Entanglement Cost and Quantum Channel Simulation

Entanglement cost and quantum channel simulation

Mark M. Wilde Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA (Dated: November 2, 2018) This paper proposes a revised definition for the entanglement cost of a quantum channel N . In particular, it is defined here to be the smallest rate at which entanglement is required, in addition to free classical communication, in order to simulate n calls to N , such that the most general discriminator cannot distinguish the n calls to N from the simulation. The most general discriminator is one who tests the channels in a sequential manner, one after the other, and this discriminator is known as a quantum tester [Chiribella et al., Phys. Rev. Lett., 101, 060401 (2008)] or one who is implementing a quantum co-strategy [Gutoski et al., Symp. Th. Comp., 565 (2007)]. As such, the proposed revised definition of entanglement cost of a quantum channel leads to a rate that cannot be smaller than the previous notion of a channel’s entanglement cost [Berta et al., IEEE Trans. Inf. Theory, 59, 6779 (2013)], in which the discriminator is limited to distinguishing parallel uses of the channel from the simulation. Under this revised notion, I prove that the entanglement cost of certain teleportation-simulable channels is equal to the entanglement cost of their underlying resource states. Then I find single-letter formulas for the entanglement cost of some fundamental channel models, including dephasing, erasure, three-dimensional Werner–Holevo channels, epolarizing channels (complements of depolarizing channels), as well as single-mode pure-loss and pure-amplifier bosonic Gaussian channels. These examples demonstrate that the resource theory of entanglement for quantum channels is not reversible. Finally, I discuss how to generalize the basic notions to arbitrary resource theories.

I. INTRODUCTION in the same paper that introduced the resource theory of entanglement [BDSW96], the authors there appreciated the relevance of this point and proposed that the dis- The resource theory of entanglement [BDSW96] has tillation question could be extended to quantum chan- been one of the richest contributions to quantum infor- nels. The distillation question for channels is then as mation theory [Hol12, Hay06, Wil17, Wat18], and these follows: given n uses of a quantum channel A→B con- days, the seminal ideas coming from it are inﬂuencing di- necting a sender Alice to a receiver Bob, alongN with the verse areas of physics [CG18]. A fundamental question in assistance of free classical communication, what is the entanglement theory is to determine the smallest rate at optimal rate at which these channels can produce ebits which Bell states (or ebits) are needed, along with the as- reliably [BDSW96]? By invoking the teleportation pro- sistance of free classical communication, in order to gen- tocol [BBC+93] and the fact that free classical commu- erate n copies of an arbitrary bipartite state ρAB reliably nication is allowed, this rate is also equal to the rate at (in this introduction, n should be understood to be an which arbitrary qubits can be reliably communicated by arbitrarily large number) [BDSW96]. The optimal rate is using the channel n times [BDSW96]. The optimal rate known as the entanglement cost of ρAB [BDSW96], and a is known as the distillable entanglement of the channel formal expression is known for this quantity in terms of a [BDSW96], and various lower bounds [DW05] and up- regularization of the entanglement of formation [HHT01]. per bounds [TGW14a, TGW14b, Wil16, BW18] are now An upper bound in terms of entanglement of formation known for it, strongly related to the bounds for distillable has been known for some time [BDSW96, HHT01], while entanglement of states, as given above. a lower bound in terms of a semi-deﬁnite programming Some years after the distillable entanglement of a chan- quantity has been determined recently [WD17]. Con-

arXiv:1807.11939v3 [quant-ph] 31 Oct 2018 nel was proposed in [BDSW96], the question converse to versely, a related fundamental question is to determine it was proposed and addressed in [BBCW13]. The au- the largest rate at which one can distill ebits reliably from thors of [BBCW13] defined the entanglement cost of a n copies of ρAB, again with the assistance of free classical quantum channel A→B as the smallest rate at which communication [BDSW96]. This optimal rate is known entanglement is required,N in addition to the assistance as the distillable entanglement, and various lower bounds of free classical communication, in order to simulate [DW05] and upper bounds [Rai99, Rai01, CW04, WD16] n uses of A→B. Key to their definition of entangle- are known for it. ment costN is the particular notion of simulation consid- The above resource theory is quite rich and interest- ered. In particular, the goal of their simulation protocol ing, but soon after learning about it, one might imme- is to simulate n parallel uses of the channel, written as ⊗n diately question its operational significance. How are ( A→B) . Furthermore, they considered a simulation N the bipartite states ρAB established in the first place? protocol An→Bn to have the following form: Of course, a quantum communication channel, such as a P

ﬁber-optic or free-space link, is required. Consequently, An→Bn (ωAn ) n n (ωAn Φ ), (1) P ≡ LA A0B0→B ⊗ A0B0 2

where ωAn is an arbitrary input state, n n is inator to prepare an arbitary state ρR A , call the first LA A0B0→B 1 1 a free channel, whose implementation is restricted to channel use A1→B1 or its simulation, apply an arbi- consist of local operations and classical communication N (1) trary channel R B →R A , call the second channel use (LOCC) [BDSW96, CLM+14], and Φ is a maximally A 1 1 2 2 A0B0 or its simulation, etc. After the nth call is made, the entangled resource state. For ε [0, 1], the simulation is discriminator then performs a joint measurement on the ∈ ⊗n then considered ε-distinguishable from ( A→B) if the remaining quantum systems. See Figure1 for a visual N following condition holds depiction. If the simulation is good, then the probability for the discriminator to distinguish the n channels 1 ⊗n 1 ( A→B) An→Bn ε, (2) from the simulation should be no larger than 2 (1 + ε), 2 N − P ♦ ≤ for small ε. where denotes the diamond norm [Kit97]. The phys- In this paper, I propose a new definition for the en- k·k♦ tanglement cost of a channel A→B, such that it is the ical meaning of the above inequality is that it places a N limitation on how well any discriminator can distinguish smallest rate at which ebits are needed, along with the ⊗n assistance of free classical communication, in order to the channel ( A→B) from the simulation An→Bn in a guessing game.N Such a guessing game consistsP of the simulate n uses of A→B, in such a way that a discriminator performingN the most stringest test, as described discriminator preparing a quantum state ρRAn , the ref- ⊗n above, cannot distinguish the simulation from n actual eree picking ( A→B) or An→Bn at random and then N n P calls of A→B (SectionIIB). Here I denote the opti- applying it to the A systems of ρRAn , and the discrimi- N nator finally performing a quantum measurement on the mal rate by EC ( ), and the prior quantity defined in (Np) systems RBn. If the inequality in (2) holds, then the [BBCW13] by E ( ), given that the simulation there C N probability that the discriminator can correctly distin- was only required to pass a less stringent parallel discrim- guish the channel from its simulation is bounded from ination test, as discussed above. Due to the fact that it 1 above by (1 + ε), regardless of the particular state ρRAn is more difficult to pass the simulation test as specified 2 (p) and final measurement chosen for his distinguishing strat- by the new definition, it follows that EC ( ) EC ( ) egy [Kit97, Hel69, Hol73, Hel76]. Thus, if ε is close to (discussed in more detail in what follows).N After≥ estab-N zero, then this probability is not much better than ran- lishing definitions, I then prove a general upper bound dom guessing, and in this case, the channels are con- on the entanglement cost of a quantum channel, using sidered nearly indistinguishable and the simulation thus the notion of teleportation simulation (SectionIIIA). I reliable. prove that the entanglement cost of certain “resource- In parallel to the above developments in entanglement seizable,” teleportation-simulable channels takes on a theory, there have indubitably been many advances in particularly simple form (SectionIIIB), which allows for the theory of quantum channel discrimination [CDP08b, concluding single-letter formulas for the entanglement CDP08a, CDP09b, DFY09, HHLW10, CMW16] and re- cost of dephasing, erasure, three-dimensional Werner– lated developments in the theory of quantum interac- Holevo channels, and epolarizing channels (complements tive proof systems [GW07, Gut09, Gut12, GRS18]. No- of depolarizing channels), as detailed in SectionIV. Note tably, the most general method for distinguishing a that the result about entanglement cost of dephasing quantum memory channel from another one consists channels solves an open question from [BBCW13]. I then of a quantum-memory-assisted discrimination protocol extend the results to the case of bosonic Gaussian chan- [CDP08a, CDP09b]. In the language of quantum interac- nels (SectionV), proving single-letter formulas for the tive proof systems, memory channels are called strategies entanglement cost of fundamental channel models, in- and memory-assisted discrimination protocols are called cluding pure-loss and pure-amplifier channels (Theorem2 co-strategies [GW07, Gut09, Gut12]. For a visual illus- in SectionVG). These examples lead to the conclusion tration of the physical setup, please consult [CDP08a, that the resource theory of entanglement for quantum Figure 2] or [GW07, Figure 2]. In subsequent work after channels is not reversible. I also prove that the entan- [GW07, CDP08a], a number of theoretical results listed glement cost of thermal, amplifier, and additive-noise above have been derived related to memory channel dis- bosonic Gaussian channels is bounded from below by the crimination or quantum strategies. entanglement cost of their “Choi states.” In SectionVI, The aforementioned developments in the theory of I discuss how to generalize the basic notions to other re- quantum channel discrimination indicate that the notion source theories. Finally, SectionVII concludes with a of channel simulation proposed in [BBCW13] is not the summary and some open questions. most general notion that could be considered. In particular, if a simulator is claiming to have simulated n uses of the channel A→B, then the discriminator should be II. NOTIONS OF QUANTUM CHANNEL able to test thisN assertion in the most general way possi- SIMULATION ble, as given in [GW07, CDP08a, CDP09b]. That is, we would like for the simulation to pass the strongest pos- In this section, I review the definition of entanglement sible test that could be performed to distinguish it from cost of a quantum channel, as detailed in [BBCW13], the n uses of A→B. Such a test allows for the discrim- and I also review the main theorem from [BBCW13]. N 3

