[hyperref,natbib,gensymb]

# Postquantum common-cause channels: the resource theory of local operations
and shared entanglement

David Schmid  Perimeter Institute for Theoretical Physics, 31 Caroline St. N,
Waterloo, Ontario, N2L 2Y5, Canada  Institute for Quantum Computing and Dept.
of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1,
Canada<EMAIL_ADDRESS>Haoxing Du  Perimeter Institute for
Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada
Maryam Mudassar  Perimeter Institute for Theoretical Physics, 31 Caroline St.
N, Waterloo, Ontario, N2L 2Y5, Canada  Ghi Coulter-de Wit  Perimeter Institute
for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada
Denis Rosset  Perimeter Institute for Theoretical Physics, 31 Caroline St. N,
Waterloo, Ontario, N2L 2Y5, Canada  Matty J. Hoban  Department of Computing,
Goldsmiths, University of London, New Cross, London SE14 6NW, United Kingdom

###### Abstract

We define the type-independent resource theory of local operations and shared
entanglement (LOSE). This allows us to formally quantify postquantumness in
common-cause scenarios such as the Bell scenario. Any nonsignaling bipartite
quantum channel which cannot be generated by LOSE operations requires a
postquantum common cause to generate, and constitutes a valuable resource. Our
framework allows LOSE operations that arbitrarily transform between different
types of resources, which in turn allows us to undertake a systematic study of
the different manifestations of postquantum common causes. Only three of these
have been previously recognized, namely postquantum correlations, postquantum
steering, and ‘non-localizable’ channels, all of which are subsumed as special
cases of resources in our framework. Finally, we prove several fundamental
results regarding how the type of a resource determines what conversions into
other resources are possible, and also places constraints on the resource’s
ability to provide an advantage in distributed tasks such as nonlocal games,
semiquantum games, steering games, etc.

##  1 Introduction

In space-like separated experiments, it is well-known that quantum theory can
generate correlations which violate Bell inequalities [Brunner  _et
al._(2014a)Brunner, Cavalcanti, Pironio, Scarani, and Wehner], witnessing the
fact that they cannot be explained by a classical common-cause process [Wood
and Spekkens(2015)]. In a common-cause scenario, such correlations can only be
explained by a common cause described by a more general theory, such as a
quantum common cause [Allen  _et al._(2017)Allen, Barrett, Horsman, Lee, and
Spekkens, Barrett  _et al._(2019)Barrett, Lorenz, and Oreshkov, Costa and
Shrapnel(2016)] (which for our purposes can be viewed simply as a composite
quantum system in an entangled state).

However, it is also well-known that in common-cause scenarios, quantum theory
cannot generate correlations which are maximally nonclassical, as was pointed
out by Tsirelson [Cirel’son(1980)] and by Popescu and Rohrlich [Popescu and
Rohrlich(1994)]. That is, nonsignaling correlations which achieve the
logically maximal violation of a Bell inequality generally cannot be achieved
by local measurements on entangled states, and yet are achievable by
postquantum common causes [Barrett(2007)], that is, those described by a
generalized probabilistic theory [Hardy(2001), Barrett(2007)] (GPT) beyond
quantum theory.

Correlations, often termed box-type resources, or simply ‘boxes’ [Wolfe  _et
al._(2020)Wolfe, Schmid, Sainz, Kunjwal, and Spekkens], are not the only
postquantum resources. More general types of postquantum
resources111Throughout this work, the adjective postquantum refers to the
common cause required to generate the given resource, while the resources
themselves are always channels taking quantum states to quantum states (as
opposed to channels which act on postquantum states, which we do not
consider).  have been studied in a few previous works [Beckman  _et
al._(2001)Beckman, Gottesman, Nielsen, and Preskill, Piani  _et
al._(2006)Piani, Horodecki, Horodecki, and Horodecki, D’Ariano  _et
al._(2011)D’Ariano, Facchini, and Perinotti, Hoban and Sainz(2018), Belén
Sainz  _et al._(2019)Belén Sainz, Hoban, Skrzypczyk, and Aolita], which found
instances of channels which cannot be generated by local operations and shared
entanglement (LOSE) [Gutoski(2008)].

Ref. [Beckman  _et al._(2001)Beckman, Gottesman, Nielsen, and Preskill] was
the first to show that the set of nonsignaling channels is strictly larger
than the set of channels realizable by local operations and shared
entanglement. The authors refer to the former set of channels as ‘causal’ and
to the latter set of channels as ‘localizable’, but we will not adopt this
terminology (largely because the term ‘causal’ is ambiguous and overloaded). A
key motivation in Ref. [Beckman  _et al._(2001)Beckman, Gottesman, Nielsen,
and Preskill] (which also applies to our work) is to better understand the
restrictions on operations that are imposed by special relativity.
Subsequently, Refs. [Piani  _et al._(2006)Piani, Horodecki, Horodecki, and
Horodecki, D’Ariano  _et al._(2011)D’Ariano, Facchini, and Perinotti] made
some progress in characterizing the set of postquantum channels, and Ref.
[D’Ariano  _et al._(2011)D’Ariano, Facchini, and Perinotti] showed that there
exist postquantum channels that are not entanglement-breaking.

Additionally, Ref. [Belén Sainz  _et al._(2019)Belén Sainz, Hoban, Skrzypczyk,
and Aolita] studied the set of postquantum resources which arise in
generalized steering scenarios in the bipartite case. First, Ref. [Sainz  _et
al._(2015)Sainz, Brunner, Cavalcanti, Skrzypczyk, and Vértesi] noted that all
logically possible bipartite quantum steering assemblages can be generated
using local operations and shared entanglement, and hence that postquantum
common causes do not give an advantage for steering of quantum states.
Initially this motivated the study of multipartite postquantum steering [Sainz
_et al._(2015)Sainz, Brunner, Cavalcanti, Skrzypczyk, and Vértesi, Sainz  _et
al._(2018)Sainz, Aolita, Piani, Hoban, and Skrzypczyk, Hoban and Sainz(2018)].
However, returning to the bipartite setting, Ref. [Belén Sainz  _et
al._(2019)Belén Sainz, Hoban, Skrzypczyk, and Aolita] considered a generalized
type of steering scenario in which the steered party (Bob) has a classical
input system. These were termed Bob-with-input steering assemblages, and Ref.
[Belén Sainz  _et al._(2019)Belén Sainz, Hoban, Skrzypczyk, and Aolita]
provides examples of such resources which cannot be generated using local
operations and shared entanglement.

In fact, there are a wide variety of nontrivial postquantum resources,
including boxes, distributed measurements, and Bob-with-input steering
assemblages as special cases, as well as measurement-device-independent
steering assemblages, channel-steering assemblages, ensemble-preparing
channels, and so on.

We introduce a unified framework for the study of postquantum common cause
channels, which subsumes all of these types of resources as special cases.
This framework is the resource theory [Coecke  _et al._(2016)Coecke, Fritz,
and Spekkens] of local operations and shared entanglement. Any common-cause
process which cannot be realized by LOSE operations is a valuable resource and
a signature of a postquantum common cause. Furthermore, by considering
transformations between resources that can be enacted using LOSE operations,
one can quantitatively characterize the postquantumness of any given resource.

The study of such postquantum resources sheds light on the space of logically
conceivable processes which are not realizable with quantum common causes. As
evidenced by the example of the Popescu-Rohrlich (PR) box, such processes
serve as useful foils [Chiribella and Spekkens(2016)] for teaching us about
quantum theory. Furthermore, postquantum processes provide clues for where one
might find physics beyond quantum theory [Hardy(2005)]. Conversely, a recent
line of research involves the search for reasonable physical principles which
distinguish nonlocal quantum correlations from postquantum correlations:
macroscopic locality [Navascués and Wunderlich(2010)], information causality
[Pawłowski  _et al._(2009)Pawłowski, Paterek, Kaszlikowski, Scarani, Winter,
and Żukowski, Pawłowski and Scarani(2016)], limits on communication complexity
[Brassard  _et al._(2006)Brassard, Buhrman, Linden, Méthot, Tapp, and Unger]
or nonlocal computation [Linden  _et al._(2007)Linden, Popescu, Short, and
Winter, Broadbent(2016)], local orthogonality [Fritz  _et al._(2013)Fritz,
Sainz, Augusiak, Brask, Chaves, Leverrier, and Acín] or the Specker principle
[Cabello(2012), Gonda  _et al._(2018)Gonda, Kunjwal, Schmid, Wolfe, and
Sainz]. Almost quantum correlations [Navascués  _et al._(2015)Navascués,
Guryanova, Hoban, and Acín, Henson and Sainz(2015)] were originally developed
to provide an example of a postquantum theory that nevertheless satisfies
several of the reasonable physical principles above. This postquantum theory
was first studied in the context of nonlocal boxes, and was then extended to
generalized steering scenarios [Sainz  _et al._(2015)Sainz, Brunner,
Cavalcanti, Skrzypczyk, and Vértesi, Belén Sainz  _et al._(2019)Belén Sainz,
Hoban, Skrzypczyk, and Aolita] and beyond [Hoban and Sainz(2018)].
Discriminating quantum from postquantum (as opposed to discriminating
nonclassical from classical) can also be useful in the field of device-
independent quantum information [Navascués  _et al._(2007)Navascués, Pironio,
and Acín], such as for self-testing [Yang  _et al._(2014)Yang, Vértesi,
Bancal, Scarani, and Navascués, Šupić  _et al._(2019)Šupić, Hoban, Colomer,
and Acín], random certification [Colbeck(2009), Colbeck and Kent(2011),
Pironio  _et al._(2010)Pironio, Acín, Massar, de la Giroday, Matsukevich,
Maunz, Olmschenk, Hayes, Luo, Manning, and Monroe, Šupić  _et al._(2017)Šupić,
Skrzypczyk, and Cavalcanti], and the study of nonlocal games [Doherty  _et
al._(2008)Doherty, Liang, Toner, and Wehner]. We also note recent experimental
explorations of the boundary of quantum theory [Mazurek  _et
al._(2019)Mazurek, Pusey, Resch, and Spekkens, Liang and Zhang(2019)].

