#  Path-entangling evolution and quantum gravitational interaction

Akira Matsumura<EMAIL_ADDRESS>Department of Physics,
Kyushu University, Fukuoka, 819-0395, Japan

###### Abstract

We explore a general feature of the interaction mediated by the gravitational
fields of spatially superposed masses. For this purpose, based on quantum
information theory, we characterize the evolution of two particles each in a
superposition state of paths. The evolution is assumed to be given by a
completely positive trace-preserving (CPTP) map. We further assume that the
probability of particle being on each path is unchanged during the evolution.
This property is called population-preserving. We examine when a population-
preserving CPTP map can create entanglement in terms of separable operations,
which form a large class of local operations and classical communication
(LOCC). In general, entanglement is not always generated by inseparable or
non-LOCC operations, and one can consider a model of gravity described by an
inseparable operation which does not create entanglement. However, we find
that a population-preserving CPTP map is inseparable if and only if it can
create entanglement. This means that the above model of gravity is
incompatible with the possible evolution of spatially superposed masses.

###### Contents

  1. I Introduction
  2. II Population-preserving dynamics of superposed particles
  3. III Demonstration of gravity-induced entanglement
  4. IV Physical operations on spatially superposed particles
    1. IV.1 Characterization of completely positive trace-preserving maps
    2. IV.2 Characterization of separable operations
    3. IV.3 Equivalence between inseparable and entangling operations
  5. V Conclusion

##  I Introduction

Exploring the quantum nature of gravity is a fundamental task for modern
physics. Recently, much of the theoretical and experimental efforts has been
made to test the quantum superposition of the gravitational field caused by a
spatially superposed mass. This phenomenon may be interpreted as the quantum
superposition of a spacetime curvature. To capture such an interesting
phenomenon, gravity-induced entanglement has attracted attentions.

The gravity-induced entanglement is expected to be tested in table-top
experiments. In Refs.Bose2017 ; Marletto2017 , it was shown that the
gravitational interaction of two masses generates the entanglement between
them when each mass travels through an individual matter-wave interferomenter.
The works Bose2017 ; Marletto2017  has been stimulated advanced research and
other proposals using matter-wave interferometers Nguyen2020 ; Chevalier2020 ;
vandeKamp2020 ; Toros2021 ; Miki2021a , levitated nanoparticles or mechanical
oscillators Qvafort2020 ; Krisnanda2020 , optomechanical systems Balushi2018 ;
Miao2020 ; Matsumura2020 ; Miki2021b  and their hybrid model Carney2021a ;
Carney2022 ; Pedernales2021 ; Matsumura2021b . Also, the field theoretical
aspects of gravity-induced entanglement have been discussed in Belenchia2018 ;
Marshman2020 ; Matsumura2021a ; Carney2021b ; Danielson2021 ; Bose2022 ;
Christodoulou2022 .

The theoretical importance of gravity-induced entanglement mainly relies on
quantum information theory. To present it explicitly, the paradigm of local
operations and classical communication (LOCC) plays a crucial role Nielsen2002
; Horodecki2009 . An LOCC operation is a physical process in two distant
systems, which describes the local evolution of each system and the causal
transfer of classical information between them. In order to capture general
properties of LOCC operations, one often introduces the notion of separable
operations Rains1997 ; Vedral1998  including all LOCC operations as a proper
subset Chitambar2014 . It is known that any separable (or LOCC) operation
cannot generate entanglement in non-entangled (separable) states Horodecki2009
. Gravity-induced entanglement is an evidence that a gravitational interaction
is described by an inseparable or non-LOCC operation. In terms of LOCC, if the
interaction is local, then the gravity-induced entanglement is incompatible
with classical communication. This means that the gravitational field carries
quantum information.

As mentioned above, entanglement tells us the description of gravity in
quantum information theory, that is, whether gravity is described by an
inseparable (or a non-LOCC) operation. However, the relation between
entanglement and separable operations is not simple: there is an inseparable
operation not to generate entanglement Cirac2001 ; Harrow2003 . In principle,
one can consider a model of the gravitational interaction represented by an
inseparable operation without entanglement creation.

In this paper, to discuss such a possibility, we examine a general feature of
the quantum interaction between two particles. The tools in quantum
information theory make the analysis independent of the details of
interaction. As in Ref.Bose2017 ; Marletto2017 , we consider the setting that
each particle is in a superposition of paths. The evolution of the particles
is assumed to be given by a completely positive trace-preserving (CPTP) map
Nielsen2002 ; Horodecki2009 , which describes a large class of physical
dynamics. We further suppose that the probability of particle being on each
path is preserved. This property is called population-preserving in
Carney2021a . We can derive a representation theorem for a population-
preserving CPTP map. To characterize the entanglement evolution of the
particles, we use the notion of a separable operation and a non-entangling
operation Harrow2003 . The latter is defined as a CPTP map which transforms
every separable state to another separable state (that is, entanglement does
not occur). It is known that those operations are not equivalent to each other
Harrow2003 . However, we show the equivalence between inseparable operation
and entangling operation if they are population-preserving CPTP. Within the
above framework and assumptions, this means that gravity described by an
inseparable operation without entanglement creation is incompatible with the
evolution of spatially superposed masses.

The structure of this paper is as follows. In Sec. II, the population-
preserving dynamics of superposed particles is presented. In Sec. III, we
demonstrate the entanglement generation by the gravitational potential between
two massive particles. In Sec. IV, we obtain several key properties of a
population-preserving CPTP map and show the equivalence between inseparable
and entangling operations. Sec. V presents the conclusion of this paper. We
use the unit $\hbar=1$ in this paper.

##  II Population-preserving dynamics of superposed particles

In this section, to show the population-preserving property, we consider the
dynamics of two particles A and B described by the Schrödinger equation with
the Hamiltonian

$\hat{H}=\hat{H}_{\text{A}}+\hat{H}_{\text{B}}+\hat{V},$ (1)

where $\hat{H}_{\text{A}}$ and $\hat{H}_{\text{B}}$ are each Hamiltonian of
the particles A and B, and $\hat{V}$ is the interaction between them. Each
particle is assumed to be superposed on two paths (see Fig.1). Here, we also
assume that each path is determined by the Hamiltonians $\hat{H}_{\text{A}}$
and $\hat{H}_{\text{B}}$ and is almost undisturbed by the interaction
$\hat{V}$.

Figure 1: An example of the configuration of two particles A and B.

Each particle is initially in the following superposed state,

$|\Psi_{\text{in}}\rangle=\sum_{a,b=\text{L},\text{R}}\psi_{a}|a\,;\text{in}\rangle_{\text{A}}\,\phi_{b}|b\,;\text{in}\rangle_{\text{B}},$
(2)

where $|a\,;\text{in}\rangle_{\text{A}}\,(|b\,;\text{in}\rangle_{\text{B}})$
satisfying the orthonormal condition ${}_{\text{A}}\langle
a^{\prime};\text{in}|a\,;\text{in}\rangle_{\text{A}}\approx\delta_{a^{\prime}a}\,(\,{}_{\text{B}}\langle
b^{\prime};\text{in}|b\,;\text{in}\rangle_{\text{B}}\approx\delta_{b^{\prime}b}\,)$
represents the state of the particle A (B) with the localized wave packet
whose peak is at the position $a\,(b)$ at an initial time $t_{\text{in}}$, and
$\psi_{a}\,(\phi_{b})$ gives the probability
$|\psi_{a}|^{2}\,(|\phi_{b}|^{2})$ which the particle A (B) is in
$|a\,;\text{in}\rangle_{\text{A}}\,(\,|b\,;\text{in}\rangle_{\text{B}}\,)$.
The evolved state of the particles A and B is

