# Quantum Bell Nonlocality is Entanglement

Kuntal Sengupta<EMAIL_ADDRESS>Department of Mathematics and
Statistics, Institute for Quantum Science and Technology, University of
Calgary, AB, Canada T2N 1N4  Rana Zibakhsh<EMAIL_ADDRESS>Department of Physics and Astronomy, University of Calgary, AB, Canada T2N 1N4
Eric Chitambar<EMAIL_ADDRESS>Department of Electrical and Computer
Engineering, Coordinated Science Laboratory, University of Illinois at Urbana-
Champaign, Urbana, IL 61801  Gilad Gour<EMAIL_ADDRESS>Department of
Mathematics and Statistics, Institute for Quantum Science and Technology,
University of Calgary, AB, Canada T2N 1N4  Department of Physics and
Astronomy, University of Calgary, AB, Canada T2N 1N4

###### Abstract

Bell nonlocality describes a manifestation of quantum mechanics that cannot be
explained by any local hidden variable model. Its origin lies in the nature of
quantum entanglement, although understanding the precise relationship between
nonlocality and entanglement has been a notorious open problem. In this paper,
we resolve this problem by developing a dynamical framework in which quantum
Bell nonlocality emerges as special form of entanglement, and both are unified
as resources under local operations and classical communication (LOCC). Our
framework is built on the notion of quantum processes, which are abstract
quantum channels mapping elements between fixed intervals in space and time.
Entanglement is then identified as a quantum process that cannot be generated
by LOCC while Bell nonlocality is the subset of these processes that have an
instantaneous input-output delay time. LOCC pre-processing is a natural set of
free operations in this theory, thereby enabling all entangled states to
activate some form of Bell nonlocality. In addition, we generalize the CHSH
witnesses from the state domain to the domain of entangled quantum
measurements, and provide a systematic method to quantify the Bell nonlocality
of a bipartite quantum channel.

##  I Introduction

Entanglement and Bell nonlocality represent two of the most stunning non-
classical features of quantum mechanics. Entanglement is traditionally
understood as the property of non-separability Werner (1989); Horodecki  _et
al._ (2009). More precisely, a bipartite quantum state $\rho^{AB}$ is
entangled if it cannot be separated into a statistical mixture of product
states,

$\rho^{AB}\not=\sum_{\lambda}p(\lambda)\rho_{\lambda}^{A}\otimes\rho_{\lambda}^{B}.$
(1)

In contrast, quantum Bell nonlocality is the property that enables certain
quantum states to generate nonlocal classical channels. The latter refers to
classical channels that cannot be described by a local hidden variable (LHV)
model; i.e the transition probabilities $W(a,b|x,y)$ cannot be separated into
a statistical mixture of product channels,

$W(a,b|x,y)\not=\sum_{\lambda}p(\lambda)W_{\lambda}(a|x)W_{\lambda}(b|y).$ (2)

Building on the pioneering insight of J.S. Bell Bell (1964), one can test
whether a given channel satisfies a LHV model by checking whether the
probabilities $W(a,b|x,y)$ satisfy a finite family of inequalities, known as
Bell inequalities Brunner  _et al._ (2014). The well-known Clauser-Horne-
Shimony-Holt (CHSH) Inequality is one such inequality Clauser  _et al._
(1969), and it completely characterizes the set of nonlocal channels for
binary inputs and outputs Froissart (1981); Fine (1982).

The most direct way to generate a nonlocal channel from a quantum state
$\rho^{AB}$ is by performing local measurements. The channel generated by such
a process has transition probabilities determined by Born’s rule,

$W(a,b|x,y)={\rm Tr}[\rho^{AB}(M^{x}_{a}\otimes N^{y}_{b})],$ (3)

where $\\{M^{x}_{a}\\}_{a}$ represent measurement operators on Alice’s system
($A$) for measurement choice $x$ and outcome $a$, and similarly for the
operators $\\{N^{y}_{b}\\}_{b}$ on Bob’s system ($B$). For example, by
choosing suitable pairs of local measurements, a bipartite maximally entangled
state $|\phi^{+}\rangle=\sqrt{1/2}(|00\rangle+|11\rangle)$ can be used to
generate a classical channel that violates the CHSH Inequality.

The similarity between Eqns. (1) and (2) suggests that quantum entanglement
and Bell nonlocality can perhaps be unified in some fundamental way. By
inspection, we can conclude that entangled states and nonlocal classical
channels are both objects that cannot be built using local operations and
shared randomness (LOSR). This has led some to conclude that LOSR should
provide the underlying fabric to weave together entanglement and Bell
nonlocality Schmid  _et al._ (2020a). However, as we argue below and
throughout this paper, a restriction to LOSR misses key features of both
entanglement and nonlocality. Another approach is needed.

While bipartite entanglement is well understood Horodecki  _et al._ (2009),
its relationship to Bell nonlocality is far more elusive Werner (1989); Hardy
(1993); Eberhard (1993); Popescu (1995); Barrett (2002); Wiseman  _et al._
(2007); Methot and Scarani (2007); Masanes  _et al._ (2008); Hirsch  _et al._
(2013); Quintino  _et al._ (2015); Augusiak  _et al._ (2015); Hirsch  _et al._
(2016). Most notable is the existence of entangled states that cannot violate
any Bell inequality when generating a classical channel according to Eq. (3)
Werner (1989); Barrett (2002). The discovery of such states seems to cast some
doubt on whether it is possible to formulate any unified theory of both
entanglement and Bell nonlocality. At the same time, however, quantum Bell
nonlocality is known to demonstrate the remarkable effect of super-activation.
There exists two states $\rho^{A_{0}B_{0}}$ and
$\sigma^{A_{0}^{\prime}B_{0}^{\prime}}$, each of which is unable to violate a
Bell inequality. Yet when combined and measured jointly, the state
$\rho^{A_{0}B_{0}}\otimes\sigma^{A_{0}^{\prime}B_{0}^{\prime}}$ can violate
some Bell inequality Palazuelos (2012). An even more important collection of
results for the purposes of this work shows that certain states have hidden
nonlocality that becomes revealed through the use of local filters Popescu
(1995); Hirsch  _et al._ (2013). In fact, every bipartite entangled state
$\rho^{A_{0}B_{0}}$ can violate the CHSH Inequality when combined with another
state $\sigma^{A_{0}^{\prime}B_{0}^{\prime}}$ that itself cannot violate the
CHSH Inequality, after the two are allowed to undergo processing by local
operations and classical communication (LOCC) prior to receiving any classical
inputs for the resultant channel (see Fig. 1) Masanes  _et al._ (2008). That
is, a CHSH-violating classical channel can be generated by
$\rho^{A_{0}B_{0}}\otimes\sigma^{A_{0}^{\prime}B_{0}^{\prime}}$ when Eq. (3)
is modified to have the form

$\displaystyle W(x_{1},y_{1}|x_{0},y_{0})$ $\displaystyle={\rm
Tr}\left[\mathcal{L}_{\text{pre}}(\rho^{A_{0}B_{0}}\otimes\sigma^{A_{0}^{\prime}B_{0}^{\prime}})(M_{x_{1}}^{x_{0}}\otimes
N^{y_{0}}_{y_{1}})\right],$ (4)

where $\mathcal{L}_{\text{pre}}$ is a so-called pre-processing LOCC map that
is applied to $\rho^{A_{0}B_{0}}\otimes\sigma^{A_{0}^{\prime}B_{0}^{\prime}}$
before the choice of measurements $(x_{0},y_{0})$. A similar result also holds
for multipartite entangled states Liang  _et al._ (2012).

LOCC is the natural class of operations to consider when studying quantum
entanglement since it models quantum information processing in the “distant
lab” setting Plenio and Virmani (2007). The fact that LOCC is needed to fully
reveal the nonlocal features of quantum states, as in Eq. (I), indicates that
LOCC is somehow also fundamental to quantum Bell nonlocality. However, a
precise and physically-motivated connection between the theories of quantum
entanglement and Bell nonlocality has thus far been lacking. Here we address
this void by describing a framework in which Bell nonlocality emerges as a
subtheory of entanglement and both ultimately belong to the same species,
resources under LOCC.

Figure 1: A state $\rho$ that cannot violate a Bell inequality can be
transformed into a state that violates one by performing a pre-LOCC map. The
key property we identify in this work is that after the LOCC map, what remains
is a bipartite classical channel with essentially zero delay time between the
inputs $(x_{0},y_{0})$ and outputs $(x_{1},y_{1})$. Dots with the same colour
correspond to the same time.

We establish our claim within the domain of dynamical resource theories
(DRTs), which have been extensively studied in recent years Gour and Winter
(2019); Wang and Wilde (2019); Wang  _et al._ (2019); Gour and Scandolo
(2020); Bäuml  _et al._ (2019); Fang  _et al._ (2020); Liu and Yuan (2020);
Liu and Winter (2019); Gour and Wilde (2018); Kaur and Wilde (2017); Katariya
and Wilde (2020a); Fang and Fawzi (2019a). In fact, the whole field of quantum
Shannon theory Wilde (2013) can be seen as a DRT Devetak  _et al._ (2008). In
a general quantum resource theory, one identifies a restricted subset of
quantum operations as being “free”, and objects that cannot be generated by
these free operations are deemed to possess a resource Coecke  _et al._
(2016); Chitambar and Gour (2019). The essential difference between a static
and a dynamic resource theory is the element of time. While a quantum state
represents a quantum system at a given snapshot in time, a quantum channel
typically requires a non-zero time for any signal to propagate from the inputs
to the outputs. This delay time is particularly relevant for bipartite
channels shared between two spatially-separated parties since the finite speed
of information propagation limits how fast the input information of one party
can influence the output information of the other.

The input-output delay time of bipartite channels has been a notorious
consideration in almost all Bell experiments. The so-called locality loophole
describes how the CHSH Inequality can be violated under a local hidden
variable model when the measurement settings on Alice’s side can propagate to
Bob’s laboratory in a time shorter than the input-output delay time of Bob’s
measurement device Aspect  _et al._ (1982); Bell (2004). For this reason, when
modelling an ideal Bell experiment it is usually assumed that Alice’s
measurement choice (i.e. her channel input $x_{0}$) is spacelike separated
from Bob’s measurement outcome (i.e. his channel output $y_{1}$), and vice-
versa. Hence, what is crucial here is not that Alice and Bob are able to
generate the particular bipartite classical channel capable of violating the
CHSH Inequality, but rather that they are able to generate this channel using
local physical processes having input-output delay times shorter than the time
it takes light to travel between the two laboratories. While previous efforts
to construct a resource theory of Bell nonlocality Gallego  _et al._ (2012);
de Vicente (2014); Horodecki  _et al._ (2015); Gallego and Aolita (2017);
Wolfe  _et al._ (2020); Schmid  _et al._ (2020b) may have implicitly
acknowledged this crucial fact or used it to motivate the set of free
operations, we take the approach of building the input-output delay time
directly into the resource theory.

