Amortization does not enhance the max-Rains information of a quantum channel

Given an entanglement measure E, the entanglement of a quantum channel is defined as the largest amount of entanglement E that can be generated from the channel, if the sender and receiver are not allowed to share a quantum state before using the channel. The amortized entanglement of a quantum channel is defined as the largest net amount of entanglement E that can be generated from the channel, if the sender and receiver are allowed to share an arbitrary state before using the channel. Our main technical result is that amortization does not enhance the entanglement of an arbitrary quantum channel, when entanglement is quantified by the max-Rains relative entropy. We prove this statement by employing semi-definite programming (SDP) duality and SDP formulations for the max-Rains relative entropy and a channel’s max-Rains information, found recently in Wang et al (arXiv:1709.00200). The main application of our result is a single-letter, strong converse, and efficiently computable upper bound on the capacity of a quantum channel for transmitting qubits when assisted by positive-partial-transpose preserving (PPT-P) channels between every use of the channel. As the class of local operations and classical communication (LOCC) is contained in PPT-P, our result establishes a benchmark for the LOCC-assisted quantum capacity of an arbitrary quantum channel, which is relevant in the context of distributed quantum computation and quantum key distribution.


Introduction
One of the main goals of quantum information theory is to understand the fundamental limitations on communication when a sender and receiver are connected by a quantum communication channel [Hol12,Hay06,Wil16a]. Since it might be difficult to transmit information reliably by making use of a channel just once, a practically relevant setting is when the sender and receiver use the channel multiple times, with the goal being to maximize the rate of communication subject to a constraint on the error probability.The capacity of a quantum channel is defined to be the maximum rate of reliable communication, such that the error probability tends to zero in the limit when the channel is utilized an arbitrary number of times.
Among the various capacities of a quantum channel N , the LOCC-assisted quantum capacity Q ↔ (N ) [BDSW96] is particularly relevant for tasks such as distributed quantum computation.In Figure 1: A protocol for PPT-assisted quantum communication that uses a quantum channel n times.Every channel use is interleaved by a PPT-preserving channel.The goal of such a protocol is to produce an approximate maximally entangled state in the systems M A and M B , where Alice possesses system M A and Bob system M B .
the setting corresponding to this capacity, the sender and receiver are allowed to perform arbitrary LOCC (local operations and classical communication) between every use of the channel, and the capacity is equal to the maximum rate, measured in qubits per channel use, at which qubits can be transmitted reliably from the sender to the receiver [BDSW96].Due to the teleportation protocol [BBC + 93], this rate is equal to the maximum rate at which shared entangled bits (Bell pairs) can be generated reliably between the sender and the receiver [BDSW96].The LOCC-assisted quantum capacity of certain channels such as the quantum erasure channel has been known for some time [BDS97], but in general, it remains an open question to characterize Q ↔ (N ).One can address this question by establishing either lower bounds or upper bounds on Q ↔ (N ).
In this paper, we are interested in placing upper bounds on the LOCC-assisted quantum capacity, and one way of simplifying the mathematics behind this task is to relax the class of free operations that the sender and receiver are allowed to perform between each channel use.With this in mind, we follow the approach of [Rai99,Rai01] and relax the set LOCC to a larger class of operations known as PPT-preserving, standing for channels that are positive partial transpose preserving.The resulting capacity is then known as the PPT-assisted quantum capacity Q PPT,↔ (N ), and it is equal to the maximum rate at which qubits can be communicated reliably from a sender to a receiver, when they are allowed to use a PPT-preserving channel in between every use of the actual channel N .Figure 1 provides a visualization of such a PPT-assisted quantum communication protocol.Due to the containment LOCC ⊂ PPT [Rai99,Rai01], the inequality holds for all channels N .Thus, if we find an upper bound on Q PPT,↔ (N ), then by (1), such an upper bound also bounds the physically relevant LOCC-assisted quantum capacity Q ↔ (N ).
A general approach for bounding these assisted capacities of a quantum channel has been developed recently in [KW17] (see [BHLS03, LHL03, CMH17, BDGDMW17, RKB + 17] for related notions).The starting point is to consider an entanglement measure E(A; B) ρ [HHHH09], which is evaluated for a bipartite state ρ AB .Given such an entanglement measure, one can define the entanglement E(N ) of a channel N in terms of it by taking an optimization over all pure, bipartite states that could be input to the channel: where ω RB = N A→B (ψ RA ).The channel's entanglement E(N ) characterizes the amount of entanglement that a sender and receiver can generate by using the channel if they do not share entanglement prior to its use.Due to the properties of an entanglement measure and the well known Schmidt decomposition theorem, it suffices to take system R isomorphic to the channel input system A and furthermore to optimize over pure states ψ RA .One can alternatively consider the amortized entanglement E A (N ) of a channel N as the following optimization [KW17]: where τ A BB = N A→B (ρ A AB ) and ρ A AB is a state.The supremum is with respect to all states ρ A AB and the systems A B are finite-dimensional but could be arbitrarily large (so that the supremum might never be achieved for any particular finite-dimensional A B , but only in the limit of unbounded dimension).Thus, E A (N ) is not known to be computable in general.The amortized entanglement quantifies the net amount of entanglement that can be generated by using the channel N , if the sender and receiver are allowed to begin with some initial entanglement in the form of the state ρ A AB .That is, E(A A; B ) ρ quantifies the entanglement of the initial state ρ A AB , and E(A ; BB ) τ quantifies the final entanglement of the state after the channel acts.As observed in [KW17], the inequality always holds for any entanglement measure E and for any channel N , simply because one could take the B system trivial in the optimization for E A (N ), which is the same as not allowing entanglement between the sender and receiver before the channel acts.It is nontrivial if the opposite inequality holds, which is known to occur generally for certain entanglement measures [TGW14, CMH17, KW17] or for certain channels with particular symmetries [KW17].
One of the main observations of [KW17], connected to earlier developments in [BHLS03, LHL03, CMH17, BDGDMW17, RKB + 17], is that the amortized entanglement of a channel serves as an upper bound on the entanglement of the final state ω AB generated by an LOCC-or PPT-assisted quantum communication protocol that uses the channel n times: The basic intuition for why this bound holds is that, after a given channel use, the sender and receiver are allowed to perform a free operation such as LOCC or PPT, and thus the state that they share before the next channel use could have some entanglement.So the amount of entanglement generated by each channel use cannot exceed the amortized entanglement E A (N ), and if the channel is used n times in such a protocol, then the entanglement of the final state ω AB cannot exceed the channel's amortized entanglement multiplied by the number n of channel uses.Such a general bound can then be used to derive particular upper bounds on the assisted quantum capacities, such as strong converse bounds.Clearly, if the inequality in (5) holds, then E A (N ) = E(N ) and the upper bound becomes much simpler because the channel entanglement E(N ) is simpler than the amortized entanglement E A (N ).Thus, one of the main contributions of [KW17] was to reduce the physical question of determining meaningful upper bounds on the assisted capacities of N to a purely mathematical question of whether amortization can enhance the entanglement of a channel, i.e., whether the equality E A (N ) holds for a given entanglement measure E and/or channel N .
In this paper, we solve the mathematical question posed above for the max-Rains information R max (N ) of a quantum channel N , by proving that amortization does not enhance it; i.e., we prove that for all channels N , where R max,A (N ) denotes the amortized max-Rains information.Note that R max (N ) and R max,A (N ) are respectively defined by taking the entanglement measure E in (2) and (3) to be the max-Rains relative entropy, which we define formally in the next section.We note here that the equality in ( 8) solves an open question posed in the conclusion of [CMH17], and we set our result in the context of the prior result of [CMH17] and other literature in Section 6.The max-Rains information of a quantum channel is a special case of a quantity known as the sandwiched Rényi-Rains information [TWW17] and was recently shown to be equal to an information quantity discussed in [WD16b,WFD17] and based on semi-definite programming.To prove our main technical result (the equality in (8)), we critically make use of the tools and framework developed in the recent works [WD16b,WD16a,WFD17].In particular, we employ semi-definite programming duality [BV04] and the well known Choi isomorphism to establish our main result, with the proof consisting of just a few lines once the framework from [WD16b, WD16a, WFD17] is set in place.
The main application of the equality in ( 8) is an efficiently computable, single-letter, strong converse bound on Q PPT,↔ (N ), the PPT-assisted quantum capacity of an arbitrary channel N .Due to (1), this is also an upper bound on the physically relevant LOCC-assisted quantum capacity Q ↔ (N ).To arrive at this result, we simply apply the general inequality in (6) along with the equality in (8).For the benefit of the reader, we give technical details of this application in Section 4. The quantity R max (N ) has already been shown in [WFD17] to be efficiently computable via a semi-definite program, and in Section 4, we explain how R max (N ) is both "single-letter" and "strong converse." Our paper is organized as follows.In the next section, we review some background material before starting with the main development.Section 3 gives a short proof of our main technical result, and Section 4 discusses its application as a efficiently computable, single-letter, strong converse bound on Q PPT,↔ (N ).In Section 5, we revisit a result from [CMH17], in which it was shown that amortization does not enhance a channel's max-relative entropy of entanglement.The authors of [CMH17] proved this statement by employing complex interpolation theory [BL76].We prove the main inequality underlying this statement using a method different from that used in [CMH17], but along the lines of that given for our proof of (8) (i.e., convex programming duality), and we suspect that our alternative approach could be useful in future applications.In Section 6, we discuss how our result fits into the prior literature on assisted quantum capacities and strong converses.We conclude with a brief summary in Section 7.