$\displaystyle|\Psi_{\text{out}}\rangle$
$\displaystyle=e^{-i\hat{H}(t_{\text{out}}-t_{\text{in}})}|\Psi_{\text{in}}\rangle$
$\displaystyle=e^{-i(\hat{H}_{\text{A}}+\hat{H}_{\text{B}})(t_{\text{out}}-t_{\text{in}})}\text{T}\exp[-i\int^{t_{\text{out}}}_{t_{\text{in}}}dt\hat{V}_{\text{I}}(t)\Big{]}|\Psi_{\text{in}}\rangle$
$\displaystyle=e^{-i(\hat{H}_{\text{A}}+\hat{H}_{\text{B}})(t_{\text{out}}-t_{\text{in}})}\text{T}\exp[-i\int^{t_{\text{out}}}_{t_{\text{in}}}dt\hat{V}_{\text{I}}(t)\Big{]}\sum_{a,b=\text{L},\text{R}}\psi_{a}|a\,;\text{in}\rangle_{\text{A}}\,\phi_{b}|b\,;\text{in}\rangle_{\text{B}}$
$\displaystyle\approx
e^{-i(\hat{H}_{\text{A}}+\hat{H}_{\text{B}})(t_{\text{out}}-t_{\text{in}})}\sum_{a,b=\text{L},\text{R}}e^{i\Phi_{ab}}\,\psi_{a}|a\,;\text{in}\rangle_{\text{A}}\,\phi_{b}|b\,;\text{in}\rangle_{\text{B}}$
$\displaystyle=\sum_{a,b=\text{L},\text{R}}e^{i\Phi_{ab}}\,\psi_{a}|a\,;\text{out}\rangle_{\text{A}}\,\phi_{b}|b\,;\text{out}\rangle_{\text{B}},$
(3)

where $\hat{V}_{\text{I}}(t)$ is the interaction operator in the interaction
picture with respect to $\hat{H}_{\text{A}}+\hat{H}_{\text{B}}$, and the out
states

$\displaystyle|a\,;\text{out}\rangle_{\text{A}}=e^{-i\hat{H}_{\text{A}}(t_{\text{out}}-t_{\text{in}})}|a\,;\text{in}\rangle_{\text{A}},\quad|b\,;\text{out}\rangle_{\text{B}}=e^{-i\hat{H}_{\text{B}}(t_{\text{out}}-t_{\text{in}})}|b\,;\text{in}\rangle_{\text{B}}$
(4)

describe the wave packets of the particles A and B with the peaks at the
positions $a$ and $b$ at the time $t_{\text{out}}$, respectively. In the
fourth line of (3), since the motion of each particle is unperturbed, the
interaction potential $\hat{V}$ is evaluated along each path, which leads to
the phase factors $e^{i\Phi_{ab}}$ with

$\Phi_{ab}=-\int^{t_{\text{out}}}_{t_{\text{in}}}dtV_{ab}(t).$ (5)

The kinetic and other potential terms in the local Hamiltonians
$\hat{H}_{\text{A}}$ and $\hat{H}_{\text{B}}$ may give another accumulated
phases, which are included in the out states
$|a\,;\text{out}\rangle_{\text{A}}$ and $|b\,;\text{out}\rangle_{\text{B}}$.
Since such phases are given by a local unitary evolution, they do not affect
the entanglement between the particles. However, the fluctuation of the phases
may cause dephasing or decoherence effects. The detailed analysis of such
effects was performed in Toros2021 .

We consider two particles with masses $m_{\text{A}}$ and $m_{\text{B}}$. The
gravitational potential between the massive particles is

$\hat{V}=-\frac{Gm_{\text{A}}m_{\text{B}}}{|\hat{\bm{x}}_{\text{A}}-\hat{\bm{x}}_{\text{B}}|},$
(6)

where $\hat{\bm{x}}_{\text{A}}$ and $\hat{\bm{x}}_{\text{B}}$ are the position
operators of each particle. The potential energy $V_{ab}(t)$ in the
accumulated phase (5) is given as

$V_{ab}(t)=-\frac{Gm_{\text{A}}m_{\text{B}}}{|\bm{x}^{a}_{\text{A}}(t)-\bm{x}^{b}_{\text{B}}(t)|}.$
(7)

Here, $\bm{x}^{a}_{\text{A}}(t)$ and $\bm{x}^{b}_{\text{B}}(t)$ describe the
trajectories of the massive particles A and B which initially sit at the
positions $a$ and $b$, respectively. One notes that the gravitational
interaction is instantaneous and the locality (causality) of theory seems not
to satisfy. We should consider that the gravitational interaction is just
given in the non-relativistic limit and the locality holds in principle. This
concern was recently discussed in Bose2022 ; Christodoulou2022 . In Sec. III,
the gravity-induced entanglement between A and B will be demonstrated.

From the evolution given in Eq.(3), we find the equality

$\langle\Psi_{\text{out}}|\hat{\Pi}^{\text{out}}_{a}\otimes\hat{\Pi}^{\text{out}}_{b}|\Psi_{\text{out}}\rangle=\langle\Psi_{\text{in}}|\hat{\Pi}^{\text{in}}_{a}\otimes\hat{\Pi}^{\text{in}}_{b}|\Psi_{\text{in}}\rangle,$
(8)

where $\hat{\Pi}^{\text{in}}_{a}=|a\,;\text{in}\rangle_{\text{A}}\langle
a\,;\text{in}|$,
$\hat{\Pi}^{\text{out}}_{a}=|a\,;\text{out}\rangle_{\text{A}}\langle
a\,;\text{out}|$,
$\hat{\Pi}^{\text{in}}_{b}=|b\,;\text{in}\rangle_{\text{B}}\langle
b\,;\text{in}|$ and
$\hat{\Pi}^{\text{out}}_{b}=|b\,;\text{out}\rangle_{\text{B}}\langle
b\,;\text{out}|$. Namely, the probability which we find the particles on each
path is preserved. This property is called population-preserving in
Ref.Carney2021a . In Sec. IV, we will introduce a completely positive trace-
preserving map, a separable operation and a non-entangling operation. Imposing
the population-preserving property, we will examine how the map and the
operations are represented.

##  III Demonstration of gravity-induced entanglement

In this section, the basics of entanglement are introduced, and the
entanglement due to gravitational interaction is demonstrated. To quantify
entanglement, we introduce the positive partial transpose criterion Peres1996
; Horodecki1996  and the negativity Vidal2002  as follows. For a given density
operator $\rho$ of a bipartite system AB, we define the partial transposition
$\rho^{\text{T}_{\text{A}}}$ with the components

${}_{\text{A}}\langle a|{}_{\text{B}}\langle
b|\rho^{\text{T}_{\text{A}}}|a^{\prime}\rangle_{\text{A}}|b^{\prime}\rangle_{\text{B}}={}_{\text{A}}\langle
a^{\prime}|{}_{\text{B}}\langle
b|\rho|a\rangle_{\text{A}}|b^{\prime}\rangle_{\text{B}}$ (9)

for a basis $\\{|a\rangle_{\text{A}}|b\rangle_{\text{B}}\\}_{a,b}$ of the
Hilbert space $\mathcal{H}_{\text{A}}\otimes\mathcal{H}_{\text{B}}$ of the
bipartite system AB. One can show that if the density operator $\rho$ is
separable (non-entangled) Werner1989 , that is,

$\rho=\sum_{k}p_{k}\rho_{k}\otimes\sigma_{k},$ (10)

where $p_{k}$ is a probability, and $\rho_{k}$ and $\sigma_{k}$ are the
density operators of each subsystem, then the partial transposition
$\rho^{\text{T}_{\text{A}}}$ has only non-negative eigenvalues. Hence, if the
partial transposition $\rho^{\text{T}_{\text{A}}}$ has a negative eigenvalue
then the given density operator $\rho$ is non-separable (entangled). This is
called the positive partial transpose (PPT) criterion Peres1996 ;
Horodecki1996 . In particular, the PPT criterion becomes the necessary and
sufficient condition for a two-qubit system
($\mathbb{C}^{2}\otimes\mathbb{C}^{2}$) and a qubit-qutrit system
($\mathbb{C}^{2}\otimes\mathbb{C}^{3}$) Horodecki1996 . The negativity

$\mathcal{N}=\sum_{\lambda_{i}<0}|\lambda_{i}|$ (11)

with the eigenvalues $\lambda_{i}$ of the partial transposition
$\rho^{\text{T}_{\text{A}}}$ is an entanglement measure (quantifier) Vidal2002
. The PPT criterion means that a density operator $\rho$ with nonzero
negativity is entangled.

