I}=\dim A_{O}=\dim B_{I}=\dim B_{O}=d$ and $\dim P=\dim F=D$.
(1) If $D=d$, then $U$ is causally ordered and $U$ can be decomposed into

$U^{PA_{O}B_{O}\to A_{I}B_{I}F}=U_{0}^{P\to A_{I}}\otimes U_{1}^{A_{O}\to
B_{I}}\otimes U_{2}^{B_{O}\to F}$ (26)

for unitary operators $U_{0}:P\to A_{I}$, $U_{1}:A_{O}\to B_{I}$,
$U_{2}:B_{O}\to F$ or $U$ can be decomposed into

$U^{PA_{O}B_{O}\to A_{I}B_{I}F}=V_{0}^{P\to B_{I}}\otimes V_{1}^{B_{O}\to
A_{I}}\otimes V_{2}^{A_{O}\to F}$ (27)

for unitary operators $V_{0}:P\to B_{I}$, $V_{1}:B_{O}\to A_{I}$,
$V_{2}:A_{O}\to F$.
(2) If $D=2d$, then $U$ is causally ordered or $U$ is called switch-like, that
is, $U$ is decomposed into

$U^{PA_{O}B_{O}\to A_{I}B_{I}F}=\ket{0}^{F_{c}}\bra{0}^{P_{c}}\otimes
U_{0}^{P_{t}\to A_{I}}\otimes U_{1}^{A_{O}\to B_{I}}\otimes U_{2}^{B_{O}\to
F_{t}}\\\ +\ket{1}^{F_{c}}\bra{1}^{P_{c}}\otimes V_{0}^{P_{t}\to B_{I}}\otimes
V_{1}^{B_{O}\to A_{I}}\otimes V_{2}^{A_{O}\to F_{t}},$ (28)

for unitary operators $U_{0}:P_{t}\to A_{I}$, $U_{1}:A_{O}\to B_{I}$,
$U_{2}:B_{O}\to F_{t}$, $V_{0}:P_{t}\to B_{I}$, $V_{1}:B_{O}\to A_{I}$,
$V_{2}:A_{O}\to F_{t}$ where $P=P_{c}\otimes P_{t}$, $F=F_{c}\otimes F_{t}$,
$\dim P_{c}=\dim F_{c}=2$ and $\dim P_{t}=\dim F_{t}=d$. The Choi vector form
of Eq. (28) is given by

$\ket{U}\rangle=\ket{0}^{P_{c}}\ket{U_{0}}\rangle^{P_{t}A_{I}}\ket{U_{1}}\rangle^{A_{O}B_{I}}\ket{U_{2}}\rangle^{B_{O}F_{t}}\ket{0}^{F_{c}}\\\
+\ket{1}^{P_{c}}\ket{V_{0}}\rangle^{P_{t}B_{I}}\ket{V_{1}}\rangle^{B_{O}A_{I}}\ket{V_{2}}\rangle^{A_{O}F_{t}}\ket{1}^{F_{c}}.$
(29)

We refer to pure superchannels of the form of Eq. (28) as switch-like because,
similarly to the quantum switch, the linear spaces $P$ and $F$ admit the
decomposition $P=P_{c}\otimes P_{t}$ and $F=F_{c}\otimes F_{t}$ where $P_{c}$
and $F_{c}$ play the role of a control system, and $P_{t}$ and $F_{t}$ play
the role of a target system.

Finally, we show that not all pure superchannels are “switch-like”. That is,
we may not be able to decompose $P$ and $F$ as $P=P_{c}\otimes P_{t}$ and
$F=F_{c}\otimes F_{t}$. In Ref. [38] the authors ask whether any indefinite
causally ordered two-slot superchannels with a rank-1 Choi operator666In Ref.
[38] the authors define pure process as superchannels where its Choi
representation is a rank-1 projector. Although this is not equivalent to the
definition of pure superchannels used here, it holds true that all pure
superchannels are represented by rank-1 Choi operators. has a control
subsystem of causal orders in $P$ and $F$. While Cor. 5 ensures this is true
in the case of $D=2d$, a general pure superchannel with two slots may not
admit a separate control system. For instance, when $D=3d$, there exists a
unitary operator representing a quantum superchannel with two slots which
cannot be written in such a form with the control system. For example, suppose
a direct sum of pure combs $U=U^{A\prec B}\oplus U^{B\prec A}$ such that

$U^{A\prec
B}\left(\ket{c,t}^{P}\ket{a}^{A_{O}}\ket{b}^{B_{O}}\right)=\ket{t}^{A_{I}}\ket{c\oplus
a}^{B_{I}}\ket{a,b}^{F},$ (30) $U^{B\prec
A}\left(\ket{2,t}^{P}\ket{a}^{A_{O}}\ket{b}^{B_{O}}\right)=\ket{t}^{B_{I}}\ket{b}^{A_{I}}\ket{2,a}^{F}$
(31)

(note the order of the superscripts) for each $c=0,1$, $t=0,1$, $a=0,1$,
$b=0,1$ where $\\{\ket{c,t}^{P}\\}_{c=0,1,\,t=0,1,2}$ and
$\\{\ket{c,t}^{F}\\}_{c=0,1,\,t=0,1,2}$ are orthonormal bases in $P$ and $F$,
respectively, and

$\displaystyle P^{A\prec B}$
$\displaystyle=\text{SPAN}\left[\ket{c,t}^{P}\mathrel{}\middle|\mathrel{}c=0,1,t=0,1\right],$
(32) $\displaystyle P^{B\prec A}$
$\displaystyle=\text{SPAN}\left[\ket{2,t}^{P}\mathrel{}\middle|\mathrel{}t=0,1\right],$
(33) $\displaystyle F^{A\prec B}$
$\displaystyle=\text{SPAN}\left[\ket{c,t}^{F}\mathrel{}\middle|\mathrel{}c=0,1,t=0,1\right],$
(34) $\displaystyle F^{B\prec A}$
$\displaystyle=\text{SPAN}\left[\ket{2,t}^{F}\mathrel{}\middle|\mathrel{}t=0,1\right].$
(35)

In Figure 18 and Figure 19, the corresponding quantum circuits for $U^{A\prec
B}$ and $U^{B\prec A}$ are shown, respectively. In this case, $U$ cannot have
a separate control subsystem of causal orders in $P$ and $F$, since $U$ is not
causal and the “control parameter” $c$ of $U^{A\prec B}$ depends on the output
$a$ from $A$. Note however that we can always embed a pure superchannel in a
larger space where this superchannel becomes switch-like. For instance, we can
embed $U^{B\prec A}$ of Eq. (31) in a larger linear space by defining the
unitary $U^{\prime B\prec A}:=U^{B\prec A}\oplus 1$. With we have ${U^{A\prec
B}\oplus U^{\prime B\prec A}=\ket{0}^{F_{c}}\bra{0}^{P_{c}}\otimes U^{A\prec
B}+\ket{1}^{F_{c}}\bra{1}^{P_{c}}\otimes U^{\prime B\prec A}}$.

Figure 18: The circuit of $U^{A\prec B}$ in Eq. (30). Figure 19: The circuit
of $U^{B\prec A}$ in Eq. (31).

##  5 Conclusions

Quantum superchannels with indefinite causal order have revealed fundamental
properties of quantum theory and have proved useful to enhance performance in
several information theoretic tasks. Nevertheless, when compared to quantum
channels, we are still taking our first steps towards understanding
superchannels. For instance, despite superchannels without a definite causal
order being allowed by quantum theory, it is still not clear if there is a
fair universal procedure to physically implement such objects. In addition,
apart from the quantum switch and some straightforward generalizations, we
still lack a simple interpretation for indefinite causal order in quantum
theory.

In this paper, we have presented a simple mathematical decomposition for two-
slot pure superchannels, that is, two-slot superchannels which preserve the
reversibility of quantum input operations. This characterization allows us to
understand every indefinite causally ordered two-slot pure superchannels as a
coherent superposition of ordered quantum circuit, this does not hold true for
general superchannels. Roughly speaking, our results show that all two-slot
pure superchannels are similar to the quantum switch and admit a similar
physical interpretation. We believe that our findings may contribute in to the
debate of interpretations and physical realization of quantum superchannels.

We have also shown that purifiable two-slot superchannels, also referred as
purifiable bipartite processes, cannot violate causal inequalities, even if
the parties share additional unlimited entanglement, and that purifiable
bipartite processes with no global future are necessarily causally separable.
These implications provide partial answers to the questions raised in Ref.
[7].

A clear future direction is to investigate when superchannels with more than
two slots admit a simple decomposition and which methods/results can be
generalized. We know however that our main results cannot be generalized in a
straightforward way. That is, there exist pure superchannels with three slots
which violate causal inequalities [39, 7, 40], showing that pure superchannels
with more than two slots are not always a direct sum of pure combs. A more
technical future direction is to investigate if condition (2) in Thm. 5 can be
proved without exploiting condition (3) of Thm. 5. If that is the case, our
methods may be applied to more general scenarios and superchannels with
multiple slots.

## Acknowledgements

We acknowledge Alessandro Bisio and Michal Sedlák for useful discussions. This
work was supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant
Number JPMXS0118069605, Japan Society for the Promotion of Science (JSPS) by
KAKENHI grant No. 15H01677, 16F16769, 17H01694, 18H04286 and 18K13467.

## Note added

After completion of this work we became aware of a related result presented in
Section IX of Ref. [41] by Barrett et al. which was obtained independently. We
also became aware of a related result by David Trillo, forthcoming.

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##  Appendix A Relevant properties of composite Hilbert space

In our proofs of the main theorems, we perform various operations on composite
Hilbert spaces. In A.1 and A.2, we respectively provide notations and
properties for elementary linear algebra relevant to the operations. In A.3
and A.4, we provide notations particular to this paper. In A.5, we introduce
the notion of “reduced subspace” generated from composite Hilbert spaces. In
A.6 we derive properties of unitary operators representing quantum
superchannels. In A.7, we focus on such properties about pure superchannels
with two slots.

###  A.1 Notations for linear algebra

We consider only finite-dimensional Hilbert spaces. Let
$\mathcal{X},\mathcal{Y}$ be subspaces of a finite-dimensional Hilbert space
$\mathcal{H}$ and let $\ket{x},\ket{y}$ be vectors (which are not necessarily
normalized) in $\mathcal{H}$.

  * •

$\left(\ket{x},\ket{y}\right)$ and $\braket{x}{y}$ denote the inner product of
$\ket{x}$ and $\ket{y}$.

  * •

$\|\ket{x}\|$ denotes the norm $\sqrt{\braket{x}{x}}$ of $\ket{x}$.

  * •

$\ket{x}$ and $\ket{y}$ are orthogonal (denoted by $\ket{x}\perp\ket{y}$) if
$\braket{x}{y}=0$.

  * •

$\ket{x}$ and $\mathcal{X}$ are orthogonal (denoted by
$\ket{x}\perp\mathcal{X}$) if $\ket{x}\perp\ket{y}$ for all
$\ket{y}\in\mathcal{X}$.

  * •

$\mathcal{X}$ and $\mathcal{Y}$ are orthogonal (denoted by
$\mathcal{X}\perp\mathcal{Y}$) if $\ket{x}\perp\ket{y}$ for all
$\ket{x}\in\mathcal{X}$ and for all $\ket{y}\in\mathcal{Y}$.

  * •

$\mathcal{X}+\mathcal{Y}$ denotes the sum of $\mathcal{X}$ and $\mathcal{Y}$,
which is the subspace of $\mathcal{H}$ defined as follows:

$\mathcal{X}+\mathcal{Y}:=\left\\{\ket{x}+\ket{y}\mathrel{}\middle|\mathrel{}\ket{x}\in\mathcal{X},\ket{y}\in\mathcal{Y}\right\\}.$
(36)

Since the operation $+$ is associative and commutative, this definition has a
straightforward extension to the case of more than two subspaces.

  * •

When the linear subspaces $\mathcal{X}$ and $\mathcal{Y}$ are orthogonal, it
holds that $\mathcal{X}+\mathcal{Y}=\mathcal{X}\oplus\mathcal{Y}$, where
$\oplus$ stands for the orthogonal direct sum777The orthogonal direct sum is
also referred as internal direct sum and it is isomorphic to the external
orthogonal direct sum, which constructs a Hilbert space from two or more
Hilbert spaces [42, 43]. [44]. In order to emphasize the orthogonal relations
of subspaces, when $\mathcal{X}\perp\mathcal{Y}$ we will often write
$\mathcal{X}\oplus\mathcal{Y}$ instead of $\mathcal{X}+\mathcal{Y}$.

  * •

$\mathcal{X}^{\perp}$ denotes the orthogonal complement of $\mathcal{X}$,
which is the subspace of $\mathcal{H}$ defined as follows:

$\mathcal{X}^{\perp}\coloneqq\left\\{\ket{v}\in\mathcal{H}\mathrel{}\middle|\mathrel{}\ket{v}\perp\mathcal{X}\right\\}.$
(37)

Let $(\mathcal{X}_{\lambda})_{\lambda\in\Lambda}$ be a family of subspaces of
$\mathcal{H}$ indexed by a set $\Lambda$, which may be continuous.

  * •

$+_{\lambda\in\Lambda}\mathcal{X}_{\lambda}$ denotes the sum of
$(\mathcal{X}_{\lambda})_{\lambda\in\Lambda}$. In this work we will also
consider the case where the sum runs over continuous indices, i.e., $\Lambda$
is not countable. Note however that, since
$+_{\lambda\in\Lambda}\mathcal{X}_{\lambda}$ is, by definition, a subset of
the finite linear space $\mathcal{H}$, this sum can always be restricted to
have only finite terms

888In order to overcome this technicality, when $\Lambda$ has infinity
elements we define $+_{\lambda\in\Lambda}\mathcal{X}_{\lambda}$ as follows:
$\displaystyle\operatorname*{\scalerel*{+}{\sum}}_{\lambda\in\Lambda}\mathcal{X}_{\lambda}$
$\displaystyle:=\left\\{\sum_{i=0}^{n-1}\ket{x_{i}}\mathrel{}\middle|\mathrel{}n\in\mathbb{N},\
(\lambda_{i})_{i=0}^{n-1}\in\Lambda^{n}\
,\ket{x_{i}}\in\mathcal{X}_{\lambda_{i}}\right\\},$ (38) where $\Lambda^{n}$
is the set of ordered n-tuples of elements of $\Lambda$ or the set of
functions from the set $\\{0,1,\ldots,n-1\\}$ to the set $\Lambda$.

We have some properties about this $\operatorname*{\scalerel*{+}{\sum}}$. The
subset inclusion
$\mathcal{X}_{\lambda^{\prime}}\subset\operatorname*{\scalerel*{+}{\sum}}_{\lambda\in\Lambda}\mathcal{X}_{\lambda}$
holds true for every $\lambda^{\prime}\in\Lambda$. Also, if
$\mathcal{X}_{\lambda}\subset\mathcal{Y}$ for all $\lambda\in\Lambda$, then
$\operatorname*{\scalerel*{+}{\sum}}_{\lambda\in\Lambda}\mathcal{X}_{\lambda}\subset\mathcal{Y}$.

###  A.2 Useful properties

######  Lemma 2.

Let $\mathcal{X},\mathcal{Y}$ be subspaces of a finite-dimensional Hilbert
space $\mathcal{H}$ and suppose $\mathcal{X}_{0},\mathcal{X}_{1}$ are
subspaces of $\mathcal{H}$ such that
$\mathcal{X}=\mathcal{X}_{0}\oplus\mathcal{X}_{1}$. If
$\mathcal{X}_{0}\subset\mathcal{Y}$ and $\mathcal{X}_{1}\perp\mathcal{Y}$,
then $\mathcal{X}_{0}=\mathcal{X}\cap\mathcal{Y}$ and
$\mathcal{X}_{1}=\mathcal{X}\cap\mathcal{Y}^{\perp}$.

###### Proof.

Assume $\mathcal{X}_{0}\subset\mathcal{Y}$ and
$\mathcal{X}_{1}\perp\mathcal{Y}$. From
$\mathcal{X}\cap\mathcal{Y}\subset\mathcal{X}$ and
$\mathcal{X}\cap\mathcal{Y}\perp\mathcal{X}_{1}$,
$\mathcal{X}\cap\mathcal{Y}\subset\mathcal{X}_{0}$.
$\mathcal{X}_{0}\subset\mathcal{X}\cap\mathcal{Y}$ is trivial so
$\mathcal{X}_{0}=\mathcal{X}\cap\mathcal{Y}$. From
$\mathcal{X}\cap\mathcal{Y}^{\perp}\subset\mathcal{X}$ and
$\mathcal{X}\cap\mathcal{Y}^{\perp}\perp\mathcal{X}_{0}$,
$\mathcal{X}\cap\mathcal{Y}^{\perp}\subset\mathcal{X}_{1}$.
$\mathcal{X}_{1}\subset\mathcal{X}\cap\mathcal{Y}^{\perp}$ is trivial so
$\mathcal{X}_{1}=\mathcal{X}\cap\mathcal{Y}^{\perp}$. ∎

######  Lemma 3.

Let $\mathcal{X}$, $\mathcal{Y}$ be subspaces of a finite-dimensional Hilbert
space $\mathcal{H}$. If $\mathcal{X}\perp\mathcal{Y}$, then
$\mathcal{H}=\mathcal{X}\oplus(\mathcal{X}^{\perp}\cap\mathcal{Y}^{\perp})\oplus\mathcal{Y}$.