The framework introduced here considers every quantum channel on the abstract
level as a completely-positive trace-preserving (CPTP) map between an input
and output system separated by a specific fixed distance and time interval.
Hence every quantum channel is associated with a family of dynamical objects,
which we refer to as quantum processes and which differ in their input-output
spacetime separations. Quantum processes constitute the resource objects in
this theory, and the free operations are LOCC superprocesses, which are LOCC-
implementable protocols that map one process to another. Accounting for the
input-output delay time of a channel allows us to draw a distinction between
two dynamical resources under LOCC:  instantaneous and non-instantaneous.
Using this distinction, one can identify quantum Bell nonlocality as an
instantaneous classical process that can be obtained by LOCC only when Alice
and Bob are initially supplied with an entangled state, as in Fig. 1. One of
the key features of our model is that pre-LOCC maps are naturally free
operations; i.e. static-to-dynamic conversions having the form of Eq. (I) are
allowed in this framework. At the same time, the use of LOCC maps after the
classical inputs $(x_{0},y_{0})$ in Eq. (I) is automatically prohibited as it
leads to a non-instantaneous resource.

In this paper we develop the theory of quantum processes and study
interconversions among them. When extended to bipartite processes and LOCC
interconversions, what emerges is a resource theory that encompasses both
entanglement and Bell nonlocality. Within this resource theory, we show that
all entangled states and all entangled bipartite channels can exhibit some
features of Bell nonlocality using LOCC. We also introduce a systematic method
to quantify the Bell nonlocality of bipartite quantum states and channels
within the LOCC paradigm. In addition, in the appendix we generalize the CHSH
witnesses from the state domain to the domain of entangled quantum
measurements.

###  I.1 The Shortcomings of an LOSR Theory

Before we unpack our framework in more detail, let us comment on the role that
LOSR plays in our theory and explain why LOSR itself is insufficient to
formulate a physically motivated resource theory of Bell nonlocality, despite
claims to the contrary Schmid  _et al._ (2020a). Unlike LOCC, LOSR lacks a
clear operational interpretation when understood in the context of quantum
information protocols. LOSR is typically justified using a common-cause model
Wolfe  _et al._ (2020) in which a helper distributes shared randomness to
spatially separated parties. This common randomness, however, is useless for
carrying out some protocol without the parties first agreeing on some pre-
established strategy, an agreement which inevitably will require some
communication. For example, the only way that Alice, Bob, and perhaps some
other party, can recognize which physical system encodes the shared randomness
is by prior interactive communication. A second issue is that LOSR misses
certain important aspects of Bell nonlocality such as it being “hidden” in
some states; i.e. certain channels generated in Eq. (I) cannot also be
realized when the pre-LOCC map $\mathcal{L}_{\text{pre}}$ is restricted to
LOSR. The framework presented here overcomes these problems since it starts
with the physically-motivated class of LOCC and then arrives at a restricted
subset of operations, which includes pre-LOCC, by incorporating the
practically-relevant property of input-output delay time. LOSR still plays an
important role in this theory; however it emerges not by appealing to some
common cause, but rather by considering bipartite LOCC channels that are
implemented with zero input-output delay time.

###  I.2 Notations

In this paper, quantum physical systems and their corresponding Hilbert spaces
will be denoted by $A$ and $B$, while classical systems will be denoted by $X$
and $Y$. We will make a distinction between _static_ vs _dynamic_ systems.
Static systems will be denoted with subscripts such as
$A_{0},\;A_{1},\;B_{0},\;B_{1}$. Dynamic systems will be denoted without
subscripts; for example, $A$ refers to an input-output system $(A_{0},A_{1})$,
where the zero subscript will always refer to the input subsystem and the
subscript one to the output subsystem. Similarly, $B=(B_{0},B_{1})$,
$X=(X_{0},X_{1})$, etc. One exception of this notation is the auxiliary
systems (e.g. environment system, reference systems) $E$ and $R$ which will
always correspond to static systems.

We will also use the notation $A_{1}B_{1}\equiv A_{1}\otimes B_{1}$ to
represent a bipartite (static) system. We use the notation
$|A|=|A_{0}||A_{1}|$ to denote the dimension of a dynamic system $A$, where
$|A_{0}|$ and $|A_{1}|$ are the dimensions of its corresponding input and
output subsystems. The space of bounded linear operators describing
endomorphisms on $A_{1}$ is denoted by $\mathfrak{B}(A_{1})$. We denote by
$\mathfrak{D}(A_{1})\subset\mathfrak{B}(A_{1})$ the set of all density
matrices (i.e. positive semidefinite matrices with trace one) acting on
$A_{1}$. As customary, we will use $\rho$ and $\sigma$ to represent mixed
quantum states and $\psi$ and $\phi$ to represent pure states. A maximally
entangled state in $\mathfrak{D}(A_{1}B_{1})$ is denoted by
$\phi_{+}^{A_{1}B_{1}}$.

We denote by $\mathfrak{L}(A)\coloneqq\mathfrak{L}(A_{0}\to A_{1})$ the set of
all linear maps from $\mathfrak{B}(A_{0})$ to $\mathfrak{B}(A_{1})$. The set
of all completely positive and trace preserving (CPTP) maps in
$\mathfrak{L}(A)$ is denoted by ${\rm CPTP}(A)\coloneqq{\rm CPTP}(A_{0}\to
A_{1})$. Similarly, we will use the notation ${\rm CPTP}(AB)$ in short for
${\rm CPTP}(A_{0}B_{0}\to A_{1}B_{1})$. Quantum channels will be denoted with
calligraphic letters $\mathcal{M},\mathcal{N},\mathcal{E},\mathcal{F}$. We
will use the superscripts $\mathcal{N}^{A}$ and $\mathcal{N}^{AB}$ to indicate
$\mathcal{N}\in{\rm CPTP}(A)$ and $\mathcal{N}\in{\rm CPTP}(AB)$,
respectively. The identity channel in ${\rm CPTP}(A_{0}\to A_{0})$ is denoted
by $\mathsf{id}^{A_{0}}$.

Linear maps from $\mathfrak{L}(A)$ to $\mathfrak{L}(B)$ are called supermaps
Gour (2019). Supermaps that takes quantum channels to quantum channels in a
complete sense (i.e. even when tensored with the identity supermap) are called
superchannels Chiribella  _et al._ (2008); Gour (2019). The collection of all
superchannels from $\mathfrak{L}(A)$ to $\mathfrak{L}(B)$ will be denoted by
${\rm SC}(A\to B)$. We use the letter $\Theta,\Upsilon$ to denote
superchannels. In Chiribella  _et al._ (2008); Gour (2019) it was shown that
the action of every superchannel $\Theta\in{\rm SC}(A\to A^{\prime})$ on a
channel $\mathcal{N}\in{\rm CPTP}(A)$ can be represented as

$\Theta^{A\to
A^{\prime}}\left[\mathcal{N}^{A}\right]=\mathcal{E}_{\text{post}}^{EA_{1}\to
A_{1}^{\prime}}\circ\mathcal{N}^{A_{0}\to
A_{1}}\circ\mathcal{E}_{\text{pre}}^{A_{0}^{\prime}\to EA_{0}},$ (5)

where $\mathcal{E}_{\text{pre}}\in{\rm CPTP}(A_{0}^{\prime}\to EA_{0})$ and
$\mathcal{E}_{\text{post}}\in{\rm CPTP}(EA_{1}\to A_{1}^{\prime})$ are fixed
quantum channels corresponding to pre-processing and post-processing of the
channel $\mathcal{N}^{A}$. This result indicates that superchannels are not
just mathematical abstractions, but in fact have a physical realization.

Crucial to this work is the set of superchannels that are built using local
quantum operations and classical communication (LOCC) Chitambar  _et al._
(2014). A general LOCC superchannel transforms one quantum channel
$\mathcal{N}^{AB}$ into another by an application of pre- and post- LOCC
channels, as depicted in Fig. 2. For given input-output systems, we denote the
set of LOCC superchannels as ${\rm LOCC}(AB\to A^{\prime}B^{\prime})$.
Throughout this paper, double lines will represent classical systems, whereas
single lines are reserved for quantum systems. Classical channels are denoted
by $\mathcal{N}^{XY}\in{\rm CPTP}(X_{0}Y_{0}\to X_{1}Y_{1})$, and we write
$\mathcal{N}_{\text{CHSH}}^{XY}$ to denote a bipartite classical channel that
gives the maximal violation of the CHSH Inequality. Note that every classical
channel belongs to LOCC.

Figure 2: The action of an LOCC superchannel on a bipartite channel
$\mathcal{N}$.

##  II Quantum Processes

Mathematically, a quantum channel $\mathcal{N}\in{\rm CPTP}(A)$ is a map that
sends positive operators of system $A_{0}$ to positive operators of system
$A_{1}$. Usually this abstraction is sufficient to probe the foundations and
theoretical limitations of quantum information processing. However, to
properly characterize Bell nonlocality in quantum mechanics, we need to
understand quantum channels as objects that generate transformations across
fixed intervals in space and time, something we broadly refer to as quantum
processes. A quantum process, denoted by $(\mathcal{N},\Delta\mathbf{x},\Delta
t)$, is a quantum channel $\mathcal{N}\in{\rm CPTP}(A)$ that transforms any
state $\rho\in\mathfrak{D}(A_{0})$ at arbitrary spacetime point
$(\mathbf{x}_{A_{0}},t_{0})$ into state
$\mathcal{N}(\rho)\in\mathfrak{D}(A_{1})$ at spacetime point
$(\mathbf{x}_{A_{1}},t_{1})$, where
$\mathbf{x}_{A_{1}}=\mathbf{x}_{A_{0}}+\Delta\mathbf{x}$ and
$t_{1}=t_{0}+\Delta t$ (see Fig. 3). Here we assume for simplicity that
systems $A_{0}$ and $A_{1}$ are at rest in the same inertial frame and the
coordinates are measured with respect to this frame. The time interval $\Delta
t\geqslant 0$ in a quantum process $(\mathcal{N},\Delta\mathbf{x},\Delta t)$
is called the input-output delay time of $\mathcal{N}$ (or simply the “delay
time” of $\mathcal{N}$), and it quantifies how quickly the channel input
propagates to the output.

Figure 3: A quantum process $(\mathcal{N},\Delta\mathbf{x},\Delta t)$ takes an
input $\rho$ at arbitrary spacetime point $(\mathbf{x}_{A_{0}},t_{0})$ and
generates an output $\mathcal{N}(\rho)$ at spacetime point
$(\mathbf{x}_{A_{1}}=\mathbf{x}_{A_{0}}+\Delta\mathbf{x},t_{1}=t_{0}+\Delta
t)$. The input-output delay time of $\mathcal{N}$ in this process is $\Delta
t$.