Most of the results in this article refer not to specific instances of
postquantum resources, but rather to what can be said about a postquantum
resource merely from knowing its type (e.g. whether it is a box, an
assemblage, a distributed measurement, etc). Most of our analysis and results
have close analogues in Ref. [Schmid  _et al._(2020a)Schmid, Rosset, and
Buscemi], which undertook a similar study for the type-independent resource
theory of local operations and shared randomness (LOSR), which is the
appropriate resource theory for quantifying nonclassicality of common-cause
processes [Wolfe  _et al._(2020)Wolfe, Schmid, Sainz, Kunjwal, and Spekkens]
(to be contrasted with the resource theory of LOSE studied here, which is
appropriate for quantifying postquantumness of common-cause processes). Both
of these works are grounded conceptually in the idea that correlations should
be deemed classical or nonclassical depending on what sort of causal (and
inferential) theories are required to reproduce them while respecting the
conservative network structure [Wood and Spekkens(2015), Wolfe  _et
al._(2020)Wolfe, Schmid, Sainz, Kunjwal, and Spekkens, Schmid  _et
al._(2020b)Schmid, Selby, and Spekkens].

In Section 2, we introduce the types of resources we consider; that is, the
enveloping set of resources in our resource theory. In Section 3, we introduce
the free resources and free transformations. We then characterize the types of
bipartite resources which can possibly be nonfree—that is, which can exhibit
postquantumness. In Section 3.1, we discuss how the free transformations
induce a preorder over all resources, which quantitatively determines how
valuable each resource is. In Section 3.2, we state some results (proved in
Appendix A) about the relative value of various postquantum resources that
have been considered in the literature. In Section 4, we discuss type
encodings, which formalize the idea that some resource types are able to
exhibit all of the same manifestations of postquantumness as another type of
resource. We begin a systematic study of which types of resources can encode
the postquantumness of all resources of other types, and we show that the
distributed measurement type can encode the postquantumness of all resources
of all types. In Section 5, we recall the framework of distributed games,
introduced in Ref. [Schmid  _et al._(2020a)Schmid, Rosset, and Buscemi], which
unifies and generalizes many traditional distributed tasks and games,
including entanglement witnessing, nonlocal games, semiquantum games, steering
games, teleportation games, and so on. We then discuss how players can
implement type-changing LOSE operations on whatever shared resources they have
access to in order to generate an optimal strategy for the game they are
playing. In Section 5.1, we discuss implications from the type of a shared
resource to how well it can perform at various games. In Section 6, we
conclude with some interesting open questions.

##  2 The enveloping theory

Following Ref. [Schmid  _et al._(2020a)Schmid, Rosset, and Buscemi, Rosset
_et al._(2020)Rosset, Schmid, and Buscemi], the set of resources we consider
in this work are the multipartite, completely-positive [Nielsen and
Chuang(2010), Schmid  _et al._(2019a)Schmid, Ried, and Spekkens] trace
preserving (CPTP) maps that are nonsignaling [Rosset  _et al._(2020)Rosset,
Schmid, and Buscemi] between every pair of parties. We will focus on the
bipartite case for simplicity; however, most of our framework and results
generalize immediately to arbitrarily many parties.

Different types of resources are distinguished by whether the input and output
systems of each party are trivial, classical, or quantum. As in Ref. [Rosset
_et al._(2020)Rosset, Schmid, and Buscemi], a single party’s system is said to
be trivial if it has dimension one, is said to be classical if all operators
on its Hilbert space are diagonal, and is otherwise said to be quantum. If a
single party has a collection of input (output) systems, we will consider them
as a single effective input (output) system. We will focus on bipartite
resources shared between two parties, Alice and Bob, and will denote Alice’s
input and output systems by $X$ and $A$, respectively, and Bob’s input and
output systems by $Y$ and $B$, respectively. Each of these systems has trivial
($\mathsf{I}$), classical ($\mathsf{C}$), or quantum ($\mathsf{Q}$) type, and
so we can denote the global type of the resource by
$T:=\mathbf{Type}[\mathsf{X}]\mathbf{Type}[\mathsf{Y}]\\!\\!\to\\!\\!\mathbf{Type}[\mathsf{A}]\mathbf{Type}[\mathsf{B}]$.
The partition-type of a single party is specified by the types of its inputs
and outputs, e.g. in Alice’s case we write
$T_{A}:=\mathbf{Type}[\mathsf{X}]\\!\\!\to\\!\\!\mathbf{Type}[\mathsf{A}]$.

The most commonly studied types of resources, depicted in Figure 1 and
discussed in more detail in Refs. [Schmid  _et al._(2020a)Schmid, Rosset, and
Buscemi, Rosset  _et al._(2020)Rosset, Schmid, and Buscemi], are quantum
states, of type $\mathsf{II}\\!\\!\to\\!\\!\mathsf{QQ}$, steering assemblages
[Einstein  _et al._(1935)Einstein, Podolsky, and Rosen, Schrodinger(1935),
Wiseman  _et al._(2007)Wiseman, Jones, and Doherty, Skrzypczyk  _et
al._(2014)Skrzypczyk, Navascués, and Cavalcanti, Gallego and Aolita(2015),
Piani and Watrous(2015), Cavalcanti and Skrzypczyk(2017), Uola  _et
al._(2019)Uola, Costa, Nguyen, and Gühne, Pusey(2013)], of type
$\mathsf{CI}\\!\\!\to\\!\\!\mathsf{CQ}$, teleportages [Cavalcanti  _et
al._(2017)Cavalcanti, Skrzypczyk, and Šupić, Šupić  _et al._(2019)Šupić,
Skrzypczyk, and Cavalcanti, Hoban and Sainz(2018)], of type
$\mathsf{QI}\\!\\!\to\\!\\!\mathsf{CQ}$, nonsignaling boxes [Barrett  _et
al._(2005)Barrett, Linden, Massar, Pironio, Popescu, and Roberts, Brunner  _et
al._(2014b)Brunner, Cavalcanti, Pironio, Scarani, and Wehner], of type
$\mathsf{CC}\\!\\!\to\\!\\!\mathsf{CC}$, distributed measurements (or
semiquantum channels) [Buscemi(2012), Šupić  _et al._(2017)Šupić, Skrzypczyk,
and Cavalcanti, Hoban and Sainz(2018)], of type
$\mathsf{QQ}\\!\\!\to\\!\\!\mathsf{CC}$, MDI-steering channels [Cavalcanti
_et al._(2013)Cavalcanti, Hall, and Wiseman], of type
$\mathsf{CQ}\\!\\!\to\\!\\!\mathsf{CC}$, channel steering assemblages
[Piani(2015)], of type $\mathsf{CQ}\\!\\!\to\\!\\!\mathsf{CQ}$, Bob-with-input
steering assemblages [Hoban and Sainz(2018), Belén Sainz  _et al._(2019)Belén
Sainz, Hoban, Skrzypczyk, and Aolita], of type
$\mathsf{CC}\\!\\!\to\\!\\!\mathsf{CQ}$, distributed ensemble-preparing
channels [Beckman  _et al._(2001)Beckman, Gottesman, Nielsen, and Preskill],
of type $\mathsf{CC}\\!\\!\to\\!\\!\mathsf{QQ}$, and generic nonsignaling
bipartite quantum channels, of type $\mathsf{QQ}\\!\\!\to\\!\\!\mathsf{QQ}$.
There are also five types of nontrivial bipartite resources that have not (to
our knowledge) been previously studied, namely
$\mathsf{QC}\\!\\!\to\\!\\!\mathsf{CQ}$,
$\mathsf{CQ}\\!\\!\to\\!\\!\mathsf{QQ}$,
$\mathsf{IQ}\\!\\!\to\\!\\!\mathsf{QQ}$,
$\mathsf{QQ}\\!\\!\to\\!\\!\mathsf{CQ}$, and
$\mathsf{CI}\\!\\!\to\\!\\!\mathsf{QQ}$.

Figure 1: Common types of nonsignaling resources, where classical systems are
represented by single wires and quantum systems are represented by double
wires. As we will show, every resource of the three shaded types is LOSE-free.
From left to right, top to bottom, we depict: quantum states, steering
assemblages, teleportages, boxes, distributed measurements (or semiquantum
channels), measurement-device-independent steering assemblages, channel
steering assemblages, Bob-with-input steering assemblages, distributed
ensemble-preparations, and generic nonsignaling bipartite quantum channels.

We denote the set of all nonsignaling resources (of all types) as
$\mathbf{E}_{\rm ns}$, and we take this as our enveloping theory. One could
consider a different set of multipartite channels as one’s enveloping theory,
such as the set $\mathbf{E}_{\rm gcc}$ of multipartite CPTP maps which can be
realized by GPT common causes, or the set $\mathbf{E}_{\rm sig}$ of all CPTP
maps (including signaling maps). Note that $\mathbf{E}_{\rm
gcc}\subset\mathbf{E}_{\rm ns}\subset\mathbf{E}_{\rm sig}$.

It is an interesting open question whether $\mathbf{E}_{\rm gcc}$ is a strict
subset of $\mathbf{E}_{\rm ns}$, or whether the two are equal.

######  Open Question 1.

Do there exist bipartite nonsignaling quantum channels which cannot be
realized by GPT common causes?

If the two sets coincide, then one would have a simple characterization of the
set of quantum channels realizable by a GPT common-cause, which a priori
appears to be a difficult set to characterize. For the special case of box-
type resources, it was shown in Ref. [Barrett(2007)] that GPT common-cause
processes can generate all nonsignaling boxes. On the other hand, it is known
that in generic causal scenarios, the set of correlations (over some observed
classical variables) that can be generated by unobserved GPT common causes is
constrained not only by equality constraints, but also by nontrivial
inequality constraints [Henson  _et al._(2014)Henson, Lal, and Pusey] (beyond
positivity, normalization, and nonsignaling constraints). This makes it
plausible that even in the simple common-cause scenario we are considering,
the set of processes over quantum systems might also be constrained beyond the
nonsignaling constraint.