A B A B A B N N N R R R Q

Vs. B B B

A A A A B A B A B

R R R Q

FIG. 1. The top part of the ﬁgure displays a three-round interaction between the discriminator and the simulator in the case that the actual channel NA→B is called three times. The bottom part of the ﬁgure displays the interaction between the discriminator and the simulator in the case that the simulation of three channel uses is called.

After that, I propose the revised deﬁnition of a channel’s success probability psucc( , ) for any physical experi- entanglement cost. ment, of the kind discussedR S after (2), to distinguish the Before starting, let us deﬁne a maximally entangled channels and : state Φ of Schmidt rank d as R S AB 1 1 psucc( , ) = 1 + . (6) d 2 2 ♦ 1 X R S kR − Sk ΦAB i j A i j B, (3) ≡ d | ih | ⊗ | ih | i,j=1 A. Entanglement cost of a quantum channel where i A i and i B i are orthonormal bases. An from [BBCW13] {| i } {| i } LOCC channel A0B0→AB is a bipartite channel that can L be written in the following form: Let us now review the notion of entanglement cost from

X y y [BBCW13]. Fix n, M N, ε [0, 1], and a quantum A0B0→AB = 0 0 , (4) ∈ ∈ L EA →A ⊗ FB →B channel A→B. According to [BBCW13], an (n, M, ε) y (parallel)N LOCC-assisted channel simulation code con-

y y sists of an LOCC channel n n and a maxi- A A0B0→B where 0 and 0 are sets of completely A →A y B →B y mally entangled resource stateL Φ of Schmidt rank M, positive,{E trace-non-increasing} {F maps,} such that the sum A0B0 P y y such that together they implement a simulation channel map 0 0 is a quantum channel (com- y A →A B →B n n E ⊗ F + A →B , as defined in (1). In this model, to be clear, we pletely positive and trace preserving) [CLM 14]. How- assumeP that Alice has access to all systems labeled by A, ever, not every channel of the form in (4) is an LOCC Bob has access to all systems labeled by B, and they are channel (there are separable channels of the form in (4) n n + in distant laboratories. The simulation A →B is con- that are not implementable by LOCC [BDF 99]). The P ⊗n sidered ε-distinguishable from n parallel calls ( A→B) diamond norm of the difference of two channels A→B N R of the actual channel A→B if the condition in (2) holds. and A→B is defined as [Kit97] N S Note here again that the condition in (2) corresponds to a discriminator who is restricted to performing only a sup A→B(ψRA) A→B(ψRA) , (5) ♦ 1 kR − Sk ≡ ψRA kR − S k parallel test to distinguish the n calls of A→B from its simulation. Let us also note here that theN condition in (2) where the optimization is with respect to all pure bi- can be understood as the simulation An→Bn providing P ⊗n partite states ψRA with system R isomorphic to sys- an approximate teleportation simulation of ( A→B) , tem A and the trace norm of an operator X is defined as in the language of the later work of [KW18]. N † X 1 Tr √X X . The operational interpretation of A rate R is said to be achievable for (parallel) chan- k k ≡ { } the diamond norm is that it is related to the maximum nel simulation of A→B if for all ε (0, 1], δ > 0, and N ∈ 4 sufficiently large n, there exists an (n, 2n[R+δ], ε) LOCC- To begin with, let us fix n, M N, ε [0, 1], and a ∈ ∈ assisted channel simulation code. The (parallel) entan- quantum channel A→B. We define an (n, M, ε) (sequen- (p) tial) LOCC-assistedN channel simulation code to consist of glement cost EC ( ) of the channel is equal to the N N a maximally entangled resource state Φ of Schmidt infimum of all achievable rates, with the superscript (p) A0B0 indicating that the test of the simulation is restricted to rank M and a set be a parallel discrimination test. (i) n The main result of [BBCW13] is that the channel’s i=1 (11) {LAiAi−1Bi−1→BiAiBi } entanglement cost E(p)( ) is equal to the regularization C N of its entanglement of formation. To state this result of LOCC channels. Note that the systems AnBn of the precisely, recall that the entanglement of formation of a final LOCC channel (n) can be taken LAnAn−1Bn−1→BnAnBn bipartite state ρAB is defined as [BDSW96] trivial without loss of generality. As before, Alice has access to all systems labeled by A, Bob has access to EF (A; B)ρ all systems labeled by B, and they are in distant lab- ≡ ( ) oratories. The structure of this simulation protocol is X X x inf pX (x)H(A)ψx : ρAB = pX (x)ψAB , (7) intended to be compatible with a discrimination strategy x x that can test the actual n channels versus the above simulation in a sequential way, along the lines discussed in where the infimum is with respect to all convex decom- x [CDP08a, CDP09b] and [Gut12]. I later show how this positions of ρAB into pure states ψAB and encompasses the parallel tests discussed in the previous x x section. H(A)ψx Tr ψ log ψ (8) ≡ − { A 2 A} A sequential discrimination strategy consists of an ini- (i) n−1 x tial state ρR A , a set of adaptive is the quantum entropy of the marginal state ψA = 1 1 RiBi→Ri+1Ai+1 i=1 x {A } TrB ψ . The entanglement of formation does not in- channels, and a quantum measurement QRnBn ,IRnBn { AB} { − crease under the action of an LOCC channel [BDSW96]. QR B . Let us employ the shorthand ρ, ,Q to ab- n n } { A } A channel’s entanglement of formation EF ( ) is then breviate such a discrimination strategy. Note that, in defined as N performing a discrimination strategy, the discriminator has a full description of the channel A→B and the sim- N EF ( ) sup EF (R; B)ω, (9) ulation protocol, which consists of Φ and the set in N ≡ A0B0 ψRA (11). If this discrimination strategy is performed on the n uses of the actual channel A→B, the relevant states where ωRB A→B(ψRA), and it suffices to take the op- N ≡ N involved are timization with respect to a pure state input ψRA, with system R isomorphic to system A, due to purification, the (i) ρR A (ρR B ), (12) Schmidt decomposition theorem, and the LOCC mono- i+1 i+1 ≡ ARiBi→Ri+1Ai+1 i i tonicity of entanglement of formation [BDSW96]. We can for i 1, . . . , n 1 and now state the main result of [BBCW13] described above: ∈ { − } ρ (ρ ), (13) (p) 1 ⊗n RiBi Ai→Bi RiAi E ( ) = lim EF ( ). (10) ≡ N C N n→∞ n N for i 1, . . . , n . If this discrimination strategy is per- The regularized formula on the right-hand side may be formed∈ { on the simulation} protocol discussed above, then difficult to evaluate in general, and thus can only be the relevant states involved are considered a formal expression, but if the additivity re- 1 ⊗n (1) lation EF ( ) = EF ( ) holds for a given chan- τR B A B (τR1A1 ΦA B ), n N N 1 1 1 1 ≡ LA1A0B0→B1A1B1 ⊗ 0 0 nel for all n 1, then it simplifies significantly as (i) (p)N ≥ τ (τ ), (14) R A A B RiBi→Ri+1Ai+1 R B A B EC ( ) = EF ( ). i+1 i+1 i i ≡ A i i i i N N for i 1, . . . , n 1 , where τR A = ρR A , and ∈ { − } 1 1 1 1 B. Proposal for a revised notion of entanglement (i) cost of a channel τR B A B (τR A A B ), (15) i i i i ≡ LAiAi−1Bi−1→BiAiBi i i i−1 i−1 Now I propose the new or revised definition for entan- for i 2, . . . , n . The discriminator then performs the ∈ { } glement cost of a channel. As motivated in the intro- measurement QR B ,IR B QR B and guesses “ac- { n n n n − n n } duction, a parallel test of channel simulation is not the tual channel” if the outcome is QRnBn and “simulation” most general kind of test that can be considered. Thus, if the outcome is IRnBn QRnBn . Figure1 depicts the dis- the new definition proposes that the entanglement cost crimination strategy in− the case that the actual channel of a channel should incorporate the most stringent test is called n = 3 times and in the case that the simulation possible. is performed. 5