Some quantum processes can be physically implemented and some cannot. For
example, the no-signaling principle of special relativity prohibits
$(\mathcal{N},\Delta\mathbf{x},\Delta t)$ from being physically realizable
whenever $\mathcal{N}$ is able to transmit information and $\Delta
t<\Delta\mathbf{x}/c$, where $c$ is the speed of light. Quantum processes with
zero input-output delay time are known as instantaneous, and they play an
important role in this theory. Whenever $\Delta\mathbf{x}>0$, an instantaneous
process $(\mathcal{N},\Delta\mathbf{x},0)$ is physically realizable if and
only if $\mathcal{N}$ is a replacement channel, i.e. $\mathcal{N}$ has the
form $\mathcal{N}_{\rho}(X):={\rm Tr}[X]\rho$ for
$\rho\in\mathfrak{D}(A_{1})$. An instantaneous implementation of
$\mathcal{N}_{\rho}$ is done as follows: knowing that Alice will receive the
channel input at time $t_{0}$, Bob simply prepares the state $\rho$ at
$t_{0}$.

The abstraction of quantum processes also applies to bipartite channels
$\mathcal{N}\in{\rm CPTP}(AB)$. In this case, however, the input and output of
the channel are each distributed across two points in space. Hence, a
bipartite process
$(\mathcal{N},\Delta\mathbf{x}_{A},\Delta\mathbf{x}_{B},\Delta t)$ is the
channel $\mathcal{N}$ that transforms a bipartite state $\rho^{A_{0}B_{0}}$
held at $(\mathbf{x}_{A_{0}},t_{0})$ and $(\mathbf{x}_{B_{0}},t_{0})$ into the
state $\mathcal{N}(\rho^{A_{0}B_{0}})$, held at $(\mathbf{x}_{A_{1}},t_{1})$
and $(\mathbf{x}_{B_{1}},t_{1})$, where
$\mathbf{x}_{A_{1}}=\mathbf{x}_{A_{0}}+\Delta\mathbf{x}_{A}$,
$\mathbf{x}_{B_{1}}=\mathbf{x}_{B_{0}}+\Delta\mathbf{x}_{B}$, and
$t_{1}=t_{0}+\Delta t$. Note that the number of inputs and outputs can differ
by treating one of the systems as trivial. Multipartite processes are defined
in the same way: all inputs at spatial coordinates
$(\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{n})$ evolve to outputs at
spatial coordinates
$(\mathbf{x}_{1}+\Delta\mathbf{x}_{1},\mathbf{x}_{2}+\Delta\mathbf{x}_{2},\cdots,\mathbf{x}_{n}+\Delta\mathbf{x}_{n})$
in the same time interval $\Delta t$. By composing processes together, we
obtain a multiprocess. The only difference between a process and a
multiprocess is that the latter can have differing time delays for different
subsystems due to the composition, whereas the former cannot. The general idea
is depicted in Fig. 4.

Figure 4: The bipartite channel
$\mathcal{E}_{2}\circ\mathcal{N}\circ\mathcal{E}_{1}$ can have differing delay
times for the different subsystems if it is built by composing multiple
processes (top). The result is a multiprocess (bottom). Dots with the same
colour correspond to the same time.

We next go one step forward and define a quantum superprocess as a
superchannel that transforms one process $(\mathcal{N},\Delta\mathbf{x},\Delta
t)$ into another $(\mathcal{N}^{\prime},\Delta\mathbf{x}^{\prime},\Delta
t^{\prime})$. We denote such objects by
$(\Theta,\Delta\mathbf{x}\to\Delta\mathbf{x}^{\prime},\Delta t\to\Delta
t^{\prime})$. A superprocess is constructed by invoking Eq. (5) and
considering $\mathcal{E}_{\text{pre}}$ and $\mathcal{E}_{\text{post}}$ as
processes, as they would be in any physical implementation of a superchannel.
A general superprocess then has the form of Fig. 5. The wires entering and
leaving each channel might be multiple systems bundled together.

Figure 5: A quantum superprocess
$(\Theta,\Delta\mathbf{x}\to\Delta\mathbf{x}^{\prime},\Delta t\to\Delta
t^{\prime})$ converts process $(\mathcal{N},\Delta\mathbf{x},\Delta t)$ to
process $(\mathcal{N}^{\prime},\Delta\mathbf{x}^{\prime},\Delta t^{\prime})$.
The shaded yellow area represents the action of the superprocess. Since
$\mathcal{E}_{\rm pre}$ and $\mathcal{E}_{\rm post}$ might belong to
multiprocess, the times $s_{0}$ and $s_{1}$ are not necessarily equal to
$t_{0}$ or $t_{1}$. For example, in Fig. 6 we will see an example in which
$t_{0}<t_{0}^{\prime}=s_{0}=s_{1}=t_{1}$.

If the overall process transformation is $(\mathcal{N},\Delta\mathbf{x},\Delta
t)\to(\mathcal{N}^{\prime},\Delta\mathbf{x}^{\prime},\Delta t^{\prime})$, then
there are four possibilities for how
$\mathcal{E}_{\text{pre}}^{A_{0}^{\prime}\to EA_{0}}$ and
$\mathcal{E}_{\text{post}}^{EA_{1}\to A_{1}^{\prime}}$ can facilitate this
transformation.

  1. 1.

The input to $\mathcal{N}$ at the point $(\mathbf{x}_{A},t_{0})$ depends on
the input to $\mathcal{E}_{\text{pre}}$ at the point
$(\mathbf{x}_{A_{0}^{\prime}},t_{0}^{\prime})$ (and hence
$t_{0}^{\prime}\leqslant t_{0}$) and the output of $\mathcal{E}_{\text{post}}$
at $(\mathbf{x}_{A_{1}^{\prime}},t_{1}^{\prime})$ depends on the output of
$\mathcal{N}$ at $(\mathbf{x}_{A_{1}},t_{1})$ (and hence $t_{1}\leqslant
t_{1}^{\prime}$). In this case the input-output delay time is always non-
decreasing, i.e. $\Delta t\leqslant\Delta t^{\prime}$.

  2. 2.

The input to $\mathcal{N}$ at the point $(\mathbf{x}_{A},t_{0})$ does not
depend on the input to $\mathcal{E}_{\text{pre}}$ at the point
$(\mathbf{x}_{A_{0}^{\prime}},t_{0}^{\prime})$ but the output of
$\mathcal{E}_{\text{post}}$ at $(\mathbf{x}_{A_{1}^{\prime}},t_{1}^{\prime})$
depends on the output of $\mathcal{N}$ at $(\mathbf{x}_{A_{1}},t_{1})$ (and
hence $t_{1}\leqslant t_{1}^{\prime}$). This means that $\mathcal{E}_{\rm
pre}^{A_{0}^{\prime}\to A_{0}E}$ as appear in Fig. 5 is a quantum channel
whose output at $A_{0}$ is fixed and independent on the input of the channel.
This is a special case of a semi-causal channel Beckman  _et al._ (2001) and
was proven in Eggeling  _et al._ (2002) to have for all
$\omega\in\mathfrak{D}(A_{0}^{\prime})$ the form

$\mathcal{E}_{\text{pre}}^{A_{0}^{\prime}\to
EA_{0}}\left(\omega^{A_{0}^{\prime}}\right)=\mathcal{L}^{A_{0}^{\prime}A_{2}\to
E}\left(\rho^{A_{0}A_{2}}\otimes\omega^{A_{0}^{\prime}}\right)\;,$ (6)

where $A_{2}$ is some auxiliary system, $\rho\in\mathfrak{D}(A_{0}A_{2})$ is a
fixed quantum state, and $\mathcal{L}\in{\rm CPTP}(A_{0}^{\prime}A_{2}\to E)$
is a quantum channel. However, the form above implies that
$\mathcal{L}^{A_{0}^{\prime}A_{2}\to E}$ can be “absorbed” into
$\mathcal{E}_{{\rm post}}$ (of Fig. 5) so that w.l.o.g. in Fig. 6 we replaced
$\mathcal{E}_{\rm post}^{EA_{1}\to
A_{1}^{\prime}}\circ\mathcal{L}^{A_{0}^{\prime}A_{2}\to E}$ with
$\mathcal{E}_{\rm post}^{A_{0}^{\prime}A_{1}A_{2}\to A_{1}^{\prime}}$. Hence
when $t_{0}$ is sufficiently earlier than $t_{0}^{\prime}=t_{1}$, one can
attain $\Delta t=t_{1}-t_{0}>\Delta t^{\prime}=t_{1}^{\prime}-t_{0}^{\prime}$.
Note that the difference $\Delta t^{\prime}$ is determined entirely by the
delay time of $\mathcal{E}_{\text{post}}$, whereas $\Delta t$ is the delay
time of the input process $(\mathcal{N},\Delta\mathbf{x},\Delta t)$.

Figure 6: When the input to $\mathcal{N}$ does not depend on the input to
$\mathcal{E}_{\text{pre}}$ it is possible to construct a superprocess
$(\Theta,\Delta\mathbf{x}\to\Delta\mathbf{x}^{\prime},\Delta t\to\Delta
t^{\prime})$ that decrease the input-output delay time, $\Delta
t^{\prime}<\Delta t$. Observe in particular that we absorbed
$\mathcal{L}^{A_{0}^{\prime}\to E}$ that appears in the expression (6) of
$\mathcal{E}_{\text{pre}}$ into $\mathcal{E}_{\text{post}}$. In this case the
delay time $\Delta t^{\prime}$ depends only on the implementation of
$\mathcal{E}_{\text{post}}$ and in general can be much smaller than $\Delta
t$. Dots with the same colour correspond to the same time.

  3. 3.

The input to $\mathcal{N}$ at the point $(\mathbf{x}_{A},t_{0})$ depends on
the input to $\mathcal{E}_{\text{pre}}$ at the point
$(\mathbf{x}_{A_{0}^{\prime}},t_{0}^{\prime})$ but the output of
$\mathcal{E}_{\text{post}}$ at $(\mathbf{x}_{A_{1}^{\prime}},t_{1}^{\prime})$
does not depend on the output of $\mathcal{N}$ at
$(\mathbf{x}_{A_{1}},t_{1})$. The superprocess in this case acts as
_replacement_ superchannel, always outputting the same channel

$\Theta[\mathcal{N}]=\mathcal{E}_{\text{post}}^{E\to
A_{1}^{\prime}}\circ\mathcal{E}_{\text{pre}}^{A_{0}^{\prime}\to E}$ (7)

irrespective of the input channel $\mathcal{N}$. Here,
$\mathcal{E}_{\text{pre}}^{A_{0}^{\prime}\to E}\coloneqq{\rm
Tr}_{A_{0}}\circ\mathcal{E}_{\text{pre}}^{A_{0}^{\prime}\to EA_{0}}$ and for
all $\rho\in\mathfrak{D}(E)$ $\mathcal{E}_{\text{post}}^{E\to
A_{1}^{\prime}}(\rho^{E})\coloneqq\mathcal{E}_{\text{post}}^{EA_{1}\to
A_{1}^{\prime}}(\rho^{E}\otimes\omega^{A_{1}})$ for some arbitrary fixed
$\omega\in\mathfrak{D}(A_{1})$ . Therefore, while this case might result with
$\Delta t>\Delta t^{\prime}$ 111e.g. remove $\mathcal{E}_{\rm post}$ from Fig.
5, discard the output to $\mathcal{N}$ at $(\mathbf{x}_{A_{1}},t_{1})$, and
extend $(\mathbf{x}_{E},t_{0})$ with a straight line to the output
$(\mathbf{x}_{A_{1}^{\prime}},t_{1}^{\prime})$ so that $t_{0}=t_{1}^{\prime}$
it is somewhat a trivial case and will not play an important role in our
formalism.