Apart from its intrinsic interest, characterizing the set $\mathbf{E}_{\rm
gcc}$ is worth exploring because it constitutes the most natural choice for an
enveloping set of processes in our resource theory. Given a nonsignaling
channel that cannot be generated by LOSE operations, there are two possible
explanations: either the channel was generated by a postquantum common cause,
or it was generated by fine-tuned cause-effect influences.222The former view
is in the spirit of Ref. [Wolfe  _et al._(2020)Wolfe, Schmid, Sainz, Kunjwal,
and Spekkens, Allen  _et al._(2017)Allen, Barrett, Horsman, Lee, and
Spekkens], while the latter, endorsed (e.g.) in Ref. [Piani  _et
al._(2006)Piani, Horodecki, Horodecki, and Horodecki], seems to be more
traditional. If the causal structure in a given scenario is a common-cause
structure (with no cause-effect relations between the parties), then only the
postquantum common-cause explanation is viable. Such an assumption on the
causal structure can be grounded, for example, by relativity theory in the
context of space-like separated parties. We will take this approach, and will
refer to valuable LOSE channels as resources of postquantum common cause, or
simply ‘postquantum resources’. As such, the enveloping theory we are really
interested in is $\mathbf{E}_{\rm gcc}$. If there do exist any processes in
$\mathbf{E}_{\rm ns}$ that are not in $\mathbf{E}_{\rm gcc}$, then these could
only be explained by a resource of fine-tuned cause-effect influence, not by a
resource of postquantum common cause, and hence our interpretation of such
resources would be inconsistent. As there is no evidence that such processes
exist, we will for the purpose of this paper assume they do not, allowing us
to take the well-characterized set $\mathbf{E}_{\rm ns}$ as our enveloping
theory.333Subsequent work [John Selby()] by Selby et. al. has shown that
indeed, no such processes exist, answering Open Question 1 in the negative.

The third option for the enveloping set, $\mathbf{E}_{\rm sig}$, is also
sensible, as it is the most general set of channels that is natural in our
context. However, this added generality makes the resulting resource theory
more difficult to characterize. In any case, the choice of the enveloping set
is not especially consequential, since the value of resources depends only on
the free transformations, not on the enveloping set of resources. Hence, we
have chosen the set which leads to the most interesting insights, namely
$\mathbf{E}_{\rm ns}$.

##  3  LOSE resources and transformations

The key component of any resource theory is a set of free processes, that is,
resources and transformations which can be implemented at no cost. Here, we
wish to quantify the notion of postquantum common causes as resources.
Consequently, the set of free processes are taken to be all and only those
that can be generated by quantum common causes—that is, by shared
entanglement. We imagine that all local quantum operations are free, but that
all forms of communication between parties are inaccessible. In other words,
the free processes are defined by the set of local operations and shared
entanglement, or LOSE operations.

A generic free resource is depicted in Fig. 2(a), while free resources of
three specific types are shown in Fig. 2(b), Fig. 2(c), and Fig. 2(d). Any
resource in the enveloping theory that cannot be generated in this manner is
nonfree: a valuable resource which requires a postquantum common-cause to
generate. We provide various explicit examples of valuable resources (of
several different types) in Appendix A.

A generic free transformation is depicted in Fig. 4. We elaborate on the set
of free transformations in Section 3.1, as well as how these free
transformations generate the essential structure of our resource theory, that
is, the preorder which determines the value of every resource in the
enveloping theory. We give three examples of (type-changing) free
transformations in Appendix A.

Figure 2: Dashed wires depict systems with arbitrary type. A free resource (of
arbitrary type) is one which can be generated as in (a), by shared
entanglement (in purple) and local operations (in black). Free boxes are those
which decompose as in (b), and free distributed measurements are those which
decompose as in (c). As a consequence of Theorem 1, all teleportages are free,
since they admit of a decomposition as in (d).

Many types of resources are trivial, in the sense that quantum theory can
generate every resource of that kind, which in turn implies that there are no
postquantum resources of that type.

######  Definition 1.

A resource type is LOSE-trivial if every resource of that type can be
generated by LOSE operations.

Of the resource types in Fig. 1, entangled states are all free (by
definition), while steering assemblages and teleportages are (less obviously)
also all free. We now provide an elegant characterization of the bipartite
resource types that can give a postquantum advantage.

######  Theorem 1.

A bipartite resource type is LOSE-trivial if and only if it has at least one
trivial input or output.

###### Proof.

First, note that there exist examples of boxes which are not LOSE-free; a
well-known example of such a box is the PR box [Popescu and Rohrlich(1994)],
defined in Appendix A. Since every type of bipartite resource whose inputs and
outputs are all nontrivial has the PR box as a special case, it follows that
none of these types is LOSE-trivial.

Conversely, we prove that every bipartite type which has a trivial input or
output is LOSE-trivial. The argument relies on the fact, proven in Ref.
[Eggeling  _et al._(2002)Eggeling, Schlingemann, and Werner] that channels
which are nonsignaling from one party to a second can always be realized
without any causal influences from the first to the second. Consider the case
where an output of the bipartite channel is trivial; taking this to be Alice’s
output without loss of generality, the channel (denoted $\mathcal{E}$) is
depicted on the left-hand-side of Fig. 3(a). As we are assuming that
$\mathcal{E}$ is nonsignaling from Alice’s input to Bob’s output, the result
of Ref. [Eggeling  _et al._(2002)Eggeling, Schlingemann, and Werner] states
that this channel can be decomposed into two channels—one of which has no
output—as shown. But the only CPTP channel with no outputs is the trace, which
factorizes, as shown in the second equality. The resulting channel is
manifestly constructed from LOSE operations, and hence $\mathcal{E}$ can be
generated by LOSE operations. Next, consider the case where an input of the
bipartite channel is trivial; taking this to be Alice’s input without loss of
generality, the channel (denoted $\mathcal{E}^{\prime}$) is depicted on the
left-hand-side of Fig. 3(b). As we are assuming that $\mathcal{E}^{\prime}$ is
nonsignaling from Bob’s input to Alice’s output, the result of Ref. [Eggeling
_et al._(2002)Eggeling, Schlingemann, and Werner] states that this channel can
be decomposed as shown. But this decomposition requires only LOSE operations,
and hence $\mathcal{E}^{\prime}$ can also be generated by LOSE operations.

Figure 3:  (a) Every channel $\mathcal{E}$ of the form on the left-hand-side
which is nonsignaling from Alice to Bob can be decomposed into LOSE
operations, as on the right-hand-side. (b) Every channel
$\mathcal{E}^{\prime}$ of the form on the left-hand-side which is nonsignaling
from Bob to Alice can be decomposed into LOSE operations, as on the right-
hand-side.

Since this sequence of operations is manifestly LOSE-free, it follows that
$\mathcal{E}$ is necessarily LOSE-free, merely by virtue of its type.

∎

Theorem 1 has the following corollary, which is formalized in Section 5.1.

######  Corollary 2.

No GPT can outperform quantum theory at steering or teleporting quantum
states; that is, at any steering games [Schmid  _et al._(2020a)Schmid, Rosset,
and Buscemi, Cavalcanti and Skrzypczyk(2017)] or teleportation games [Schmid
_et al._(2020a)Schmid, Rosset, and Buscemi, Cavalcanti  _et
al._(2017)Cavalcanti, Skrzypczyk, and Šupić, Lipka-Bartosik and
Skrzypczyk(2019)].

###  3.1 Type-changing LOSE operations

In keeping with the discussion above, the set of free transformations are all
and only those that can be generated by LOSE operations. We denote this set by
LOSE, an element of this set by $\tau\in\textup{LOSE}$, and a generic resource
of arbitrary type by $R$, where the action of $\tau$ on $R$ is denoted
$\tau\circ R$.

Using the fact that the most general map taking a quantum channel to a quantum
channel is a quantum comb [Chiribella  _et al._(2008)Chiribella, D’Ariano, and
Perinotti, Chiribella  _et al._(2009)Chiribella, D’Ariano, and Perinotti], we
can decompose an arbitrary (bipartite) free transformation as in Fig. 4. That
is, any free transformation can be generated by each party locally
implementing some quantum comb, where furthermore each local comb has as input
one subsystem of an arbitrary shared entangled state. (Note that this
entangled state can also be fed down the side channel of each local comb, and
hence is effectively an input to both the pre-processings and the post-
processings that constitute the local comb.) The set of LOSE operations is
manifestly convex.

Figure 4:  A generic LOSE transformation (in green) on a bipartite resource
(in black) consists of a local comb for each party, where each local comb may
have as input one subsystem of an arbitrary shared entangled state.

Critically, note that the local combs can change the type of a resource. That
is, the types of the input and output systems of the transformed resource (in
green in Fig. 4) need not be the same types as the input and output systems of
the source resource (in black in Fig. 4). The type of systems of the source
and target resource merely determine what sort of pre-and-post-processings
make up the local combs. We give three specific examples of type-changing
transformations in Appendix A; in particular, see Fig. 9, Fig. 11, Fig. 14,
and the surrounding discussions.

This set of free transformations induces a preorder over the set of resources
in the enveloping theory.

######  Definition 2.

For resources $R$ and $R^{\prime}$ of two arbitrary types, we say that
$R\succeq_{\rm LOSE}R^{\prime}$, or equivalently that
$R\xmapsto{\textup{LOSE}}R^{\prime}$, iff there exists some
$\tau\in\textup{LOSE}$ such that $R^{\prime}=\tau\circ R$.

In such cases, we say that $R$ is at least as postquantum (or ‘as valuable’)
as $R^{\prime}$. This is indeed a preorder, since the identity transformation
is LOSE-free (reflexivity) and since LOSE transformations are closed under
composition (transitivity).

Resources $R$ and $R^{\prime}$ are equally postquantum if they are freely
interconvertible under LOSE transformations; that is, if
$R\xmapsto{\textup{LOSE}}R^{\prime}$ and $R^{\prime}\xmapsto{\textup{LOSE}}R$.
We denote this $R\xleftrightarrow{\textup{LOSE}}R^{\prime}$ and say that $R$
and $R^{\prime}$ are in the same LOSE equivalence class.

Resources $R$ and $R^{\prime}$ are incomparable if neither can convert to the
other under LOSE transformations.

###  3.2 Comparing postquantumness across resource types

With this resource theory framework in hand, one can go about quantitatively
comparing the postquantumness of resources, even across different types. In
this section, we discuss what can be said about the relative value of some
known examples of postquantum resources (of various different types) from the
literature.