###### Proof.

Let $\ket{x^{\perp}}\in\mathcal{X}^{\perp}$. $\ket{x^{\perp}}$ can be
decomposed into $\ket{x^{\perp}}=:\ket{y}+\ket{y^{\perp}}$ with some
$\ket{y}\in\mathcal{Y}$ and some $\ket{y^{\perp}}\in\mathcal{Y}^{\perp}$.
Since $\mathcal{Y}\subset\mathcal{X}^{\perp}$, $\ket{y}\in\mathcal{X}^{\perp}$
and then we obtain
$\ket{y^{\perp}}=\ket{x^{\perp}}-\ket{y}\in\mathcal{X}^{\perp}$. Therefore,
$\ket{x^{\perp}}\in(\mathcal{X}^{\perp}\cap\mathcal{Y}^{\perp})\oplus\mathcal{Y}$,
which implies
$\mathcal{X}^{\perp}\subset(\mathcal{X}^{\perp}\cap\mathcal{Y}^{\perp})\oplus\mathcal{Y}$.
Clearly
$\mathcal{X}^{\perp}\supset(\mathcal{X}^{\perp}\cap\mathcal{Y}^{\perp})\oplus\mathcal{Y}$
and then we obtain
$\mathcal{H}=\mathcal{X}\oplus\mathcal{X}^{\perp}=\mathcal{X}\oplus(\mathcal{X}^{\perp}\cap\mathcal{Y}^{\perp})\oplus\mathcal{Y}$.
∎

######  Lemma 4.

Let $f(\ket{x},\,\ket{x^{\prime}})$ be a sesquilinear form on a Hilbert space
$\mathcal{H}$. If for all $\ket{x},\,\ket{x^{\perp}}\in\mathcal{H}$ such that
$\ket{x}\perp\ket{x^{\perp}}$,

$f(\ket{x},\,\ket{x^{\perp}})=0,$ (39)

then for all $\ket{x_{0}},\,\ket{x_{1}}\in\mathcal{H}$ such that
$\|\ket{x_{0}}\|=\|\ket{x_{1}}\|$,

$f(\ket{x_{0}},\,\ket{x_{0}})=f(\ket{x_{1}},\,\ket{x_{1}}).$ (40)

###### Proof.

When $\dim\mathcal{H}=1$, it is trivial. In the following, we assume
$\dim\mathcal{H}\geq 2$. First, we show the case of
$\ket{x_{0}}^{B_{O}}\perp\ket{x_{1}}^{B_{O}}$. Take the computational basis in
$\mathcal{H}$ and assume $\ket{x_{0}}^{B_{O}}=\ket{0}^{B_{O}}$ and
$\ket{x_{1}}^{B_{O}}=\ket{1}^{B_{O}}$ without loss of generality. In Eq. (39),
substitute $\ket{0}^{B_{O}}+\ket{1}^{B_{O}}$ for $\ket{x}^{B_{O}}$ and
$\ket{0}^{B_{O}}-\ket{1}^{B_{O}}$ for $\ket{x^{\perp}}^{B_{O}}$ and then using
Eq. (39) we obtain

$f(\ket{0},\,\ket{0})-f(\ket{0},\,\ket{1})+f(\ket{1},\,\ket{0})-f(\ket{1},\,\ket{1})=0.$
(41) $\therefore
f(\ket{0},\,\ket{0})=f(\ket{1},\,\ket{1}).\quad(\because\mathrm{Eq.\,\eqref{eq:xxperp}})$
(42)

Next, we show the general case where $\ket{x_{0}}$ and $\ket{x_{1}}$ may not
be orthogonal. Without loss of generality, assume $\ket{x_{0}}=\ket{0}$ and
$\ket{x_{1}}=a\ket{0}+b\ket{1}$ where $a$, $b$ are arbitrary complex numbers
such that $|a|^{2}+|b|^{2}=1$. Then,

$\displaystyle f(\ket{x_{1}},\,\ket{x_{1}})$ $\displaystyle=$
$\displaystyle|a|^{2}$ $\displaystyle
f(\ket{0},\,\ket{0})+a^{*}bf(\ket{0},\,\ket{1})+b^{*}af(\ket{1},\,\ket{0})+|b|^{2}f(\ket{1},\,\ket{1})$
$\displaystyle=$ $\displaystyle|a|^{2}$ $\displaystyle
f(\ket{0},\,\ket{0})+|b|^{2}f(\ket{1},\,\ket{1})\quad(\because\text{Eq.\,\eqref{eq:xxperp}})$
$\displaystyle=$ $\displaystyle|a|^{2}$ $\displaystyle
f(\ket{0},\,\ket{0})+|b|^{2}f(\ket{0},\,\ket{0})\quad(\because\text{Eq.\,\eqref{eq:x0x0=x1x1}})$
$\displaystyle=$ $\displaystyle f(\ket{0},\,\ket{0}).$ (43)

∎

###  A.3 Notation of multiple-ket

$\ket{\phi}^{\mathcal{X}}$ denotes a vector in a Hilbert space $\mathcal{X}$.
The superscript $\mathcal{X}$ of $\ket{\phi}^{\mathcal{X}}$ represents the
space where it belongs. $\ket{\Phi}\rangle^{\mathcal{X}\mathcal{Y}}$ denotes a
vector in the tensor product of Hilbert spaces $\mathcal{X}$ and
$\mathcal{Y}$. We use $\ket{\Phi}\rangle^{\mathcal{X}\mathcal{Y}}$, not with a
single-ket, to emphasize the fact that it belongs to the tensor product of two
Hilbert spaces. Similarly, we use
$\ket{\eta}\rangle\rangle^{\mathcal{X}\mathcal{Y}\mathcal{Z}}$, not with a
single-ket, to emphasize on the fact that it belongs to the tensor product of
three Hilbert spaces. Please note that $\Phi$ in
$\ket{\Phi}\rangle^{\mathcal{X}\mathcal{Y}}$ is just a label and
$\ket{\Phi}\rangle^{\mathcal{X}\mathcal{Y}}$ is not a Choi vector of some
operator $\Phi$, that is, $\ket{\Phi}\rangle^{\mathcal{X}\mathcal{Y}}\neq
I\otimes\Phi\ket{I}\rangle$.

###  A.4 Function from subspaces to a subspace

In this paper, we deal with functions mapping linear subspaces to linear
subspaces where some particular spaces are often used. We define two useful
subspaces.

Let $\ket{\alpha}^{A_{O}}\in A_{O}$, we define the one-dimensional subspace
spanned by $\ket{\alpha}$ as

$\alpha:=\mathop{\mathrm{SPAN}}\left[\ket{\alpha}^{A_{O}}\right]$ (44)

and its orthogonal complement as

$\overline{\alpha}:=\mathop{\mathrm{SPAN}}\left[\ket{\alpha}^{A_{O}}\right]^{\perp}.$
(45)

Analogously, if $\ket{\beta}^{B_{O}}\in B_{O}$, we define

$\beta:=\mathop{\mathrm{SPAN}}\left[\ket{\beta}^{B_{O}}\right],\quad\overline{\beta}:=\mathop{\mathrm{SPAN}}\left[\ket{\beta}^{B_{O}}\right]^{\perp}.$
(46)

Let ${\mathcal{V}}$ be a function which takes a pair of linear space as
arguments such that ${\mathcal{V}}\left(A_{O},B_{O}\right)$ is a linear space.
For simplicity of notation, we will often write the argument of such functions
by sub-indexes, for example, ${\mathcal{V}}_{A_{O}B_{O}}$ for
${\mathcal{V}}\left(A_{O},B_{O}\right)$, ${\mathcal{V}}_{A_{O}\beta}$ for
${\mathcal{V}}(A_{O},\beta)$, ${\mathcal{V}}_{\alpha\overline{\beta}}$ for
${\mathcal{V}}\left(\alpha,\overline{\beta}\right)$, etc.

We call ${\mathcal{V}}$ bi-additive if
${\mathcal{V}}(A_{0}+A_{1},B_{0}+B_{1})={\mathcal{V}}(A_{0},B_{0})+{\mathcal{V}}(A_{0},B_{1})+{\mathcal{V}}(A_{1},B_{0})+{\mathcal{V}}(A_{1},B_{1})$
for all subspaces $A_{0},A_{1}$ in $A_{O}$ and $B_{0},B_{1}$ in $B_{O}$.

###  A.5 Reduced subspace

Let $\mathcal{E},\mathcal{F}$ be two finite-dimensional Hilbert spaces and
$\mathcal{W}$ be a subspace of $\mathcal{E}\otimes\mathcal{F}$. Define the
following notation ${{}^{\mathcal{E}}\\!{\bullet}}$ which generate some
subspace of $\mathcal{F}$:

${{}^{\mathcal{E}}\\!{\mathcal{W}}}:=\mathop{\mathrm{SPAN}}\left[{}^{\mathcal{E}}\braket{\epsilon}{\eta}\rangle^{\mathcal{EF}}\mathrel{}\middle|\mathrel{}\ket{\eta}\rangle^{\mathcal{EF}}\in\mathcal{W},\ket{\epsilon}^{\mathcal{E}}\in\mathcal{E}\right].$
(47)

For example, let $\mathcal{E}\cong\mathcal{F}\cong\mathbb{C}^{2}$. If

$\mathcal{W}=\mathop{\mathrm{SPAN}}\left[\ket{0}^{\mathcal{E}}\ket{0}^{\mathcal{F}}+2\ket{1}^{\mathcal{E}}\ket{1}^{\mathcal{F}}\right],$
(48)

then

${{}^{\mathcal{E}}\\!{\mathcal{W}}}=\mathop{\mathrm{SPAN}}\left[\ket{0}^{\mathcal{F}},\,\ket{1}^{\mathcal{F}}\right].$
(49)

${{}^{\mathcal{E}}\\!{\mathcal{W}}}$ consists of “$\mathcal{F}$ parts” of
$\mathcal{W}$ in some sense. Note that the subspace
${{}^{\mathcal{E}}\\!{\mathcal{W}}}$ is equal to the sum of the supports of
the reduced density operators
$\rho_{\eta}^{\mathcal{F}}:=\mathrm{Tr}_{\mathcal{E}}\ket{\eta}\rangle\langle\bra{\eta}$
for system $\mathcal{F}$ corresponding to each state
$\ket{\eta}\rangle^{\mathcal{EF}}\in\mathcal{E}\otimes\mathcal{F}$:

${{}^{\mathcal{E}}\\!{\mathcal{W}}}=\operatorname*{\scalerel*{+}{\sum}}_{\ket{\eta}\rangle^{\mathcal{EF}}\in\mathcal{E}\otimes\mathcal{F}}\operatorname{supp}\left(\rho_{\eta}^{\mathcal{F}}\right).$
(50)

It is also interesting to remark that the simple union
$\bigcup_{\ket{\eta}\rangle^{\mathcal{EF}}\in\mathcal{E}\otimes\mathcal{F}}\operatorname{supp}\left(\rho_{\eta}^{\mathcal{F}}\right)$
is not a linear space in general. One of the counter examples is in the case
$\mathcal{W}=\mathop{\mathrm{SPAN}}\left[\ket{0}^{\mathcal{E}}\ket{0}^{\mathcal{F}},\ket{0}^{\mathcal{E}}\ket{1}^{\mathcal{F}}+\ket{1}^{\mathcal{E}}\ket{2}^{\mathcal{F}}\right]$.)
Since this operation ${{}^{\mathcal{E}}\\!{\bullet}}$, which transforms
tensored subspaces into subspaces of the reduced space, is like the partial
trace, which transforms density operators into reduced density operators, we
give the name “reduced subspace” to its product
${{}^{\mathcal{E}}\\!{\mathcal{W}}}$.