  4. 4.

The input to $\mathcal{N}$ at the point $(\mathbf{x}_{A},t_{0})$ does not
depend on the input to $\mathcal{E}_{\text{pre}}$ at the point
$(\mathbf{x}_{A_{0}^{\prime}},t_{0}^{\prime})$ and the output of
$\mathcal{E}_{\text{post}}$ at $(\mathbf{x}_{A_{1}^{\prime}},t_{1}^{\prime})$
does not depend on the output of $\mathcal{N}$ at
$(\mathbf{x}_{A_{1}},t_{1})$. As in the previous case, also here the
superprocess acts as the _replacement_ superchannel that always output the
channel given in (7).

The superprocess depicted in Fig. 6 can also produce $\Delta t^{\prime}=0$
irrespective of the delay time $\Delta t$ of the quantum process
$(\mathcal{N},\Delta\mathbf{x},\Delta t)$. This happens if $\mathcal{E}_{{\rm
post}}$ corresponds to a quantum process with a zero delay time, or more
generally, if $\mathcal{E}_{{\rm post}}$ has the form (see Fig. 7)

$\mathcal{E}_{{\rm post}}^{A_{0}^{\prime}A_{1}A_{2}\to
A_{1}^{\prime}}=\mathcal{F}_{2}^{A_{0}^{\prime}R\to
A_{1}^{\prime}}\circ\mathcal{F}_{1}^{A_{1}A_{2}\to R}$ (8)

where $\mathcal{F}_{1}^{A_{1}A_{2}\to R}$ corresponds to a quantum process
with arbitrary delay time, whereas $\mathcal{F}_{2}^{A_{0}^{\prime}R\to
A_{1}^{\prime}}$ corresponds to an instantaneous quantum process. Since
$\mathcal{F}_{2}$ is instantaneous the delay time $\Delta t^{\prime}=0$
irrespective of the delay times associated with $\mathcal{N}$ and
$\mathcal{F}_{1}$. In Fig. 7 we depicted the most general superprocess with
the property that it converts a non-instantaneous quantum process into an
instantaneous one.

Figure 7: The most general superprocess that can convert a quantum process
$(\mathcal{N}^{A},\Delta\mathbf{x},\Delta t)$ with $\Delta t>0$ into an
instantaneous quantum process. Irrespective of the time delays of
$\mathcal{N}$ and $\mathcal{F}_{1}$, since $\mathcal{F}_{2}$ is instantaneous
we have $t_{1}^{\prime}=t_{0}^{\prime}\geqslant t_{1}>t_{0}$. In particular,
$\Delta t^{\prime}=t_{1}^{\prime}-t_{0}^{\prime}=0$.

To isolate the essential features of this theory, going forward we will assume
that the spatial intervals $\Delta\mathbf{x}$ and $\Delta\mathbf{x}^{\prime}$
are given and fixed. Then, the only processes we consider are specified by the
channel and its delay time, $(\mathcal{N},\Delta t)$. The relevant
superprocesses in this case are characterized by a superchannel and its
induced change in delay times, $(\Theta,\Delta t\to\Delta t^{\prime})$.

##  III Entanglement Theory with Instantaneous Resources, alias Bell
Nonlocality

We are interested here in quantum superprocesses that can be realized by
performing LOCC. These are superprocesses $(\Theta,\Delta t\to\Delta
t^{\prime})$ with $\Theta\in{\rm LOCC}(AB\to A^{\prime}B^{\prime})$, and they
constitute the free operations in this resource theory. Every LOCC
superchannel has the form given in Fig. 2, and to isolate the essential
features of this approach, we will make the simplifying assumption that all
local operations are instantaneous, a reasonable assumption to make when Alice
and Bob’s laboratories are separated relatively far apart. Consequently, the
delay time $\Delta t$ of an LOCC channel will always be proportional to the
number of communication exchanges conducted in the particular implementation
of the channel.

The fundamental question is whether one quantum process can be converted to
another using an LOCC superprocess. When $(\mathcal{N},\Delta
t)\to(\mathcal{N}^{\prime},\Delta t^{\prime})$ is achievable by LOCC, we write

$\displaystyle(\mathcal{N},\Delta
t)\xrightarrow{\text{LOCC}}(\mathcal{N}^{\prime},\Delta t^{\prime})\;.$ (9)

The free objects in this resource theory are the quantum processes that can be
generated by LOCC “from scratch” (i.e. from the trivial process),

$(\mathsf{id}^{1},0)\xrightarrow{\text{LOCC}}(\mathcal{N},\Delta t)\;,$ (10)

where $\mathsf{id}^{1}$ represents here the trivial state/channel; i.e. the
only element of $\mathfrak{D}(\mathbb{C})\cong{\rm
CPTP}(\mathbb{C}\to\mathbb{C})$. Notice that every LOCC channel $\mathcal{L}$
belongs to some free process $(\mathcal{L},\Delta t)$ with $\Delta
t\in[0,+\infty]$. As a special case, a bipartite LOCC channel $\mathcal{L}$ is
LOSR if it belongs to an instantaneous LOCC process $(\mathcal{L},0)$. In
other words, LOSR corresponds precisely to the family of instantaneous LOCC
processes.

There are non-LOCC bipartite channels $\mathcal{N}$ whose instantaneous
process $(\mathcal{N},0)$ can also be physically implemented. These are
channels that belong to the class of local operations and shared entanglement
(LOSE). We say that $\mathcal{N}\in{\rm LOSE}(AB)$ if there exists some
entangled state $\omega\in\mathfrak{D}(A_{2}B_{2})$ such that

$\mathcal{N}^{AB}=\Upsilon\left[\omega^{A_{2}B_{2}}\right]\;,$ (11)

where $\Upsilon$ is an LOSR superchannel comprising of the two local channels
$\mathcal{E}\in{\rm CPTP}(A_{0}A_{2}\to A_{1})$ and $\mathcal{F}\in{\rm
CPTP}(B_{0}B_{2}\to B_{1})$ such that for any
$\rho\in\mathfrak{D}(A_{0}B_{0})$

$\displaystyle\Upsilon\left[\omega^{A_{2}B_{2}}\right]\left(\rho^{A_{0}B_{0}}\right)$
(12) $\displaystyle\coloneqq\mathcal{E}^{A_{0}A_{2}\to
A_{1}}\otimes\mathcal{F}^{B_{0}B_{2}\to
B_{1}}\left(\rho^{A_{0}B_{0}}\otimes\omega^{A_{2}B_{2}}\right)\;.$

Figure 8: Implementation of an LOSE quantum process with an entangled state
$\omega\in\mathfrak{D}(A_{2}B_{2})$ and local superprocess $\Upsilon$ that
takes the quantum state $\omega^{A_{2}B_{2}}$ as its input and then outputs
the bipartite quantum channel
$\mathcal{N}^{AB}\coloneqq\Upsilon[\omega^{A_{2}B_{2}}]$.

This framework captures all of entanglement theory. For instance, all
bipartite quantum states (or more generally all bipartite replacement
channels) can be implemented with instantaneous processes: the process
$(\rho,0)$ with $\rho\in\mathfrak{D}(A_{1}B_{1})$ can be viewed as a bipartite
quantum process with a one-dimensional input and zero input-output delay time
since if Alice and Bob hold $\rho^{A_{1}B_{1}}$ and they know they will
receive an input at time $t_{0}$, they can immediately output
$\rho^{A_{1}B_{1}}$ at time $t_{0}$. Hence, entangled states can be
implemented with instantaneous processes that are non-LOCC, and separable
states can be implemented with the instantaneous LOCC processes (i.e. LOSR).
Converting one entangled state $\rho^{A_{1}B_{1}}$ to another
$\sigma^{A_{1}B_{1}}$ by LOCC then becomes a special case of converting one
instantaneous process to another by an LOCC superprocess,

$(\rho^{A_{1}B_{1}},0)\xrightarrow{\text{LOCC}}(\sigma^{A_{1}B_{1}},0).$ (13)

The central thesis of this paper is that the framework of LOCC superprocesses
also encompasses the resource theory of quantum Bell nonlocality. Every
experimental test for Bell nonlocality involves the conversion of an entangled
bipartite state $\rho^{A_{1}B_{1}}$ into a classical channel
$\mathcal{N}^{XY}$, like in Eq. (3) or Eq. (I). This is sometimes described as
transforming a static resource ($\rho^{A_{1}B_{1}}$) into a dynamical
classical resource ($\mathcal{N}^{XY}$). However, this description does not
make explicit the resource-theoretic aspect of nonlocality. While every
classical channel can be implemented by LOCC, not every instantaneous
classical process admits such an implementation. Hence, in the resource theory
developed here, $(\mathcal{N}^{XY},\Delta t)$ is a free object for $\Delta t$
sufficiently large (which depends on the distance between Alice and Bob), but
$(\mathcal{N}^{XY},0)$ is a resource if $\mathcal{N}^{XY}$ is non-LOSR, i.e.
not admitting a LHV model. We can now see how Bell nonlocality emerges as just
a special type of entanglement. Entanglement, whether it be static or dynamic,
can be defined as any process $(\mathcal{N}^{AB},\Delta t)$ that cannot be
generated by LOCC,

$\displaystyle\text{Entanglement:}\;\;(\text{id}^{1},0)\;$ $\displaystyle\kern
6.00006pt\not\kern-8.99994pt\xrightarrow{\text{LOCC}}(\mathcal{N}^{AB},\Delta
t),$ (14)

and it is the resource in this theory. Bell nonlocality is just a particular
type of this resource in that it is a classical instantaneous process not
reachable by LOCC,

$\displaystyle\text{Bell nonlocality:}\;\;(\text{id}^{1},0)\;$
$\displaystyle\kern
6.00006pt\not\kern-8.99994pt\xrightarrow{\text{LOCC}}(\mathcal{N}^{XY},0).$
(15)

When referring to quantum Bell nonlocality, we mean some Bell nonlocal process
that can be obtained from a quantum process using LOCC,

$\displaystyle(\mathcal{N}^{AB},\Delta t)$
$\displaystyle\xrightarrow{\text{LOCC}}(\mathcal{N}^{XY},0)$ (16)
$\displaystyle\text{but}\qquad\;\;(\text{id}^{1},0)\;$ $\displaystyle\kern
6.00006pt\not\kern-8.99994pt\xrightarrow{\text{LOCC}}(\mathcal{N}^{XY},0).$
(17)

Eq. (16) describes a resource conversion just like Eq. (13) except that it
considers more general channels and a possible change in input-output delay
time. For example, the quantum Bell nonlocality possessed by the maximally
entangled state $|\phi_{+}^{A_{1}B_{1}}\rangle$ (with $|A_{1}|=|B_{1}|=2$) can
be expressed as the resource conversion

$(\phi_{+}^{A_{1}B_{1}},0)\xrightarrow{\text{LOCC}}(\mathcal{N}^{XY}_{\text{CHSH}},0),$
(18)

where $\phi_{+}^{A_{1}B_{1}}:=|\phi_{+}\rangle\langle\phi_{+}|^{A_{1}B_{1}}$.