Background and notation
In this section, we provide background on the Choi isomorphism, partial transpose, positive partial transpose (PPT) states, separable states, PPT-preserving channels, max-relative entropy, max-Rains relative entropy, and max-Rains information.For basic concepts and standard notation used in quantum information theory, we point the reader to [Wil16a].
The Choi isomorphism represents a well known duality between channels and states, often employed in quantum information theory.Let N A→B be a quantum channel, and let |Υ RA denote the maximally entangled vector where the Hilbert spaces H R and H A are of the same dimension and {|i R } i and {|i A } i are fixed orthonormal bases.The Choi operator for a channel N A→B is defined as where id R denotes the identity map on system R.One can recover the action of the channel N A→B on an arbitrary input state ρ SA as follows: where A is a system isomorphic to the channel input A. The above identity can be understood in terms of a postselected variant [HM04,Ben05] of the quantum teleportation protocol [BBC + 93].
Another identity we recall is that for an operator X SR acting on H S ⊗ H R .For a fixed basis {|i B } i , the partial transpose is the following map: where X AB is an arbitrary operator acting on a tensor-product Hilbert space H A ⊗ H B .For simplicity we often employ the abbreviation T B (X AB ) = (id A ⊗T B )(X AB ).The partial transpose map plays a role in the following well known transpose trick identity: The partial transpose map plays another important role in quantum information theory because a separable (unentangled) state for a distribution p(x) and states τ x A and ω x B , stays within the set of separable states under this map [HHH96,Per96]: This motivates defining the set of PPT states, which are those states σ AB for which T B (σ AB ) ≥ 0. This in turn motivates defining the more general set of positive semi-definite operators [ADMVW02]: where we have employed the trace norm, defined for an operator X as We then have the containments SEP ⊂ PPT ⊂ PPT .An LOCC quantum channel N AB→A B consists of an arbitrarily large but finite number of compositions of the following: 1. Alice performs a quantum instrument, which has both a quantum and classical output.She forwards the classical output to Bob, who then performs a quantum channel conditioned on the classical data received.This sequence of actions corresponds to a channel of the following form: x where {F x A→A } x is a collection of completely positive maps such that x F x A→A is a quantum channel and {G x B→B } x is a collection of quantum channels.
2. The situation is reversed, with Bob performing the initial instrument, who forwards the classical data to Alice, who then performs a quantum channel conditioned on the classical data.This sequence of actions corresponds to a channel of the form in (18), with the A and B labels switched.
If supp(ρ) ⊆ supp(σ), then D max (ρ σ) = ∞.The max-relative entropy is monotone non-increasing under the action of a quantum channel N [Dat09], in the sense that The max-Rains relative entropy of a state ρ AB is defined as and it is monotone non-increasing under the action of a PPT-preserving quantum channel for ω A B = N AB→A B (ρ AB ).The max-Rains information of a quantum channel N A→B is defined by replacing E in (2) with the max-Rains relative entropy R max ; i.e., where ω SB = N A→B (φ SA ) and φ SA is a pure state, with |S| = |A|.The amortized max-Rains information of a channel, denoted as R max,A (N ), is defined by replacing E in (3) with the max-Rains relative entropy R max .
Recently, in [WD16a, Eq. ( 8)] (see also [WFD17]), the max-Rains relative entropy of a state ρ AB was expressed as where W (A; B) ρ is the solution to the following semi-definite program: Similarly, in [WFD17, Eq. ( 21)], the max-Rains information of a quantum channel N A→B was expressed as where Γ(N ) is the solution to the following semi-definite program: These formulations of R max (A; B) ρ and R max (N ) are the tools that we use to prove our main technical result, Proposition 1.It is worthwhile to mention that the formulations above follow by employing the theory of semi-definite programming and its duality.

