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Appendix A Preliminaries
We begin here by establishing some notation and reviewing some definitions needed in the rest of the supplementary material.
a.1 States, channels, isometries, and k-extendibility
Let denote the algebra of bounded linear operators acting on a Hilbert space . For the majority of our developments, we restrict to finite-dimensional Hilbert spaces. However, some of the claims apply to separable, infinite-dimensional Hilbert spaces, and in what follows, we clarify which ones do. The subset of containing all positive semi-definite operators is denoted by . We denote the identity operator as and the identity superoperator as . The Hilbert space of a quantum system is denoted by . The state of a quantum system is represented by a density operator , which is a positive semi-definite operator with unit trace. Let denote the set of density operators, i.e., all elements such that . The Hilbert space for a composite system is denoted as where . The density operator of a composite system is defined as , and the partial trace over gives the reduced density operator for system , i.e., such that . The notation indicates a composite system consisting of subsystems, each of which is isomorphic to Hilbert space . A pure state of a system is a rank-one density operator, and we write it as for
a unit vector in. A purification of a density operator is a pure state such that , where is known as a purifying system. denotes the maximally mixed state. The fidelity of is defined as , where denotes the trace norm.
The adjoint of a linear map is the unique linear map that satisfies
where is the Hilbert-Schmidt inner product. An isometry is a linear map such that .
The evolution of a quantum state is described by a quantum channel. A quantum channel is a completely positive, trace-preserving (CPTP) map . Let denote an isometric extension of a quantum channel , which by definition means that
along with the following conditions for to be an isometry:
where is a projection onto a subspace of the Hilbert space .
The Choi isomorphism represents a well known duality between channels and states. Let be a quantum channel, and let denote the following maximally entangled vector:
where , and and are fixed orthonormal bases. We extend this notation to multiple parties with a given bipartite cut as
The maximally entangled state is denoted as
where . The Choi operator for a channel is defined as
where denotes the identity map on . For , the following identity holds
for an operator .
Let denote the set of all separable states , which are states that can be written as
where is a probability distribution, , and for all . This set is closed under the action of the partial transpose maps or [69, 70]. Generalizing the set of separable states, we can define the set of all bipartite states that remain positive after the action of the partial transpose . A state is also called a PPT (positive under partial transpose) state. We then have the containment .
A local operations and classical communication (LOCC) channel can be written as
where and are sets of completely positive (CP) maps such that is trace preserving.
A special kind of LOCC channel is a one-way (1W-) LOCC channel from to , in which Alice performs a quantum instrument, sends the classical outcome to Bob, who then performs a quantum channel conditioned on the classical outcome received from Alice. As such, any 1W-LOCC channel takes the form in (18), except that is a set of CP maps such that the sum map is trace preserving, while is a set of quantum channels.
a.2 Entropies and information
The quantum entropy of a density operator is defined as 
The quantum relative entropy of two quantum states is a measure of their distinguishability. For and , it is defined as 
The quantum relative entropy is non-increasing under the action of positive trace-preserving maps , which is the statement that for any two density operators and and a positive trace-preserving map (this inequality applies to quantum channels as well , since every completely positive map is also a positive map by definition).
a.3 Generalized divergence and relative entropies
It is a measure of distinguishability of the states and . As a direct consequence of the above inequality, any generalized divergence satisfies the following two properties for an isometry and a state :
but it is set to for if . The sandwiched Rényi relative entropy obeys the following “monotonicity in ” inequality :
The following lemma states that the sandwiched Rényi relative entropy is a particular generalized divergence for certain values of .
and if then .
for , , and .
a.4 Entanglement measures
Let denote an entanglement measure  that is evaluated for a bipartite state . The basic property of an entanglement measure is that it should be an LOCC monotone , i.e., non-increasing under the action of an LOCC channel. Given such an entanglement measure, one can define the entanglement of a channel in terms of it by optimizing over all pure, bipartite states that can be input to the channel:
where . Due to the properties of an entanglement measure and the well known Schmidt decomposition theorem, it suffices to optimize over pure states such that (i.e., one does not achieve a higher value of by optimizing over mixed states with an unbounded reference system ). In an information-theoretic setting, the entanglement of a channel characterizes the amount of entanglement that a sender and a receiver can generate by using the channel if they do not share entanglement prior to its use.
where for a state , with and reference systems. The supremum is with respect to all input states , and the systems are finite-dimensional but could be arbitrarily large. Thus, in general, need not be computable. The amortized entanglement quantifies the net amount of entanglement that can be generated by using the channel , if the sender and the receiver are allowed to begin with some initial entanglement in the form of the state . That is, quantifies the entanglement of the initial state , and quantifies the entanglement of the final state produced after the action of the channel.
a.5 Channels with symmetry
Consider a finite group . For every , let and be projective unitary representations of acting on the input space and the output space of a quantum channel , respectively. A quantum channel is covariant with respect to these representations if the following relation is satisfied [88, 89, 90]:
In our paper, we define covariant channels in the following way:
Definition 1 (Covariant channel)
A quantum channel is covariant if it is covariant with respect to a group for which each has a unitary representation acting on , such that is a unitary one-design; i.e., the map always outputs the maximally mixed state for all input states.
The notion of teleportation simulation of a quantum channel first appeared in , and it was subsequently generalized in [91, Eq. (11)] to include general LOCC channels in the simulation. It was developed in more detail in  and used in the context of private communication in  and [61, 94].