Let us now examine Eq. (16) closer and consider which LOCC superprocesses can
convert a quantum process into an instantaneous classical one. Since the delay
time of the target process is zero, there are two possibilities for the delay
time $\Delta t$ in the initial process $(\mathcal{N}^{AB},\Delta t)$. In the
first case, we have $\Delta t=0$ and we need an LOCC superprocess constructed
like either Fig. 5 or Fig. 7 that is instantaneous-preserving. The other
possibility is that $\Delta t>0$ so that the LOCC superprocess decreases the
delay time. As outlined above, every superprocess that decreases delay time
has the form of Fig. 7. The following theorem characterizes the structure of
LOCC superprocesses for each of these cases.

######  Theorem 1.

Let $(\Theta^{AB\to A^{\prime}B^{\prime}},\Delta t\to\Delta t^{\prime})$ be an
LOCC superprocess. 1. If $\Delta t=\Delta t^{\prime}=0$, then the superprocess
has the form of either Fig. 9 or Fig. 10. 2. If $\Delta t>\Delta
t^{\prime}=0$, then the superprocess has the form of Fig. 10.

###### Proof of 1.

Recall that _any_ superprocess must have the form of Fig. 5. Since we consider
here the bipartite case, we have to replace in Fig. 5 the system $A_{0}$ with
$A_{0}B_{0}$ and $A_{1}$ with $A_{1}B_{1}$, since the input channel of the
superchannel is a bipartite channel $\mathcal{N}^{AB}$. Similarly, since the
output of the superchannel $\Theta$ is $A^{\prime}B^{\prime}$ we have to
replace system $A_{0}^{\prime}$ with $A_{0}^{\prime}B_{0}^{\prime}$ and
$A_{1}^{\prime}$ with $A_{1}^{\prime}B_{1}^{\prime}$. Consider now Fig. 5
under these replacements.

Recall the four cases that we considered above on how $\mathcal{E}_{\rm pre}$
and $\mathcal{E}_{\rm post}$ can facilitate a superprocess. In the first of
these cases we got that $t_{0}^{\prime}\leqslant t_{0}$ and $t_{1}\leqslant
t_{1}^{\prime}$. Since we assume first that $\Delta t^{\prime}=\Delta t=0$ we
must have that $t_{0}=t_{1}=t_{0}^{\prime}=t_{1}^{\prime}$ so that
$\mathcal{E}_{\rm pre}$ and $\mathcal{E}_{\rm post}$ belong to instantaneous
quantum processes. Since we also have that $\Theta$ is LOCC, $\mathcal{E}_{\rm
pre}$ and $\mathcal{E}_{\rm post}$ must be both LOCC _and_ correspond to
instantaneous quantum processes; i.e. both $\mathcal{E}_{\rm pre}$ and
$\mathcal{E}_{\rm post}$ are LOSR which is precisely the form given in Fig. 9.

In the second case (out of the four cases) the superprocess has the form of
Fig. 6 adjusted to the bipartite case as discussed above. Now, since $\Delta
t^{\prime}=0$ the superprocess has the form of Fig. 7 (again adjusted to the
bipartite case). Note that since $\Theta$ is LOCC, $\rho$, $\mathcal{F}_{1}$,
and $\mathcal{F}_{2}$ of Fig. 7 all must be LOCC since we want the
superprocess to be both LOCC _and_ instantaneous preserving. Hence, $\rho$ is
is precisely the LOCC process/state that appears on the left side of Fig. 10,
$\mathcal{F}_{1}$ is the LOCC process that appears at the output of
$\mathcal{N}^{AB}$ in Fig. 10, and since also $\mathcal{F}_{2}$ corresponds to
instantaneous quantum process it must be an LOSR channel corresponding to the
two post-LO channels that appear on the right side on Fig. 10. Note that the
shared randomness of $\mathcal{F}_{2}$ can be absorbed into the pre-LOCC
channel preceding it. Since the third and fourth cases do not lead to new
implementations of the superprocess, the proof is concluded for the case
$\Delta t^{\prime}=\Delta t=0$.

Consider now the second part of the theorem in which $\Delta t>\Delta
t^{\prime}=0$. As discussed above, the only implementation (out of the four)
of the superprocess for this case is given in Fig. 7 adjusted to the bipartite
case. As discussed in the first part of the proof, an LOCC superprocess of
this form is given in Fig. 10. This completes the proof. ∎

Figure 9: An instantaneous LOCC channel taking the form of LOSR wherein the
delay time of the input channel $\mathcal{N}$ is preserved. Figure 10: An
instantaneous LOCC superchannel containing pre-processing LOCC simulating an
instantaneous dynamical resource.

One of the most important features of this framework is that it provides a
physical justification for allowing pre-LOCC processes in experiments of Bell
nonlocality. If the channel $\mathcal{N}$ is replaced by a quantum state
$\rho^{A_{1}B_{1}}$, then Fig. 10 has precisely the same form as Fig. 1. Thus,
hidden Bell nonlocality and superactivation of nonlocality are both features
that are operationally accessible in this resource theory. More precisely,
channels having the form of Eq. (I) reflect the resource conversion

$(\rho^{A_{1}B_{1}}\otimes\sigma^{A_{1}^{\prime}B_{1}^{\prime}},0)\xrightarrow{\text{LOCC}}(\mathcal{N}^{XY},0),$
(19)

and so the result of Masanes  _et al._ (2008) implies that every entangled
state can activate some Bell nonlocal resource using the free operations of
this theory.

Furthermore, all so-called “anomalies” of bipartite entanglement Methot and
Scarani (2007) vanish under this resource theory since partially entangled
states cannot be “more resourceful” than maximally entangled ones. This is
because every partially entangled bipartite state can be obtained from a
maximally entangled one under LOCC. Hence whatever process can be freely
generated by a weakly entangled state can also be generated by a maximally
entangled one.

Let us also remark that Theorem 1 unifies previous resource-theoretic
formulations of Bell nonlocality. Figure 9 describes the transformation of
channels by LOSR superchannels. This model has been studied in the abstract
setting of nonlocal “boxes” that are governed by a common cause Wolfe  _et
al._ (2020); Schmid  _et al._ (2020b, a). An alternative operational model has
been proposed in Ref. Gallego  _et al._ (2012) which is known as wiring and
prior-to-input classical communication (WPICC) (see also Gallego and Aolita
(2017)). In the context of quantum resources, WPICC amounts to applying a pre-
processing map implementable by LOCC, as in Fig. 10. Our resource theory
introduces the notion of input-output delay time as a way to physically
motivate both of these types of channel transformations in a unified way.

Returning to Eq. (16), we can consider the range of such transformations. That
is, what instantaneous classical processes can be generated using entanglement
as a resource and LOCC superprocessing? The following theorem shows that
dynamical entangled resources (i.e. channels) have no greater ability to
generate instantaneous classical processes than static entanglement (i.e.
states).

Channels and States are equi-resourceful

######  Theorem 2.

Let $(\mathcal{N}^{AB},\Delta t)$ be a bipartite quantum process with delay
time $\Delta t\in[0,\infty]$ and let $(\mathcal{M}^{XY},0)$ be a bipartite
instantaneous classical process. If $(\mathcal{N}^{AB},\Delta
t)\xrightarrow{\text{LOCC}}(\mathcal{M}^{XY},0)$, then there exists some
bipartite state $\rho^{A_{1}B_{1}}$ such that
$(\rho^{A_{1}B_{1}},0)\xrightarrow{\text{LOCC}}(\mathcal{M}^{XY},0)$.

###### Proof.

We will divide the proof into two parts. Consider first the case that $\Delta
t>0$. In this case we saw that the LOCC superprocess must have the form of
Fig. 10. Let $\rho^{\tilde{A}_{1}\tilde{B}_{1}}$ be the state at the output
$\tilde{A}_{1}\tilde{B}_{1}$ of the second pre-LOCC process in Fig. 10. Then,
the proof is concluded with the observation that Fig. 10 can be viewed as an
LOSR process on the state $\rho^{\tilde{A}_{1}\tilde{B}_{1}}$.

Consider next the case $\Delta t=0$. In this case, the LOCC superprocess can
be either the one given in Fig. 9 or the one given in Fig. 10. Since for the
latter corresponds to the previous case, we assume now that the supperprocess
has the form of Fig. 9. In this case, since $(\mathcal{N}^{AB},0)$ is an
instantaneous quantum process, it follows that $\mathcal{N}^{AB}$ is LOSE
channel of the form (11). By definition, every LOSE channel can be obtained by
applying an LOSR superchannel $\Upsilon$ on an entanglement state
$\omega^{A_{2}B_{2}}$ (see (11)). We therefore conclude that when the LOSR
superchannel $\Theta$ of Fig. 9 acts on an LOSE channel
$\mathcal{N}^{AB}=\Upsilon[\omega^{A_{2}B_{2}}]$ it produce the channel
$\mathcal{M}^{XY}=\Theta\circ\Upsilon[\omega^{A_{2}B_{2}}]$. Since the
superchannel $\Theta\circ\Upsilon$ is also LOSR, we conclude that
$\omega^{A_{2}B_{2}}$ can be converted to $(\mathcal{M}^{XY},0)$ with an LOSR
superchannel $\Theta\circ\Upsilon$. This concludes the proof. ∎

##  IV Quantification of Bell Nonlocality

We next turn to the question of quantifying Bell nonlocality in this resource
theory. A functional ${\rm E}:\cup_{A,B}{\rm
CPTP}(AB)\to\mathbb{R}_{+}\cup\\{0\\}$ was defined in Gour and Scandolo (2020)
as a dynamical entanglement monotone if ${\rm
E}(\Theta[\mathcal{N}])\leqslant{\rm E}(\mathcal{N})$ for any
$\mathcal{N}\in{\rm CPTP}(AB)$ and for any $\Theta\in{\rm LOCC}(AB\to
A^{\prime}B^{\prime})$. Since in our model the resources are quantum
processes, we will need to adjust this definition to incorporate the new
aspect of input-output delay time associated with each quantum channel.