Let us demonstrate the entanglement generation due to the gravitational
potential (6) assuming the concrete trajectories as

$\bm{x}^{a}_{\text{A}}(t)=[x^{a}(t),v_{y}t,0]^{\text{T}},\quad\bm{x}^{b}_{\text{B}}(t)=[x^{b}(t)+D,v_{y}t,0]^{\text{T}},$
(12)

where $v_{y}>0$ is the velocity in the y-direction and

$x^{a}(t)=\epsilon_{a}\left\\{\begin{array}[]{lll}vt&t_{\text{in}}\leq
t\leq\tau\\\ v\tau&\tau\leq t\leq T+\tau\\\ v(T+\tau-t)+v\tau&T+\tau\leq t\leq
t_{\text{out}}\end{array}\right.$ (13)

with $\epsilon_{\text{L}}=-\epsilon_{\text{R}}=-1$ and the velocity $v(>0)$ in
the x-direction. The accumulated phase (5) is computed as

$\displaystyle\Phi_{ab}=\int^{t_{\text{out}}}_{t_{\text{in}}}dt\frac{Gm_{\text{A}}m_{\text{B}}}{|\bm{x}^{a}_{\text{A}}(t)-\bm{x}^{b}_{\text{B}}(t)|}\approx\int^{T+\tau}_{\tau}dt\frac{Gm_{\text{A}}m_{\text{B}}}{|\bm{x}^{a}_{\text{A}}(t)-\bm{x}^{b}_{\text{B}}(t)|}=\frac{Gm_{\text{A}}m_{\text{B}}T}{D+(\epsilon_{b}-\epsilon_{a})L/2},$
(14)

where $L=2v\tau$. In the approximation, assuming a large $T$, we evaluated the
phase given in the intermediate stage $\tau\leq t\leq T+\tau$. The
configuration of the particle paths is shown in Fig.2.

Figure 2: The concrete configuration to demonstrate the entanglement behavior
of two particles. $L$ is the typical length scale of superposition of each
particle, and $D$ is the distance between the two particles at the initial
time $t_{\text{in}}$

Assuming the initial state (2) with $\psi_{a}=\phi_{b}=1/\sqrt{2}$, we obtain
the following negativity

$\mathcal{N}=\frac{1}{2}\Big{|}\sin[\frac{\Phi_{\text{RL}}+\Phi_{\text{LR}}-\Phi_{\text{LL}}-\Phi_{\text{RR}}}{2}\Big{]}\Big{|}=\frac{1}{2}\Big{|}\sin[\frac{Gm_{\text{A}}m_{\text{B}}T}{2}\Big{(}\frac{1}{D+L}+\frac{1}{D-L}-\frac{2}{D}\Big{)}\Big{]}\Big{|}.$
(15)

Fig.3 shows the negativity as a function of $Gm_{\text{A}}m_{\text{B}}T/\pi D$
for a fixed $D$. For a small $D$, the accumulated phases frequently change,
and the entanglement generation due to the interaction is promoted.

Figure 3: The behavior of the negativity between the two massive particles.
The vertical axis represents the negativity, which is a dimensionless
quantity. The horizontal axis describes the time $T$ in the unit $\pi
D/Gm_{\text{A}}m_{\text{B}}$.

##  IV Physical operations on spatially superposed particles

In this section, the main theorems for the evolution of two particles in a
path superposition are devoted. In the subsection IV.1, assuming the evolution
is described by a completely positive trace-preserving and population-
preserving map, we will obtain a representation theorem for the evolution. In
the subsection IV.2, introducing the notion of separable operations, we will
find the separability criterion on the evolution. In the subsection IV.3, it
will be shown that inseparable operations on the particles are equivalent to
entangling operations.

###  IV.1 Characterization of completely positive trace-preserving maps

We consider a map $\Phi$ from the set of density operators on a Hilbert space
$\mathcal{H}$ to the set of density operators on another Hilbert space
$\mathcal{K}$. The map $\Phi$ is assumed to satisfy the following properties:
$\text{Tr}\big{[}\Phi[\rho]\big{]}=\text{Tr}[\rho]$ (trace preserving),
$\Phi[\lambda\rho+(1-\lambda)\sigma]=\lambda\Phi[\rho]+(1-\lambda)\Phi[\sigma]$
(convex linear) and $\Phi\otimes\mathcal{I}_{\text{E}}[\Omega]\geq 0$
(completely positive), where $\rho$ and $\sigma$ are density operators on the
Hilbert space $\mathcal{H}$, $\lambda$ is a probability ($0\leq\lambda\leq
1$), $\mathcal{I}_{\text{E}}$ is the identity map on an extra system E with a
Hilbert space $\mathcal{H}_{\text{E}}$, and $\Omega$ is an arbitrary density
operator on the Hilbert space $\mathcal{H}\otimes\mathcal{H}_{\text{E}}$. The
map $\Phi$ is called a completely positive trace-preserving (CPTP) map
Nielsen2002 ; Horodecki2009 . The CPTP map $\Phi$ can describe a unitary
evolution ($\Phi[\rho]=\hat{U}\rho\hat{U}^{\dagger}$) and the coupled dynamics
with measurement apparatuses or environments. It is known that every CPTP map
is represented as

$\Phi[\rho]=\sum_{\ell}\hat{K}_{\ell}\,\rho\,\hat{K}^{\dagger}_{\ell},$ (16)

where the operators $\hat{K}_{\ell}:\mathcal{H}\rightarrow\mathcal{K}$
satisfying
$\sum_{\ell}\hat{K}^{\dagger}_{\ell}\hat{K}_{\ell}=\hat{\mathbb{I}}_{\mathcal{H}}$
( $\hat{\mathbb{I}}_{\mathcal{H}}$ is the indentity operator on $\mathcal{H}$)
are called Kraus operators. Conversely, the map defined by Eq.(16) is a CPTP
map. The representation Eq.(16) of a CPTP map is called the operator-sum
representation Nielsen2002 .

In the setting presented in Sec.II (for example, see Fig.1), the population
preserving holds, that is, the probability of particles being on each path
does not change. We assume that the evolution of the particles is given by a
population-preserving CPTP map $\Phi$. The latter condition means that the
following equality holds for any initial density operator $\rho_{\text{in}}$:

$\text{Tr}\big{[}\hat{\Pi}^{\text{out}}_{a}\otimes\hat{\Pi}^{\text{out}}_{b}\rho_{\text{out}}\big{]}=\text{Tr}\big{[}\hat{\Pi}^{\text{in}}_{a}\otimes\hat{\Pi}^{\text{in}}_{b}\rho_{\text{in}}\big{]},$
(17)

where $\hat{\Pi}^{\text{in}}_{a}=|a\,;\text{in}\rangle_{\text{A}}\langle
a\,;\text{in}|$,
$\hat{\Pi}^{\text{out}}_{a}=|a\,;\text{out}\rangle_{\text{A}}\langle
a\,;\text{out}|$,
$\hat{\Pi}^{\text{in}}_{b}=|b\,;\text{in}\rangle_{\text{B}}\langle
b\,;\text{in}|$,
$\hat{\Pi}^{\text{out}}_{b}=|b\,;\text{out}\rangle_{\text{B}}\langle
b\,;\text{out}|$ and $\rho_{\text{out}}=\Phi[\rho_{\text{in}}]$. In Sec.II,
each particle was assumed to be superposed in two paths. One can consider the
setting of a multipath superposition, which is adopted to test gravity-induced
entanglement Tilly2021 . Taking into account multiple paths of each particle,
we can show the following theorem.

######  Theorem 1.

Let
$\mathcal{H}^{\text{in}}_{\text{A}}\otimes\mathcal{H}^{\text{in}}_{\text{B}}$
and
$\mathcal{H}^{\text{out}}_{\text{A}}\otimes\mathcal{H}^{\text{out}}_{\text{B}}$
be the Hilbert space with the complete bases
$\\{|a\,;\text{in}\rangle_{\text{A}}|b\,;\text{in}\rangle_{\text{B}}\\}_{a=1,2,\dots,d_{\text{A}},b=1,2,\dots,d_{\text{B}}}$
and
$\\{|a\,;\text{out}\rangle_{\text{A}}|b\,;\text{out}\rangle_{\text{B}}\\}_{a=1,2,\dots,d_{\text{A}},b=1,2,\dots,d_{\text{B}}}$,
respectively. A map
$\Phi:\rho_{\text{in}}\mapsto\rho_{\text{out}}=\Phi[\rho_{\text{in}}]$, where
$\rho_{\text{in}}$ and $\rho_{\text{out}}$ are density operators each on
$\mathcal{H}^{\text{in}}_{\text{A}}\otimes\mathcal{H}^{\text{in}}_{\text{B}}$
and
$\mathcal{H}^{\text{out}}_{\text{A}}\otimes\mathcal{H}^{\text{out}}_{\text{B}}$,
is CPTP and population-preserving if and only if the map $\Phi$ is represented
by using $\hat{M}_{a}=|a\,;\text{out}\rangle_{\text{A}}\langle a\,;\text{in}|$
and $\hat{N}_{b}=|b\,;\text{out}\rangle_{\text{B}}\langle b\,;\text{in}|$ as