If the a priori probabilities for the actual channel or be understood as a particular strategy for discriminat- simulation are equal, then the success probability of the ing A0→B0 from a simulation of A0→B0 , due to the discriminator in distinguishing the channels is given by structuralM decomposition in (20). FollowingM definitions, a simple consequence is the following LOCC monotonicity 1 inequality for the entanglement cost of these channels: [Tr QR B ρR B + Tr (IR B QR B ) τR B ] 2 { n n n n } { n n − n n n n } 1 1 EC ( ) EC ( ). (21) 1 + ρR B τR B , (16) N ≤ M ≤ 2 2 k n n − n n k1 Thus, it takes more or the same entanglement to sim- where the latter inequality is well known from the theory ulate the channel than it does to simulate . Fur- of quantum state discrimination [Hel69, Hol73, Hel76]. thermore, the decompositionM in (20) and the boundN in For this reason, we say that the n calls to the actual (21) can be used to bound the entanglement cost of a channel A→B are ε-distinguishable from the simulation channel from below. Note that the structure in (20) if the followingN condition holds for the respective final was discussedM recently in the context of general resource states theories [CG18, Section III-D-5]. 1 ρR B τR B ε. (17) 2 k n n − n n k1 ≤ D. Parallel tests as a special case of sequential tests If this condition holds for all possible discrimination strategies ρ, ,Q , i.e., if { A } A parallel test of the form described in SectionIIA is 1 a special case of the sequential test outlined above. One sup ρRnBn τRnBn ε, (18) can see this in two seemingly different ways. First, we can 2 k − k1 ≤ {ρ,A} think of the sequential strategy taking a particular form. The state ξ is prepared, and here we identify then the simulation protocol constitutes an (n, M, ε) RA1A2···An systems RA A with system R of ρ in an adap- channel simulation code. It is worthwhile to remark: If 2 n 1 R1A1 tive protocol··· and system A of ξ with system we ascribe the shorthand ( )n for the n uses of the chan- 1 RA1A2···An Nn A1 of ρR A . Then the channel A →B or its simula- nel and the shorthand ( ) for the simulation, then the 1 1 N 1 1 condition in (18) can beL understood in terms of the n- tion is called. After that, the action of the first adaptive channel is simply to swap in system A of ξ to round strategy norm of [CDP08a, CDP09b, Gut12]: 2 RA1A2···An the second call of the channel A →B or its simulation, N 2 2 1 n n while keeping systems RB1A3 An as part of the refer- ( ) ( ) ,n ε. (19) ence R of the state ρ . Then··· this iterates and the 2 k N − L k♦ ≤ 2 R2A2 final measurement is performed on all of the remaining As before, a rate R is achievable for (sequential) chan- systems. nel simulation of if for all ε (0, 1], δ > 0, and suffi- The other way to see how a parallel test is a special N ∈ n[R+δ] ciently large n, there exists an (n, 2 , ε) (sequential) kind of sequential test is to rearrange the simulation pro- channel simulation code for . We define the (sequen- N tocol as has been done in Figure2. Here, we see that tial) entanglement cost EC ( ) of the channel to be N N the simulation protocol has a memory structure, and it the infimum of all achievable rates. Due to the fact that is clear that the simulation protocol can accept as in- this notion is more general, we sometimes simply refer put a state ξRA1A2···An and outputs a state on systems to EC ( ) as the entanglement cost of the channel in N N RB1 Bn, which can subsequently be measured. what follows. As··· a consequence of this reduction, any (n, M, ε) sequential channel simulation protocol can serve as an (n, M, ε) parallel channel simulation protocol. Further- C. LOCC monotonicity of the entanglement cost more, if R is an achievable rate for sequential channel simulation, then it is also an achievable rate for parallel Let us note here that if a channel A→B can be real- channel simulation. Finally, these reductions imply the N ized from another channel A0→B0 via a preprocessing following inequality: pre M LOCC channel 0 and a postprocessing LOCC A→A AM BM post L (p) channel B0A B →B as EC ( ) E ( ). (22) L M M N ≥ C N post pre = 0 0 0 0 , (20) A→B B AM BM →B A →B A→A AM BM Intuitively, one might sometimes require more entangle- N L ◦ M ◦ L ment in order to pass the more stringest test that occurs then it follows that any (n, M, ε) protocol for sequential in sequential channel simulation. As a consequence of channel simulation of A0→B0 realizes an (n, M, ε) pro- M (10) and (22), we have that tocol for sequential channel simulation of A→B. This N is an immediate consequence of the fact that the best 1 E ( ) lim E ( ⊗n). (23) strategy for discriminating A→B from its simulation can C F N N ≥ n→∞ n N 6

The first inequality follows from the condition in (18), B B as well as from the continuity bound for entanglement of formation from [Win16b, Corollary 4]. The second A B B inequality follows from the LOCC monotonicity of the entanglement of formation [BDSW96], here thinking of A A B the person who possesses systems RA1 An to be in the same laboratory as the one possessing··· the systems A A Ai, while the person who possesses the Bi systems is in a different laboratory. The first equality follows from the A B fact that ψ is in tensor product with Φ , so RA1···An A0B0 that by a local channel, one may remove ψRA1···An or R Q append it for free. The final equality follows because the entanglement of formation of the maximally entangled state is equal to the logarithm of its Schmidt rank. Since FIG. 2. The simulation protocol from the bottom part of Fig- the bound holds uniformly regardless of the input state ure1 rewritten to clarify that it can participate in a parallel ψRA1···An , after an optimization and a rearrangement we channel simulation test. conclude the stated lower bound on the non-asymptotic 1 entanglement cost n log2 M of the protocol.