The resources we discuss here are defined explicitly in Appendix A. There are
five of these, of four different types, and we will refer to them based on
their type and the initials of the authors who first considered them. These
include the PR box [Popescu and Rohrlich(1994)], the PHHH ensemble-preparation
[Piani  _et al._(2006)Piani, Horodecki, Horodecki, and Horodecki], the SHSA
Bob-with-input steering assemblage [Belén Sainz  _et al._(2019)Belén Sainz,
Hoban, Skrzypczyk, and Aolita], and two nonsignaling channels of type
$\mathsf{QQ}\\!\\!\to\\!\\!\mathsf{QQ}$, namely the BGNP channel [Beckman  _et
al._(2001)Beckman, Gottesman, Nielsen, and Preskill] and the DFP channel
[D’Ariano  _et al._(2011)D’Ariano, Facchini, and Perinotti]. In Appendix A, we
prove that four conversions between these resources are possible using LOSE
operations, which in turn teaches us about their relative postquantumness. For
instance, we show that the PR box is interconvertible with the PHHH ensemble-
preparation; hence, the two are equally postquantum. Additionally, we show
that the PHHH ensemble-preparation can be converted into the SHSA Bob-with-
input-assemblage and the DFP channel. By transitivity, of course, this also
implies that the PR box can be converted into either of these resources.

These results (and some remaining open questions) are summarized in Fig. 5.

Figure 5:  We prove that the PR box and the PHHH ensemble-preparation are
interconvertible, and either of these can be converted into the SHSA Bob-with-
input assemblage or into the DFP assemblage. The remaining conversion
relations are unknown (although see the main text for some relevant facts).

We were not able to resolve the remaining conversion relations, indicated by
the blue question marks in the figure. However, there are some hints as to
whether or not some of these conversions are possible. For example, the form
of the BGNP channel444E.g., the fact that the dephasing that is applied must
be identical in three subspaces but distinct in the fourth. suggests that it
can also be obtained from the PR box. Also, it was shown in Ref. [D’Ariano
_et al._(2011)D’Ariano, Facchini, and Perinotti] that the DFP channel exhibits
a non-maximal violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality,
which suggests that one might be able to construct a monotone from the CHSH
functional (analogous to the constructions in Ref. [Wolfe  _et
al._(2020)Wolfe, Schmid, Sainz, Kunjwal, and Spekkens]), such that the value
achieved by the PR box is higher than that achieved by the DFP channel. If
this were the case, it would establish that the DFP channel could not be
converted to the PR box, which in turn would prove that the PR box is strictly
more postquantum. Finally, we note that Ref. [Belén Sainz  _et al._(2019)Belén
Sainz, Hoban, Skrzypczyk, and Aolita] establishes that the SHSA Bob-with-input
assemblage cannot be converted to any postquantum box (including the PR box)
using operations which can be extended to include all local operations and
shared randomness. This implies that, if the SHSA Bob-with-input assemblage
can be converted to the PR box, the conversion will definitely require shared
entanglement.

These results give us some insight into the structure of the type-independent
resource theory of LOSE. First of all, it shows that the property of
entanglement-breaking is not critical for postquantumness: the PR box, despite
being completely entanglement-breaking, is at least as postquantum as the DFP
channel, which is not entanglement-breaking. Additionally, we see that there
are no examples of postquantum channels that have been proven to be
inequivalent to the PR box, nor that have been proven to be strictly more
resourceful than the PR box. Hence, it remains a possibility that all
postquantum channels can be generated from box-type postquantum channels. This
would be a striking result, implying that all manifestations of
postquantumness in nature are extremely simple—and limited. It would also
imply that the box-type encodes all other types, according to the definition
of encoding in the next section. While we suspect that not all postquantum
channels can be generated from postquantum boxes, this remains a critical open
question.

######  Open Question 2.

Do there exist LOSE-nonfree channels which cannot be freely generated using
LOSE operations and arbitrary nonsignaling boxes?

Finally, we learn that resources of more limited type (like boxes) can
sometimes exhibit more postquantumness than resources of a more general type
(like generic quantum channels). We elaborate on this in the next section.

##  4 Encoding postquantumness of one type into postquantumness of another
type

It is interesting to consider transformations from one type of resource to
another which do not degrade the value of the initial resource. We will prove
some general results that do not rely on the particular details of the initial
resource, but depend only on its type. To do so, we require the following
definition, which defines a preorder over types.

######  Definition 3.

Global type $T$ encodes global type $T^{\prime}$, denoted $T\succeq_{\rm
type}T^{\prime}$, if for every resource $R^{\prime}$ of type $T^{\prime}$,
there exists at least one resource $R$ of type $T$ such that
$R\xleftrightarrow{\textup{LOSE}}R^{\prime}$.

In other words, there exists at least one resource of the higher type in every
equivalence class of resources of the lower type. The higher type is able to
exhibit all the same manifestations of postquantum common-cause
resourcefulness as the lower type. Furthermore, we will show that some
resource of the higher type is always able to perform at least as well as any
given resource of the lower type at any distributed game.

###  4.1 Semiquantum channels encode all types of postquantumness

We now show that resources with only classical outputs are able to encode the
postquantumness of all other types of resources. We focus on the bipartite
case for simplicity, but the result generalizes immediately to arbitrarily
many parties.

######  Theorem 3.

Every bipartite resource (of arbitrary type) can be converted to a bipartite
distributed measurement (of type $\mathsf{QQ}\\!\\!\to\\!\\!\mathsf{CC}$)
which is in the same equivalence class.

The proof is constructive, and is given in Appendix B.1. Furthermore, one can
immediately see from the proof that the transformation from the given resource
to a distributed measurement in its equivalence class can be achieved by a
single fixed transformation, independent of the specific initial resource.

This fact implies that all resources of arbitrary types can have their
postquantumness quantified in a measurement-device-independent [Cavalcanti
_et al._(2013)Cavalcanti, Hall, and Wiseman] fashion, since classical systems
may be probed even without well-characterized quantum measurement devices.

###  4.2 Partition-type encodings

There are a large variety of global types, especially in multipartite
scenarios. A useful tool for studying the preorder over these is the notion of
a preorder over partition-types, that is, over the nine types
$T_{A}:=\mathbf{Type}[\mathsf{X}]\\!\\!\to\\!\\!\mathbf{Type}[\mathsf{A}]$ of
(e.g.) Alice’s share of a resource.

######  Definition 4.

Partition-type $T_{1}$ is above partition-type $T_{1}^{\prime}$, denoted
$T_{1}\succeq_{\rm type}T_{1}^{\prime}$, if for every resource $R^{\prime}$ of
type $T_{1}^{\prime}T_{2}...T_{n}$ (as one ranges over all $T_{2},...,T_{n}$
and all $n$), there exists a resource $R$ of type $T_{1}T_{2}...T_{n}$
satisfying $R^{\prime}\xleftrightarrow{\textup{LOSE}}R$.

This is useful because if every partition-type of a given global type is
higher than the corresponding partition-type of a second global type, then the
first global type is higher in the preorder over global types. Hence, it is
useful to systematically characterize which partition-types encode which
others. We summarize the known (and unknown) encodings in Table 6.

Figure 6:  A green check mark indicates that the column partition-type $T$ is
higher in the order than the row partition-type $T^{\prime}$, that is,
$T\succeq_{\rm type}T^{\prime}$. The question marks indicate open questions.
In two cases (indicated by $\Leftrightarrow$), a pair of these must have the
same answer.

Clearly, partition-type $\mathsf{Q}\\!\\!\to\\!\\!\mathsf{Q}$ is at the top of
the preorder over partition-types, since every other partition-type is a
special case of this partition-type. We refer to such trivial encodings as
embeddings of one type into another.

An immediate consequence of Theorem 1 is that the partition-types
$\mathsf{I}\\!\\!\to\\!\\!\mathsf{I}$, $\mathsf{I}\\!\\!\to\\!\\!\mathsf{C}$,
$\mathsf{I}\\!\\!\to\\!\\!\mathsf{Q}$, $\mathsf{C}\\!\\!\to\\!\\!\mathsf{I}$,
and $\mathsf{Q}\\!\\!\to\\!\\!\mathsf{I}$ are all at the bottom of the
preorder over partition-types.

Noting that the proof of Theorem 3 required only local operations on each
individual party, it follows immediately that the distributed measurement (or
semiquantum) partition-type is at the top of the preorder. Since partition-
types $\mathsf{Q}\\!\\!\to\\!\\!\mathsf{Q}$ and
$\mathsf{Q}\\!\\!\to\\!\\!\mathsf{C}$ encode each other, they are in the same
equivalence class at the top of the preorder.

There remain several open questions, as indicated in the table by question
marks.

First, consider the questions of whether partition-type
$\mathsf{C}\\!\\!\to\\!\\!\mathsf{C}$ encodes either
$\mathsf{Q}\\!\\!\to\\!\\!\mathsf{C}$ or
$\mathsf{Q}\\!\\!\to\\!\\!\mathsf{Q}$. Recalling that
$\mathsf{Q}\\!\\!\to\\!\\!\mathsf{C}$ and
$\mathsf{Q}\\!\\!\to\\!\\!\mathsf{Q}$ are in the same equivalence class, the
answer to these two questions must be the same. This question has deep
implications for the task of characterizing resources in practice, since a
positive answer would imply the possibility of device-independent
[Scarani(2019)] characterizations of the postquantumness of arbitrary
resources.

Second, consider the questions of whether partition-type
$\mathsf{C}\\!\\!\to\\!\\!\mathsf{Q}$ encodes either
$\mathsf{Q}\\!\\!\to\\!\\!\mathsf{C}$ or
$\mathsf{Q}\\!\\!\to\\!\\!\mathsf{Q}$. For the reason given just above, the
answer to these two questions must again be the same. A positive answer to
this question would imply the possibility of preparation-device-independent
characterizations of the postquantumness of arbitrary resources.

Finally, consider the question of whether partition-type
$\mathsf{C}\\!\\!\to\\!\\!\mathsf{C}$ encodes partition-type
$\mathsf{C}\\!\\!\to\\!\\!\mathsf{Q}$. A positive answer to this question
would imply the possibility of device-independent characterizations of the
postquantumness of any resources whose inputs are all classical. It is also
relevant for the question of whether or not postquantum Bob-with-input
steering assemblages constitute a novel form of postquantumness, a question
first raised in Refs. [Hoban and Sainz(2018), Belén Sainz  _et al._(2019)Belén
Sainz, Hoban, Skrzypczyk, and Aolita]. There, the authors argued that some
Bob-with-input assemblages are indeed novel, in the sense that they cannot be
converted into any postquantum box by applying sets of quantum measurements to
the states in the Bob-with-input assemblage.