Main technical result
The following proposition constitutes our main technical result, and an immediate corollary of it is that amortization does not enhance the max-Rains information of a quantum channel: Proposition 1 Let ρ A AB be a state and let N A→B be a quantum channel.Then where Proof.By removing logarithms and applying (24) and (26), the desired inequality is equivalent to the following one: and so we aim to prove this one.Exploiting the identity in (25), we find that subject to the constraints while the identity in (27) gives that subject to the constraints The identity in (25) implies that the left-hand side of (30) is equal to subject to the constraints With these SDP formulations in place, we can now establish the inequality in (30) by making judicious choices for E A BB and F A BB .Let C A AB and D A AB be optimal for W (A A; B ) ρ , and let Y SB and V SB be optimal for Γ(N ).Let |Υ SA be the maximally entangled vector, as defined in (9).Pick We note that these choices are somewhat similar to those made in the proof of [WFD17, Proposition 6], and they can be understood roughly via (11) as a postselected teleportation of the optimal operators of W (A A; B ) ρ through the optimal operators of Γ(N ), with the optimal operators of W (A A; B ) ρ being in correspondence with the input state ρ A AB through (33) and the optimal operators of Γ(N ) being in correspondence with the Choi operator J N SB through (36).We then have that The inequality follows from (33) and (36), and the last equality follows from (11).Also consider that The second equality follows from ( 14) and ( 12).The inequality is a consequence of Hölder's inequality.The final equality follows because the spectrum of an operator is invariant under the action of a (full) transpose (note, in this case, that T A is a full transpose because the operator Tr B {V AB + Y AB } acts only on system A).Thus, we can conclude that our choices of E A BB and F A BB are feasible for W (A ; BB ) ω .Since W (A ; BB ) ω involves a minimization over all E A BB and F A BB satisfying (38) and (39), this concludes our proof of (30).
An immediate corollary of Proposition 1 is the following: Corollary 2 Amortization does not enhance the max-Rains information of a quantum channel N A→B ; i.e., the following equality holds Proof.The inequality R max,A (N ) ≥ R max (N ) always holds, as reviewed in (4).The other inequality is an immediate consequence of Proposition 1. Letting ρ A AB denote an arbitrary input state, Proposition 1 implies that where ω A BB = N A→B (ρ A AB ).Since the inequality holds for any state ρ A AB , it holds for the supremum over all such input states, leading to R max,A (N ) ≤ R max (N ).

Application to PPT-assisted quantum communication
We now give our main application of Proposition 1, which is that the max-Rains information is a single-letter, strong-converse upper bound on the PPT-assisted quantum capacity of any channel.
The term "single-letter" refers to the fact that the max-Rains information requires an optimization over a single use of the channel.As we remarked previously, the max-Rains information is efficiently computable via semi-definite programming, as observed in [WD16b,WFD17].Finally, the bound is a strong converse bound because, as we will show, if the rate of a sequence of PPT-assisted quantum communication protocols exceeds the max-Rains information, then the error probability of these protocols necessarily tends to one exponentially fast in the number of channel uses.

Protocol for PPT-assisted quantum communication
We begin by reviewing the structure of a PPT-assisted quantum communication protocol, along the lines discussed in [KW17].In such a protocol, a sender Alice and a receiver Bob are spatially separated and connected by a quantum channel N A→B .They begin by performing a PPT channel P (1) , which leads to a PPT state ρ (1) , where A 1 and B 1 are systems that are finitedimensional but arbitrarily large.The system A 1 is such that it can be fed into the first channel use.Alice sends system A 1 through the first channel use, leading to a state σ ).