A channel is teleportation-simulable if there exists a resource state such that for all
where is an LOCC channel (a particular example of an LOCC channel could be a generalized teleportation protocol ).
Appendix B Framework for the resource theory of -unextendibility
Any quantum resource theory is framed around two ingredients : the free states and the restricted set of free channels. The resource states by definition are those that are not free. The resource states or channels are useful and needed to carry out a given task. Resource states cannot be obtained by the action of the free channels on the free states. Free channels are incapable of increasing the amount of resourcefulness of a given state, whereas free states can be generated for free (without any resource cost).
b.1 -extendible states
To develop a framework for the quantum resource theory of -unextendibility, specified with respect to a fixed subsystem () of a bipartite system (), let us first recall the definition of a -extendible state [10, 11, 12]:
Definition 3 (-extendible state)
For integer , a state is -extendible if there exists a state that satisfies the following two criteria:
The state is permutation invariant with respect to the systems, in the sense that
where is the unitary permutation channel associated with .
The state is the marginal of , i.e.,
Note that, due to the permutation symmetry, the second condition above is equivalent to
where , and for all , and .
Definition 4 (Unextendible state)
A state that is not -extendible by Definition 3 is called -unextendible.
For simplicity and throughout this work, if we mention “extendibility,” “extendible,” “unextendibility,” or “extendible,” then these terms should be understood as -extendibility, -extendible, -unextendibility, or -unextendible, respectively, with an implicit dependence on .
Let denote the set of all states that are -extendible with respect to system . A -extendible state is also -extendible, where . This follows trivially from the definition.
b.2 -extendible channels
In order to define -extendible channels, we need to generalize the notions of permutation invariance and marginals of quantum states to quantum channels. First, permutation invariance of a state gets generalized to permutation covariance of a channel. Next, the marginal of a state gets generalized to the marginal of a channel, which includes a no-signaling constraint, in the following sense:
Definition 5 (-extendible channel)
A bipartite channel is -extendible if there exists a quantum channel that satisfies the following two criteria:
The channel is permutation covariant with respect to the systems. That is, and for all states , the following equality holds
where is the unitary permutation channel associated with .
The channel is the marginal of in the following sense:
A channel satisfying the above conditions is called a -extension of .
Equivalently, the condition in (39) can be formulated as
Classical -extendible channels were defined in a somewhat similar way in , and so our definition above represents a quantum generalization of the classical notion. We also note here that -extendible channels were defined in a slightly different way in , but our definitions reduce to the same class of channels in the case that the input systems through and the output systems are trivial.
We can reformulate the constraints on the -extendible channels in terms of the Choi state of the extension channel of as follows:
where is an arbitrary Hermitian operator and the last constraint need only be verified on a Hermitian matrix basis of . The key to deriving these constraints is the following well known “transpose trick”:
where is the transpose of with respect to the basis in (11).
The following theorem is the key statement that makes the resource theory of unextendibility, as presented above, a consistent resource theory:
For a bipartite -extendible channel and a -extendible state , the output state is -extendible.
Proof. Let be a -extension of . Let be a channel that extends . Then the following state is a -extension of :
To verify this statement, consider that , the following holds by applying (38) and the fact that is a -extension of :
Due to (39), it follows that is a marginal of .
With the above framework in place, we note here that postulates I–V of  apply to the resource theory of unextendibility. The -extendible channels are the free channels, and the -extendible states are the free states.
An important and practically relevant class of -extendible channels are 1W-LOCC channels:
Example 1 (1w-Locc)
An example of a -extendible channel is a one-way local operations and classical communication (W-LOCC) channel. Consider that a W-LOCC channel can be written as
where is a collection of completely positive maps such that is a quantum channel and is a collection of quantum channels. A -extension of the channel can be taken as follows:
A W-LOCC channel can also be represented as
where is an arbitrary channel, is a measurement channel, is a preparation channel, such that is a classical system, and is an arbitrary channel. A measurement channel followed by a preparation channel realizes an entanglement breaking (EB) channel .
b.2.1 Subclass of extendible channels
We now define a subclass of -extendible channels. These channels are inspired by 1W-LOCC channels and are realized as follows: Alice performs a quantum channel on her system and obtains systems . Then, Alice sends to Bob over a -extendible channel . This channel is a special case of the bipartite -extendible channel considered in Definition 5, in which we identify the input with of , the output with of and the systems and are trivial. Finally, Bob applies the channel on system and his local system to get . Denoting the overall channel by , it is realized as follows:
Due to their structure, we can place an upper bound on the distinguishability of a channel in the subclass described above and the set of 1W-LOCC channels, as quantified by the diamond norm . This upper bound allows us to conclude that the subclass of channels discussed above converges to the set of 1W-LOCC channels in the limit . Before stating it, recall that the diamond norm of the difference of two channels and is given by
where the optimization is with respect to pure-state inputs , with a reference system isomorphic to the channel input system .
The diamond distance between the channel in (52) and a W-LOCC channel is bounded from above as
where 1W-LOCC denotes the set of all 1W-LOCC channels acting on input systems and with output systems .
Proof. Letting denote an extension channel for , observe that
The first inequality follows from (52), by choosing a particular W-LOCC and from the monotonicity of trace norm with respect to quantum channels. The first equality follows from the definition of diamond distance. The second inequality follows from the definition of diamond distance, which has an implicit maximization over all the input states. We now observe that