In light of our framework, this argument is not conclusive. Our framework
provides a more principled notion of what it means for a given type $T$ to be
‘truly novel’ with respect to another type (in this case, box-type): there
should exist at least one resource of type $T$ which is in an LOSE-equivalence
class which is not shared by any box-type resource. In other words, it should
be that the box-type does not encode type $T$. Refs. [Hoban and Sainz(2018),
Belén Sainz  _et al._(2019)Belén Sainz, Hoban, Skrzypczyk, and Aolita] do not
resolve this open question, since they do not establish that there exist
postquantum Bob-with-input assemblages that cannot be converted to postquantum
boxes by LOSE operations. Rather, the set of operations allowed in their proof
(sets of quantum measurements on Bob’s steered state) is a strict subset of
the full set of LOSE operations from Bob-with-input assemblages to boxes. It
is not difficult to extend their argument to show that the nonfree Bob-with-
input assemblage of Eq. (3)) cannot generate any nonfree box using local
operations and shared randomness. However, it remains possible that under LOSE
operations, one could generate some nonfree box.

Indeed, we prove in Appendix A that the PR box is at least as postquantum as
the primary example of a postquantum Bob-with-input assemblage in Ref. [Belén
Sainz  _et al._(2019)Belén Sainz, Hoban, Skrzypczyk, and Aolita]. From this
fact and the point of view endorsed above, it remains an open question whether
or not there exist forms of postquantum steering that are truly novel.

######  Open Question 3.

Do there exist Bob-with-input steering assemblages that constitute genuinely
novel forms of postquantumness—that is, assemblages whose LOSE-equivalence
class contains no postquantum boxes?

##  5 Distributed games

Resources of various types are often used to perform distributed information-
processing tasks, such as teleportation, certified randomness generation via
violation of Bell inequalities, and so on. Conversely, distributed experiments
such as entanglement witnessing, nonlocal games, semiquantum games, steering
experiments, and teleportation games are often used to characterize the sense
in which various processes are resources. Ref. [Schmid  _et al._(2020a)Schmid,
Rosset, and Buscemi] defined an abstract framework of distributed games which
subsumes all of these types of tasks as special cases. Within this framework,
there is a natural set of games associated to every global type $T$.

######  Definition 5.

For a given global type $T$, we define a distributed $\mathbf{T}$-game as a
linear map from resources of type $T$ to the real numbers. A resource of type
$T$ is then naturally said to be a strategy for a $T$-game.

The referee and players are all assumed to know the complete specification of
the game (that is, the types of inputs and outputs, as well as the particular
linear map to the real numbers) in advance. To play the game, the players
process their initial shared resources (which may be of arbitrary types) using
arbitrary LOSE processes, ultimately generating a resource of the appropriate
type for the game—that is, a strategy for the game. Based only on the strategy
they generate, the referee assigns them a score. We depict three types of
$T$-games abstractly in Fig. 7, where a closed diagram is to be interpreted as
a real number, which in this case is the score assigned to the strategy (in
light grey) by the particular game (in black). Note that the processing of the
player’s initial resource into a strategy is not depicted here.

Figure 7:  Some games and their strategies. (a) Boxes are strategies for
nonlocal games. (b) Distributed measurements are strategies for semiquantum
games. (c) Teleportages are strategies for teleportation games. As we prove,
all strategies for teleportation games are LOSE-free.

See Ref. [Schmid  _et al._(2020a)Schmid, Rosset, and Buscemi] for a
prescription by which one can experimentally implement any game of any type,
as well as for a discussion of how one can interpret these games less
abstractly.555In Ref. [Schmid  _et al._(2020a)Schmid, Rosset, and Buscemi],
the players are only allowed LOSR operations to generate their strategies.
Since the resources we consider are ultimately just quantum channels, one can
also give a convenient mathematical decomposition of the map corresponding to
any such game in terms of the trace of the given resource’s Choi state with an
operator in the dual space in which that Choi state lives; see e.g. Refs.
[Rosset  _et al._(2018)Rosset, Buscemi, and Liang, Takagi and Regula(2019),
Uola  _et al._(2020)Uola, Kraft, and Abbott]. For our purposes, however, we
merely note that the abstract definition of games above subsumes most of the
usual distributed tasks, including entanglement witnesses, Bell tests,
steering witnesses, teleportation games, and so on.

Given a shared resource of any type, the distributed parties can apply
arbitrary free operations to transform that resource into a strategy for the
game they are playing, with the aim of optimizing their score. Denote a
particular game of type $T$ by $G_{\\!T}$, and the score it assigns to a
strategy $\mathcal{E}$ (a specific resource of type $T$) by
$G_{\\!T}(\mathcal{E})$.

######  Definition 6.

The optimal performance of a resource $R$ (of arbitrary type) with respect to
a game $G_{\\!T}$ of arbitrary type $T$ is given by

$\omega_{G_{\\!T}}(R)=\max_{\tau:\mathbf{Type}[R]\\!\to\\!T}G_{\\!T}(\tau\circ
R),$ (1)

where $\tau\in\textup{LOSE}$ takes resources of the given type
$\mathbf{Type}[R]$ to type $T$.

Next, we define a second preorder over resources, based on their performance
at games of a given type. Denote the set of all games of type $T$ by
$\mathcal{G}_{\\!T}$.

######  Definition 7.

For resources $R$ and $R^{\prime}$ of different and arbitrary type, we say
that $R\succeq_{\mathcal{G}_{\\!T}}R^{\prime}$ iff
$\omega_{G_{\\!T}}(R)\geq\omega_{G_{\\!T}}(R^{\prime})$ for every
$G_{\\!T}\in\mathcal{G}_{\\!T}$.

This is not to be confused with the preorder over resources induced by the
free LOSE-transformations, which is generally a distinct preorder. However,
Theorem 4 below provides a condition under which the two preorders coincide.

###  5.1 Implications from the type of a resource to its performance at games

We now establish various connections between the type of a resource and how
well it can perform at distributed games when the players make use of LOSE
operations.

First, consider games of a given type $T$ that is LOSE-trivial. Since every
resource of this type (and hence every strategy for such games) can be
generated freely, one can achieve the optimal score for any such game at no
cost. Hence, postquantum resources are not useful for them, and conversely, no
game of type $T$ can be used to witness the postquantumness of any resource.
The two most interesting examples of this were stated in Corollary 2, which
stated that there is no postquantum advantage for steering or teleporting
quantum states, as made quantitative by steering games [Schmid  _et
al._(2020a)Schmid, Rosset, and Buscemi, Cavalcanti and Skrzypczyk(2017)] or
teleportation games [Schmid  _et al._(2020a)Schmid, Rosset, and Buscemi,
Cavalcanti  _et al._(2017)Cavalcanti, Skrzypczyk, and Šupić, Lipka-Bartosik
and Skrzypczyk(2019)]. These two facts were pointed out in Refs. [Hoban and
Sainz(2018), Šupić  _et al._(2019)Šupić, Skrzypczyk, and Cavalcanti]. They
follow more formally as an immediate consequence of Theorem 4 below.

Note that if it were the case that the task of teleportation were only
concerned with establishing an identity channel between two parties who share
entanglement, then it is clear that quantum theory can achieve this task
perfectly, and the observation that postquantum common causes give no
advantage would be trivial. However, this observation is not trivial, since
many teleportation tasks are not of this form, as pointed out in Refs.
[Cavalcanti  _et al._(2017)Cavalcanti, Skrzypczyk, and Šupić, Lipka-Bartosik
and Skrzypczyk(2019)].

On the other hand, any nonsignaling resource of any type $T$ which is nonfree
in our resource theory provides a postquantum advantage for at least one game
of type $T$. This follows as an immediate consequence of the following (more
general) theorem.

######  Theorem 4.

If $T\succeq_{\rm type}T^{\prime}$, then for resources $R_{1},R_{2}$ of type
$T^{\prime}$, $R_{1}\succeq_{\rm LOSE}R_{2}$ iff
$R_{1}\succeq_{\mathcal{G}_{\\!T}}R_{2}$.

The proof is given in Appendix C.

In other words, if type $T$ is above type $T^{\prime}$, then for resources of
type $T^{\prime}$, the preorder over resources defined by LOSE-conversion
coincides with the preorder over resources defined by performance at all games
of type $T$. This implies that games of a higher type perfectly characterize
the postquantumness of resources of a lower type. Conversely, it implies that
every nonfree resource is useful for some game of the same type, and also for
some game of every higher type.

Combining Theorem 4 with Theorem 1, it follows that there are postquantum
advantages for nonlocal games, semiquantum games, measurement-device-
independent steering games, channel steering games, Bob-with-input steering
games, and so on. To our knowledge, these advantages have only been studied in
the context of nonlocal games [Brunner  _et al._(2014a)Brunner, Cavalcanti,
Pironio, Scarani, and Wehner] and Bob-with-input-steering [Belén Sainz  _et
al._(2019)Belén Sainz, Hoban, Skrzypczyk, and Aolita], while the rest remain
to be studied.

Combining Theorem 4 with Theorem 3, it follows that the set $\mathcal{G}_{\rm
SQ}$ of all semiquantum games witnesses (and exactly quantifies) the
resourcefulness of any nonfree resource.

######  Corollary 5.

For any resources $R$ and $R^{\prime}$ (which may be of arbitrary and
different types), $R\succeq_{\rm LOSE}R^{\prime}$ if and only if
$R\succeq_{\mathcal{G}_{\rm SQ}}R^{\prime}$.

We close by noting a useful result (e.g., it is used in the Appendix for
proving Theorem 4), which states that if one resource outperforms a second at
all possible games of a given type, then it can also generate any specific
strategy of that type which the second resource can generate.