It is an interesting question (not addressed here) to de- Remark 1 Let us note here that the entanglement cost termine if there exists a channel such that the inequality of a quantum channel is equal to zero if and only if in (22) is strict. the channel is entanglement-breaking [HSR03, Hol08]. If desired, it is certainly possible to obtain a non- The “if-part” follows as a straightforward consequence of asymptotic, weak-converse bound that implies the above deﬁnitions and the fact that these channels can be im- bound after taking limits. Let us state this bound as plemented as a measurement followed by a preparation follows: [HSR03, Hol08], given that this measure-prepare procedure is a particular kind of LOCC and thus allowed for Proposition 1 Let A→B be a quantum channel, and free (without any cost) in the above model. The “only-if” N let n, M N and ε [0, 1]. Set d = min A , B , i.e., part follows from (22) and [BBCW13, Corollary 18], the the minimum∈ of the∈ input and output dimensions{| | | |} of the latter of which depends on the result from [YHHSR05]. channel A→B. Then the following bound holds for any (n, M, ε)Nsequential channel simulation code: III. BOUNDS FOR THE ENTANGLEMENT 1 1 ⊗n 1 COST OF TELEPORTATION-SIMULABLE log2 M EF ( ) √ε log d g2(√ε), (24) n ≥ n N − − n CHANNELS

1 where n log2 M is understood as the non-asymptotic entanglement cost of the protocol and the bosonic entropy A. Upper bound on the entanglement cost of teleportation-simulable channels function g2(x) is deﬁned for x 0 as ≥ The most trivial method for simulating a channel is g2(x) (x + 1) log2(x + 1) x log2 x. (25) ≡ − to employ the teleportation protocol [BBC+93] directly. Proof. To see this, suppose that there exists an (n, M, ε) In this method, Alice and Bob could use the teleporta- protocol for sequential channel simulation. Then by the tion protocol so that Alice could transmit the input of above reasoning (also see Figure2), it can be thought the channel to Bob, who could then apply the channel. of as a parallel channel simulation protocol, such that Repeating this n times, this trivial method would implement an (n, A n , 0) simulation protocol in either the the criterion in (2) holds. Suppose that ψRA1···An is a n parallel or sequential| | model. Alternatively, Alice could test input state, with R = A , leading to ωRB1···Bn = ⊗n | | | | apply the channel ﬁrst and then teleport the output to ( A→B) (ψRA1···An ) when the actual channels are ap- pliedN and σ when the simulation is applied. Then Bob, and repeating this n times would implement an RB1···Bn n we have that (n, B , 0) simulation protocol in either the parallel or sequential| | model. Thus, they could always achieve a rate EF (R; B1 Bn)ω of log2(min A , B ) using this approach, and this rea- ··· soning establishes{| | | a|} simple dimension upper bound on EF (R; B1 Bn)σ + n√ε log d + g2(√ε) ≤ ··· the entanglement cost of a channel: EF (RA1 AnA0; B0)ψ⊗Φ + n√ε log d + g2(√ε) ≤ ··· EC ( A→B) log (min A , B ). (27) = EF (A0; B0)Φ + n√ε log d + g2(√ε) N ≤ 2 {| | | |} = log2 M + n√ε log d + g2(√ε). (26) In this context, also see [KW18, Proposition 9]. 7

A less trivial approach is to exploit the fact that some holds [BD11, Theorem 2] channels of interest could be teleportation-simulable with 1 δ n n 1 associated resource state ωA0B0 , in which the resource lim lim EF,0(C ; D )τ ⊗n = lim EF (C; D)τ , (33) state need not be a maximally entangled state (see δ→0 n→∞ n n→∞ n [BDSW96, Section V] and [HHH99, Eq. (11)]). Re- = EC (τCD). (34) call from these references that a channel A→B is where the latter quantity denotes the entanglement cost teleportation-simulable with associated resourceN state of the state τCD [HHT01]. ωA0B0 if there exists an LOCC channel AA0B0→B such L that the following equality holds for all input states ρA: Proof of Proposition2. The approach for an (n, M, ε) sequential channel simulation consists of the following A→B(ρA) = AA0B0→B(ρA ωA0B0 ). (28) steps: N L ⊗ First, employ the one-shot entanglement cost protocol If a channel possesses this structure, then we arrive at from [BD11, Theorem 1], which consumes a maximally the following upper bound on the entanglement cost: entangled state Φ of Schmidt rank M along with an A0B0 LOCC channel 0n 0n to generate n approximate A0B0→A B Proposition 2 Let A→B be a quantum channel that P N copies of the resource state ωA0B0 . In particular, using is teleportation-simulable with associated resource state the maximally entangled state Φ with A0B0 ωA0B0 , as defined in (28). Let n, M N and ε (0, 1). Then there exists an (n, M, √ε) sequential∈ channel∈ sim- ε/2 0n 0n log2 M = E (A ; B )ω⊗n , (35) ulation code satisfying the following bound F,0 one can achieve the following approximation [BD11, The- 1 1 ε/2 0n 0n orem 1]: log M E (A ; B ) ⊗n , (29) n 2 ≤ n F,0 ω 1 ⊗n ω 0 0 ω 0n 0n √ε, (36) 1 A B eA B 1 where n log2 M is understood as the non-asymptotic en- 2 − ≤ ε/2 0n 0n tanglement cost of the protocol, and EF,0 (A ; B )ω⊗n is where the ε/2-smooth entanglement of formation (EoF) [BD11] ωA0nB0n 0n 0n (Φ ). (37) recalled in Definition1 below. e ≡ PA0B0→A B A0B0 Next, at the first instance in which the channel should Definition 1 (Smooth EoF [BD11]) Let δ (0, 1) be simulated, Alice and Bob apply the LOCC chan- and τ be a bipartite state. Let = p (x), φ∈x de- 0 0 CD X CD nel AA0B0→B from (28) to the A and B systems of E { } L 1 1 note a pure-state ensemble decomposition of τCD, mean- ω 0n 0n . For the second instance, they apply the LOCC P x x eA B ing that τCD = pX (x)φ , where φ is a pure state 0 0 x CD CD channel AA0B0→B from (28) to the A2 and B2 systems and p is a probability distribution. Define the condi- L X of ωeA0nB0n . This continues for the next n 2 rounds of tional entropy of order zero H0(K L)ω of a bipartite state − | the sequential channel simulation. ωKL as By the data processing inequality for trace distance,

ω it is guaranteed that the following bound holds on the H0(K L)ω max log2 Tr ΠKL(IK σL) , (30) performance of this protocol for sequential channel sim- | ≡ σL { ⊗ } ulation: ω where ΠKL denotes the projection onto the support of 1 n n 1 ⊗n ω and σ is a density operator. Then the δ-smooth ( ) ( ) ω 0 0 ωA0nB0n √ε. KL L 2 k N − L k♦,n ≤ 2 A B − e 1 ≤ entanglement of formation of τCD is given by (38)

δ This follows because the distinguishability of the sim- EF,0(C; D)τ min H0(C X)τ , (31) E,τ ∈Bδ (τ ) e ulation from the actual channel uses is limited by the ≡ eXC cq XC | ⊗n distinguishability of the states ωA0B0 and ωeA0nB0n , due to where the minimization is with respect to all pure- the assumed structure of the channel in (28), as well as the structure of the sequential channel simulation. state ensemble decompositions of τCD, τXCD = P x E By applying definitions, the bound in Proposition2, x pX (x) x x X φCD is a labeled pure-state extension | ih | ⊗ δ taking the limits n and then ε 0 (with M = of τCD, and the δ-ball Bcq(τXC ) of cq states for a cq state 2n[R+δ] for a fixed rate→ ∞R and arbitrary→ δ > 0), and τXC is defined as applying (33), we conclude the following statement:

δ n X x B (τ ) ω : ω 0, ω = x x ω , Corollary 1 Let A→B be a quantum channel that cq XC XC XC XC C N ≡ ≥ x | ih |⊗ is teleportation-simulable with associated resource state o ωA0B0 , as deﬁned in (28). Then the entanglement cost of ωXC τXC δ . (32) k − k1 ≤ the channel is never larger than the entanglement cost N of the resource state ω 0 0 : The δ-smooth entanglement of formation has the property A B ⊗n that, for a tensor-power state τ , the following limit EC ( ) EC (ωA0B0 ). (39) CD N ≤ 8