$\Phi[\rho_{\text{in}}]=\sum^{d_{\text{A}}}_{a,a^{\prime}=1}\sum^{d_{\text{B}}}_{b,b^{\prime}=1}\mathcal{E}_{aba^{\prime}b^{\prime}}\hat{M}_{a}\otimes\hat{N}_{b}\,\rho_{\text{in}}\,\hat{M}^{\dagger}_{a^{\prime}}\otimes\hat{N}^{\dagger}_{b^{\prime}},$
(18)

where the coefficients $\mathcal{E}_{aba^{\prime}b^{\prime}}$ satisfy
$\mathcal{E}_{abab}=1$, and the square matrix $\mathcal{E}$ with the
components $\mathcal{E}_{aba^{\prime}b^{\prime}}$ has only non-negative
eigenvalues, whose characteristic equation with an eigenvector
$\bm{v}=[v_{11},v_{12},\dots,v_{1d_{\text{B}}},v_{21},v_{22},\dots,v_{d_{\text{A}}d_{\text{B}}}]^{\text{T}}$
and an eigenvalue $\nu$ is

$(\mathcal{E}\bm{v})_{ab}=\sum^{d_{\text{A}}}_{a^{\prime}=1}\sum^{d_{\text{B}}}_{b^{\prime}=1}\mathcal{E}_{aba^{\prime}b^{\prime}}v_{a^{\prime}b^{\prime}}=\nu
v_{ab}.$ (19)

The latter condition on the matrix $\mathcal{E}$ is equivalent to that the
components $\mathcal{E}_{aba^{\prime}b^{\prime}}$ satisfy
$\sum_{a,b,a^{\prime},b^{\prime}}Z_{ab}\mathcal{E}_{aba^{\prime}b^{\prime}}Z^{*}_{a^{\prime}b^{\prime}}\geq
0$ for all complex numbers $Z_{ab}$.

###### Proof.

Let us first prove the sufficient condition. A population-preserving CPTP map
$\Phi$ satisfies

$\displaystyle\text{Tr}\big{[}\hat{\Pi}^{\text{in}}_{a}\otimes\hat{\Pi}^{\text{in}}_{b}\rho_{\text{in}}\big{]}$
$\displaystyle=\text{Tr}\big{[}\hat{\Pi}^{\text{out}}_{a}\otimes\hat{\Pi}^{\text{out}}_{b}\Phi[\rho_{\text{in}}]\big{]}$
$\displaystyle=\text{Tr}\big{[}\hat{\Pi}^{\text{out}}_{a}\otimes\hat{\Pi}^{\text{out}}_{b}\sum_{\ell}\hat{K}_{\ell}\,\rho_{\text{in}}\,\hat{K}^{\dagger}_{\ell}\big{]}$
$\displaystyle=\text{Tr}\Big{[}\Big{(}\sum_{\ell}\hat{K}^{\dagger}_{\ell}\,\hat{\Pi}^{\text{out}}_{a}\otimes\hat{\Pi}^{\text{out}}_{b}\,\hat{K}_{\ell}\Big{)}\,\rho_{\text{in}}\Big{]},$
(20)

where $\hat{\Pi}^{\text{in}}_{a}=|a\,;\text{in}\rangle_{\text{A}}\langle
a\,;\text{in}|$,
$\hat{\Pi}^{\text{out}}_{a}=|a\,;\text{out}\rangle_{\text{A}}\langle
a\,;\text{out}|$,
$\hat{\Pi}^{\text{in}}_{b}=|b\,;\text{in}\rangle_{\text{B}}\langle
b\,;\text{in}|$,
$\hat{\Pi}^{\text{out}}_{b}=|b\,;\text{out}\rangle_{\text{B}}\langle
b\,;\text{out}|$, and the first equality holds by the population preserving,
and in the second line the operator-sum representation (16) of $\Phi$ was
used. The above equality holds for any initial states. Choosing the initial
state as $\rho_{\text{in}}=|a^{\prime}\,;\text{in}\rangle_{\text{A}}\langle
a^{\prime}\,;\text{in}|\otimes|b^{\prime};\text{in}\rangle_{\text{B}}\langle
b^{\prime};\text{in}|$, we get the following equation

$\delta_{aa^{\prime}}\delta_{bb^{\prime}}=\sum_{\ell}|{}_{\text{A}}\langle
a\,;\text{out}|{}_{\text{B}}\langle
b\,;\text{out}|\hat{K}_{\ell}|a^{\prime}\,;\text{in}\rangle_{\text{A}}|b^{\prime};\text{in}\rangle_{\text{B}}|^{2},$
(21)

where $\delta_{aa^{\prime}}$ is the Kronecker’s delta. For $a\neq a^{\prime}$
or $b\neq b^{\prime}$, the above equation yields ${}_{\text{A}}\langle
a\,;\text{out}|{}_{\text{B}}\langle
b\,;\text{out}|\hat{K}_{\ell}|a^{\prime}\,;\text{in}\rangle_{\text{A}}|b^{\prime};\text{in}\rangle_{\text{B}}=0$.
Hence, the Kraus operator $\hat{K}_{\ell}$ has the form

$\hat{K}_{\ell}=\sum^{d_{\text{A}}}_{a=1}\sum^{d_{\text{B}}}_{b=1}k^{(\ell)}_{ab}\,\hat{M}_{a}\otimes\hat{N}_{b},$
(22)

where $\hat{M}_{a}=|a\,;\text{out}\rangle_{\text{A}}\langle a\,;\text{in}|$,
$\hat{N}_{b}=|b\,;\text{out}\rangle_{\text{B}}\langle b\,;\text{in}|$ and a
complex number $k^{(\ell)}_{ab}$. Then, the map $\Phi$ is represented as

$\Phi[\rho]=\sum_{\ell}\hat{K}_{\ell}\rho\hat{K}^{\dagger}_{\ell}=\sum^{d_{\text{A}}}_{a,a^{\prime}=1}\sum^{d_{\text{B}}}_{b,b^{\prime}=1}\mathcal{E}_{aba^{\prime}b^{\prime}}\hat{M}_{a}\otimes\hat{N}_{b}\,\rho\,\hat{M}^{\dagger}_{a^{\prime}}\otimes\hat{N}^{\dagger}_{b^{\prime}},$
(23)

where we defined the coefficients
$\mathcal{E}_{aba^{\prime}b^{\prime}}=\sum_{\ell}k^{(\ell)}_{ab}k^{(\ell)*}_{a^{\prime}b^{\prime}}$,
which satisfy $\mathcal{E}_{abab}=1$ by the trace preserving and
$\sum_{a,b,a^{\prime},b^{\prime}}Z_{ab}\mathcal{E}_{aba^{\prime}b^{\prime}}Z^{*}_{a^{\prime}b^{\prime}}\geq
0$ for all complex numbers $Z_{ab}$ by the definition itself. Next, we show
the necessary condition. We assume that a map $\Phi$ is given as

$\Phi[\rho]=\sum^{d_{\text{A}}}_{a,a^{\prime}=1}\sum^{d_{\text{B}}}_{b,b^{\prime}=1}\mathcal{E}_{aba^{\prime}b^{\prime}}\hat{M}_{a}\otimes\hat{N}_{b}\,\rho\,\hat{M}^{\dagger}_{a^{\prime}}\otimes\hat{N}^{\dagger}_{b^{\prime}},$
(24)

where $\mathcal{E}_{aba^{\prime}b^{\prime}}$ satisfies $\mathcal{E}_{abab}=1$
and
$\sum_{a,b,a^{\prime},b^{\prime}}Z_{ab}\mathcal{E}_{aba^{\prime}b^{\prime}}Z^{*}_{a^{\prime}b^{\prime}}\geq
0$ for all complex numbers $Z_{ab}$. From the condition $\mathcal{E}_{abab}=1$
and the form of the map $\Phi$, it is easy to check that $\Phi$ is a trace-
preserving and population-preserving map. By the other condition
$\sum_{a,b,a^{\prime},b^{\prime}}Z_{ab}\mathcal{E}_{aba^{\prime}b^{\prime}}Z^{*}_{a^{\prime}b^{\prime}}\geq
0$, we can always find the complex numbers $k^{(\ell)}_{ab}$ such that
$\mathcal{E}_{aba^{\prime}b^{\prime}}=\sum_{\ell}k^{(\ell)}_{ab}k^{(\ell)*}_{a^{\prime}b^{\prime}}$.
Thus,