Some useful properties are following where ${\mathcal{W}}$,
${\mathcal{W}}^{\prime}$ are subspaces of $\mathcal{E}\otimes\mathcal{F}$ and
$\mathcal{X}$ is a subspace of $\mathcal{F}$:

  * •

${{\vphantom{\mathcal{E}\otimes\mathcal{X}}}^{\mathcal{E}}\\!\\!\left(\mathcal{E}\otimes\mathcal{X}\right)}=\mathcal{X}$.

  * •

$\mathcal{W}\subset\mathcal{E}\otimes{{}^{\mathcal{E}}\\!{\mathcal{W}}}$.

  * •

If ${\mathcal{W}}\subset{\mathcal{W}}^{\prime}$, then
${{}^{\mathcal{E}}\\!{{\mathcal{W}}}}\subset{{}^{\mathcal{E}}\\!{{\mathcal{W}}^{\prime}}}$.

  * •

${{}^{\mathcal{E}}\\!{\mathcal{W}}}\subset\mathcal{X}$ if and only if
$\mathcal{W}\subset\mathcal{E}\otimes\mathcal{X}$ .

  * •

If
${{}^{\mathcal{E}}\\!{{\mathcal{W}}}}\perp{{}^{\mathcal{E}}\\!{{\mathcal{W}}^{\prime}}}$,
then ${\mathcal{W}}\perp{\mathcal{W}}^{\prime}$.

  * •

${{}^{\mathcal{E}}\\!{\mathcal{W}}}\perp\mathcal{X}$ if and only if
$\mathcal{W}\perp\mathcal{E}\otimes\mathcal{X}$.

  * •

${{\vphantom{{\mathcal{W}}+{\mathcal{W}}^{\prime}}}^{\mathcal{E}}\\!\\!\left({\mathcal{W}}+{\mathcal{W}}^{\prime}\right)}={{}^{\mathcal{E}}\\!{{\mathcal{W}}}}+{{}^{\mathcal{E}}\\!{{\mathcal{W}}^{\prime}}}$.

######  Lemma 5.

Let $\mathcal{E}$, $\mathcal{F}$ be two finite-dimensional Hilbert spaces and
${\mathcal{W}}_{0}$, ${\mathcal{W}}_{1}$ be two subspaces of
$\mathcal{E}\otimes\mathcal{F}$ such that
$\mathcal{E}\otimes\mathcal{F}={\mathcal{W}}_{0}\oplus{\mathcal{W}}_{1}$.
Suppose $\mathcal{X}$ is a subspace of $\mathcal{F}$. Then,
$\mathcal{X}\perp{{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}}$ if and only if
$\mathcal{E}\otimes\mathcal{X}\subset{\mathcal{W}}_{0}$.

###### Proof.

$\mathcal{X}\perp{{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}}\iff\mathcal{E}\otimes\mathcal{X}\perp{\mathcal{W}}_{1}\iff\mathcal{E}\otimes\mathcal{X}\subset{\mathcal{W}}_{0}$.
∎

######  Corollary 6.

Let $\mathcal{E}$, $\mathcal{F}$ be two finite-dimensional Hilbert spaces and
${\mathcal{W}}_{0}$, ${\mathcal{W}}_{1}$ be two subspaces of
$\mathcal{E}\otimes\mathcal{F}$ such that
$\mathcal{E}\otimes\mathcal{F}={\mathcal{W}}_{0}\oplus{\mathcal{W}}_{1}$.
${{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}}\perp{{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}}$
if and only if
$\mathcal{E}\otimes{{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}}={\mathcal{W}}_{0}$.

######  Lemma 6.

Let $\mathcal{E}$, $\mathcal{F}$ be two finite-dimensional Hilbert spaces and
${\mathcal{W}}_{0}$, ${\mathcal{W}}_{1}$ be subspaces of
$\mathcal{E}\otimes\mathcal{F}$ such that
$\mathcal{E}\otimes\mathcal{F}={\mathcal{W}}_{0}\oplus{\mathcal{W}}_{1}$. Then
$\mathcal{F}=({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}})^{\perp}\oplus({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}}\cap{{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}})\oplus({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}})^{\perp}$
holds.

###### Proof.

First, we show
$({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}})^{\perp}\perp({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}})^{\perp}$.
From ${\mathcal{W}}_{0}\oplus{\mathcal{W}}_{1}=\mathcal{E}\otimes\mathcal{F}$,
$({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}})^{\perp}\perp{{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}}$
and Lem. 5, we obtain
$\mathcal{E}\otimes({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}})^{\perp}\subset{\mathcal{W}}_{0}$.
Similarly, we can obtain
$\mathcal{E}\otimes({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}})^{\perp}\subset{\mathcal{W}}_{1}$.
Thus, since ${\mathcal{W}}_{0}\perp{\mathcal{W}}_{1}$, we obtain
$\mathcal{E}\otimes({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}})^{\perp}\perp\mathcal{E}\otimes({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}})^{\perp}$,
which implies
$({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}})^{\perp}\perp({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}})^{\perp}$.
Therefore, from Lem. 3, we obtain

$\mathcal{F}=({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}})^{\perp}\oplus({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}}\cap{{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}})\oplus({{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}})^{\perp}.$
(51)

∎

######  Corollary 7.

Let $\mathcal{E}$, $\mathcal{F}$ be two finite-dimensional Hilbert spaces and
${\mathcal{W}}_{0}$, ${\mathcal{W}}_{1}$ be subspaces of
$\mathcal{E}\otimes\mathcal{F}$ such that
$\mathcal{E}\otimes\mathcal{F}={\mathcal{W}}_{0}\oplus{\mathcal{W}}_{1}$.
${{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}}$ is decomposed into

${{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}}=\mathcal{F}_{0}^{\mathrm{share}}\oplus\mathcal{F}_{0}^{\mathrm{perp}}$
(52)

by using the subspaces $\mathcal{F}_{0}^{\mathrm{share}}$,
$\mathcal{F}_{0}^{\mathrm{perp}}$ of
${{}^{\mathcal{E}}\\!{{\mathcal{W}}_{0}}}$ satisfying

$\mathcal{F}_{0}^{\mathrm{share}}\subset{{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}},$
(53)
$\mathcal{F}_{0}^{\mathrm{perp}}\perp{{}^{\mathcal{E}}\\!{{\mathcal{W}}_{1}}}.$
(54)

Note that $\mathcal{F}_{0}^{\mathrm{share}},\mathcal{F}_{0}^{\mathrm{perp}}$
are uniquely determined from Lem. 2.

Let $\mathcal{A}$, $\mathcal{B}$ be two finite-dimensional Hilbert spaces and
${\mathcal{W}}$ be a subspace of
$\mathcal{A}\otimes\mathcal{B}\otimes\mathcal{F}$.

  * •

${{}^{\mathcal{A}\otimes\mathcal{B}}\\!{{\mathcal{W}}}}={{}^{\mathcal{AB}}\\!{{\mathcal{W}}}}={{}^{\mathcal{BA}}\\!{{\mathcal{W}}}}$.

  * •

${{\vphantom{B\otimes{\mathcal{W}}}}^{\mathcal{A}}\\!\\!\left(B\otimes{\mathcal{W}}\right)}=B\otimes{{}^{\mathcal{A}}\\!{{\mathcal{W}}}}$.

###  A.6 Unitary operators representing pure superchannels

In this section, we construct from Def. 3 and Def. 5 the conditions that
unitary operators represent a superchannel and a quantum comb, respectively.
In addition, we provide the theorem about decomposition of pure quantum combs.

######  Lemma 7.

Let
$U:\bigotimes_{n=0}^{N}\mathcal{H}_{2n}\to\bigotimes_{n=0}^{N}\mathcal{H}_{2n+1}$
be a unitary operator. $U$ represents a quantum superchannel if and only if
for all auxiliary Hilbert spaces $\mathcal{H}_{m}^{\prime}$,
$m\in\\{1,\cdots,2N\\}$, and unitary operators
$U_{n}:\mathcal{H}_{2n-1}\otimes\mathcal{H}_{2n-1}^{\prime}\to\mathcal{H}_{2n}\otimes\mathcal{H}_{2n}^{\prime}$,
$n\in\\{1,\cdots,N\\}$,
$\ket{U}\rangle\langle\bra{U}\ast(\bigotimes_{n=1}^{N}\ket{U_{n}}\rangle\langle\bra{U_{n}})$
is the Choi operator of a unitary channel, that is,

$U_{G}:=\mathrm{Tr}_{\mathcal{H}_{2}\mathcal{H}_{4}\cdots\mathcal{H}_{2N}}\left[\left(\mbox{1}\mbox{l}^{\mathcal{H}_{2N+1}}\otimes\bigotimes_{n=1}^{N}U_{n}\right)\left(U\otimes\bigotimes_{n=1}^{N}\mbox{1}\mbox{l}^{\mathcal{H}_{2n-1}^{\prime}}\right)\right]$
(55)

is a unitary operator.