The above corollary captures the intuitive idea that Theorem 1 Let A→B be a resource-seizable, if a single instance of the channel can be simulated teleportation-simulableN channel with associated re- N via LOCC starting from a resource state ωA0B0 , then the source state ωA0B0 , as given in Deﬁnition2. Then the entanglement cost of the channel should not exceed the entanglement cost of the channel A→B is equal to its entanglement cost of the resource state. The idea of the parallel entanglement cost, which inN turn is equal to the above proof is simply to prepare a large number n of entanglement cost of the resource state ωA0B0 : copies of ωA0B0 approximately and then use these to sim- (p) ulate n uses of the channel , such that the simulation EC ( ) = EC ( ) = EC (ωA0B0 ). (41) could not be distinguished fromN n uses of the channel N N in any sequential test. N Proof. Consider from (22) that

(p) 1 ⊗n EC ( ) E ( ) = lim EF ( ). (42) N ≥ C N n→∞ n N B. The entanglement cost of resource-seizable, n teleportation-simulable channels Let ψRA ψRA1···An be an arbitrary pure input state to consider≡ at the input of the tensor-power channel ⊗n ( A→B) , leading to the state In this section, I deﬁne teleportation-simulable chan- N nels that are resource-seizable, meaning that one can ⊗n σRBn ( A→B) (ψRA ···A ). (43) seize the channel’s underlying resource state by the fol- ≡ N 1 n lowing procedure: From the assumption that the channel is teleportation- simulable with associated resource state ωA0B0 , we have 1. prepare a free, separable state, from (28) that

⊗n ⊗n σRBn = ( AA0B0→B) (ψRAn ω 0 0 ) (44) 2. input one of its systems to the channel, and then L ⊗ A B Then 3. post-process with a free, LOCC channel. n n 0n 0n EF (R; B )σ EF (RA A ; B )ψ⊗ω⊗n (45) This procedure is indeed related to the channel process- ≤ 0n 0n = E (A ; B ) ⊗n , (46) ing described earlier in (20). After that, I prove that the F ω entanglement cost of a resource-seizable channel is equal where the inequality follows from LOCC monotonicity of to the entanglement cost of its underlying resource state. the entanglement of formation. Since the bound holds for an arbitrary input state, we conclude that the following Deﬁnition 2 (Resource-seizable channel) Let inequality holds for all n N: be a teleportation-simulable channel with associ- ∈ A→B 1 1 N 0 0 ⊗n 0n 0n ated resource state ωA B , as deﬁned in (28). Suppose EF ( ) EF (A ; B ) ⊗n . (47) n N ≤ n ω that there exists a separable input state ρAM ABM to the channel and a postprocessing LOCC channel Now taking the limit n , we conclude that 0 0 0 0 → ∞ AM BBM →A B such that the resource state ωA B can D (p) be seized from the channel A→B as follows: E ( ) EC (ωA0B0 ). (48) N C N ≤

A BB →A0B0 ( A→B(ρA AB )) = ωA0B0 . (40) To see the other inequality, let a decomposition of the D M M N M M separable input state ρAM ABM be given by Then we say that the channel is a resource-seizable, X ρ = p (x)ψx φx . (49) teleportation-simulable channel. AM ABM X AM A BM x ⊗ In AppendixA, I discuss how resource-seizable chan- Considering that [ψx ]⊗n is a particular input to the AM A nels are related to those that are “implementable from ⊗n tensor-power channel ( A→B) , we conclude that their image,” as defined in [CMH17, Appendix A]. In N SectionVI, I also discuss how to generalize the notion ⊗n n n EF ( ) EF (AM ; B )[N (ψx)]⊗n . (50) of a resource-seizable channel to an arbitrary resource N ≥ theory. Since this holds for all x, we have that The main result of this section is the following simplify- ⊗n X n n ing form for the entanglement cost of a resource-seizable EF ( ) pX (x)EF (A ; B )[N (ψx)]⊗n N ≥ M channel (as defined above), establishing that its entan- x glement cost in the asymptotic regime is the same as the X n n n = pX (x)EF (AM ; B BM )[N (ψx)⊗φx]⊗n entanglement cost of the underlying resource state. Fur- x thermore, for these channels, the entanglement cost is n n n EF (AM ; B BM )[N (ρ)]⊗n not increased by the need to pass a more stringest test ≥ 0n 0n EF (A ; B ) ⊗n , (51) for channel simulation as required in a sequential test. ≥ ω 9 where the equality follows because introducing a prod- of a depolarizing channel, is not known. In the next few uct state locally does not change the entanglement, the subsections, I detail some example channels for which it second inequality follows from convexity of entanglement is possible to characterize their entanglement cost. of formation [BDSW96], and the last inequality follows from the assumption in (40) and the LOCC monotonicity of the entanglement of formation. Since the inequality A. Erasure channels holds for all n N, we can divide by n and take the limit n to conclude∈ that → ∞ A simple example of a channel that is covariant is the (p) quantum erasure channel, defined as [GBP97] E ( ) EC (ωA0B0 ), (52) C N ≥ q(ρ) (1 q)ρ + q e e , (55) and in turn, from (48), that E ≡ − | ih | (p) where ρ is a d-dimensional input state, q [0, 1] is the EC ( ) = EC (ωA0B0 ). (53) ∈ N erasure probability, and e e is a pure erasure state or- | ih | Combining this equality with the inequalities in (39) and thogonal to any input state, so that the output state (42) leads to the statement of the theorem. has d + 1 dimensions. By the remark above, we conclude q q that EC ( ) = EC ( A→B(ΦRA)), and so determining the entanglementE cost boilsE down to determining the entan- IV. EXAMPLES glement cost of the Choi state

q IR The equality in Theorem1 provides a formal expres- A→B(ΦRA) = (1 q)ΦRA + e e . (56) sion for the entanglement cost of any resource-seizable, E − d ⊗ | ih | teleportation-simulable channel, given in terms of the en- q An obvious pure-state decomposition for A→B(ΦRA) tanglement cost of the underlying resource state ωA0B0 . (see [BBCW13, Eqs. (93)–(95)]) leads to E Due to the fact that the entanglement cost of a state is generally not equal to its entanglement of forma- q q EC ( (ΦRA)) EF ( (ΦRA)) (57) tion [Has09], it could still be a signiﬁcant challenge to EA→B ≤ EA→B (1 q) log d. (58) compute the entanglement cost of these special chan- ≤ − 2 nels. However, for some special states, the equality 0 0 As it turns out, these inequalities are tight, due to E (ω 0 0 ) = E (A ; B ) does hold, and I discuss sev- C A B F ω an operational argument. In particular, the distill- eral of these examples and related channels here. q able entanglement of (ΦRA) is exactly equal to Let us begin by recalling the notion of a covariant chan- EA→B (1 q) log2 d [BDS97], and due to the operational fact nel A→B [Hol02]. For a group G with unitary channel − N g g that the distillable entanglement of a state cannot ex- representations g and g acting on the input sys- {UA} {VB} ceed its entanglement cost [BDSW96], we conclude that tem A and output system B of the channel A→B, the q N EC ( A→B(ΦRA)) = (1 q) log2 d, and in turn that channel A→B is covariant with respect to the group G E − N if the following equality holds q (p) q EC ( ) = EC ( ) = (1 q) log2 d. (59) g g E E − A→B A = B A→B. (54) N ◦ U V ◦ N This result generalizes the ﬁnding from [BBCW13], 1 P g (p) If the averaging channel is such that (X) = which is that E ( q) = (1 q) log d, and so we con- |G| g UA C 2 Tr[X]I/ A (implementing a unitary one-design), then clude that for erasureE channels,− the entanglement cost | | we simply say that the channel A→B is covariant. of these channels is not increased by the need to pass Then from [CDP09a, SectionN 7] (see also [WTB17, a more stringest test for channel simulation, as posed Appendix A]), we conclude that any covariant channel by a sequential test. Note also that the distillable en- q is teleportation-simulable with associated resource state tanglement of the erasure channel is given by ED( ) = E given by the Choi state of the channel, i.e., ω 0 0 = (1 q) log d, due to [BDS97]. A B − 2 A→B(ΦA0A). As such, covariant channels are resource- The fact that the distillable entanglement of an era- seizable,N so that the equality in Theorem1 applies to sure channel is equal to its entanglement cost, implies all covariant channels. Thus, the entanglement cost that, if we restrict the resource theory of entanglement of a covariant channel is equal to the entangle- for quantum channels to consist solely of erasure chan- ment cost of its Choi state. In spite of this reduction, nels, then it is reversible. By this, we mean that, in the it could still be a great challenge to compute formulas limit of many channel uses, if one begins with an erasure for the entanglement cost of these channels, due to the channel of parameter q and distills ebits from it at a rate fact that the entanglement of formation is not necessar- (1 q) log2 d, then one can subsequently use these dis- ily equal to the entanglement cost for the Choi states of tilled− ebits to simulate the same erasure channel again. these channels. For example, the entanglement cost of As we see below, this reversibility breaks down when con- an isotropic state [Wer89, HH99], which is the Choi state sidering other channels. 10