$\displaystyle\Phi[\rho]$
$\displaystyle=\sum^{d_{\text{A}}}_{a,a^{\prime}=1}\sum^{d_{\text{B}}}_{b,b^{\prime}=1}\mathcal{E}_{aba^{\prime}b^{\prime}}\hat{M}_{a}\otimes\hat{N}_{b}\,\rho\,\hat{M}^{\dagger}_{a^{\prime}}\otimes\hat{N}^{\dagger}_{b^{\prime}}$
$\displaystyle=\sum^{d_{\text{A}}}_{a,a^{\prime}=1}\sum^{d_{\text{B}}}_{b,b^{\prime}=1}\sum_{\ell}k^{(\ell)}_{ab}k^{(\ell)*}_{a^{\prime}b^{\prime}}\hat{M}_{a}\otimes\hat{N}_{b}\,\rho\,\hat{M}^{\dagger}_{a^{\prime}}\otimes\hat{N}^{\dagger}_{b^{\prime}}$
$\displaystyle=\sum_{\ell}\Big{(}\sum^{d_{\text{A}}}_{a=1}\sum^{d_{\text{B}}}_{b=1}k^{(\ell)}_{ab}\hat{M}_{a}\otimes\hat{N}_{b}\Big{)}\,\rho\,\Big{(}\sum^{d_{\text{A}}}_{a^{\prime}=1}\sum^{d_{\text{B}}}_{b^{\prime}=1}k^{(\ell)}_{a^{\prime}b^{\prime}}\hat{M}_{a^{\prime}}\otimes\hat{N}_{b^{\prime}}\Big{)}^{\dagger}.$
(25)

This corresponds to the operator-sum representation. Hence, $\Phi$ is a
population-preserving CPTP map. ∎

Theorem 1 gives the representation of CPTP maps on two particles each in a
superposition of multiple paths (two paths when
$d_{\text{A}}=d_{\text{B}}=2$). In the next subsection, using Theorem 1, we
characterize separable operations on spatially superposed particles.

###  IV.2 Characterization of separable operations

We define a separable operation $\Phi_{\text{SEP}}$ Rains1997 ; Vedral1998  on
a bipartite system AB as

$\Phi_{\text{SEP}}[\rho]=\sum_{\ell}\hat{A}_{\ell}\otimes\hat{B}_{\ell}\,\rho\,\hat{A}^{\dagger}_{\ell}\otimes\hat{B}^{\dagger}_{\ell},$
(26)

where $\rho$ is a density operator on the Hilbert space
$\mathcal{H}_{\text{A}}\otimes\mathcal{H}_{\text{B}}$, and the operators
$\hat{A}_{\ell}$ and $\hat{B}_{\ell}$ locally act on each Hilbert space and
satisfy
$\sum_{\ell}\hat{A}^{\dagger}_{\ell}\hat{A}_{\ell}\otimes\hat{B}^{\dagger}_{\ell}\hat{B}_{\ell}=\hat{\mathbb{I}}$.
Since $\hat{A}_{\ell}$ and $\hat{B}_{\ell}$ depend on the same parameter
$\ell$ (in general, it may be continuous), the systems A and B are correlated.
However, quantum entanglement is not generated by separable operations, since
one can show that separable operations transform a separable state into an
another separable state. It is known that every LOCC operation is a separable
operation Horodecki2009 . In terms of LOCC, the parameter $\ell$ plays a role
of classical information (e.g. measurement outcomes) exchanged through
classical channels.

The crucial theorem to determine whether a given CPTP map is separable was
proved in Cirac2001 . To explain the theorem, let me introduce the following
matrix (operator)

$\mathcal{E}_{\Phi}=\Phi\otimes\mathcal{I}_{\text{A}^{\prime}\text{B}^{\prime}}\Big{[}|\Psi\rangle_{\text{AA}^{\prime}}\langle\Psi|\otimes|\tilde{\Psi}\rangle_{\text{BB}^{\prime}}\langle\tilde{\Psi}|\Big{]},$
(27)

where $\Phi$ is a CPTP map for a bipartite system AB with a Hilbert space
$\mathcal{H}_{\text{A}}\otimes\mathcal{H}_{\text{B}}$, and
$\mathcal{I}_{\text{A}^{\prime}\text{B}^{\prime}}$ is the identity map for two
extra systems $\text{A}^{\prime}$ and $\text{B}^{\prime}$. Here,
$|\Psi\rangle_{\text{AA}^{\prime}}$ and
$|\tilde{\Psi}\rangle_{\text{BB}^{\prime}}$ are the maximally entangled
(unnormalized) states as

$|\Psi\rangle_{\text{AA}^{\prime}}=\sum^{d}_{a=1}|a\rangle_{\text{A}}|a\rangle_{\text{A}^{\prime}},\quad|\tilde{\Psi}\rangle_{\text{BB}^{\prime}}=\sum^{d^{\prime}}_{b=1}|b\rangle_{\text{B}}|b\rangle_{\text{B}^{\prime}},$
(28)

where $d=\dim\mathcal{H}_{\text{A}}$ and
$d^{\prime}=\dim\mathcal{H}_{\text{B}}$. The properties of the map $\Phi$ is
translated into those of the matrix $\mathcal{E}_{\Phi}$. In Ref. Cirac2001 ,
the matrix $\mathcal{E}_{\Phi}$ was introduced as an extension of the Choi-
Jamiolkowski isomorphism Jamiolkowski1972 ; Choi1975 , which leads to the
following statement.

######  Theorem 2.

(the operation separability Cirac2001 ) a CPTP map $\Phi$ is a separable
operation if and only if the matrix $\mathcal{E}_{\Phi}$ given in Eq.(27) is a
separable density matrix with respect to $\text{AA}^{\prime}$ and
$\text{BB}^{\prime}$ (up to the normalization constant).

Before applying Theorem 2, it is useful to state the following lemma obtained
from Theorem 1.

######  Lemma 1.

Let $\Phi$ be the population-preserving CPTP map in Theorem 1. The matrix
$\mathcal{E}_{\Phi}$ given in Eq.(27) is the square matrix $\mathcal{E}$
defined in Theorem 1.

###### Proof.

By Theorem 1, a population-preserving CPTP map $\Phi$ is represented as

$\Phi[\rho_{\text{in}}]=\sum^{d_{\text{A}}}_{a,a^{\prime}=1}\sum^{d_{\text{B}}}_{b,b^{\prime}=1}\mathcal{E}_{aba^{\prime}b^{\prime}}\hat{M}_{a}\otimes\hat{N}_{b}\,\rho_{\text{in}}\,\hat{M}^{\dagger}_{a^{\prime}}\otimes\hat{N}^{\dagger}_{b^{\prime}},$
(29)

with $\hat{M}_{a}=|a\,;\text{out}\rangle_{\text{A}}\langle a\,;\text{in}|$,
$\hat{N}_{b}=|b\,;\text{out}\rangle_{\text{B}}\langle b\,;\text{in}|$, the
conditions $\mathcal{E}_{abab}=1$ and
$\sum_{a,b,a^{\prime},b^{\prime}}Z_{ab}\mathcal{E}_{aba^{\prime}b^{\prime}}Z^{*}_{a^{\prime}b^{\prime}}\geq
0$ for all complex numbers $Z_{ab}$. The corresponding Choi matrix is