######  Theorem 6.

For resources $R$ and $R^{\prime}$ of arbitrary types and a resource
$\mathcal{E}_{T}$ of arbitrary type $T$,
$R\succeq_{\mathcal{G}_{\\!T}}R^{\prime}$ iff
$R^{\prime}\xmapsto{\textup{LOSE}}\mathcal{E}_{T}\implies
R\xmapsto{\textup{LOSE}}\mathcal{E}_{T}$, that is, iff any strategy
$\mathcal{E}_{T}$ for games of type $T$ that can be freely generated from
$R^{\prime}$ can also be freely generated from $R$.

##  6 Open questions and Conclusion

We have presented the type-independent resource theory of local operations and
shared entanglement, or LOSE operations. We showed how this unifies various
types of postquantum resources, and discussed how these resources can be
transformed into each other, regardless of their types. This allows us to
rigorously quantify the amount of postquantumness in a given resource, and to
quantitatively compare the postquantumness of arbitrary resources of arbitrary
types. Next, we began the systematic characterization of which types of
resources are able to express all the same manifestations of postquantumness
as which other types, as made formal by the idea of encodings of types of
resources. We then discussed how players can use any given resource to
generate strategies for distributed games of arbitrary types, and how the type
of a given resource helps determine what advantage, if any, the players can
hope to gain over strategies that are achievable by quantum common causes.

There remain a great deal of interesting questions to be understood about this
resource theory; we have already highlighted some of these in the main text.
Attaining a complete understanding of postquantum common cause resources will
require answering all of the usual resource theoretic questions: what is the
structure of the single-copy deterministic preorder, what monotones and
witnesses provide valuable information about this preorder, what asymptotic
and catalytic conversions are possible, and so on. A modest starting point
would be to finish characterizing which conversions between known postquantum
resources (such as those in Appendix A) are possible, and to find type-
independent monotones that witness the interconversions that are not possible.
Note that for the type-independent LOSR resource theory, Ref. [Rosset  _et
al._(2020)Rosset, Schmid, and Buscemi] developed technical tools for answering
such questions, and it seems likely that many of these techniques (such as the
robustness monotones defined therein) would be transferrable with minor
modifications to the LOSE resource theory. It would also be interesting to
study detailed properties of the LOSE preorder even for specific types of
resources, in analogy to what was done in Ref. [Wolfe  _et al._(2020)Wolfe,
Schmid, Sainz, Kunjwal, and Spekkens] for box-type resources in the resource
theory of LOSR. It would also be helpful to find some algorithmic methods for
determining if a given channel is LOSE-free or not; see Appendix C for three
partial results along these lines from Ref. [Beckman  _et al._(2001)Beckman,
Gottesman, Nielsen, and Preskill]. See also Appendix A.6 for an example of a
well-known channel which we conjecture is LOSE-nonfree. Another interesting
question is whether the generalized notion of self-testing of arbitrary types
of resources introduced in Ref. [Schmid  _et al._(2019b)Schmid, Fraser,
Kunjwal, Sainz, Wolfe, and Spekkens] could be applied in this resource theory.
Finally, it is an interesting question whether or not resource theories like
ours will be of use at constraining the set of quantum correlations (or more
general processes) from physical principles.

##  7 Acknowledgments

D.S. is supported by a Vanier Canada Graduate Scholarship. Research at
Perimeter Institute is supported in part by the Government of Canada through
the Department of Innovation, Science and Economic Development Canada and by
the Province of Ontario through the Ministry of Colleges and Universities. MJH
acknowledges the FQXi large grant The Emergence of Agents from Causal Order.
This publication was made possible through the support of a grant from the
John Templeton Foundation. The opinions expressed in this publication are
those of the authors and do not necessarily reflect the views of the John
Templeton Foundation.

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##  Appendix A Explicit examples of nonfree resources

The majority of this article focuses on what can be said about resources by
considering only their type, and hence does not focus on particular instances
of resources. Nonetheless, it is illustrative to see some concrete examples of
postquantum resources. We now describe five explicit examples of four
different types. We denote Alice’s input (output) system by $A$ ($X$) and
Bob’s input (output) system by $B$ ($Y$). If any of these systems are
classical, then we will identify the corresponding capital letter with the
classical variable corresponding to that system, and will denote the possible
values that it can take by $a,x,b,y$, respectively, where
$a,x,b,y\in\\{0,1\\}$ unless otherwise specified. We will also denote the
three Pauli matrices by $\sigma_{1},\sigma_{2},$ and $\sigma_{3}$.

###  A.1 The PR box

Our first explicit example is the celebrated Popescu-Rohrlich (PR) box
[Popescu and Rohrlich(1994)], defined as the resource that achieves the
logically maximal violation of the (CHSH) inequality [Clauser  _et
al._(1969)Clauser, Horne, Shimony, and Holt]. The PR box is a resource of type
$\mathsf{CC}\\!\\!\to\\!\\!\mathsf{CC}$, and is completely specified by the
following conditional probability distribution:

$\mbox{p}(ab|xy)=\begin{cases}1/2\quad&\text{if }a\oplus b=xy,\\\
0\quad&\text{otherwise}.\end{cases}$ (2)

The PR box is well-known precisely because it cannot be generated by local
operations and shared entanglement. More specifically, it achieves a value of
4 for the CHSH functional, while the largest value achievable by local
measurements on a shared entangled state is $2\sqrt{2}$.

###  A.2 The PHHH ensemble-preparation

Our second example of a postquantum resource is an ensemble-preparing channel
introduced in Ref. [Piani  _et al._(2006)Piani, Horodecki, Horodecki, and
Horodecki], which we refer to as the PHHH ensemble-preparation.666 Ref. [Piani
_et al._(2006)Piani, Horodecki, Horodecki, and Horodecki] presented this
resource as one with quantum inputs, but since these are immediately dephased
in the computational basis, one can simply treat them as classical. Note also
that an incoherent version of this channel was shown to be postquantum in Ref.
[Beckman  _et al._(2001)Beckman, Gottesman, Nielsen, and Preskill]. This
resource has type $\mathsf{CC}\\!\\!\to\\!\\!\mathsf{QQ}$, and is constructed
as follows. If the product of the inputs $xy=0$, then the channel outputs the
Bell state $\ket{\phi^{+}}=(\ket{00}+\ket{11})/\sqrt{2}$, but if $xy=1$ it
outputs the Bell state $\ket{\psi^{+}}=(\ket{01}+\ket{10})/\sqrt{2}$.

A quantum circuit that realizes the PHHH ensemble-preparation using classical
communication from Bob to Alice is shown in Fig. 8.

Figure 8:  A quantum circuit that realizes the PHHH ensemble-preparation. Note
that this circuit requires communication from Bob to Alice, and hence does not
constitute an LOSE realization of the channel. Here and throughout, controlled
operations are denoted by a horizontal line ending in a dot on the target
system.

Converting the PR box into the PHHH ensemble-preparation and back— To see that
the PHHH ensemble-preparation is LOSE-nonfree, we show that it is in the same
equivalence class as the PR box; that is, the two are interconvertible using
LOSE operations.

To obtain a PR box from the PHHH ensemble-preparation, it suffices for each
party to locally dephase their output system in the computational basis,
resulting in a $\mathsf{CC}\\!\\!\to\\!\\!\mathsf{CC}$ channel with exactly
the PR box correlations in the computation basis. To obtain the PHHH ensemble-
preparation from the PR box, the two parties begin with maximally entangled
state $\ket{\phi^{+}}$, and each party performs a local Pauli $\sigma_{1}$
transformation if the output they receive from the PR box is $1$. When $xy=0$,
the PR box outputs perfectly correlated bits, and hence either both parties
perform the transformation, or neither party does; in either case, the state
$\ket{\phi^{+}}$ is invariant. When $xy=1$, however, the PR box outputs
perfectly anticorrelated bits, and so only one party performs the Pauli
$\sigma_{1}$ transformation; in either case, the resulting state is
$\ket{\psi^{+}}$. The overall channel, depicted in Fig. 9, is exactly the PHHH
ensemble-preparation.

Figure 9:  Generating the PHHH ensemble-preparation from the PR box using LOSE
operations (green).

###  A.3 The SHSA Bob-with-input steering assemblage

Our third example is a postquantum Bob-with-input (BWI) steering assemblage,
of type $\mathsf{CC}\\!\\!\to\\!\\!\mathsf{CQ}$, that was introduced in Ref.
[Belén Sainz  _et al._(2019)Belén Sainz, Hoban, Skrzypczyk, and Aolita]. We
refer to it as the SHSA BWI-assemblage. A Bob-with-input steering scenario
generalizes the traditional steering scenario by allowing Bob a classical
(rather than trivial) input. A BWI-assemblage is fully specified by the set
$\\{\rho_{a|xy}\\}_{a,x,y}$ of unnormalized quantum states (on system $B$)
that Bob can be steered into, indexed by the values $X=x$ ($A=a$) of Alice’s
classical input (output) and the value $Y=y$ of Bob’s classical input.

The SHSA BWI-assemblage is defined in Ref. [Belén Sainz  _et al._(2019)Belén
Sainz, Hoban, Skrzypczyk, and Aolita] by

$\rho_{a|xy}=\frac{1}{4}\left(I+(-1)^{a}\sigma_{x+1}\right)^{T^{y}}.$ (3)

Here, $\sigma_{x}$ are the three Pauli matrices, depending on the value
$x\in\\{0,1,2\\}$, and $T^{y}$ is the transpose (in the computational basis)
on Bob’s state when $y=1$ and the identity map when $y=0$.

Mathematically (but not physically), this assemblage can be depicted as in
Fig. 10, where a measurement of the Pauli observable $(-1)^{x}\sigma_{x+1}$
(depending on the input value $x$, and with outcomes $0$ and $1$ corresponding
to eigenvalues $+1$ and $-1$, respectively) is performed on half of a Bell
state $\ket{\phi^{+}}$, and $T^{y}$ is applied on the other half, depending on
the input value $y$. The set of states on $B$ conditioned on particular values
$X=x$, $A=a$, and $Y=y$ is given by Eq. (3). Crucially, the transpose is not a
valid quantum channel, since it is not completely positive. Hence, Fig. 10
does not represent a valid quantum circuit, much less a set of LOSE
operations.

Figure 10:  An (unphysical) depiction of the assemblage of Eq. (3). Note that
this is not a quantum circuit, much less an LOSE process, since the transpose
is not a CPTP map.