Monotones of Bell nonlocality for classical channels have been previously
explored Wolfe  _et al._ (2020), where the domain of the functionals
presented, is restricted to no-signalling bipartite classical channels.
Interestingly, since in our model signalling classical processes are non-
instantaneous and therefore free, we do not consider them as well. In fact,
any classical Bell nonlocality monotone will take the zero value on all non-
instantaneous classical cprocesses (since classical resources can only be
instantaneous). For instance, the Bell nonlocality of the _classical_ swap
channel, which in previous works were considered as a maximal resource, is
zero. For this reason, some monotones of Bell nonlocality that were previously
studied in literature need to be adjusted in order to capture the input-output
delay time of the quantum process.

Each quantum precess $(\mathcal{N}^{AB},\Delta t)$ is realizable if the input-
output delay time $\Delta t$ is large enough so that the channel
$\mathcal{N}^{A_{0}B_{0}\to A_{1}B_{1}}$ can be physically implemented (e.g.
the swap operation between Alice and Bob cannot be implemented with $\Delta
t=0$). For simplicity, we only consider quantum processes with
$\mathbf{x}_{A_{0}}=\mathbf{x}_{A_{1}}\equiv\mathbf{x}_{A}$ and
$\mathbf{x}_{B_{0}}=\mathbf{x}_{B_{1}}\equiv\mathbf{x}_{B}$. On the other
hand, the distance between Alice and Bob
$\Delta\mathbf{x}=\mathbf{x}_{B}-\mathbf{x}_{A}>0$ will be assumed to be
strictly positive and kept fixed. Therefore, as in the previous section, there
will be no need to include $\Delta\mathbf{x}$ in the notation of quantum
processes, and this also enables us, as before, to classify resources as being
either instantaneous or non-instantaneous.

In the following definition we denote by $\mathfrak{Q}(AB)$ the set of all
realizable bipartite quantum processes of the form $(\mathcal{N}^{AB},\Delta
t)$ over all $\Delta t\in[0,\infty]$, where $\mathcal{N}\in{\rm CPTP}(AB)$.
Similarly, we denote by $\mathfrak{Q}(XY)$ the set of all such classical
processes $(\mathcal{N}^{XY},\Delta t)$. Note that any instantaneous resource,
$(\mathcal{N}^{AB},0)\in\mathfrak{Q}(AB)$, must be implementable by LOSE.
Otherwise, any communication from Alice to Bob will create a non-zero input-
output delay time.

Dynamical Entanglement Measure

######  Definition IV.1.

The function $E:\bigcup_{A,B}\mathfrak{Q}(AB)\to\mathbb{R}\;,$ where the union
is over all finite dynamical systems $A$ and $B$, is called a _measure of
dynamical entanglement_ if on the trivial process $(\mathsf{id}^{1},\Delta
t)$, $E\left(\mathsf{id}^{1},\Delta t\right)=0$, and for any two bipartite
processes $(\mathcal{N},\Delta t)$ and $(\mathcal{N}^{\prime},\Delta
t^{\prime})$ such that $(\mathcal{N},\Delta
t)\xrightarrow{\text{LOCC}}(\mathcal{N}^{\prime},\Delta t^{\prime})$ (20) we
have $E(\mathcal{N},\Delta t)\geqslant E(\mathcal{N}^{\prime},\Delta
t^{\prime})\;.$

When $\mathcal{N}$ and $\mathcal{N}^{\prime}$ are bipartite quantum states the
definition above reduces to a measure of (static) entanglement on bipartite
states. When $\mathcal{N}$ and $\mathcal{N}^{\prime}$ are classical bipartite
channels and $\Delta t=\Delta t^{\prime}=0$ (i.e. instantaneous processes) the
definition above reduces to a measure of Bell non-locality. This means that
any measure of dynamical entanglement, $E$, reduces to a measure of Bell non-
locality when the domain is restricted to instantaneous classical processes.
We denote by $E_{cl}$ the restriction of $E$ to the classical domain, and call
it a classical entanglement measure. Note that if $\Delta t>0$ then
$E_{cl}(\mathcal{N}^{XY},\Delta t)=0$ for any classical bipartite channel
$\mathcal{N}\in{\rm CPTP}(XY)$. Therefore, we will only consider instantaneous
classical processes and write in short $E_{cl}(\mathcal{N}^{XY})$ for
$E_{cl}(\mathcal{N}^{XY},0)$.

As an example, we introduce a classical Bell nonlocality monotone that is
based on channel divergences Cooney  _et al._ (2016); Leditzky  _et al._
(2018); Berta  _et al._ (2020); Gour (2019); Fang and Fawzi (2019b); Katariya
and Wilde (2020b); Gour (2020). Typically, in resource theories such monotones
are constructed as the “distance” (as measured by the divergence) of the
resource to the set of free objects. However, in our case, there are two types
of free processes: non-instantaneous LOCC quantum processes, and instantaneous
LOSR processes. Since divergences are not defined on quantum processes they
are insensitive to the distinction between instantaneous vs non-instantaneous
resources. For this reason we first define $E$ on classical systems and then
extend the domain of $E$ to all channels using the extension techniques
developed in Gour and Tomamichel (2020).

Let $D$ be the Kullback-Leibler divergence (relative entropy) defined on any
two $n$-dimensional probability vectors $\mathbf{p}$ and $\mathbf{q}$ as

$D(\mathbf{p}\|\mathbf{q})\coloneqq\sum_{x=1}^{n}p_{x}(\log p_{x}-\log
q_{x})\;.$ (21)

Its extension to classical channels is defined as Gour (2020)

$D(\mathcal{N}\|\mathcal{M})\coloneqq\max_{x\in[|X_{0}|]}D\left(\mathcal{N}\left(|x\rangle\langle
x|^{X_{0}}\right)\big{\|}\mathcal{M}\left(|x\rangle\langle
x|^{X_{0}}\right)\right)\;,$

for all classical channels $\mathcal{N},\mathcal{M}\in{\rm CPTP}(X)$. The
above function is called a channel divergence since it satisfies the data
processing inequality; i.e. for all $\Theta\in{\rm SC}(X\to Y)$ we have
$D\big{(}\Theta[\mathcal{N}]\big{\|}\Theta[\mathcal{M}]\big{)}\leqslant
D(\mathcal{N}\|\mathcal{M})$. In Gour (2020) it was shown that the above
classical channel relative entropy is the only channel relative entropy that
reduces to the Kullback-Leibler divergence on classical states (i.e. when
$|X_{0}|=1$). We define the relative entropy of Bell nonlocality as

$E_{cl}^{rel}\left(\mathcal{N}^{XY}\right)\coloneqq\min_{\mathcal{M}\in{\rm
LOSR}(XY)}D\left(\mathcal{N}^{XY}\big{\|}\mathcal{M}^{XY}\right)\;.$ (22)

Note that the function above quantifies the dynamical entanglement of an
instantaneous bipartite classical channel (i.e. it quantifies quantum Bell
non-locality). In particular, since the channel divergence $D$ satisfies the
DPI it follows that for any $\Theta\in{\rm LOSR}(XY\to X^{\prime}Y^{\prime})$

$E_{cl}^{rel}\left(\Theta\left[\mathcal{N}^{XY}\right]\right)\leqslant
E_{cl}^{rel}\left(\mathcal{N}^{XY}\right)\;,$ (23)

where ${\rm LOSR}(XY\to X^{\prime}Y^{\prime})$ denotes the set of all
instantaneous super processes of the form depicted in Fig. 9.

We would like now to extend the above measure to all instantaneous bipartite
quantum processes; i.e. to all channels $\mathcal{N}\in{\rm LOSE}(AB)$ as
given in (11). We focus in this paper on the quantification of instantaneous
resources since the quantification of non-instantaneous entanglement, both
static and dynamic, are well understood Plenio and Virmani (2007); Horodecki
_et al._ (2009); Gour and Scandolo (2020). Recall that LOSE also include all
bipartite quantum states, so that the extensions of $E_{cl}^{rel}$ to all
instantaneous bipartite channels, will provide us a way to quantify the
quantum Bell nonlocality of a bipartite state. We follow the extension
approach recently put forward in Gour and Tomamichel (2020) for generalized
resource theories.

Minimal and Maximal Extensions

######  Definition IV.2.

Let $E:\bigcup_{X,Y}\mathfrak{Q}(XY)\to\mathbb{R}$ be a classical entanglement
measure. Then, for any bipartite quantum channel $\mathcal{N}\in{\rm
LOSE}(AB)$, the minimal and maximal extensions of $E_{cl}$ to all
instantaneous quantum processes, denoted by $\overline{E_{cl}}$ and
$\underline{E_{cl}}$, respectively, are given by
$\displaystyle\underline{E_{cl}}(\mathcal{N})\coloneqq\sup
E_{cl}\big{(}\Theta[\mathcal{N}]\big{)}\quad\text{and}$ (24)
$\displaystyle\overline{E_{cl}}(\mathcal{N}):=\inf E_{cl}(\mathcal{C})$ where
the supremum and infimum are taken over all LOCC superchannels $\Theta\in{\rm
LOCC}(AB\to XY)$ and all bipartite classical channels
$\mathcal{C}\in\cup_{X,Y}{\rm CPTP}(XY)$ that satisfy
$\mathcal{N}=\Upsilon[\mathcal{C}]$ for some superchannel $\Upsilon\in{\rm
LOSR}(XY\to AB)$.

######  Remark.

Since we are only considering here instantaneous resources, the superchannels
$\Theta$ and $\Upsilon$ above are themselves instantaneous LOCC operations. In
particular, $\Theta$ has either the form of Fig. 9 (i.e. LOSR) or the form of
Fig. 10, while $\Upsilon$ can only have the LOSR form of Fig. 9 since its
domain is classical (note that if systems $\tilde{A}_{1}$ and $\tilde{B}_{1}$
of Fig. 10 were classical then they would only carry classical shared
randomness, and therefore in this case the superchannel depicted in Fig. 10
becomes a special case of the LOSR superchannel of Fig. 9).

The following theorem follows directly from the general formalism introduced
in Gour and Tomamichel (2020) for the extension of resource measures from one
domain to a larger one.

######  Theorem 3.

Let $E_{cl}$ be a classical entanglement measure, and $\underline{E_{cl}}$ and
$\overline{E_{cl}}$ be as above. Then, $\underline{E_{cl}}$ and
$\overline{E_{cl}}$ are entanglement measures for instantaneous bipartite
quantum processes. Moreover, any other such measure $E_{cl}^{\prime}$ that
reduces to $E_{cl}$ on classical instantaneous processes satisfies
$\underline{E_{cl}}(\mathcal{N})\leqslant
E_{cl}^{\prime}(\mathcal{N})\leqslant\overline{E_{cl}}(\mathcal{N})\quad\forall\;\mathcal{N}\in{\rm
LOSE}(AB)\;.$

Note that from its definition, $\underline{E_{cl}}$ cannot be increased even
under pre-LOCC operations as depicted in Fig. 10 that results in an
instantaneous classical channel.