###### Proof.

Sufficiency is trivial from Thm. 1. To show necessity, take $N$ arbitrary CPTP
maps
$\tilde{E_{n}}:\mathcal{L}(\mathcal{H}_{2n-1}\otimes\mathcal{H}_{2n-1}^{\prime})\to\mathcal{L}(\mathcal{H}_{2n}\otimes\mathcal{H}_{2n}^{\prime})$,
$n\in\\{1,\cdots,N\\}$, as the input of $W_{U}:=\ket{U}\rangle\langle\bra{U}$.
Next, extend these CPTP maps into a unitary transformation with suitable
auxiliary systems and then we obtain a unitary operation with the auxiliary
systems as the output of $W_{U}$ since $W_{U}$ is pure. Finally, reduce the
auxiliary systems then we obtain a CPTP map as the output of $W_{U}$ from the
arbitrary CPTP maps $\tilde{E_{n}}$, $n\in\\{1,\cdots,N\\}$. ∎

######  Lemma 8.

Let
$U^{(N+1)}:\bigotimes_{n=0}^{N}\mathcal{H}_{2n}\to\bigotimes_{n=0}^{N}\mathcal{H}_{2n+1}$
be a unitary operator. $U^{(N+1)}$ represents a quantum comb with $N$ slots if
and only if $U^{(N+1)}$ satisfies for each $n=1\cdots N$, for all
$\ket{\alpha}^{2n}\in\mathcal{H}_{2n}$,

${{}^{13\cdots 2n-1}\\!{{\mathcal{V}}^{(n)}_{\alpha}}}\perp{{}^{13\cdots
2n-1}\\!{{\mathcal{V}}^{(n)}_{\overline{\alpha}}}}$ (56)

where

${\mathcal{V}}^{(n)}(\mathcal{H}_{2n,\,\mathrm{sub}}):=U^{(N+1)}\left(\left(\bigotimes_{k\neq
n}\mathcal{H}_{2k}\right)\otimes\mathcal{H}_{2n,\,\mathrm{sub}}\right)$ (57)

for a given subspace $\mathcal{H}_{2n,\,\mathrm{sub}}$ of $\mathcal{H}_{2n}$.
We have used abbreviations similarly to the one introduced in A.4 as follows:

$\displaystyle{\mathcal{V}}^{(n)}_{\alpha}$
$\displaystyle:={\mathcal{V}}^{(n)}\left(\mathop{\mathrm{SPAN}}\left[\ket{\alpha}^{2n}\right]\right)$
(58) $\displaystyle{\mathcal{V}}^{(n)}_{\overline{\alpha}}$
$\displaystyle:={\mathcal{V}}^{(n)}\left(\left(\mathop{\mathrm{SPAN}}\left[\ket{\alpha}^{2n}\right]\right)^{\perp}\right)$
(59)

(Notations ${{\vphantom{\bullet}}^{m}\\!\\!\left(\bullet\right)}$ and
$\ket{\bullet}^{m}$ are shorthands respectively for
${{\vphantom{\bullet}}^{\mathcal{H}_{m}}\\!\\!\left(\bullet\right)}$ and
$\ket{\bullet}^{\mathcal{H}_{m}}$.)

###### Proof.

We show only the only-if part. Let $n$ be any integer in $1...N$. From Def. 5,
we have

$\displaystyle\mathrm{Tr}_{2N+1\ 2N-1\ \cdots\
2n+1}\ket{U^{(N+1)}}\rangle\langle\bra{U^{(N+1)}}=\mbox{1}\mbox{l}^{2N\ 2N-2\
\cdots\ 2n}\otimes R^{(n)}$ (60)

for some Choi operator $R^{(n)}$. Let
$\ket{\pi}^{\widehat{2n}},\ket{\pi^{\prime}}^{\widehat{2n}}\in\mathcal{H}_{\widehat{2n}}$,
$\ket{\alpha^{\perp}}^{2n}\in\mathcal{H}_{2n}$ such that
$\ket{\alpha^{\perp}}^{2n}\perp\ket{\alpha}^{2n}$ and $\ket{\Phi}^{13\cdots
2n-1},\ket{\Phi^{\prime}}^{13\cdots
2n-1}\in\otimes_{k=0}^{n-1}\mathcal{H}_{2k+1}$, then

$\displaystyle\mathrm{Tr}_{2N+1\ 2N-1\ \cdots\
2n+1}\left[\left({}^{\widehat{2n}}\bra{\pi^{*}}{}^{2n}\bra{\alpha^{*}}{}^{13\cdots
2n-1}\bra{\Phi}\right)\ket{U^{(N+1)}}\rangle\langle\bra{U^{(N+1)}}\left(\ket{{\pi^{\prime}}^{*}}^{\widehat{2n}}\ket{{\alpha^{\perp}}^{*}}^{2n}\ket{{\Phi^{\prime}}}^{13\cdots
2n-1}\right)\right]=0.$ $\therefore\left({}^{13\cdots
2n-1}\bra{\Phi}U^{(N+1)}\left(\ket{\pi}^{\widehat{2n}}\ket{\alpha}^{2n}\right),{}^{13\cdots
2n-1}\bra{\Phi^{\prime}}U^{(N+1)}\left(\ket{\pi^{\prime}}^{\widehat{2n}}\ket{\alpha^{\perp}}^{2n}\right)\right)=0.$
(61)

Therefore, we obtain

$\displaystyle{{}^{13\cdots 2n-1}\\!{{\mathcal{V}}^{(n)}_{\alpha}}}$
$\displaystyle\perp{{}^{13\cdots
2n-1}\\!{{\mathcal{V}}^{(n)}_{\alpha^{\perp}}}}.$ (62)

Thus we have

${{}^{13\cdots 2n-1}\\!{{\mathcal{V}}^{(n)}_{\alpha}}}\perp{{}^{13\cdots
2n-1}\\!{{\mathcal{V}}^{(n)}_{\overline{\alpha}}}}.$ (63)

∎

In A.7, the special case of this lemma where $N=2$ is given. In Eq. (56),
${{\vphantom{\bullet}}^{13\cdots 2n-1}\\!\\!\left(\bullet\right)}$ means
reducing the subsystems of $\mathcal{H}_{1}$, $\mathcal{H}_{3}$, $\cdots$,
$\mathcal{H}_{2n-1}$ and extracting the states on the subsystem of
$\mathcal{H}_{2n+1}$, $\mathcal{H}_{2n+3}$, $\cdots$, $\mathcal{H}_{2N+1}$.
This lemma just rephrases the condition that a Choi operator satisfies the
causality condition in terms of unitary operators.

The previous lemma can be understood as follows. For any subspace
$\mathcal{H}_{2n}$, if the two orthogonal vectors
$\ket{\alpha}^{2n},\ket{\alpha^{\perp}}^{2n}\in\mathcal{H}_{2n}$ are used as
input-states for $U^{N+1}$, the corresponding output-states are orthogonal on
the reduced states on the subsystems of
$\bigotimes_{k=n}^{N}\mathcal{H}_{2k+1}$ .

From this lemma, the condition that $U^{(N+1)}$ represents a quantum comb is
formulated by using the reduced subspaces ${{}^{13\cdots
2n-1}\\!{{\mathcal{V}}^{(n)}_{\alpha}}}$ generated from $U^{(N+1)}$.

In the following, we present our results in terms of unitary operators, not in
terms of Choi matrices as the proof of our main statement (Prop. 4) is written
in terms of unitary operators.

###  A.7 Quantum combs with two slots

In the case of a quantum comb of $A\prec B$, Lem. 8 is in the following form:

######  Corollary 8.

Let $U:P\otimes A_{O}\otimes B_{O}\to A_{I}\otimes B_{I}\otimes F$ be a
unitary operator. $U$ represents a quantum comb of $A\prec B$ if and only if
for all $\ket{\alpha}^{A_{O}}\in A_{O}$ and $\ket{\beta}^{B_{O}}\in B_{O}$,

${{}^{A_{I}B_{I}}\\!{{\mathcal{V}}_{A_{O}\beta}}}\perp{{}^{A_{I}B_{I}}\\!{{\mathcal{V}}_{A_{O}\overline{\beta}}}},$
(64) ${{}^{A_{I}}\\!{{\mathcal{V}}_{\alpha
B_{O}}}}\perp{{}^{A_{I}}\\!{{\mathcal{V}}_{\overline{\alpha}B_{O}}}}$ (65)

where

${\mathcal{V}}(A_{\mathrm{sub}},B_{\mathrm{sub}}):=\left\\{U\left(\ket{\pi}^{P}\ket{\alpha}^{A_{O}}\ket{\beta}^{B_{O}}\right)\mathrel{}\middle|\mathrel{}\ket{\pi}^{P}\in
P,\,\ket{\alpha}^{A_{O}}\in A_{\mathrm{sub}},\,\ket{\beta}^{B_{O}}\in
B_{\mathrm{sub}}\right\\}.$ (66)

for a subspace $A_{\mathrm{sub}}$, $B_{\mathrm{sub}}$ of $A_{O}$, $B_{O}$,
respectively.