Alice and Bob then perform the PPT channel P
(2) , which leads to the state ). (44) Alice sends system A 2 through the second channel use N A 2 →B 2 , leading to the state σ ).This process iterates: the protocol uses the channel n times.In general, we have the following states for all i ∈ {2, . . ., n}: where is a PPT channel.The final step of the protocol consists of a PPT channel P (n+1) A n BnB n →M A M B , which generates the systems M A and M B for Alice and Bob, respectively.The protocol's final state is as follows: Figure 1 depicts such a protocol.The goal of the protocol is that the final state ω M A M B is close to a maximally entangled state.Fix n, M ∈ N and ε ∈ [0, 1].The original protocol is an (n, M, ε) protocol if the channel is used n times as discussed above, where the fidelity A rate R is achievable for PPT-assisted quantum communication if for all ε ∈ (0, 1], δ > 0, and sufficiently large n, there exists an (n, 2 n(R−δ) , ε) protocol.The PPT-assisted quantum capacity of a channel N , denoted as Q PPT,↔ (N ), is equal to the supremum of all achievable rates.
On the other hand, a rate R is a strong converse rate for PPT-assisted quantum communication if for all ε ∈ [0, 1), δ > 0, and sufficiently large n, there does not exist an (n, 2 n(R+δ) , ε) protocol.The strong converse PPT-assisted quantum capacity Q PPT,↔ † (N ) is equal to the infimum of all strong converse rates.We say that a channel obeys the strong converse property for PPT-assisted quantum communication if We can also consider the whole development above when we only allow the assistance of LOCC channels instead of PPT channels.In this case, we have similar notions as above, and then we arrive at the LOCC-assisted quantum capacity Q ↔ (N ) and the strong converse LOCC-assisted quantum capacity Q ↔ † (N ).It then immediately follows that because every LOCC channel is a PPT channel.