##  V Conclusions

In this paper we have demonstrated that Bell nonlocality is a property of
bipartite quantum systems that can be studied as a special form of
entanglement. This was accomplished by constructing a resource theory based on
the abstract notion of a quantum processes. The important physical element
captured in this approach is the input-output time delay of a quantum channel.
Resources in quantum information science can thus be diversely classified as
as (1) classical or quantum, (2) static or dynamic, (3) noisy or noiseless,
(4) private or public, and as we add in this paper, (5) instantaneous or non-
instantaneous.

Despite claims that LOSR is the correct operational foundation to study Bell
nonlocality in quantum states Schmid  _et al._ (2020a), we have established an
operationally consistent quantum resource theory in which the inclusion of
LOCC pre-processing is physically motivated and follows naturally from the
identification of Bell nonlocality as an instantaneous quantum process. There
are also merits to allowing pre-LOCC processing in the resource theory of Bell
nonlocality. First, a well-defined partial order among static and dynamic
resource convertibility can be established. If one allows for LOCC pre-
processing, then a maximally entangled state is a resource for generating all
forms of quantum correlations, including those present in a Hardy-like Bell
test Hardy (1993). More generally, under the use of LOCC pre-processing,
maximally entangled states will necessarily have maximal nonlocality with
respect to any pre-LOCC monotone. This reflects our overall conclusion that
Bell nonlocality belongs to the same resource theory as quantum entanglement.

Furthermore, pre-LOCC is able to identify a source of Bell nonlocal states
whereas LOSR cannot. As argued in Ref. Theurer  _et al._ (2019), any
physically meaningful quantum resource theory should allow for the possibility
of detecting resource states using the free operations of the theory along
with the possible consumption of some auxiliary resource state. LOSR fails to
satisfy this criterion while pre-LOCC does not. Suppose Alice and Bob have
access to an i.i.d. source of some state $\rho^{AB}$ which is promised to be a
particular Bell nonlocal state. Using LOSR, they cannot distinguish this state
from the product state $\rho^{A}\otimes\rho^{B}$. On the other hand, when
using pre-LOCC, measurement outcomes can be discussed between the processing
of each copy of $\rho^{AB}$, and by performing a suitable Bell test, they can
verify that $\rho^{AB}$ is indeed a resource.

We also introduced a systematic way to quantify the dynamical entanglement of
quantum processes. As an example, we introduced a measure of instantaneous
entanglement (e.g. quantum Bell non-locality) that is based on the the
Kullback-Leibler divergence. Since the latter is the only asymptotically
continuous divergence Matsumoto (2018) it has numerous applications in quantum
information. Remarkably, its extension to classical channels is unique Gour
(2020), and therefore, we expect that our specific construction of an
instantaneous entanglement measure in (22), and its quantum extensions in
(LABEL:ecl2), will have operational meaning; we leave it for future work.

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Appendix

## Additional Results

###  V.1 Fully Bell local

So far, we have captured the essence of quantum Bell nonlocality and
entanglement theory in the framework of LOCC superprocesses. A quantum
superprocess, $(\Theta^{AB\to A^{\prime}B^{\prime}},\Delta t\to\Delta
t^{\prime})$, takes a quantum process $(\mathcal{N}^{AB},\Delta t)$ to another
process $(\mathcal{M}^{A^{\prime}B^{\prime}},\Delta t^{\prime})$. A quantum
superprocess is LOCC when the underlying superchannel $\Theta$ has an LOCC
realization, as shown in Fig. 2. While the framework of superprocesses
encapsulates all of dynamical entanglement theory, Bell nonlocality emerges
out from two restrictions on these superprocesses; i) the resultant channel is
classical, i.e. $A^{\prime}B^{\prime}=XY$, ii) the superprocess has either of
the forms shown in Fig. 9 and 10;

In this section, we consider delay time preserving (DTP) LOCC (i.e. LOSR)
222Note that if $\Delta t=0$, then both the superprocesses in Fig.9 and 10
preserve the time delay. However, if $\Delta t>0$ then only the superprocesses
of the form as shown in Fig.9 preserve the delay time of the input process,
while the superprocesses of the form as shown in Fig. 10 anihilates it (see
Theorem 1). superprocesses which take bipartite quantum processes to a
bipartite quantum to classical (POVM) processes, i.e. superprocesses of the
form $(\Theta^{AB\to A^{\prime}B^{\prime}},\Delta t\to\Delta t)$, where
$\Theta$ has the form shown in Fig. 9 with $A^{\prime}_{1}$ and
$B^{\prime}_{1}$ taken to be classical systems. We show that every entangled
bipartite channel can be used to simulate, via DTP LOCC superprocesses, at
least one bipartite nonlocal POVM channel. Therefore, even on not requiring
the output channel to be classical, the nonlocal features of entanglement can
still be observed. We call separable (i.e. LOSR) processes _fully Bell local_.

Fully Bell Local Processes

######  Definition V.1.

Let $(\mathcal{N}^{AB},\Delta t)$ be a a bipartite quantum process with delay
time $\Delta t\in[0,\infty]$. $(\mathcal{N}^{AB},\Delta t)$ is said to be
_fully Bell local_ if for every DTP LOCC (i.e. LOSR) superprocess
$(\Theta^{AB\to A^{\prime}B^{\prime}},\Delta t\to\Delta t)$, of the form given
in Fig. 9, where $A^{\prime}_{1}$ and $B^{\prime}_{1}$ are restricted to
classical systems, the process $(\Theta[\mathcal{N}],\Delta t)$ is LOSR.

This means that every bipartite POVM process that can be simulated by applying
a DTP LOCC superprocess on a fully Bell local process must also be local. We
characterize the set of fully Bell local processes in our next theorem.

Figure 11:

Instantaneous LOCC channels are Fully Bell Local

######  Theorem 4.

Let $(\mathcal{N}^{AB},\Delta t)$ be a bipartite quantum process and
$(\mathcal{L}^{AB},0)$ denote the set of all instantaneous LOCC processes from
systems $A_{0},B_{0}$ to systems $A_{1},B_{1}$. $(\mathcal{N}^{AB},\Delta t)$
is fully Bell local if and only if $(\mathcal{N}^{AB},\Delta
t)\in(\mathcal{L}^{AB},0)$. Alternatively, $(\mathcal{N}^{AB},\Delta t)$ is
fully Bell local if $\mathcal{N}^{AB}$ admits the form
$\mathcal{N}^{AB}=\sum_{j}t_{j}\mathcal{E}^{A}_{j}\otimes\mathcal{F}^{B}_{j},$
(1) where $\mathcal{E}^{A}_{j}\in{\rm CPTP}(A)$ and
$\mathcal{F}^{B}_{j}\in{\rm CPTP}(B)$ for all $j\in\mathbb{R}$ and $t_{j}$ is
a probability distribution.

###### Proof.

We start by noting that the C-J matrix of a bipartite channel is separable if
and only if it is an LOSR channel (i.e. instantaneous LOCC). Clearly, if
$\mathcal{N}$ is LOSR, it is fully Bell local, since an LOSR superchannel
takes bipartite LOSR channels to bipartite LOSR channels. We therefore assume
that $\mathcal{N}$ is fully Bell local, and by contradiction, also non-LOSR,
i.e. its C-J matrix is not separable. Since every C-J matrix can be viewed as
an unnormalized density matrix, for any non-separable C-J matrix
$J_{\mathcal{N}}$ of a non-LOSR bipartite quantum channel $\mathcal{N}$, there
will always be a witness $W\in{\rm Herm}(AB\tilde{A}\tilde{B})$, such that
${\rm Tr}[J_{\mathcal{N}}W]<0$ and ${\rm Tr}[J_{\mathcal{M}}W]\geqslant 0$ for
every LOSR channel $\mathcal{M}$. Here ${\rm Herm}(AB\tilde{A}\tilde{B})$
denotes the set of hermitian operators in
$\mathfrak{B}(AB\tilde{A}\tilde{B})$.

Now, express $W=r\eta-t\zeta$, where
$\eta,\zeta\in\mathfrak{D}(A_{0}^{\prime}B_{0}^{\prime})$ and $r,t\geqslant
0$, such that $|A_{0}^{\prime}|=|A_{0}||A_{1}|$ and
$|B_{0}^{\prime}|=|B_{0}||B_{1}|$. Consider the quantum to classical (qc)
channel, $\mathcal{F}\in{\rm CPTP}(A_{0}^{\prime}B_{0}^{\prime}\to
X_{1}Y_{1})$, generated by the application of an LOSR superchannel with pre-
processing which generates maximally entangled state and a post-processing
which includes local projective measurements as shown in Fig. 11 . Hence,
there will be some outcome in $(x,y)$ in which they both project onto the
maximally entangled state. For this outcome, on the inputs $\eta^{T}$ and
$\zeta^{T}$ the channel satisfies

$\begin{split}{\rm p}(x,y|\eta^{T})&\coloneqq\langle x,y|\mathcal{F}^{AB\to
XY}(\eta^{T})|x,y\rangle\\\
&=\frac{1}{|A_{0}||B_{0}|}\langle\phi_{+}^{A_{0}^{\prime}\tilde{A_{0}^{\prime}}}\otimes\phi_{+}^{B_{0}^{\prime}\tilde{B_{0}^{\prime}}}|\mathcal{N}^{\tilde{A_{0}}\tilde{B_{0}}\to
A_{1}B_{1}}(\phi_{+}^{A_{0}\tilde{A}_{0}}\otimes\phi_{+}^{B_{0}\tilde{B}_{0}})\otimes(\eta^{A_{0}^{\prime}B_{0}^{\prime}})^{T}|\phi_{+}^{A_{0}^{\prime}\tilde{A_{0}^{\prime}}}\otimes\phi_{+}^{B_{0}^{\prime}\tilde{B_{0}^{\prime}}}\rangle\\\
&=\frac{1}{|A_{0}||B_{0}|}\langle\phi_{+}^{A_{0}^{\prime}\tilde{A_{0}^{\prime}}}\otimes\phi_{+}^{B_{0}^{\prime}\tilde{B_{0}^{\prime}}}|J_{\mathcal{N}}\otimes(\eta^{A_{0}^{\prime}B_{0}^{\prime}})^{T}|\phi_{+}^{A_{0}^{\prime}\tilde{A_{0}^{\prime}}}\otimes\phi_{+}^{B_{0}^{\prime}\tilde{B_{0}^{\prime}}}\rangle\\\
&={\rm Tr}[\rho_{J_{\mathcal{N}}}\eta^{A_{0}^{\prime}B_{0}^{\prime}}],\\\
\end{split}$ (2)

where all $\phi_{+}$ are unnormalized and
$\rho_{J_{\mathcal{N}}}\coloneqq\frac{1}{|A_{0}||B_{0}|}J$ and similarly ${\rm
p}(x,y|\zeta^{T})={\rm
Tr}[\rho_{J_{\mathcal{N}}}\zeta^{A_{0}^{\prime}B_{0}^{\prime}}]$. Therefore,