In Eq. (64) and Eq. (65), ${{}^{A_{I}B_{I}}\\!{\bullet}}$,
${{}^{A_{I}}\\!{\bullet}}$ means reducing the subsystem of $A_{I}\otimes
B_{I}$, $A_{I}$ and extracting the states on the subsystem of $F$,
$B_{I}\otimes F$, respectively. Note that Eq. (66) is in the different form
from that of Eq. (57). Just like Lem. 8, the previous corollary can be
understood as follows. If the two orthogonal vectors
$\ket{\alpha}^{A_{O}},\ket{\alpha^{\perp}}^{A_{O}}\in A_{O}$ are used as
input-states for $U$, the corresponding output-states are orthogonal on the
reduced states on the subsystems of $F$. If the two orthogonal vectors
$\ket{\beta}^{B_{O}},\ket{\beta^{\perp}}^{B_{O}}\in B_{O}$ are used as input-
states for $U$, the corresponding output-states are orthogonal on the reduced
states on the subsystems of $A_{I}\otimes F$. From this corollary, the
condition that $U$ represents a quantum comb with two slots is formulated by
using the reduced subspaces of subspaces
${\mathcal{V}}_{A_{\mathrm{sub}}B_{\mathrm{sub}}}$ generated from $U$.

##  Appendix B Proof of Thm. 3

We prove Thm. 3 inductively by showing
(i) there exists an integer $k_{N}$ such that $d_{2N}k_{N}=d_{2N+1}$,
(ii) $U^{(N+1)}$ can be decomposed as:

$U^{(N+1)}=U_{N}U^{({N})}$ (67)

for some unitary operators
$U^{(N)}\colon\bigotimes_{n=0}^{N-1}\mathcal{H}_{2n}\to\bigotimes_{n=0}^{N-1}\mathcal{H}_{2n+1}\otimes\mathcal{A}_{N}$
and $U_{N}\colon\mathcal{H}_{2N}\otimes\mathcal{A}_{N}\to\mathcal{H}_{2N+1}$
for some Hilbert space $\mathcal{A}_{N}$ such that $\dim\mathcal{A}_{N}=k_{N}$
and
(iii) $U^{(N)}$ represents a quantum comb with $N-1$ slots ($N\geq 2$).
When $N=1$, we define $k_{0}:=1$ and $U_{0}:=U^{(1)}$ then the proof
completes.

Define

$\displaystyle P$ $\displaystyle:=$ $\displaystyle\bigotimes_{n=0}^{N-1}$
$\displaystyle\mathcal{H}_{2n},$ (68) $\displaystyle A_{I}$ $\displaystyle:=$
$\displaystyle\bigotimes_{n=0}^{N-1}$ $\displaystyle\mathcal{H}_{2n+1},$ (69)
$\displaystyle A_{O}$ $\displaystyle:=$ $\displaystyle\mathcal{H}_{2N},$ (70)
$\displaystyle F$ $\displaystyle:=$ $\displaystyle\mathcal{H}_{2N+1}.$ (71)

Then we can regard $U^{(N+1)}\colon P\otimes A_{O}\to A_{I}\otimes F$ as a
unitary operator representing a quantum comb with only one slot $A$ (see
Figure 20).

Figure 20: $U^{(N+1)}$ can be seen as a unitary operator representing a
quantum comb
$[\mathcal{L}({A_{I}})\to\mathcal{L}({A_{O}})]\to[\mathcal{L}({P})\to\mathcal{L}({F})]$
with only one slot $A$.

We denote the dimension of a Hilbert space $X$ by $d_{X}$. From unitarity of
$U^{(N+1)}$, we have

$d_{P}d_{A_{O}}=d_{A_{I}}d_{F}.$ (72)

First, we show (i), that is, there exists a positive integer $k$ such that
$d_{A_{O}}k=d_{F}$. Let $\ket{\alpha}^{A_{O}}\in A_{O}$. Using
${\mathcal{V}}:={\mathcal{V}}^{(N)}$ defined in Eq. (57), we have

$A_{I}\otimes
F={\mathcal{V}}_{\alpha}\oplus{\mathcal{V}}_{\overline{\alpha}}.$ (73)

From Lem. 8, we have

${{}^{A_{I}}\\!{{\mathcal{V}}_{\alpha}}}\perp{{}^{A_{I}}\\!{{\mathcal{V}}_{\overline{\alpha}}}}.$
(74)

In Cor. 6, take $A_{I}$ as $\mathcal{E}$, $F$ as $\mathcal{F}$,
${\mathcal{V}}_{\alpha}$ as ${\mathcal{W}}_{0}$ and
${\mathcal{V}}_{\overline{\alpha}}$ as ${\mathcal{W}}_{1}$. Then from Eq. (73)
and Eq. (74), we obtain

${\mathcal{V}}_{\alpha}=A_{I}\otimes{{}^{A_{I}}\\!{{\mathcal{V}}_{\alpha}}}.$
(75) $\therefore d_{P}=d_{A_{I}}k$ (76)

holds for $k:=\dim{{}^{A_{I}}\\!{{\mathcal{V}}_{\alpha}}}$. From Eq. (72) and
Eq. (76), we obtain

$kd_{A_{O}}=d_{F}.$ (77)

Next, we prove (ii). Take the computational basis
$\\{\ket{a}^{A_{O}}\\}_{a=0}^{d_{A_{O}}-1}$ in $A_{O}$, the computational
basis $\\{\ket{i}^{A_{I}}\\}_{i=0}^{d_{A_{I}}-1}$ in $A_{I}$, and an
orthonormal basis $\\{\ket{x,\,0}^{F}\\}_{x=0}^{k-1}$ in
${{}^{A_{I}}\\!{{\mathcal{V}}_{0}}}$. By definition of ${\mathcal{V}}_{0}$,
there exists $\ket{i,\,x}^{P}\in P$ such that

$U^{(N+1)}\left(\ket{i,\,x}^{P}\ket{0}^{A_{O}}\right)=\ket{i}^{A_{I}}\ket{x,\,0}^{F}.$
(78)

Since $\\{\ket{i,\,x}^{P}\\}_{i=0,\,x=0}^{d_{A_{I}}-1,\,k-1}$ is an
orthonormal set from Eq. (78) and a complete basis in $P$ from Eq. (76),

$\\{\ket{i,\,x}^{P}\\}_{i=0,\,x=0}^{d_{A_{I}}-1,\,k-1}\text{ is an orthonormal
basis in $P$}.$ (79)

For each $a$,

$\displaystyle\left({}^{A_{I}}\bra{i=0}U^{(N+1)}\left(\ket{i=0,\,x}^{P}\ket{a}^{A_{O}}\right),{}^{A_{I}}\bra{i^{\prime}}U^{(N+1)}\left(\ket{i^{\prime},\,x^{\prime}}^{P}\ket{a}^{A_{O}}\right)\right)$
$\displaystyle=$
$\displaystyle\left({}^{A_{I}}\bra{i=0}U^{(N+1)}\left(\ket{i=0,\,x}^{P}\ket{0}^{A_{O}}\right),{}^{A_{I}}\bra{i^{\prime}}U^{(N+1)}\left(\ket{i^{\prime},\,x^{\prime}}^{P}\ket{0}^{A_{O}}\right)\right)$
$\displaystyle=$ $\displaystyle\delta_{xx^{\prime}}$ (80)

where the first equality follows from Lem. 4 and Eq. (74) (cf. the proof of
Prop. 8). Equation (80) for $x=x^{\prime}$ gives

$U^{(N+1)}\left(\ket{i,\,x}^{P}\ket{a}^{A_{O}}\right)=\ket{i}^{A_{I}}\ket{x,\,a}^{F},$
(81)

where

$\ket{x,\,a}^{F}:={}^{A_{I}}\bra{i=0}U^{(N+1)}\left(\ket{i=0,\,x}^{P}\ket{a}^{A_{O}}\right).$
(82)

Note that Eq. (82) is defined using only $\ket{i=0,\,x}^{P}$, not with general
$\ket{i,x}^{P}$ for $i\neq 0$, and $\ket{x,\,a}^{F}$ defined here for $a=0$ is
equal to $\ket{x,\,0}^{F}$ defined previously. For each $a$,
$\\{\ket{x,\,a}^{F}\\}_{x=0}^{k-1}$ is an orthonormal set as seen from Eq.
(79) and Eq. (81). Thus,
$\\{\ket{x,\,a}^{F}\\}_{x=0,\,a=0}^{k-1,\,d_{A_{O}}-1}$ is an orthonormal set
from Eq. (74) and is a complete basis in $F$ from Eq. (77). Therefore,
$\\{\ket{x,\,a}^{F}\\}_{x=0,\,a=0}^{k-1,\,d_{A_{O}}-1}$ is an orthonormal
basis in $F$.