$\displaystyle\mathcal{E}_{\Phi}$
$\displaystyle=\Phi\otimes\mathcal{I}_{\text{A}^{\prime}\text{B}^{\prime}}\Big{[}|\Psi\rangle_{\text{AA}^{\prime}}\langle\Psi|\otimes|\tilde{\Psi}\rangle_{\text{BB}^{\prime}}\langle\tilde{\Psi}|\Big{]}$
$\displaystyle=\sum^{d_{\text{A}}}_{a,a^{\prime}=1}\sum^{d_{\text{B}}}_{b,b^{\prime}=1}\mathcal{E}_{aba^{\prime}b^{\prime}}\hat{M}_{a}|\Psi\rangle_{\text{AA}^{\prime}}\langle\Psi|\hat{M}^{\dagger}_{a^{\prime}}\otimes\hat{N}_{b}|\tilde{\Psi}\rangle_{\text{BB}^{\prime}}\langle\tilde{\Psi}|\hat{N}^{\dagger}_{b^{\prime}}$
$\displaystyle=\sum^{d_{\text{A}}}_{a,a^{\prime}=1}\sum^{d_{\text{B}}}_{b,b^{\prime}=1}\mathcal{E}_{aba^{\prime}b^{\prime}}|\Psi_{a}\rangle_{\text{AA}^{\prime}}\langle\Psi_{a^{\prime}}|\otimes|\tilde{\Psi}_{b}\rangle_{\text{BB}^{\prime}}\langle\tilde{\Psi}_{b^{\prime}}|,$
(30)

where
$|\Psi_{a}\rangle_{\text{AA}^{\prime}}=\hat{M}_{a}|\Psi\rangle_{\text{AA}^{\prime}}$
and
$|\tilde{\Psi}_{b}\rangle_{\text{BB}^{\prime}}=\hat{N}_{b}|\tilde{\Psi}\rangle_{\text{BB}^{\prime}}$.
The maximally entangled (unnormalized) states
$|\Psi\rangle_{\text{AA}^{\prime}}$ and
$|\tilde{\Psi}\rangle_{\text{BB}^{\prime}}$ are defined as

$\displaystyle|\Psi\rangle_{\text{AA}^{\prime}}=\sum^{d_{\text{A}}}_{a=1}|a\,;\text{in}\rangle_{\text{A}}|a\rangle_{\text{A}^{\prime}},\quad|\tilde{\Psi}\rangle_{\text{BB}^{\prime}}=\sum^{d_{\text{B}}}_{b=1}|b\,;\text{in}\rangle_{\text{B}}|b\rangle_{\text{B}^{\prime}},$
(31)

where $|a\rangle_{\text{A}^{\prime}}$ and $|b\rangle_{\text{B}^{\prime}}$ are
the states of extra systems $\text{A}^{\prime}$ and $\text{B}^{\prime}$
satisfying ${}_{\text{A}^{\prime}}\langle
a^{\prime}|a\rangle_{\text{A}^{\prime}}=\delta_{a^{\prime}a}$ and
${}_{\text{B}^{\prime}}\langle
b^{\prime}|b\rangle_{\text{B}^{\prime}}=\delta_{b^{\prime}b}$, respectively.
Since we find the orthonormality
${}_{\text{AA}^{\prime}}\langle\Psi_{a^{\prime}}|\Psi_{a}\rangle_{\text{AA}^{\prime}}=\delta_{a^{\prime}a}$
and
${}_{\text{BB}^{\prime}}\langle\tilde{\Psi}_{b^{\prime}}|\tilde{\Psi}_{b}\rangle_{\text{BB}^{\prime}}=\delta_{b^{\prime}b}$,
the components of the matrix $\mathcal{E}_{\Phi}$ are
$\mathcal{E}_{aba^{\prime}b^{\prime}}$, that is,
$\mathcal{E}=\mathcal{E}_{\Phi}$. ∎

Using Theorem 1, Lemma 1 and Theorem 2, we can show the statement on separable
operations as follows.

######  Theorem 3.

Let $\Phi$ be the population-preserving CPTP map in Theorem 1. The map $\Phi$
is a separable operation if and only if the matrix $\mathcal{E}$ defined in
Theorem 1 with the components $\mathcal{E}_{aba^{\prime}b^{\prime}}$ has the
following separable form as

$\mathcal{E}_{aba^{\prime}b^{\prime}}=\sum_{k}\lambda_{k}\,(\mathcal{A}_{k})_{aa^{\prime}}\,(\mathcal{B}_{k})_{bb^{\prime}},$
(32)

where $\lambda_{k}\geq 0$, and the matrices $\mathcal{A}_{k}$ and
$\mathcal{B}_{k}$ with the components $(\mathcal{A}_{k})_{aa^{\prime}}$ and
$(\mathcal{B}_{k})_{bb^{\prime}}$ have only non-negative eigenvalues.

###### Proof.

We assume that a map $\Phi$ is CPTP and population-preserving. As shown in
Eq.(30), the matrix $\mathcal{E}_{\Phi}$ associated with the map $\Phi$ is
given as

$\mathcal{E}_{\Phi}=\sum^{d_{\text{A}}}_{a,a^{\prime}=1}\sum^{d_{\text{B}}}_{b,b^{\prime}=1}\mathcal{E}_{aba^{\prime}b^{\prime}}|\Psi_{a}\rangle_{\text{AA}^{\prime}}\langle\Psi_{a^{\prime}}|\otimes|\tilde{\Psi}_{b}\rangle_{\text{BB}^{\prime}}\langle\tilde{\Psi}_{b^{\prime}}|,$
(33)

where
$|\Psi_{a}\rangle_{\text{AA}^{\prime}}=\hat{M}_{a}|\Psi\rangle_{\text{AA}^{\prime}}$
and
$|\tilde{\Psi}_{b}\rangle_{\text{BB}^{\prime}}=\hat{N}_{b}|\tilde{\Psi}\rangle_{\text{BB}^{\prime}}$
with $\hat{M}_{a}=|a\,;\text{out}\rangle_{\text{A}}\langle a\,;\text{in}|$,
$\hat{N}_{b}=|b\,;\text{out}\rangle_{\text{B}}\langle b\,;\text{in}|$ and the
maximally entangled (unnormalized) states $|\Psi\rangle_{\text{AA}^{\prime}}$
and $|\tilde{\Psi}\rangle_{\text{BB}^{\prime}}$ defined in (31). If $\Phi$ is
separable, then, by Theorem 2, $\mathcal{E}_{\Phi}$ has the separable form

$\mathcal{E}_{\Phi}=\sum_{k}q_{k}\tilde{\rho}_{k}\otimes\tilde{\sigma}_{k},$
(34)

where $q_{k}\geq 0$, and $\tilde{\rho}_{k}$ and $\tilde{\sigma}_{k}$ are
density operators on
$\mathcal{H}_{\text{A}}\otimes\mathcal{H}_{\text{A}^{\prime}}$ and
$\mathcal{H}_{\text{B}}\otimes\mathcal{H}_{\text{B}^{\prime}}$, respectively.
By the orthonormality
${}_{\text{AA}^{\prime}}\langle\Psi_{a^{\prime}}|\Psi_{a}\rangle_{\text{AA}^{\prime}}=\delta_{a^{\prime}a}$
and
${}_{\text{BB}^{\prime}}\langle\tilde{\Psi}_{b^{\prime}}|\tilde{\Psi}_{b}\rangle_{\text{BB}^{\prime}}=\delta_{b^{\prime}b}$
and using Eq.(33), we have

$\mathcal{E}_{aba^{\prime}b^{\prime}}={}_{\text{AA}^{\prime}}\langle\Psi_{a}|{}_{\text{BB}^{\prime}}\langle\tilde{\Psi}_{b}|\mathcal{E}_{\Phi}|\Psi_{a^{\prime}}\rangle_{\text{AA}^{\prime}}|\tilde{\Psi}_{b^{\prime}}\rangle_{\text{BB}^{\prime}}=\sum_{k}q_{k}(\tilde{\rho}_{k})_{aa^{\prime}}(\tilde{\sigma}_{k})_{bb^{\prime}},$
(35)

where
$(\tilde{\rho}_{k})_{aa^{\prime}}={}_{\text{AA}^{\prime}}\langle\Psi_{a}|\tilde{\rho}_{k}|\Psi_{a^{\prime}}\rangle_{\text{AA}^{\prime}}$
and
$(\tilde{\sigma}_{k})_{bb^{\prime}}={}_{\text{BB}^{\prime}}\langle\tilde{\Psi}_{b}|\tilde{\sigma}_{k}|\tilde{\Psi}_{b^{\prime}}\rangle_{\text{BB}^{\prime}}$.
The form of $\mathcal{E}_{aba^{\prime}b^{\prime}}$ is nothing but the
separable form. Conversely, if $\mathcal{E}_{aba^{\prime}b^{\prime}}$ has the
following separable form

$\mathcal{E}_{aba^{\prime}b^{\prime}}=\sum_{k}\lambda_{k}\,(\mathcal{A}_{k})_{aa^{\prime}}\,(\mathcal{B}_{k})_{bb^{\prime}},$
(36)

where $\lambda_{k}\geq 0$, and the matrices $\mathcal{A}_{k}$ and
$\mathcal{B}_{k}$ with the components $(\mathcal{A}_{k})_{aa^{\prime}}$ and
$(\mathcal{B}_{k})_{bb^{\prime}}$ have only non-negative eigenvalues, then the
matrix $\mathcal{E}_{\Phi}$ given in (33) is a separable density matrix with
respect to $\text{AA}^{\prime}$ and $\text{BB}^{\prime}$ (up to normalization
constant). By Theorem 2, the population-preserving CPTP map $\Phi$ is
separable. ∎

In particular, if the dimension of the Hilbert space in Theorem 3 is six or
less (a two-qubit or qubit-qutrit system), the following corollary is
obtained.