Max-Rains information as a strong converse rate for PPT-assisted quantum communication
We now prove the following upper bound on the communication rate 1 n log 2 M (qubits per channel use) of any (n, M, ε) PPT-assisted protocol: Theorem 3 Fix n, M ∈ N and ε ∈ (0, 1).The following bound holds for an (n, M, ε) protocol for PPT-assisted quantum communication over a quantum channel N : Proof.For convenience of the reader, we give a complete proof, but we note that some of the essential steps are available in prior works [CMH17, RKB + 17, KW17].From the assumption in (49), it follows that while [Rai99, Lemma 2] implies that for all σ M A M B ∈ PPT (M A : M B ).So under an "entanglement test," i.e., a measurement of the form and applying the data processing inequality for the max-relative entropy, we find that From the monotonicity of the Rains relative entropy with respect to PPT-preserving channels [Rai01, TWW17], we find that The first equality follows because the state ρ (1) is a PPT state with vanishing max-Rains relative entropy.The second equality follows by adding and subtracting terms.The second inequality follows because R max for all i ∈ {2, . . ., n}, due to monotonicity of the Rains relative entropy with respect to PPT channels.The final inequality follows by applying Proposition 1 to each term R max (A n ; 56) and (60), we arrive at the inequality in (53).
Remark 4 The bound in (53) can also be rewritten in the following way: where we set the rate Q = 1 n log 2 M .Thus, if the communication rate Q is strictly larger than the max-Rains information R max (N ), then the fidelity of the transmission (1 − ε) decays exponentially fast to zero in the number n of channel uses.
An immediate corollary of the above is the following strong converse statement: Corollary 5 The strong converse PPT-assisted quantum capacity is bounded from above by the max-Rains information: 5 Amortization does not increase a channel's max-relative entropy of entanglement One of the main results of [CMH17] is that amortization does not increase a channel's max-relative entropy of entanglement; i.e., where E max (N ) denotes a channel's max-relative entropy of entanglement (we will define this shortly).The authors of [CMH17] proved (63) by employing the methods of complex interpolation [BL76].The main application of ( 63) is that E max (N ) is a strong converse upper bound on the secret-key-agreement capacity of a quantum channel [CMH17] (this is defined as the private capacity of the channel, when arbitrary LOCC is allowed between every channel use-see [WTB17] or [CMH17] for a definition).
In this section, we provide an alternate proof of (63), which is along the lines of the proofs of Proposition 1 and Corollary 2. We think that this approach brings a different perspective to the result of [CMH17] and could potentially be useful in future applications.
To begin with, let us recall the definition of the max-relative entropy of entanglement of a bipartite state ρ AB [Dat09]: Let − − → SEP(A : B) denote the cone of all separable operators, i.e., X AB ∈ − − → SEP(A : B) if there exists a positive integer L and positive semi-definite operators The arrow in − − → SEP(A : B) is meant to remind the reader of "cone" and is not intended to indicate any directionality between the A and B systems.In what follows, we sometimes employ the shorthands SEP and − − → SEP when the bipartite cuts are clear from the context.Then we have the following alternative expression for the max-relative entropy of entanglement: Lemma 6 Let ρ AB be a bipartite state.Then where Proof.Employing the definition in (64), consider that min This concludes the proof.
We can then define a channel's max-relative entropy of entanglement E max (N ) as in (2), by replacing E with E max .We can alternatively write E max (N ) as follows, by employing similar reasoning as given in the proof of [CMW16, Lemma 6]: where ρ S is a density operator and J N SB is the Choi operator for the channel N , as defined in (10).We now prove the following alternative expression for E max (N ): where Proof.Employing (67) and Lemma 6, we find that So our aim is to prove that the expression inside the logarithm is equal to Σ(N ).Taking the ansatz that ρ S is an invertible density operator, we find that the condition ρ The second equality follows from the Sion minimax theorem: the sets over which we are optimizing are convex, with the set of density operators additionally being compact, and the objective function Tr{ρ S Y SB } is linear in ρ S and Y SB , and so the Sion minimax theorem applies.The third equality follows from partial trace, and the fourth follows because D ∞ = max ρ Tr{Dρ}, when the optimization is with respect to density operators.Finally, we note that the ansatz may be lifted by an appropriate limiting argument.
We can now see that the expressions for E max (A; B) ρ in Lemma 6 and E max (N ) in Lemma 7 have a very similar form to those in (24) and (26) for R max (A; B) ρ and R max (N ), respectively.However, the optimization problems for E max (A; B) ρ and E max (N ) are not necessarily efficiently computable because they involve an optimization over the cone of separable operators, which is known to be difficult [HM13] in general.Regardless, due to the forms that we now have for E max (A; B) ρ and E max (N ), we can prove an inequality from [CMH17], analogous to (28), with a proof very similar to that given in the proof of Proposition 1: Proposition 8 ([CMH17]) Let ρ A AB be a state and let N A→B be a quantum channel.Then where Proof.By removing logarithms and applying Lemmas 6 and 7, the desired inequality is equivalent to the following one: and so we aim to prove this one.Exploiting the identity in Lemma 6, we find that subject to the constraints while the identity in Lemma 7 gives that subject to the constraints The identity in Lemma 6 implies that the left-hand side of (74) is equal to subject to the constraints With these optimizations in place, we can now establish the inequality in (74) by making a judicious choice for E A BB .Let C A AB be optimal for W sep (A A; B ) ρ , and let Y SB be optimal for Σ(N ).Let |Υ SA be the maximally entangled vector, as defined in (9).Pick This choice is clearly similar to that in the proof of Proposition 1.We need to prove that E A BB is feasible for W sep (A ; BB ) ω .To this end, consider that which follows from (77), (80), and (11).Now, since C A AB ∈ − − → SEP(A A : B ), it can be written as Furthermore, consider that since Y SB ∈ − − → SEP(S : B), it can be written as y L y S ⊗ M y B for positive semi-definite L y S and M y B .Then we have that The second equality follows from (14) and the third from (12).The last statement follows because ) is positive semi-definite for each x and y.Finally, consider that The reasoning for this chain is identical to that for (41).
Thus, we can conclude that our choice of E A BB is feasible for W (A ; BB ) ω .Since W (A ; BB ) ω involves a minimization over all E A BB satisfying (82) and (83), this concludes our proof of (74).
By the same reasoning employed in the proof of Corollary 2, the equality in (63) follows as a consequence of the inequality in Proposition 8.

On converses for quantum and private capacities
Here we discuss briefly how our strong converse result stands with respect to prior work on strong converses and quantum and private capacities [HW01, TWW17, MHRW16, WD16b, WTB17, CMH17, WFD17].