$r{\rm p}(x,y|\eta^{T})-t{\rm p}(x,y|\zeta^{T})={\rm
Tr}[\rho_{J_{\mathcal{N}}}W]<0\;.$ (3)

On the other hand, for any LOSR channel $\mathcal{E}\in{\rm
LOSR}(A_{0}^{\prime}B_{0}^{\prime}\to X_{1}Y_{1})$ with POVM elements
$\\{\Pi_{x|i}^{A_{0}^{\prime}}\otimes\Pi_{y|i}^{B_{0}^{\prime}}\\}$ and prior
$\\{\rm{p}_{i}\\}$ such that $\mathcal{E}(\cdot)=\sum_{i}{\rm p}_{i}{\rm
Tr}\left[(\cdot)\left(\Pi_{x|i}^{A_{0}^{\prime}}\otimes\Pi_{y|i}^{B_{0}^{\prime}}\right)\right]|xy\rangle\langle
xy|^{X_{1}Y_{1}}$, we have for _any_ $x$ and $y$,

$r{\rm p}(x,y|\eta^{T})-t{\rm p}(x,y|\zeta^{T})=\sum_{i}{\rm p}_{i}{\rm
Tr}\left[W^{T}\left(\Pi_{x|i}^{A_{0}^{\prime}}\otimes\Pi_{y|i}B_{0}^{\prime}\right)\right]={\rm
Tr}\left[W^{T}\sum_{i}{\rm
p}_{i}\Pi_{x|i}^{A_{0}^{\prime}}\otimes\Pi_{y|i}^{B_{0}^{\prime}}\right]\geqslant
0,$ (4)

where the last inequality follows from the separability of the POVM. Hence, we
get a contradiction. This completes the proof. While writing this paper, the
authors became aware of an alternative proof provided in Schmid  _et al._
(2020c). ∎

Remark- As a special case of the theorem above, one obtains that a quantum
state $\rho\in\mathfrak{D}(A_{1}B_{1})$ is fully Bell local (i.e. it only
results in Bell local qc channels) if and only if it is separable, that is
when $\mathcal{N}$ is taken to be a replacement channel (i.e.
$\mathcal{N}(\sigma)=\rho$ for all $\sigma\in\mathfrak{D}(A_{0}B_{0})$).

###  V.2 CHSH Witness

Entanglement Witness for Bipartite POVMs

######  Theorem 5.

Let $\psi_{x_{0}}\in\mathfrak{D}{(A_{0})}$ and
$\phi_{y_{0}}\in\mathfrak{D}{(B_{0})}$ be pure quantum states,
$x_{0},y_{0}\in\\{0,1\\}$, $x_{1},y_{1}\in\mathbb{Z}$ and $\oplus$ denote
addition modulo 2. Then, the hermitian matrix
$W=\sum_{x_{1},y_{1}}W_{x_{1}y_{1}}\otimes|x_{1}y_{1}\rangle\langle
x_{1}y_{1}|$ is an ${\rm LOSR}$ witness for $\mathrm{POVM}$ channels in finite
dimensions, where
$W_{x_{1}y_{1}}=\sum_{x_{0},y_{0}}\left(\frac{3}{16}-\delta_{x_{1}\oplus
y_{1}=x_{0}y_{0}}\right)(\psi_{x_{0}}\otimes\phi_{y_{0}}).$ (5)

This witness is a generalization of of the CHSH inequality to the case of
bipartite POVM channels and is independent of the choice of basis.

###### Proof.

Let $\mathcal{E}_{qc}$ be a ${\rm POVM}$ channel with
$\\{\Pi_{x_{1}y_{1}|x_{0}y_{0}}\\}_{x_{1},y_{1}}$ as the POVM elements
satisfying $\sum_{x_{1},y_{1}}\Pi_{x_{1}y_{1}|x_{0}y_{0}}=I^{A_{0}B_{0}}$.

$\begin{split}{\rm
Tr}\left[WJ_{\mathcal{E}_{qc}}\right]&=\sum_{x_{1},y_{1}}{\rm
Tr}\left[\Pi^{A_{0}B_{0}}_{x_{1}y_{1}|x_{0}y_{0}}\otimes|x_{1}y_{1}\rangle\langle
x_{1}y_{1}|)\ \left(W_{x_{1}y_{1}}\otimes|x_{1}y_{1}\rangle\langle
x_{1}y_{1}|\right)\right]\\\ &=\sum_{x_{1},y_{1}}{\rm
Tr}\left[\left(\Pi^{A_{0}B_{0}}_{x_{1}y_{1}|x_{0}y_{0}}\otimes|x_{1}y_{1}\rangle\langle
x_{1}y_{1}|\right)\sum_{x_{0},y_{0}}\left(\frac{3}{16}-\delta_{x_{1}\oplus
y_{1}=x_{0}y_{0}}\right)[\psi_{x_{0}}^{A}\otimes\phi_{y_{0}}^{B}]\otimes|x_{1}y_{1}\rangle\langle
x_{1}y_{1}|\right]\\\ &=\sum_{x_{1},y_{1},x_{0},y_{0}}{\rm
Tr}\left[\Pi^{A_{0}B_{0}}_{x_{1}y_{1}|x_{0}y_{0}}(\psi_{x_{0}}^{A}\otimes\phi_{y_{0}}^{B})\left(\frac{3}{16}-\delta_{x_{1}\oplus
y_{1}=x_{0}y_{0}}\right)\right]{\rm Tr}\left[|x_{1}y_{1}\rangle\langle
x_{1}y_{1}|\right]\\\ &=\sum_{x_{1},y_{1},x_{0},y_{0}}{\rm
p}(x_{1},y_{1}|x_{0},y_{0})\left(\frac{3}{16}-\delta_{x_{1}\oplus
y_{1}=x_{0}y_{0}}\right)\\\ \end{split}$ (6)

Now, if $\mathcal{E}_{qc}$ is an ${\rm LOSR}$ channel, then
$\Pi_{x_{1}y_{1}|x_{0}y_{0}}=\sum_{\lambda}t_{\lambda}\Pi_{x_{1}|x_{0}\lambda}\otimes\Pi_{y_{1}|y_{0}\lambda}^{\prime}$
where $\Pi_{x_{1}|x_{0}\lambda}$ and $\Pi_{y_{1}|y_{0}\lambda}^{\prime}$ are
individual ${\rm POVM}$ elements and they satisfy
$\sum_{x_{1}}\Pi_{x_{1}|x_{0}\lambda}=I^{A}_{0}$ and
$\sum_{y_{1}}\Pi_{y_{1}|y_{0}\lambda}^{\prime}=I^{B}_{0}$ and $t_{\lambda}$ is
a probability distribution function depending on the random variable
$\lambda$. For such a channel, we have

$\begin{split}{\rm
Tr}\left[WJ_{\mathcal{E}_{qc}}\right]&=\sum_{x_{1},y_{1},\lambda}{\rm
Tr}\left[t_{\lambda}(\Pi_{x_{1}|\lambda}\otimes\Pi_{y_{1}|\lambda}^{\prime}\otimes|x_{1}y_{1}\rangle\langle
x_{1}y_{1}|)\ (W_{x_{1}y_{1}}\otimes|x_{1}y_{1}\rangle\langle
x_{1}y_{1}|)\right]\\\ &=\sum_{x_{1},y_{1},x_{0},y_{0},\lambda}\ t_{\lambda}\
{\rm p}_{x_{1}|\lambda x_{0}}\ {\rm q}_{y_{1}|\lambda
y_{0}}\left(\frac{3}{16}-\delta_{x_{1}\oplus y_{1}=x_{0}y_{0}}\right)\\\
&=\sum_{x_{1},y_{1},x_{0},y_{0}}{\rm p}_{l}(x_{1},y_{1}|x_{0},y_{0})\
\left(\frac{3}{16}-\delta_{x_{1}\oplus y_{1}=x_{0}y_{0}}\right)\geqslant
0,\end{split}$ (7)

where $\mathrm{p}_{l}$ represents a Bell local probability distribution, ${\rm
p}_{x_{1}|\lambda x_{0}}={\rm Tr}[\Pi_{x_{1}|\lambda}^{A}\psi_{x_{0}}]$ and
${\rm q}_{y_{1}|\lambda y_{0}}={\rm
Tr}[{\Pi^{\prime}}_{y_{1}|\lambda}^{B}\phi_{y_{0}}]$. This is true for any
choice of ${\rm POVM}$ elements $\Pi_{x_{1}|x_{0}\lambda}$ and
$\Pi_{y_{1}|y_{0}\lambda}^{\prime}$ and is also independent on the choice of
$\psi_{x_{0}}$ and $\phi_{y_{0}}$. On the other hand, if $\mathcal{E}_{qc}$ is
not an ${\rm LOSR}$ channel, $\Pi_{x_{1}y_{1}|x_{0}y_{0}}$ cannot admit the
convex distribution above. One possible way of constructing such a channel is
with the help of a bipartite quantum state which is not fully Bell local
(Theorem 4), i.e. entangled. However, from the setup of the CHSH game for
entangled states, it is always possible to construct at least one classical
channel via the action of a quantum to classical superchannel such that the
underlying probability distribution helps the parties to win the game. In
other words, for every Bell nonlocal bipartite state, there exists at least
one choice of $\\{\Pi_{x_{1}y_{1}|x_{0}y_{0}}\\},\\{\psi_{x_{0}}\\}$ and
$\\{\phi_{y_{0}}\\}$ such that,

$\begin{split}{\rm
Tr}\left[WJ_{\mathcal{E}_{qc}}\right]&=\sum_{x_{1},y_{1}}{\rm
Tr}\left[\Pi^{A_{0}B_{0}}_{x_{1}y_{1}|x_{0}y_{0}}\otimes|x_{1}y_{1}\rangle\langle
x_{1}y_{1}|)\ (W_{x_{1}y_{1}}\otimes|x_{1}y_{1}\rangle\langle
x_{1}y_{1}|)\right]\\\ &=\sum_{x_{1},y_{1},x_{0},y_{0}}{\rm
p}_{nl}(x_{1},y_{1}|x_{0},y_{0})\left(\frac{3}{16}-\delta_{x_{1}\oplus
y_{1}=x_{0}y_{0}}\right)<0,\end{split}$ (8)

where $\rm{p}_{nl}$ represents a Bell nonlocal probability distribution.

∎