######  Corollary 1.

In Theorem 3, we further assume that the dimension of the total Hilbert space
satisfies $d_{\text{A}}d_{\text{B}}\leq 6$. Then, a population-preserving CPTP
map $\Phi$ is separable if and only if the partial transposition
$\mathcal{E}^{\text{T}_{\text{A}}}$ with the elements
$(\mathcal{E}^{\text{T}_{\text{A}}})_{aba^{\prime}b^{\prime}}=\mathcal{E}_{a^{\prime}bab^{\prime}}$
has only non-negative eigenvalues.

This corollary follows by the fact that, when $d_{\text{A}}d_{\text{B}}\leq
6$, the PPT criterion for the matrix $\mathcal{E}_{\Phi}(=\mathcal{E})$ is the
necessary and sufficient condition on the separability of
$\mathcal{E}_{\Phi}$. Corollary 1 gives us the following criterion on
inseparable operations. We have a population-preserving CPTP map $\Phi$ on two
particles. We assume that one of the particles is superposed in two paths and
the other is in two or three paths. The partial transposition
$\mathcal{E}^{\text{T}_{\text{A}}}$ of the square matrix $\mathcal{E}$ has a
negative eigenvalue if and only if the map $\Phi$ is inseparable.

To demonstrate how to use the criterion, let us focus on the dynamics of
superposed massive particles considered in Sec.II. The evolved state given by
Eq.(3) is rewritten as

$|\Psi_{\text{out}}\rangle=\sum_{a,b=\text{L},\text{R}}e^{i\Phi_{ab}}\,\psi_{a}|a\,;\text{out}\rangle_{\text{A}}\,\phi_{b}|b\,;\text{out}\rangle_{\text{B}}=\sum_{a,b=\text{L},\text{R}}e^{i\Phi_{ab}}\,\hat{M}_{a}\otimes\hat{N}_{b}|\Psi_{\text{in}}\rangle,$
(37)

or equivalently,

$|\Psi_{\text{out}}\rangle\langle\Psi_{\text{out}}|=\sum_{a,a^{\prime}=\text{L},\text{R}}\sum_{b,b^{\prime}=\text{L},\text{R}}e^{i(\Phi_{ab}-\Phi_{a^{\prime}b^{\prime}})}\,\hat{M}_{a}\otimes\hat{N}_{b}|\Psi_{\text{in}}\rangle\langle\Psi_{\text{in}}|\hat{M}^{\dagger}_{a^{\prime}}\otimes\hat{N}^{\dagger}_{b^{\prime}},$
(38)

where $|\Psi_{\text{in}}\rangle$ is the initial state (2),
$\hat{M}_{a}=|a\,;\text{out}\rangle_{\text{A}}\langle a\,;\text{in}|$ and
$\hat{N}_{b}=|b\,;\text{out}\rangle_{\text{B}}\langle b\,;\text{in}|$. Then,
we find the matrix $\mathcal{E}$ with the elements
$\mathcal{E}_{aba^{\prime}b^{\prime}}=e^{i(\Phi_{ab}-\Phi_{a^{\prime}b^{\prime}})}$
as

$\mathcal{E}=\begin{bmatrix}1&e^{i(\Phi_{\text{LL}}-\Phi_{\text{LR}})}&e^{i(\Phi_{\text{LL}}-\Phi_{\text{RL}})}&e^{i(\Phi_{\text{LL}}-\Phi_{\text{RR}})}\\\
e^{i(\Phi_{\text{LR}}-\Phi_{\text{LL}})}&1&e^{i(\Phi_{\text{LR}}-\Phi_{\text{RL}})}&e^{i(\Phi_{\text{LR}}-\Phi_{\text{RR}})}\\\
e^{i(\Phi_{\text{RL}}-\Phi_{\text{LL}})}&e^{i(\Phi_{\text{RL}}-\Phi_{\text{LR}})}&1&e^{i(\Phi_{\text{RL}}-\Phi_{\text{RR}})}\\\
e^{i(\Phi_{\text{RR}}-\Phi_{\text{LL}})}&e^{i(\Phi_{\text{RR}}-\Phi_{\text{LR}})}&e^{i(\Phi_{\text{RR}}-\Phi_{\text{RL}})}&1\\\
\end{bmatrix},$ (39)

where the 4 rows and 4 columns labeled by
$ab=\text{LL},\text{LR},\text{RL},\text{RR}$ and
$a^{\prime}b^{\prime}=\text{LL},\text{LR},\text{RL},\text{RR}$, respectively.
The partial transposition $\mathcal{E}^{\text{T}_{\text{A}}}$ with the
components
$(\mathcal{E}^{\text{T}_{\text{A}}})_{aba^{\prime}b^{\prime}}=\mathcal{E}_{a^{\prime}bab^{\prime}}$
is

$\mathcal{E}^{\text{T}_{\text{A}}}=\begin{bmatrix}1&e^{i(\Phi_{\text{LL}}-\Phi_{\text{LR}})}&e^{i(\Phi_{\text{RL}}-\Phi_{\text{LL}})}&e^{i(\Phi_{\text{RL}}-\Phi_{\text{LR}})}\\\
e^{i(\Phi_{\text{LR}}-\Phi_{\text{LL}})}&1&e^{i(\Phi_{\text{RR}}-\Phi_{\text{LL}})}&e^{i(\Phi_{\text{RR}}-\Phi_{\text{LR}})}\\\
e^{i(\Phi_{\text{LL}}-\Phi_{\text{RL}})}&e^{i(\Phi_{\text{LL}}-\Phi_{\text{RR}})}&1&e^{i(\Phi_{\text{RL}}-\Phi_{\text{RR}})}\\\
e^{i(\Phi_{\text{LR}}-\Phi_{\text{RL}})}&e^{i(\Phi_{\text{LR}}-\Phi_{\text{RR}})}&e^{i(\Phi_{\text{RR}}-\Phi_{\text{RL}})}&1\\\
\end{bmatrix}.$ (40)

In the same manner as the definition of the negativity, the measure to test
the inseparable operation is introduced as

$\mathcal{V}=\sum_{\nu_{i}<0}|\nu_{i}|,$ (41)

where $\nu_{i}$ are the eigenvalues of $\mathcal{E}^{\text{T}_{\text{A}}}$. By
a straightforward computation, the measure $\mathcal{V}$ is obtained as

$\mathcal{V}=2\Big{|}\sin\Big(\frac{\Phi_{\text{LR}}+\Phi_{\text{RL}}-\Phi_{\text{LL}}-\Phi_{\text{RR}}}{2}\Big{missing})\Big{|}=2\Big{|}\sin[\frac{Gm_{\text{A}}m_{\text{B}}T}{2}\Big{(}\frac{1}{D+L}+\frac{1}{D-L}-\frac{2}{D}\Big{)}\Big{]}\Big{|},$
(42)

where the formula (14) of $\Phi_{ab}$ was substituted. Fig.4 presents the
measure as a function of $Gm_{\text{A}}m_{\text{B}}T/\pi D$ for a fixed $D$.
This measure $\mathcal{V}$ is proportional to the negativity $\mathcal{N}$.
The inseparability of the operation due to the gravitational interaction is
consistent with the entanglement generation due to it.

Figure 4: The behavior of the measure $\mathcal{V}$ to test the inseparable
operation due to the gravitational interaction between two massive particles.
The vertical axis represents the dimensionless quantity $\mathcal{V}$. The
horizontal axis describes the time $T$ in the unit $\pi
D/Gm_{\text{A}}m_{\text{B}}$.