Quantum capacities
Let Q(N ) and Q † (N ) denote the quantum capacity and the strong converse quantum capacity of a quantum channel N .These quantities are defined similarly to Q PPT,↔ (N ) and Q PPT,↔ † (N ), but there is no PPT assistance allowed.The partial transposition bound was defined in [HW01] as follows: Q where T denotes the transpose map and • ♦ is the diamond norm.In [HW01], the following bound was established which was subsequently improved in [MHRW16] to The recent work in [WD16b,WFD17] established the following two bounds: Thus, in light of the above history, it is clear that the natural question was whether Q PPT,↔ † (N ) ≤ R max (N ), and this is the question that our paper affirmatively answers.In summary, we now have that We now mention some other related results.The Rains relative entropy R(A; B) ρ of a bipartite state ρ AB is defined as [Rai99, Rai01, ADMVW02] The following bound is known from [TWW17] and it is open to determine whether This latter inequality is known to hold if the channel N has sufficient symmetry [TWW17] and approximately if the channel possesses some symmetry approximately [KW17].
The squashed entanglement E sq (A; B) ρ of a quantum state ρ AB is defined as [CW04] where (See also discussions in [Tuc99,Tuc02] for squashed entanglement.)One can also consider the squashed entanglement of a channel E sq (N ) [TGW14], as well as the amortized squashed entanglement E sq,A (N ).Another function of a quantum channel is its entanglement cost [BBCW13], which we write as E C (N ) and for which a definition is given in [BBCW13].The following bounds and relations are known regarding these quantities: It is open to determine whether the following inequality holds

Private capacities
One can also consider various private capacities and strong converse private capacities of a quantum channel, denoted as P (N ), P ↔ (N ), P † (N ), and P ↔ † (N ).Defining the relative entropy of entanglement and the max-relative entropy of entanglement E max as we did in (64), we can also define their channel versions E R (N ) and E max (N ) and their amortized versions E R,A (N ) and E max,A (N ).For these various quantities, we have that It is not known whether but the latter inequality is known to hold for channels with sufficient symmetry [WTB17], as well as for those having approximate symmetry [KW17].

Summary: Channel measures that do not increase under amortization
In summary, we know that amortization does not increase 1. the squashed entanglement E sq (N ) [TGW14], 2. the max-relative entropy of entanglement E max (N ) [CMH17], 3. or the max-Rains information R max (N ) (Corollary 2).This is the main reason that these information quantities are single-letter converse bounds for assisted capacities.Is there any chance that the same could hold generally for E R (N ) or R(N )?If so, then the known capacity bounds could be improved.

Conclusion
The main contribution of our paper was to show that the max-Rains information of a quantum channel does not increase under amortization.That is, when entanglement is quantified by the max-Rains relative entropy, the net entanglement that a channel can generate is the same as the amount of entanglement that it can generate if the sender and receiver do not start with any initial entanglement.This result then implies a single-letter, strong-converse, and efficiently computable bound for the capacity of a quantum channel to communicate qubits along with the assistance of PPT-preserving operations between every channel use.As such, the max-Rains information can be easily evaluated and is a general benchmark for this capacity.As we emphasized previously, our upper bound is also an upper bound on the physically relevant LOCC-assisted quantum capacity.
The main tool that we used to prove our result is the formulation of the max-Rains relative entropy and max-Rains information as semi-definite programs [WD16b, WD16a, WFD17] (in particular, we employed semi-definite programming duality-we note here that this kind of approach has previously been employed successfully for multiplicativity, additivity, or parallel repetition problems in quantum information theory [BT16,BFT17,VW16]).We also compared our result to other results in the growing literature on the topic of bounds for the assisted capacities of arbitrary quantum channels [TGW14, TWW17, MHRW16, WTB17, CMH17].We also provided an alternative proof for the fact that amortization does not enhance a channel's max-relative entropy of entanglement [CMH17]: i.e., E max,A (N ) = E max (N ).This statement was proved in [CMH17] by employing the methods of complex interpolation [BL76], but here we found a different proof by establishing alternative expressions for the max-relative entropy of entanglement (Lemma 6) and a channel's max-relative entropy of entanglement (Lemma 7).These alternative expressions then allowed us to employ reasoning similar to that in our proof of Proposition 1 in order to establish a different proof for the equality E max,A (N ) = E max (N ).We suspect that our approach could be useful in future applications.
Finally, in [WFD17], it was noted that the max-Rains information does not give a good upper bound on the quantum capacity of the qubit depolarizing channel.Our result gives a compelling reason for this observation: the max-Rains information finds its natural place as an upper bound on the PPT-assisted quantum capacity of the qubit depolarizing channel, and these assisting operations allowed between every channel use could result in a significant increase in capacity.