Take a Hilbert space $\mathcal{A}$ such that $\dim\mathcal{A}=k$ and
orthonormal basis $\\{\ket{x}^{\mathcal{A}}\\}_{x=0}^{k-1}$. Define a unitary
operator $U^{(N)}\colon P\to A_{I}\otimes\mathcal{A}$ as
$U^{(N)}:=\sum_{i,\,x}\ket{i}^{A_{I}}\ket{x}^{\mathcal{A}}{}^{P}\bra{i,\,x}$
and define a unitary operator $U_{N}\colon\mathcal{A}\otimes A_{O}\to F$ as
$U_{N}:=\sum_{x,\,a}\ket{x,\,a}^{F}{}^{\mathcal{A}}\bra{x}{}^{A_{O}}\bra{a}$,
then

$\displaystyle
U_{N}U^{(N)}\ket{i,\,x}^{P}\ket{a}^{A_{O}}=\ket{i}^{A_{I}}U_{N}\left(\ket{x}^{\mathcal{A}}\ket{a}^{A_{O}}\right)=\ket{i}^{A_{I}}\ket{x,\,a}^{F}.$
(83)

Comparing Eqs. (83) and (81), we conclude that $U^{(N+1)}=U_{N}U^{(N)}$.

Finally, we prove (iii). Let $n$ be an integer in $\\{1...N-1\\}$. Define a
subspace
$\mathcal{X}^{(n)}(\ket{\alpha}^{2n})\subset\otimes_{k}\mathcal{H}_{2k}$ as

$\mathcal{X}^{(n)}(\ket{\alpha}^{2n}):=\left\\{U^{(N+1)}\left(\ket{\tilde{\pi}}^{\widehat{2n,\,2N}}\ket{\alpha}^{2n}\ket{0}^{2N}\right)\mathrel{}\middle|\mathrel{}\ket{\tilde{\pi}}^{\widehat{2n,\,2N}}\in\bigotimes_{k\neq
n,\,N}\mathcal{H}_{2k}\right\\}.$ (84)

From $\mathcal{X}^{(n)}_{\alpha}\subset{\mathcal{V}}^{(n)}_{\alpha}$, Lem. 8
leads to ${{}^{13\cdots
2n-1}\\!{\mathcal{X}^{(n)}_{\alpha}}}\perp{{}^{13\cdots
2n-1}\\!{\mathcal{X}^{(n)}_{\overline{\alpha}}}}$. This implies

${{}^{13\cdots 2n-1}\\!{\mathcal{W}^{(n)}_{\alpha}}}\perp{{}^{13\cdots
2n-1}\\!{\mathcal{W}^{(n)}_{\overline{\alpha}}}},$ (85)

because

$\displaystyle\left({}^{13\cdots
2n-1}\bra{\Phi}U^{(N+1)}\left(\ket{\pi}^{\widehat{2N}}\ket{0}^{2N}\right),\,{}^{13\cdots
2n-1}\bra{\Phi^{\prime}}U^{(N+1)}\left(\ket{\pi^{\prime}}^{\widehat{2N}}\ket{0}^{2N}\right)\right)$
$\displaystyle=$ $\displaystyle\left(U_{N}\left({}^{13\cdots
2n-1}\bra{\Phi}U^{(N)}\ket{\pi}^{\widehat{2N}}\right)\ket{0}^{2N},\,U_{N}\left({}^{13\cdots
2n-1}\bra{\Phi^{\prime}}U^{(N)}\ket{\pi^{\prime}}^{\widehat{2N}}\right)\ket{0}^{2N}\right)$
$\displaystyle=$ $\displaystyle\left({}^{13\cdots
2n-1}\bra{\Phi}U^{(N)}\ket{\pi}^{\widehat{2N}},\,{}^{13\cdots
2n-1}\bra{\Phi^{\prime}}U^{(N)}\ket{\pi^{\prime}}^{\widehat{2N}}\right),$ (86)

holds for all $\ket{\alpha}^{2n}\in\mathcal{H}_{2n}$, where

$\mathcal{W}^{(n)}(\ket{\alpha}^{2n}):=\left\\{U^{(N)}\left(\ket{\tilde{\pi}}^{\widehat{2n,\,2N}}\ket{\alpha}^{2n}\right)\mathrel{}\middle|\mathrel{}\ket{\tilde{\pi}}^{\widehat{2n,\,2N}}\in\bigotimes_{k\neq
n,\,N}\mathcal{H}_{2k}\right\\}.$ (87)

From Lem. 8, Eq. (85) indicates that $U^{(N)}$ represents a quantum comb with
$N-1$ slots. ∎

##  Appendix C The proof of (1) $\implies$ (2) in Thm. 5

We start by pointing that when we insert any unitary operator
$\ket{U_{B}}\rangle\langle\bra{U_{B}}$ into the slot $B$ of a pure two-slot
superchannel described by $\ket{U}\rangle\langle\bra{U}$, the link product
$\ket{U}\rangle\langle\bra{U}*\ket{U_{B}}\rangle\langle\bra{U_{B}}$ is a pure
one-slot pure superchannel. This holds true because if we insert any unitary
operation $\ket{U_{A}}\rangle\langle\bra{U_{A}}$ into the slot $A$ of the one-
slot supermap described by
$(\ket{U}\rangle\langle\bra{U}*\ket{U_{B}}\rangle\langle\bra{U_{B}})$, the
output operation must be unitary. Thus, for all $U_{B}\colon B_{I}\otimes
B_{I}^{\prime}\to B_{O}\otimes B_{O}^{\prime}$, the product
$\ket{U}\rangle\langle\bra{U}*\ket{U_{B}}\rangle\langle\bra{U_{B}}$ is the
Choi operator of a unitary operation.

Hence, Lem. 7 shows that $\ket{U}\rangle\langle\bra{U}$ is a process matrix
with one slot $B$. Thus, $\ket{U}\rangle\langle\bra{U}$ is the Choi operator
of a quantum comb whose global past is $P\otimes A_{O}$, whose global future
is $A_{I}\otimes F$ and whose slot has $B_{I}$ ($B_{O}$) as an input (output)
space, respectively. Thus, Lem. 8 shows
${{}^{B_{I}}\\!{{\mathcal{V}}_{A_{O}\beta}}}\perp{{}^{B_{I}}\\!{{\mathcal{V}}_{A_{O}\overline{\beta}}}}$
for all $\ket{\beta}^{B_{O}}\in B_{O}$, which implies Eq. (b). Similarly, we
can obtain ${{}^{A_{I}}\\!{{\mathcal{V}}_{\alpha
B_{O}}}}\perp{{}^{A_{I}}\\!{{\mathcal{V}}_{\overline{\alpha}B_{O}}}}$ for all
$\ket{\alpha}^{A_{O}}\in A_{O}$, which implies Eq. (c).

To show Eq. (a), we start with the case where $\dim A_{I}=\dim A_{O}=:d_{A}$
and $\dim B_{I}=\dim B_{O}=:d_{B}$. And since the dimensions of the input-
channels are greater than one, we only consider the cases $d_{A}\geq 2$ and
$d_{B}\geq 2$. We now start with the following lemma:

######  Lemma 9.

Let $U:P\otimes A_{O}\otimes B_{O}\to A_{I}\otimes B_{I}\otimes F$ be a
unitary operator representing a quantum superchannel where $\dim A_{I}=\dim
A_{O}=:d_{A}$ and $\dim B_{I}=\dim B_{O}=:d_{B}$. For all
$\ket{\pi}^{P},\,\ket{\pi^{\prime}}^{P}\in P$,
$\ket{\alpha}^{A_{O}},\,\ket{\alpha^{\perp}}^{A_{O}}\in A_{O}$ such that
$\ket{\alpha}^{A_{O}}\perp\ket{\alpha^{\perp}}^{A_{O}}$,
$\ket{\beta}^{B_{O}},\,\ket{\beta^{\perp}}^{B_{O}}\in B_{O}$ such that
$\ket{\beta}^{B_{O}}\perp\ket{\beta^{\perp}}^{B_{O}}$,
$\ket{\phi}^{A_{I}},\,\ket{\phi^{\perp}}^{A_{I}}\in A_{I}$ such that
$\ket{\phi}^{A_{I}}\perp\ket{\phi^{\perp}}^{A_{I}}$,
$\ket{\psi}^{B_{I}},\,\text{ and }\ket{\psi^{\perp}}^{B_{I}}\in B_{I}$ such
that $\ket{\psi}^{B_{I}}\perp\ket{\psi^{\perp}}^{B_{I}}$, we have

$\left(\left({}^{A_{I}}\bra{\phi}{}^{B_{I}}\bra{\psi}\right)U\left(\ket{\pi}^{P}\ket{\alpha}^{A_{O}}\ket{\beta}^{B_{O}}\right),\right.\\\
\left.\left({}^{A_{I}}\bra{\phi^{\perp}}{}^{B_{I}}\bra{\psi^{\perp}}\right)U\left(\ket{\pi^{\prime}}^{P}\ket{\alpha^{\perp}}^{A_{O}}\ket{\beta^{\perp}}^{B_{O}}\right)\right)=0.$
(88)

###### Proof.

We show only the case where $d_{A}\geq 2$ and $d_{B}\geq 2$. Take the
computational bases $\\{\ket{j}^{A_{O}}\\}_{j}$, $\\{\ket{k}^{B_{O}}\\}_{k}$
in $A_{O}$, $B_{O}$, respectively. Without loss of generality, assume
$\ket{\alpha}^{A_{O}}=\ket{0}^{A_{O}}$,
$\ket{\alpha^{\perp}}^{A_{O}}=\ket{1}^{A_{O}}$,
$\ket{\beta}^{B_{O}}=\ket{0}^{B_{O}}$ and
$\ket{\beta^{\perp}}^{B_{O}}=\ket{1}^{B_{O}}$.