###  IV.3 Equivalence between inseparable and entangling operations

Let us introduce a larger class of evolutions called non-entangling operations
Harrow2003 . A CPTP map $\Phi_{\text{NE}}$ is called a non-entangling
operation if $\Phi_{\text{NE}}[\rho]$ is a separable state for every separable
state $\rho$. It is obvious that separable operations are also non-entangling.
The converse statement is not true because of counterexamples. In general,
separable operations are not equivalent to non-entangling operations.

For example, the SWAP operation
$\hat{U}\,\rho\otimes\sigma\hat{U}^{\dagger}=\sigma\otimes\rho$ is not
separable but non-entangling. Indeed, we have

$\Phi_{U}\Big{[}\sum_{k}p_{k}\rho_{k}\otimes\sigma_{k}\Big{]}=\hat{U}\sum_{k}p_{k}\rho_{k}\otimes\sigma_{k}\hat{U}^{\dagger}=\sum_{k}p_{k}\hat{U}\rho_{k}\otimes\sigma_{k}\hat{U}^{\dagger}=\sum_{k}p_{k}\sigma_{k}\otimes\rho_{k},$
(43)

hence the SWAP operation $\Phi_{U}$ is non-entangling. On the other hand,
using (27) for $\Phi_{U}$, we find the matrix

$\displaystyle\mathcal{E}_{U}$
$\displaystyle=\Phi_{U}\otimes\mathcal{I}_{\text{A}^{\prime}\text{B}^{\prime}}\Big{[}|\Psi\rangle_{\text{AA}^{\prime}}\langle\Psi|\otimes|\Psi\rangle_{\text{BB}^{\prime}}\langle\Psi|\Big{]}$
$\displaystyle=\sum^{d}_{i,i^{\prime},j,j^{\prime}=1}\Phi_{U}\otimes\mathcal{I}_{\text{A}^{\prime}\text{B}^{\prime}}\Big{[}|i\rangle_{\text{A}}\langle
i^{\prime}|\otimes|i\rangle_{\text{A}^{\prime}}\langle
i^{\prime}|\otimes|j\rangle_{\text{B}}\langle
j^{\prime}|\otimes|j\rangle_{\text{B}^{\prime}}\langle j^{\prime}|\Big{]}$
$\displaystyle=\sum^{d}_{i,i^{\prime},j,j^{\prime}=1}|j\rangle_{\text{A}}\langle
j^{\prime}|\otimes|i\rangle_{\text{A}^{\prime}}\langle
i^{\prime}|\otimes|i\rangle_{\text{B}}\langle
i^{\prime}|\otimes|j\rangle_{\text{B}^{\prime}}\langle j^{\prime}|$
$\displaystyle=|\Psi\rangle_{\text{AB}^{\prime}}\langle\Psi|\otimes|\Psi\rangle_{\text{A}^{\prime}\text{B}}\langle\Psi|$
(44)

where $|\Psi\rangle=\sum^{d}_{\text{i}=1}|i\rangle|i\rangle$. The matrix
$\mathcal{E}_{U}$ is entangled with respect to $\text{AA}^{\prime}$ and
$\text{BB}^{\prime}$. By Theorem 2, the SWAP operation $\Phi_{U}$ is not
separable .

For realizing inseparable or non-LOCC operations on two quantum systems, a
quantum channel (a channel to transfer quantum information) between the
systems may be required. The SWAP operation is an example of a quantum channel
without entanglement generation. On the other hand, we find the following
statement on inseparable and entangling operations.

######  Theorem 4.

Let $\Phi$ be the population-preserving CPTP map in Theorem 1. The map $\Phi$
is inseparable if and only if the map $\Phi$ is entangling.

###### Proof.

Since every entangling operation is inseparable, we consider the converse
statement. Let a population-preserving CPTP map $\Phi$ be an inseparable
operation. By Theorem 3, the matrix $\mathcal{E}$ with the elements
$\mathcal{E}_{aba^{\prime}b^{\prime}}$ has no separable form. Assuming
$\rho_{\text{in}}=|\psi_{\text{in}}\rangle_{\text{A}}\langle\psi_{\text{in}}|\otimes|\phi_{\text{in}}\rangle_{\text{B}}\langle\phi_{\text{in}}|$
and using Theorem 1, we find

$\displaystyle\Phi[\rho_{\text{in}}]$
$\displaystyle=\Phi[|\psi_{\text{in}}\rangle_{\text{A}}\langle\psi_{\text{in}}|\otimes|\phi_{\text{in}}\rangle_{\text{B}}\langle\phi_{\text{in}}|]$
$\displaystyle=\sum^{d_{\text{A}}}_{a,a^{\prime}=1}\sum^{d_{\text{B}}}_{b,b^{\prime}=1}\mathcal{E}_{aba^{\prime}b^{\prime}}\hat{M}_{a}\otimes\hat{N}_{b}\,|\psi_{\text{in}}\rangle_{\text{A}}\langle\psi_{\text{in}}|\otimes|\phi_{\text{in}}\rangle_{\text{B}}\langle\phi_{\text{in}}|\,\hat{M}^{\dagger}_{a^{\prime}}\otimes\hat{N}^{\dagger}_{b^{\prime}}$
$\displaystyle=\sum^{d_{\text{A}}}_{a,a^{\prime}=1}\sum^{d_{\text{B}}}_{b,b^{\prime}=1}\mathcal{E}_{aba^{\prime}b^{\prime}}\psi^{*}_{a}\psi_{a^{\prime}}\phi^{*}_{b}\phi_{b^{\prime}}|a\,;\text{out}\rangle_{\text{A}}\langle
a^{\prime};\text{out}|\otimes|b\,;\text{out}\rangle_{\text{B}}\langle
b^{\prime};\text{out}|,$ (45)

where $\psi_{a}={}_{\text{A}}\langle
a\,;\text{in}|\psi_{\text{in}}\rangle_{\text{A}}$ and
$\phi_{b}={}_{\text{B}}\langle
b\,;\text{in}|\phi_{\text{in}}\rangle_{\text{B}}$. We can always choose
$|\psi_{\text{in}}\rangle_{\text{A}}$ and
$|\phi_{\text{in}}\rangle_{\text{B}}$ so that $\psi_{a}=1/\sqrt{d_{\text{A}}}$
and $\phi_{b}=1/\sqrt{d_{\text{B}}}$, and then the elements of the density
matrix $\Phi[\rho_{\text{in}}]$ are given by
$\mathcal{E}_{aba^{\prime}b^{\prime}}/d_{\text{A}}d_{\text{B}}$. Since
$\mathcal{E}_{aba^{\prime}b^{\prime}}$ is not separable, the density matrix
$\Phi[\rho_{\text{in}}]$ describes an entangled state, and hence $\Phi$ is an
entangling operation. ∎

Theorem 4 implies that inseparable operations on two particles in a
superposition of paths are entangling operations and that the associated
quantum channel can create entanglement.

The theorems obtained so far are independent of the details of the interaction
between two particles. Applying the theorems, we find the following
implication for quantum interaction induced by gravity. Whenever a
gravitational interaction is described by an inseparable operation on path-
superposed masses, the gravitational field mediating the interaction creates
entanglement. The quantum channel model of gravity described by an inseparable
operation without entanglement generation is incompatible with the possible
evolution of the masses.

##  V Conclusion

To understand the interaction via the gravitational field induced by
superposed particles, we investigated when entanglement occurs between two
particles in a path superposition. The present approach based on quantum
information theory allowed us to analyze the evolution of the particles
without specifying a model of the interaction between them. The representation
theorem on the evolution given by a population-preserving CPTP map identifies
separable operations on the superposed particles. As a byproduct, the simple
procedure to test the separability criterion on the evolution was obtained. We
further found that inseparable operations on the particles are nothing but
non-entangling operations. This means that the entanglement between the
particles occurs only when the quantum interaction between them induces an
inseparable operation. If the interaction is mediated by a gravitational
field, we have the robust statement: there are no quantum models of
gravitational interaction given by an inseparable operation which does not
generate entanglement between two masses in a path superposition.

In an experimental viewpoint, the present setting of quantum systems in this
paper have respected matter-wave interferometers. The studies on other
proposed systems, for example, levitated nanoparticles and macroscopic
mechanical oscillators will be left as the future works. It is interesting to
see what kind of quantum channels the gravitational field forms in such
physical systems.

###### Acknowledgements.

We thank S. Kanno, S. Kukita, Y. Kuramochi and K. Yamamoto for useful
discussions and comments related to this paper.

## References

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