B. Dephasing channels Rate 1.0 A d-dimensional dephasing channel has the following action: 0.8 Ent. Cost d−1 Dist. Ent. q X i i† 0.6 (ρ) = qiZ ρZ , (60) D i=0 0.4 where q is a vector containing the probabilities qi and Z has the following action on the computational basis Z x = e2πix/d x . This channel is covariant with respect 0.2 to| thei Heisenberg–Weyl| i group of unitaries, which are well q known to be a unitary one-design. Furthermore, as re- 0.2 0.4 0.6 0.8 1.0 marked previously (e.g., in [TWW17]), the Choi state q (ΦRA) of this channel is a maximally correlated DA→B state [Rai99, Rai01], which has the form q p FIG. 3. Entanglement cost EC (D ) = h2(1/2 + q (1 − q)) q and distillable entanglement ED(D ) = 1 − h2(q) of the qubit X q αi,j i j R i j B. (61) dephasing channel D as a function of the dephasing param- | ih | ⊗ | ih | i,j eter q ∈ [0, 1], with the shaded area demonstrating the gap between them. The units for rate on the vertical axis are ebits As such, Theorem1 applies to these channels, implying per channel use, and q on the horizontal axis is dimensionless. that

q (p) q q EC ( ) = EC ( ) = EC ( A→B(ΦRA)) (62) dephasing channel with parameter q (0, 1) and dis- D Dq D ∈ = EF ( A→B(ΦRA)), (63) tilled ebits from it at the ideal rate of 1 h2(q), and D then subsequently wanted to use these ebits− to simu- with the final equality resulting from the fact that the late a qubit dephasing channel with the same parameter, entanglement cost is equal to the entanglement of for- this is not possible, because the rate at which ebits are mation for maximally correlated states [VDC02, HSS03]. distilled is not sufficient to simulate the channel again. In [HSS03, Section VI-A], an optimization procedure is Figure3 compares the formulas for entanglement cost given for calculating the entanglement of formation of and distillable entanglement of the qubit dephasing chan- maximally correlated states, which is simpler than that nel, demonstrating that there is a noticeable gap be- needed from the definition of entanglement of formation. tween them. At q = 1/2, the qubit dephasing chan- A qubit dephasing channel with a single dephasing pa- nel is a completely dephasing, classical channel, so that rameter q [0, 1] is defined as E ( 1/2) = E ( 1/2) = 0. Thus, a reasonable approx- ∈ C D imationD to the differenceD is given by a Taylor expansion q(ρ) = (1 q) ρ + qZρZ. (64) D − about q = 1/2:

For the Choi state of this channel, there is an explicit q q formula for its entanglement of formation [Woo98], from EC ( ) ED( ) = D − D which we can conclude that 1 1 1 2 1 4 2 ln 1 1 (q 2 ) + O((q 2 ) ). (68) q (p) q p ln 2 q 2 − − − EC ( ) = E ( ) = h2(1/2 + q (1 q)), (65) | − | D C D − where C. Werner–Holevo channels h2(x) x log x (1 x) log (1 x) (66) ≡ − 2 − − 2 − is the binary entropy. The equality in (65) solves an open A particular kind of Werner–Holevo channel performs question from [BBCW13], where it had only been shown the following transformation on a d-dimensional input (p) q p state ρ [WH02]: that EC ( ) h2(1/2 + q (1 q)). The resultsD ≤ of [BDSW96, Eq.− (57)] and [Hay06, 1 Eq. (8.114)] gave a simple formula for the distillable en- (d)(ρ) (Tr ρ I T (ρ)) , (69) tanglement of the qubit dephasing channel: W ≡ d 1 { } − − q where T denotes the transpose map T ( ) = ED( ) = 1 h2(q). (67) D − P i j ( ) i j . As observed in [WH02, Section· II] i,j | ih | · | ih | Thus, this formula and the formula in (65) demonstrate and [LM15, Section VII], this channel is covariant, and that the resource theory of entanglement for these chan- so an immediate consequence of [CDP09a, Section 7] nels is irreversible. That is, if one started from a qubit is that these channels are teleportation simulable with 11 associated resource state given by their Choi state. The Rate latter fact was explicitly observed in [LM15, Sections VI 1.0 ●▲ ● and VII], as well as [CMH17, Appendix A]. Furthermore, ● Ent. Cost Lower Bound its Choi state is given by 0.8 ▲ ▲ Dist. Ent. Upper Bound

(d) 1 (ΦRA) = αd (IRB FRB) , (70) WA→B ≡ d (d 1) − 0.6 ▲ − ▲ where αd is the antisymmetric state, i.e., the maximally 0.4 ● ● ●▲ ● ● ● ● ● ● ▲ mixed state on the antisymmetric subspace of a d d ▲ ▲ P × ▲ quantum system and FRB i j R j i B denotes ▲ i,j 0.2 ▲ the unitary swap operator.≡ Theorem| ih |1 ⊗|thusih applies| to these channels, and we find that ●▲ d 2 3 4 5 6 7 8 9 10 11 12 (d) (p) (d) EC ( ) = E ( ) (71) W C W = EC (αd) (72) (d) log (4/3) 0.415, (73) FIG. 4. Lower bound on the entanglement cost EC (W ) ≥ 2 ≈ from (73) and upper bound on distillable entanglement (d) (d) with the inequality following from [CSW12, Theorem 2]. ED(W ) from (77) for the Werner–Holevo channel W We also have that as a function of the parameter d ≥ 4, with the lines con- necting the dots demonstrating the gap between them. For (d) (p) (d) d = 2, the points are exact due to (79), and reversibility holds. EC ( ) = EC ( ) = EC (αd) EF (αd) = 1, (3) W W ≤ For d = 3, the entanglement cost EC (W ) is exactly equal (74) (3) with the last equality following from the result stated in to one, as recalled in (75), while (77) applies to ED(W ), and the resource theory is irreversible. For d ∈ {4, 5, 6}, the [VW01, Section IV-C]. For d = 3, the entanglement cost bounds are not strong enough to reach a conclusion about re- EC (α3) is known to be equal to exactly one ebit [Yur03]: versibility. For d ≥ 7, the resource theory is irreversible, and (d) (d) the gap EC (W ) − ED(W ) grows at least as large as the (3) (p) (3) EC ( ) = EC ( ) = 1. (75) difference of (73) and (78). The units for rate on the vertical W W axis are ebits per channel use, and d on the horizontal axis is It was observed in [CMH17, Appendix A] (as well dimensionless. as [Win16a]) that the distillable entanglement of the Werner–Holevo channel (d) is equal to the distillable entanglement of its ChoiW state: 0.793, while the entanglement cost is equal to one, as stated in (75), so that (d) ED( ) = ED(αd). (76) (3) (3) W ED( ) 0.793 < 1 = EC ( ). (80) W ≤ W Thus, an immediate consequence of [CSW12, Theorem 1 Thus, the resource theory of entanglement is not re- and Eq. (5)] is that versible for (3). For d 4, 5, 6 , the upper bound in (77) and theW lower bound∈ in { (73)} are not strong enough ( d+2 ) (d) log2 d if d is even to make a definitive statement (interestingly, the bounds ED( ) 1 d+3 (77) W ≤ 2 log2 d−1 if d is odd in (77) and (73) are actually equal for d = 6). Then for d 7, the upper bound in (77) and the lower bound in 2 1 1 = 1 + O . (78) (73≥) are strong enough to conclude that d ln 2 − d d3 · (d) (d) ED( ) < EC ( ), (81) We can now observe that the resource theory of en- W W tanglement is generally not reversible when restricted to so that the resource theory is not reversible for (d). Werner–Holevo channels. The case d = 2 is somewhat Figure4 summarizes these observations. W trivial: in this case, one can verify that the channel (2) is a unitary channel, equivalent to acting on the inputW state with the Pauli Y unitary. Thus, for d = 2, the D. Epolarizing channels (complements of channel is a noiseless qubit channel, and we trivially have depolarizing channels) that The d-dimensional depolarizing channel is a common (2) (2) ED( ) = EC ( ) = 1, (79) model of noise in quantum information, transmitting the W W input state with probability 1 q [0, 1] and replacing it so that the resource theory of entanglement is clearly with the maximally mixed state−π ∈ I with probability q: reversible in this case. For d = 3, the upper bound on ≡ d distillable entanglement in (77) evaluates to 1 log (3) ∆q(ρ) = (1 q) ρ + qπ. (82) 2 2 ≈ − 12