Let $\\{\ket{\phi_{j}}^{A_{I}}\\}_{j}$ be an orthonormal basis in $A_{I}$ such
that $\ket{\phi_{0}}^{A_{I}}=\ket{\phi}^{A_{I}}$,
$\ket{\phi_{1}}^{A_{I}}=\ket{\phi^{\perp}}^{A_{I}}$ and let
$\\{\ket{\psi_{k}}^{B_{I}}\\}_{k}$ an orthonormal basis in $B_{I}$ such that
$\ket{\psi_{0}}^{B_{I}}=\ket{\psi}^{B_{I}}$,
$\ket{\psi_{1}}^{B_{I}}=\ket{\psi^{\perp}}^{B_{I}}$. Since $U_{A}^{A_{I}\to
A_{O}}\coloneqq\sum_{j}\ket{j}^{A_{O}}\bra{\phi_{j}}^{A_{I}}$ and
$U_{B}^{B_{I}\to B_{O}}\coloneqq\sum_{k}\ket{k}^{B_{O}}\bra{\psi_{k}}^{B_{I}}$
are unitary, the unitary operator $U_{G}$ representing the output
$\ket{U}\rangle\langle\bra{U}*\ket{U_{A}}\rangle\langle\bra{U_{A}}*\ket{U_{B}}\rangle\langle\bra{U_{B}}$
is obtained by using Eq. (12) as:

$U_{G}^{P\to
F}=\sum_{j,k}\left({}^{A_{I}}\bra{\phi_{j}}{}^{B_{I}}\bra{\psi_{k}}\right)U^{PA_{O}B_{O}\to
A_{I}B_{I}F}\left(\ket{j}^{A_{O}}\ket{k}^{B_{O}}\right)=\sum_{j,k}U_{jk}$ (89)

where
$U_{jk}:=\left({}^{A_{I}}\bra{\phi_{j}}{}^{B_{I}}\bra{\psi_{k}}\right)U\left(\ket{j}^{A_{O}}\ket{k}^{B_{O}}\right)$.
Thus we have,

$I^{P}=U_{G}^{\dagger}U_{G}=\sum_{j,\,k,\,j^{\prime},\,k^{\prime}}U_{jk}^{\dagger}U_{j^{\prime}k^{\prime}}.$
(90)

If we substitute $\ket{\phi_{0}}$ with $-\ket{\phi_{0}}$ in $U_{A}$, then
$U_{A}$ remains unitary. Then Eq. (90) becomes

$\displaystyle\sum_{j,\,k,\,j^{\prime},\,k^{\prime}}(-1)^{\delta_{0j}+\delta_{0j^{\prime}}}U_{jk}^{\dagger}U_{j^{\prime}k^{\prime}}=I^{P}.$
(91)

By subtracting Eq. (91) from Eq. (90), we obtain

$\displaystyle\sum_{k,\,k^{\prime}}\biggl{\\{}\sum_{j^{\prime}\neq
0}U_{0k}^{\dagger}U_{j^{\prime}k^{\prime}}+\sum_{j\neq
0}U_{jk}^{\dagger}U_{0k^{\prime}}\biggr{\\}}=O.$ (92)

In Eq. (92), we substitute $\ket{\phi_{0}}$ with $i\ket{\phi_{0}}$ then

$\displaystyle\sum_{k,\,k^{\prime}}\biggl{\\{}i\sum_{j^{\prime}\neq
0}U_{0k}^{\dagger}U_{j^{\prime}k^{\prime}}-i\sum_{j\neq
0}U_{jk}^{\dagger}U_{0k^{\prime}}\biggr{\\}}=O.$ (93)

By multiplying $-i$ on Eq. (93) and subtracting from Eq. (92),

$\sum_{k,\,k^{\prime}}\sum_{j^{\prime}\neq
0}U_{0k}^{\dagger}U_{j^{\prime}k^{\prime}}=O.$ (94)

We substitute $\ket{\phi_{1}}$ with $-\ket{\phi_{1}}$ then Eq. (94) becomes

$\sum_{k,\,k^{\prime}}\sum_{j^{\prime}\neq
0}(-1)^{\delta_{1j^{\prime}}+1}U_{0k}^{\dagger}U_{j^{\prime}k^{\prime}}=O.$
(95)

By adding Eq. (94) and Eq. (95),

$\sum_{k,\,k^{\prime}}U_{0k}^{\dagger}U_{1k^{\prime}}=O.$ (96)

We apply the same argument to $k$ and $k^{\prime}$ in Eq. (96) then acquire

$U_{00}^{\dagger}U_{11}=O.$ (97)

Therefore,

$\displaystyle\left(\left({}^{A_{I}}\bra{\phi}{}^{B_{I}}\bra{\psi}\right)U\left(\ket{\pi}^{P}\ket{\alpha}^{A_{O}}\ket{\beta}^{B_{O}}\right),\,\left({}^{A_{I}}\bra{\phi^{\perp}}{}^{B_{I}}\bra{\psi^{\perp}}\right)U\left(\ket{\pi^{\prime}}^{P}\ket{\alpha^{\perp}}^{A_{O}}\ket{\beta^{\perp}}^{B_{O}}\right)\right)$
$\displaystyle=$
$\displaystyle\left(\left({}^{A_{I}}\bra{\phi_{0}}{}^{B_{I}}\bra{\psi_{0}}\right)U\left(\ket{\pi}^{P}\ket{0}^{A_{O}}\ket{0}^{B_{O}}\right),\left({}^{A_{I}}\bra{\phi_{1}}{}^{B_{I}}\bra{\psi_{1}}\right)U\left(\ket{\pi^{\prime}}^{P}\ket{1}^{A_{O}}\ket{1}^{B_{O}}\right)\right)$
$\displaystyle=$
$\displaystyle\left(U_{00}\ket{\pi}^{P},\,U_{11}\ket{\pi^{\prime}}^{P}\right)={}^{P}\bra{\pi}U_{00}^{\dagger}U_{11}\ket{\pi^{\prime}}^{P}=0.$

∎

Next, we show

$\left(\left({}^{A_{I}}\bra{\phi}{}^{B_{I}}\bra{\psi}\right)U\left(\ket{\pi}^{P}\ket{\alpha}^{A_{O}}\ket{\beta}^{B_{O}}\right),\left({}^{A_{I}}\bra{\phi^{\prime}}{}^{B_{I}}\bra{\psi^{\perp}}\right)U\left(\ket{\pi^{\prime}}^{P}\ket{\alpha^{\perp}}^{A_{O}}\ket{\beta^{\perp}}^{B_{O}}\right)\right)=0$
(98)

for all $\ket{\phi}^{A_{I}},\,\ket{\phi^{\prime}}^{A_{I}}\in A_{I}$ and for
all $\ket{\psi}^{B_{I}},\,\ket{\psi^{\perp}}^{B_{I}}\in B_{I}$ such that
$\ket{\psi}^{B_{I}}\perp\ket{\psi^{\perp}}^{B_{I}}$. Define the sesquilinear
function

$\displaystyle f(\ket{\phi^{\prime}}^{A_{I}},\,\ket{\phi}^{A_{I}}):=$ (99)
$\displaystyle\quad\quad\left(\left({}^{A_{I}}\bra{\phi}{}^{B_{I}}\bra{\psi}\right)U\left(\ket{\pi}^{P}\ket{\alpha}^{A_{O}}\ket{\beta}^{B_{O}}\right),\left({}^{A_{I}}\bra{\phi^{\prime}}{}^{B_{I}}\bra{\psi^{\perp}}\right)U\left(\ket{\pi^{\prime}}^{P}\ket{\alpha^{\perp}}^{A_{O}}\ket{\beta^{\perp}}^{B_{O}}\right)\right).$

From Lem. 9, we have

$f(\ket{\phi^{\perp}}^{A_{I}},\,\ket{\phi}^{A_{I}})=0$ (100)

for all $\ket{\phi}^{A_{I}},\,\ket{\phi^{\perp}}^{A_{I}}\in A_{I}$ such that
$\ket{\phi}^{A_{I}}\perp\ket{\phi^{\perp}}^{A_{I}}$. Lem. 4 and this show
$f(\ket{\phi}^{A_{I}},\,\ket{\phi}^{A_{I}})=f(\ket{\phi^{\prime}}^{A_{I}},\,\ket{\phi^{\prime}}^{A_{I}})$
for all $\ket{\phi}^{A_{I}},\,\ket{\phi^{\prime}}^{A_{I}}\in A_{I}$ such that
$\|\ket{\phi}^{A_{I}}\|=\|\ket{\phi^{\prime}}^{A_{I}}\|$. Thus, for all
$\ket{\phi}^{A_{I}}\in A_{I},\,\neq 0$, we have

$d_{A}\frac{f(\ket{\phi}^{A_{I}},\,\ket{\phi}^{A_{I}})}{\|\ket{\phi}^{A_{I}}\|^{2}}=\sum_{j=0}^{d_{A}-1}f\left(\frac{\ket{\phi}^{A_{I}}}{\|\ket{\phi}^{A_{I}}\|},\,\frac{\ket{\phi}^{A_{I}}}{\|\ket{\phi}^{A_{I}}\|}\right)=\sum_{j=0}^{d_{A}-1}f(\ket{j}^{A_{O}},\,\ket{j}^{A_{O}}),$
(101)

where $\\{\ket{j}^{A_{I}}\\}_{j=0}^{d_{A}-1}$ is the computational basis in
$A_{I}$. Recalling the definition of $f$ and completeness relation of
$\\{\ket{j}^{A_{I}}\\}_{j}$,

Eq. (101) $\displaystyle=$