Fig. 10 provides some intuition for why the SHSA BWI-assemblage is nonfree, as
the transpose map is not physical. The formal proof that it is nonfree is more
elaborate, and is given in the supplementary material of Ref. [Belén Sainz
_et al._(2019)Belén Sainz, Hoban, Skrzypczyk, and Aolita].

Converting the PHHH ensemble-preparation into the BWI-assemblage— Next, we
prove that the PHHH ensemble-preparation can be converted into the SHSA BWI-
assemblage using LOSE operations. The conversion is shown in Fig. 11. By
transitivity, this also proves that the PR box can also be converted into the
SHSA BWI-assemblage.

Figure 11:  Generating the SHSA BWI-assemblage from the PHHH ensemble-
preparation using LOSE operations (green).

First, Alice maps her given input $x\in\\{0,1,2\\}$ to $x{\rm\ mod\ }2$ and
uses this as her input to the PHHH ensemble-preparation (but keeps a copy of
$x$ as an input to her post-processings). On her quantum output, Alice then
measures the Pauli observable $(-1)^{x}\sigma_{x+1}$, depending on her input
value $x$, where the outcomes $0$ and $1$ correspond to eigenvalues $+1$ and
$-1$, respectively. It is easy to verify that this reproduces the SHSA BWI-
assemblage, e.g. by considering the 12 possible states generated for the 12
possible tuples $(a,x,y)$.

###  A.4 The BGNP channel

Our fourth example is a generic nonsignaling channel proposed in [Beckman  _et
al._(2001)Beckman, Gottesman, Nielsen, and Preskill], which we term the BGNP
channel.777Strictly speaking, it is a family of channels indexed by a unitary
$U_{B}$. It is a resource of type $\mathsf{QQ}\\!\\!\to\\!\\!\mathsf{QQ}$,
where every input and output has dimension 4. Let
$\\{\ket{0}_{A},\ket{1}_{A},\ket{2}_{A},\ket{3}_{A}\\}$ be a basis for Alice’s
Hilbert space, and let $\\{\ket{0}_{B},\ket{1}_{B},\ket{2}_{B},\ket{3}_{B}\\}$
be a basis for Bob’s Hilbert space. For Alice as well as for Bob, we refer to
the subspace spanned by $\\{\ket{0},\ket{1}\\}$ as the first qubit, labeled
$f$, and the subspace spanned by $\\{\ket{2},\ket{3}\\}$ as the second qubit,
labeled $s$. One can form a basis for the 16-dimensional joint Hilbert space
by considering the 4 Bell states on each of the two-qubit subspaces formed by
taking one of Alice’s qubits ($f$ or $s$) together with one of Bob’s qubits
($f$ or $s$); explicitly, this is the basis

$\Big{\\{}\ket{\phi_{ff}^{\pm}}\\!,\ket{\psi_{ff}^{\pm}}\\!,\ket{\phi_{fs}^{\pm}}\\!,\ket{\psi_{fs}^{\pm}}\\!,\ket{\phi_{sf}^{\pm}}\\!,\ket{\psi_{sf}^{\pm}}\\!,\ket{\phi_{ss}^{\pm}}\\!,\ket{\psi_{ss}^{\pm}}\Big{\\}}.$
(4)

It follows that

$\displaystyle\Big{\\{}\ket{\phi_{ff}^{\pm}}\\!,$
$\displaystyle\ket{\psi_{ff}^{\pm}}\\!,\ket{\phi_{fs}^{\pm}}\\!,\ket{\psi_{fs}^{\pm}}\\!,\ket{\phi_{sf}^{\pm}}\\!,\ket{\psi_{sf}^{\pm}}\\!,$
(5) $\displaystyle(I_{A}\otimes U_{B})\ket{\phi_{ss}^{\pm}}\\!,(I_{A}\otimes
U_{B})\ket{\psi_{ss}^{\pm}}\Big{\\}}$

is also a basis, since it amounts to merely rotating the Bell basis of the
$ss$ subspace by a unitary $U_{B}$ acting (only) on Bob’s second qubit. This
‘twisted’ basis is depicted schematically in Fig. 12.

Figure 12:  The basis of 16 states defined by Eq. (5).

The BGNP channel is defined as the channel which is fully dephasing in this
latter basis (that is, as the channel whose Kraus operators are the projectors
onto the states in this basis). It is proved in Ref. [Beckman  _et
al._(2001)Beckman, Gottesman, Nielsen, and Preskill] that this channel cannot
be performed using LOSE operations. The proof relies on a particular necessary
condition for a channel to be LOSE-free (which we repeat in Appendix C) which
is not satisfied by the channel.

Ref. [Beckman  _et al._(2001)Beckman, Gottesman, Nielsen, and Preskill] also
provides an intuition for why this channel cannot be achieved via LOSE
operations. The authors note that the channel which is dephasing in the Bell
basis for two qubits can be performed using only local operations and shared
randomness, as can the Bell measurement rotated by $U_{B}$. However, to
implement the measurement defined by Eq. (5), the parties would have to know
whether or not the state lies in the $ss$ subspace (to determine which of
these two Bell measurements to perform), and neither Alice nor Bob has access
to this information in the absence of communication.

As an aside: this channel is defined by a projective measurement, and it
always outputs a quantum state which is diagonal in a fixed basis. As such,
one might expect that it can be viewed as a distributed measurement, that is,
a resource of type $\mathsf{QQ}\\!\\!\to\\!\\!\mathsf{CC}$. However, it is not
natively a resource of that type. This is because the output state is not
diagonal in the computational basis, nor even in a product basis from which
Alice and Bob can read off a classical outcome locally.

###  A.5 The DFP channel

All of the previous examples are entanglement-breaking. Our fifth example,
which we refer to as the DFP channel, is a postquantum channel of type
$\mathsf{QQ}\\!\\!\to\\!\\!\mathsf{QQ}$ that was introduced in Ref. [D’Ariano
_et al._(2011)D’Ariano, Facchini, and Perinotti] as an example of an LOSE-
nonfree resource which is not entanglement-breaking. (Note that all bipartite
resources of types other than $\mathsf{QQ}\\!\\!\to\\!\\!\mathsf{QQ}$ are
necessarily entanglement-breaking.) The example is essentially a coherent
mixture of a bipartite identity channel with the PHHH ensemble-preparation
defined above (but where the bipartite quantum state used to implement the
coherent mixture is also an output of the channel).

Explicitly, the DFP channel can be defined by the circuit in Fig. 13 for
$\alpha=\frac{1}{6}$.

Figure 13:  A quantum circuit that realizes the DFP channel when
$\alpha=\frac{1}{6}$. Note that this circuit requires communication from Bob
to Alice, and hence does not constitute an LOSE realization of the channel.

Here, the Pauli unitary $\sigma_{1}$ is controlled on three systems; namely,
it is only applied if the outcomes of both computational basis measurements
(labeled by ‘$\sigma_{3}$ mmt’) are $1$ and also if the quantum system on
which it is controlled is in state $\ket{1}$. More details on this channel,
including its Kraus representation, can be found in Ref. [D’Ariano  _et
al._(2011)D’Ariano, Facchini, and Perinotti].

For $\alpha=\frac{1}{6}$, Ref. [D’Ariano  _et al._(2011)D’Ariano, Facchini,
and Perinotti] proves that this channel can be freely converted into a box-
type resource that violates the CHSH inequality beyond the quantum bound of
$2\sqrt{2}$, and hence that it is LOSE-nonfree, and also proves that its Choi
state has a partial transpose with negative eigenvalues, and hence that it is
not entanglement-breaking. Intuitively, the fact that the channel is LOSE-
nonfree is inherited from the component of the PHHH ensemble-preparation,
while the fact that the channel is not entanglement-breaking is inherited from
the component of the identity channel. They also prove that it is extremal in
the space of channels.

Converting the PR box to the DFP channel

Fig. 14 presents a construction for converting the PHHH ensemble-preparation
into the DFP channel using LOSE operations.

Figure 14:  Generating the DFP channel from the PHHH ensemble-preparation (in
black) using LOSE operations (in green).

Here, the process labeled $0$ represents a fixed preparation of state
$\ket{0}$. That the overall channel in Fig. 14 is equivalent to the DFP
channel in Fig. 13 is straightforward but tedious to check by linear algebra.
Intuitively, one can see that the construction works because the controlled
swap operations ensure that either both Alice and Bob apply the identity
transformation to their inputs, or they measure them in the $\sigma_{3}$ basis
and implement the PHHH ensemble-preparation, controlled on the state of their
quantum system being $\ket{0}$ or $\ket{1}$, respectively.

###  A.6 The Bennett channel

Our final example is a resource that we suspect is postquantum, though we have
not yet been able to prove this. The resource is defined by dephasing in the
well-known product basis considered in Ref. [Bennett  _et al._(1999)Bennett,
DiVincenzo, Fuchs, Mor, Rains, Shor, Smolin, and Wootters]. We will refer to
it as the Bennett channel. It is a resource of type
$\mathsf{QQ}\\!\\!\to\\!\\!\mathsf{QQ}$, where every input and output has
dimension 3.

Ref. [Bennett  _et al._(1999)Bennett, DiVincenzo, Fuchs, Mor, Rains, Shor,
Smolin, and Wootters] considers a particular basis of product states for two
qutrits. With the definitions

$\displaystyle\ket{\alpha_{1}}=\ket{1}$
$\displaystyle\ket{\beta_{1}}=\ket{1},$
$\displaystyle\ket{\alpha_{2}}=\ket{0}$
$\displaystyle\ket{\beta_{2}}=\ket{0}+\ket{1}$
$\displaystyle\ket{\alpha_{3}}=\ket{0}$
$\displaystyle\ket{\beta_{3}}=\ket{0}-\ket{1}$
$\displaystyle\ket{\alpha_{4}}=\ket{2}$
$\displaystyle\ket{\beta_{4}}=\ket{1}+\ket{2}$
$\displaystyle\ket{\alpha_{5}}=\ket{2}$
$\displaystyle\ket{\beta_{5}}=\ket{1}-\ket{2}$ (6)
$\displaystyle\ket{\alpha_{6}}=\ket{1}+\ket{2}$
$\displaystyle\ket{\beta_{6}}=\ket{0}$
$\displaystyle\ket{\alpha_{7}}=\ket{1}-\ket{2}$
$\displaystyle\ket{\beta_{7}}=\ket{0}$
$\displaystyle\ket{\alpha_{8}}=\ket{0}+\ket{1}$
$\displaystyle\ket{\beta_{8}}=\ket{2}$
$\displaystyle\ket{\alpha_{9}}=\ket{0}-\ket{1}$
$\displaystyle\ket{\beta_{9}}=\ket{2}$

then the relevant basis for the 9-dimensional input space defined by the two
qubits is given (up to normalization) by

$\Big{\\{}\ket{\alpha_{i}}\otimes\ket{\beta_{i}}\Big{\\}}_{i=1,2,...,9}.$ (7)

The 9 states are depicted in Fig. 15.