According to Stinespring’s theorem [Sti55], every quan- Thus, by definition, a channel complementary to ∆q is tum channel A→B can be realized by the action of some realized by N isometric channel A→BE followed by a partial trace: q † U Λ (ρA) TrA UρAU , (90) A→SG1G2 ≡ { } A→B(ρA) = TrE A→BE(ρA) . (83) N {U } and in what follows, let us refer to Λq as the A→SG1G2 Due to the partial trace and its invariance with respect epolarizing channel. to isometric channels acting exclusively on the E system, The isometry UA→SG1G2A in (87) is unitarily covari- the extending channel A→BE is not unique in general, ant, in the sense that for an arbitrary unitary VA acting but it is unique up to thisU freedom. Then given an iso- on the input, we have that metric channel A→BE extending A→B as in (83), the U c N complementary channel is defined by a partial UA→SG1G2AVA = VG1 V G2 VA UA→SG1G2A, (91) A→E ⊗ ⊗ trace over the system B andN is interpreted physically as the channel from the input to the environment: where V denotes the complex conjugate of V . The identity in (91) follows because c (ρA) = TrB A→BE(ρA) . (84) NA→E {U } UA→SG1G2AVA ψ A | i q Due to the fact that properties of the original channel = C-SWAPSG A ( φ S Φ G G VA ψ A) 1 | i | i 1 2 | i are related to properties of its complementary channel q = C-SWAPSG A φ S VG V G Φ G G VA ψ A [KMNR07, Dev05], there has been significant interest in 1 | i 1 2 | i 1 2 | i q understanding complementary channels. In this spirit, = VG1 V G2 VA C-SWAPSG1A ( φ S Φ G1G2 ψ A) and due to the prominent role of the depolarizing chan- ⊗ ⊗ | i | i | i = VG V G VA UA→SG G A ψ A. (92) nel, researchers have studied its complementary channels 1 ⊗ 2 ⊗ 1 2 | i [DFH06, LW17]. In [DFH06, Eq. (3.6)], the following The above analysis omits some tensor-product symbols q form was given for a complementary channel of ∆ : for brevity. The third equality uses the well known fact that Φ G G = VG V G Φ G G . In the fourth q q† | i 1 2 1 ⊗ 2 | i 1 2 ρ SAF (ρA IF ) SAF , (85) → ⊗ equality, we have exploited the facts that V G2 commutes with C-SWAPSG1A and that where IF is a d-dimensional identity operator and

SWAPG A (VG VA) = (VG VA) SWAPG A. (93) r 1 1 ⊗ 1 ⊗ 1 q q S IAF + AF ≡ d The covariance in (91) then implies that the epolarizing s ! channel is covariant in the following sense: q d2 1 √ √ d + 1 q −2 ΦAF . (86) q − d − d (Λ A)(ρA) A→SG1G2 ◦ V q = ( G G Λ )(ρA), (94) A channel complementary to ∆q has been called an “epo- V 1 ⊗ V 2 ◦ A→SG1G2 larizing channel” in [LW17]. where denotes the unitary channel realized by the the An alternative complementary channel, related to the unitaryV operator V . above one by an isometry acting on the output sys- As such, by the discussion after (54), the epo- tems AF , but perhaps more intuitive, is realized in the larizing channel is a resource-seizable, teleportation- following way [LW17, Eq. (28)]. Consider the isometry simulable channel with associated resource state given U deﬁned as q A→SG1G2A by Λ (Φ ). Thus, Theorem1 applies to these A→SG1G2 RA channels, implying that the ﬁrst two of the following

UA→SG1G2A ψ A equalities hold | i ≡ q C-SWAPSG A ( φ S Φ G G ψ A) , (87) 1 | i ⊗ | i 1 2 ⊗ | i q (p) q q EC (Λ ) = EC (Λ ) = EC (Λ (ΦRA)) (95) q q where the control qubit φ S √1 q 0 S + √q 1 S, = E (Λ (Φ )) (96) | i ≡ − | i | i F RA Φ G1G2 is a maximally entangled state of Schmidt q q | i rank d, and the controlled-SWAP unitary is given by = 1 q + log2 1 q + − − d − d q q C-SWAP 0 0 I + 1 1 SWAP , (d 1) log2 . (97) SG1A S G1A S G1A − − d d ≡ | ih | ⊗ | ih | ⊗ (88) with SWAPG1A denoting a unitary swap operation. By Let us now justify the ﬁnal two equalities, which give tracing over the systems SG1G2, we recover the original a simple formula for the entanglement cost of epolar- depolarizing channel izing channels. First, consider that the Choi state q ΛA0→SG G (ΦA0A) of the epolarizing channel is equal q † 1 2 ∆ (ρA) = TrSG G UρAU . (89) to the state resulting from sending in the maximally 1 2 { } 13

mixed state to the isometric channel A→SG1G2A, deﬁned from (87): U 1

q Λ 0 (ΦA0A) = A→SG G A(πA), (98) 0.8 A →SG1G2 U 1 2 where system A0 is isomorphic to A. This equality is shown in AppendixB. As such, then [MSW04, Theo- 0.6 rem 3] applies, as discussed in Example 6 therein, and Rate as a consequence, we can conclude the second and third 0.4 equalities in the following, with the bipartite cut of sys- Entanglement Cost tems taken as SG1G2 A: | 0.2 Rains relative entropy q Coherent Information EC (Λ 0 (ΦA0A)) = EC ( A→SG G A(πA)) (99) A →SG1G2 U 1 2 = E ( (π )) (100) 0 F A→SG1G2A A 0 0.2 0.4 0.6 0.8 1 U q = Hmin(∆ ). (101) q

The last line features the minimum output entropy of the depolarizing channel, which was identified in [Kin03] and FIG. 5. The figure depicts the entanglement cost, the Rains shown to be equal to (97). bound, and the coherent information of the epolarizing chan- As discussed in previous examples, it is worthwhile to nel Λq, for d = 2 and q ∈ [0, 1]. The gap between the entangle- consider the reversibility of the resource theory of en- ment cost and the Rains bound for all q ∈ (0, 1) demonstrates tanglement for epolarizing channels. In this spirit, by that the resource theory of entanglement is irreversible for invoking the covariance of Λq, the discussion after (54), epolarizing channels. The units for rate on the vertical axis are ebits per channel use, and q on the horizontal axis is di- [BDSW96, Eq. (55)], and [Rai01, Theorem 4.13], we find mensionless. the following bound on the distillable entanglement of the epolarizing channel Λq:

q recent result of [LW17], which states that the coherent ED(Λ ) R(A; SG1G2) q , (102) ≤ Λ (Φ) information is strictly greater than zero for all q (0, 1]. It is simply that the coherent information is so small∈ for where R(A; SG1G2)Λq (Φ) denotes the Rains relative en- q q . 0.18, that it is difficult to witness its strict posi- tropy of the state Λ 0 (ΦA0A). Recall that the A →SG1G2 tivity numerically. Matlab files to generate Figure5 are Rains relative entropy for an arbitrary state ρAB is defined as [Rai01] available with the arXiv posting of this paper.