Figure 15:  The basis of 9 states defined in Eq. (7) (via Eq. (A.6)).

In [Bennett  _et al._(1999)Bennett, DiVincenzo, Fuchs, Mor, Rains, Shor,
Smolin, and Wootters], it is shown that the channel defined by dephasing in
this product basis cannot be performed using local operations and classical
communication between the two parties. However, it is unclear whether or not
this channel is a postquantum resource.

######  Open Question 4.

Can the Bennett channel be realized using LOSE operations?

##  Appendix B Proofs of theorems from the main text

The three proofs in these appendices are exactly the same as the corresponding
proofs in Ref. [Schmid  _et al._(2020a)Schmid, Rosset, and Buscemi], but with
LOSR replaced by LOSE.

###  B.1 Proof of Theorem 3

We begin by proving Theorem 3.

###### Proof.

Consider a bipartite channel $\mathcal{E}$ which has a quantum output of
dimension $d$, together with arbitrary other outputs and inputs (denoted by
dashed double lines), as shown in black in Fig. 16(a). One can transform
$\mathcal{E}$ into a resource with a quantum input of dimension $d$ and a
classical output of dimension $d^{2}$ by composing $\mathcal{E}$ with a Bell
measurement as shown in green in Fig. 16(a); that is, by performing a
measurement in a maximally entangled basis on the quantum output of
$\mathcal{E}$ and a new quantum input of the same dimension $d$.

Figure 16:  (a) A free transformation (in green) that converts a quantum
output to a classical output together with a new quantum input. (b) This
transformation does not change the LOSE equivalence class, since it has a left
inverse (shown in pink) which is a free transformation.

To see that this transformation preserves LOSE equivalence class, it suffices
to note that there exists a local (and hence free) operation, shown in pink on
the left-hand side of Fig. 16(b), which takes the transformed channel back to
the original channel $\mathcal{E}$. In particular, this local operation feeds
one half of a maximally entangled state $\Phi_{\rm max}$ into the Bell
measurement, and then performs a correcting unitary operation $U$ on the other
half of the entangled state, conditioned on the classical outcome of the Bell
measurement. For the correct choice of correction operations, the overall
transformation on $\mathcal{E}$ is just the well-known teleportation protocol
[Bennett  _et al._(1993)Bennett, Brassard, Crépeau, Jozsa, Peres, and
Wootters], and so the equality shown in Fig. 16(b) holds. Hence, the channel
in Fig. 16(a) is in the same LOSE equivalence class as $\mathcal{E}$, which
implies that every partition of a resource can be transformed to a resource of
type $\mathsf{Q}\\!\\!\to\\!\\!\mathsf{C}$ in the same equivalence class. ∎

###  B.2 Proof of Theorem 6

We now prove Theorem 6, since it will be useful for the next proof.

###### Proof.

This proof refers to the decomposition of a given game into an analyzer and a
payoff function, as introduced and detailed in Ref. [Schmid  _et
al._(2020a)Schmid, Rosset, and Buscemi].

If $R^{\prime}\xmapsto{\textup{LOSE}}\mathcal{E}_{T}\implies
R\xmapsto{\textup{LOSE}}\mathcal{E}_{T}$, then $R$ can generate any strategy
for any given game $G_{\\!T}$ that $R^{\prime}$ can, and so always performs at
least as well as $R^{\prime}$ at $T$-games, and so
$R\succeq_{\mathcal{G}_{\\!T}}R^{\prime}$.

To prove the converse, consider a set of games of type $T$ defined by ranging
over all possible payoff functions $F_{\rm payoff}(abxy)$ for some fixed
analyzer $Z$—that is, a specific tomographically complete measurement for each
output system of the resource and a specific tomographically complete set of
states for each input system of the resource. Assume that
$R^{\prime}\xmapsto{\textup{LOSE}}\mathcal{E}_{T}$ for some strategy
$\mathcal{E}_{T}$, and define
$P_{Z\circ\mathcal{E}_{T}}(ab|xy)=Z\circ\mathcal{E}_{T}$. For
$R\succeq_{\mathcal{G}_{\\!T}}R^{\prime}$, it must be that
$R\xmapsto{\textup{LOSE}}\mathcal{E}_{T}^{\prime}$ for at least one strategy
$\mathcal{E}_{T}^{\prime}$ satisfying
$P_{Z\circ\mathcal{E}^{\prime}_{T}}(ab|xy)=Z\circ\mathcal{E}_{T}^{\prime}$. If
this were not the case, then the convex set $S(R)$ of all correlations which
$R$ can generate in this scenario, $S(R):=\left\\{P_{Z\circ\tau\circ
R}(ab|xy)=Z\circ\tau\circ R\right\\}_{\tau\in{\rm LOSE}}$, would not contain
$P_{Z\circ\mathcal{E}_{T}}(ab|xy)$, and the hyperplane which separated
$P_{Z\circ\mathcal{E}_{T}}(ab|xy)$ from $S$ would constitute a payoff function
$F_{\rm payoff}$ for which $R^{\prime}$ outperformed $R$, which would be in
contradiction with the claim that $R\succeq_{\mathcal{G}_{\\!T}}R^{\prime}$.
By tomographic completeness, the preimage of every correlation under $Z$
contains at most one strategy. Hence, if two strategies map to the same
correlation, then they must be the same strategy, and so it must be that
$\mathcal{E}_{T}=\mathcal{E}_{T}^{\prime}$ in argument above. That is, we have
shown that if $R\succeq_{\rm SQ}R^{\prime}$ and
$R^{\prime}\xmapsto{\textup{LOSE}}\mathcal{E}_{T}$, then
$R\xmapsto{\textup{LOSE}}\mathcal{E}_{T}$. ∎

###  B.3 Proof of Theorem 4

Finally, we prove Theorem 4.

###### Proof.

Consider the set $\mathcal{G}_{\\!T}$ of all games of type $T$ and two
resources $R_{1}$ and $R_{2}$, both of type $T^{\prime}$, where $T\succeq_{\rm
type}T^{\prime}$. Clearly $R_{1}\succeq_{\rm LOSE}R_{2}$ implies
$R_{1}\succeq_{\mathcal{G}_{\\!T}}R_{2}$, since $R_{1}\succeq_{\rm LOSE}R_{2}$
implies that $R_{1}$ can be used to freely generate $R_{2}$ and hence to
generate any strategy which can be generated using $R_{2}$. Next, we prove
that $R_{1}\succeq_{\mathcal{G}_{\\!T}}R_{2}$ implies $R_{1}\succeq_{\rm
LOSE}R_{2}$. By assumption, $T\succeq_{\rm type}T^{\prime}$, and so for
$R_{2}$ of type $T^{\prime}$, there exists a strategy $\mathcal{E}_{T}$ for
games of type $T$ such that
$R_{2}\xleftrightarrow{\textup{LOSE}}\mathcal{E}_{T}$. Since
$R_{1}\succeq_{\mathcal{G}_{\\!T}}R_{2}$, Theorem 6 tells us that
$R_{2}\xmapsto{\textup{LOSE}}\mathcal{E}_{T}$ implies
$R_{1}\xmapsto{\textup{LOSE}}\mathcal{E}_{T}$, and hence
$R_{1}\xmapsto{\textup{LOSE}}R_{2}$ by transitivity. Hence we have proven that
the two orderings are the same; that is, $R_{1}\succeq_{\rm LOSE}R_{2}$ if and
only if $R_{1}\succeq_{\mathcal{G}_{\\!T}}R_{2}$. ∎

##  Appendix C Methods for determining if a given channel is or is not LOSE-
free

Ref. [Beckman  _et al._(2001)Beckman, Gottesman, Nielsen, and Preskill] gives
(in Theorem 5 therein) a necessary condition for a channel to be LOSE-free:

######  Theorem 7.

If $\mathcal{E}$ is an LOSE-free superoperator on
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, and $\ket{\psi}$, $A\otimes
I\ket{\psi}$, and $I\otimes B\ket{\psi}$ are all eigenstates of $\mathcal{E}$
(where $A$ and $B$ are invertible operators), then $A\otimes B\ket{\psi}$ is
also an eigenstate of $\mathcal{E}$.

This theorem was leveraged in Ref. [Beckman  _et al._(2001)Beckman, Gottesman,
Nielsen, and Preskill] (as well as above) to prove that the BGNP channel is
postquantum.

Ref. [Sanders(2012)] gives (in Theorem 3 therein) a sufficient condition for a
channel to be LOSE-free:

######  Theorem 8.

Any bipartite channel $\mathcal{E}$ that can be implemented using local
operations and one-way classical communication from Alice to Bob and that has
an eigenstate of Schmidt rank equal to the dimension of Alice’s input system
is LOSE-free.

Finally, note that if one can establish that a given distributed channel
outperforms all LOSE strategies at some operational task, then it must be
nonfree. This is true even for operational tasks that are not subsumed within
our framework of distributed games, as long as one allows the players of the
game to optimize over all LOSE strategies, since the score in such tasks
constitutes an LOSE-monotone. For example, in Ref. [Beckman  _et
al._(2001)Beckman, Gottesman, Nielsen, and Preskill], the BGNP channel is
shown to be a valuable postquantum resource by virtue of reducing the
communication complexity of the inner product function below what could be
achieved by LOSE operations. In resource theoretic language: the amount of
quantum communication required to simulate a postquantum resource (optimized
over all LOSE operations) is a monotone [Gonda and Spekkens(2019)], and the
value assigned to the BGNP channel by this monotone is larger than can be
achieved by any free resource. This is another proof that it is not LOSE-free.