R(A; B)ρ min D(ρAB τAB), (103) 0 ≡ τAB ∈PPT (A;B) k V. BOSONIC GAUSSIAN CHANNELS where the quantum relative entropy is defined as [Ume62] In this section, I extend the main ideas of the paper in D(ρ τ) Tr ρ[log ρ log τ] (104) order to characterize the entanglement cost of all single- k ≡ { 2 − 2 } mode bosonic Gaussian channels [Ser17]. From a prac- and the Rains set PPT0(A; B)[ADMVW02] is given by tical perspective, we should be most interested in the single-mode thermal, amplifier, and additive-noise chan- 0 PPT (A; B) τAB : τAB 0 TB(τAB) 1 1 , nels, as these are of the greatest interest in applications, ≡ { ≥ ∧ k k ≤ (105)} as stressed in [Hol12, Section 12.6.3] and [HG12, Sec- with TB denoting the partial transpose. AppendixC tion 3.5]. However, it also turns out that these are the details a Matlab program taking advantage of recent ad- only non-trivial cases to consider among all single-mode vances in [FSP18, FF18], in order to compute the Rains bosonic Gaussian channels, as discussed below. relative entropy of any bipartite state. Figure5 plots the entanglement cost of the epolarizing channel for d = 2 (qubit input), and it also plots the A. On the definition of entanglement cost for Rains bound on distillable entanglement in (102). There infinite-dimensional channels is a gap for every value of q (0, 1), demonstrating that ∈ the resource theory of entanglement is irreversible for Before beginning, let us note that there are some sub- epolarizing channels. The figure also plots the coherent tleties involved when dealing with quantum information q s information of the state Λ 0 (Ψ 0 ), optimized A →SG1G2 A A theory in infinite-dimensional Hilbert spaces [Hol12]. For s with respect to Ψ A0A √s 00 A0A + √1 s 11 A0A example, as advised in [SH08], the direct use of the dia- for s [0, 1], which| isi known≡ to| bei a lower bound− | oni the mond norm in infinite-dimensional Hilbert spaces could distillable∈ entanglement of Λq [DW05]. Note that the co- be too strong for applications, and this observation has herent information plot is not in contradiction with the motivated some recent work [Shi18, Win17] on modifica- 14 tions of the diamond norm that take into account physi- consequence of the definitions of the underlying quanti- cal constraints such as energy limitations. On the other ties. For the same reason, the entanglement cost of the hand, the recent findings in [Wil18] suggest that the di- bosonic identity channel is also infinite. rect use of the diamond norm is reasonable when considering single-mode thermal, amplifier, and additive-noise channels, as well as some multi-mode bosonic Gaussian C. Thermal, amplifier, and additive-noise channels channels. As it turns out, we can indeed directly employ the diamond norm when analyzing the entanglement cost In light of the previous discussion, for the remainder of these channels. In fact, one of the main contributions of the paper, let us focus our attention on the thermal, of [Wil18] was to consider uniform convergence issues in amplifier, and additive-noise channels. Each of these are the teleportation simulation of bosonic Gaussian chan- defined respectively by the following Heisenberg input- nels, and due to the fact that the operational framework output relations: of entanglement cost is directly related to the approxi- p mate teleportation simulation of a channel, one should ˆb = √ηaˆ + 1 ηe,ˆ (106) expect that the findings of [Wil18] would be related to − ˆb = √Gaˆ + √G 1ê†, (107) the issues involved in the entanglement cost of bosonic − Gaussian channels. ˆb =a ˆ + (x + ip) /√2, (108) With this in mind, let us define the entanglement cost for an infinite-dimensional channel almost exactly as it wherea ˆ, ˆb, ande ˆ are the field-mode annihilation opera- has been defined in SectionIIB, with the exception that tors for the sender’s input, the receiver’s output, and the we allow for LOCC channels that have a continuous clas- environment’s input of these channels, respectively. sical index (e.g., as considered in [Shi10, Section 4]), The channel in (106) is a thermalizing channel, in thus going beyond the LOCC channels considered in (4). which the environmental mode is prepared in a thermal Specifically, let us define an (n, M, ε) sequential channel state θ(NB) of mean photon number NB 0, defined as simulation code as it has been defined in SectionIIB, ≥ ∞ n noting that the ε-error criterion is given by (18), repre- 1 X NB senting the direct generalization of the strategy norm of θ(NB) n n , (109) ≡ NB + 1 NB + 1 | ih | [CDP08a, CDP09b, Gut12] to infinite-dimensional sys- n=0 tems. Achievable rates and the entanglement cost are ∞ where n is the orthonormal, photonic number- then defined in the same way. {| i}n=0 state basis. When NB = 0, θ(NB) reduces to the vacuum state, in which case the resulting channel in (106) is called the pure-loss channel—it is said to be quantum- B. Preliminary observations about the limited in this case because the environment is injecting entanglement cost of single-mode bosonic Gaussian the minimum amount of noise allowed by quantum me- channels chanics. The parameter η (0, 1) is the transmissivity of the channel, representing the∈ average fraction of photons The starting point for our analysis of single-mode making it from the input to the output of the channel. bosonic Gaussian channels is the Holevo classification Let η,N denote this channel, and we make the further L B from [Hol07], in which canonical forms for all single-mode abbreviation η η,NB =0 when it is the pure-loss chan- bosonic Gaussian channels have been given, classifying nel. The channelL ≡ in L (106) is entanglement-breaking when them up to local Gaussian unitaries acting on the in- (1 η) NB η [Hol08], and by Remark1, the entangle- put and output of the channel. It then suffices for us to ment− cost is≥ equal to zero for these values. focus our attention on the canonical forms, as it is self- The channel in (107) is an amplifier channel, and the evident from definitions that local unitaries do not alter parameter G > 1 is its gain. For this channel, the the entanglement cost of a quantum channel. The ther- environment is prepared in the thermal state θ(NB). mal and amplifier channels form the class C discussed in If NB = 0, the amplifier channel is called the pure- [Hol07], and the additive-noise channels form the class amplifier channel—it is said to be quantum-limited for B2 discussed in the same work. The classes that remain a similar reason as stated above. Let G,N denote this A B are labeled A, B1, and D in [Hol07]. The channels in A channel, and we make the further abbreviation G A ≡ and D are entanglement-breaking [Hol08], and as a con- G,NB =0 when it is the quantum-limited amplifier chan- sequence of the “if-part” of Remark1, they have zero nel.A The channel in (107) is entanglement-breaking when entanglement cost. Channels in the class B1 are per- (G 1) NB 1 [Hol08], and by Remark1, the entangle- haps not interesting for practical applications, and as it ment− cost is≥ equal to zero for these values. turns out, they have infinite quantum capacity [Hol07]. Finally, the channel in (108) is an additive-noise chan- Thus, their entanglement cost is also infinite, because a nel, representing a quantum generalization of the clas- channel’s quantum capacity is a lower bound on its dis- sical additive white Gaussian noise channel. In (108), tillable entanglement, which is in turn a lower bound on x and p are zero-mean, independent Gaussian random its entanglement cost—these relationships are a direct variables each having variance ξ 0. Let ξ denote this ≥ T 15 channel. The channel in (108) is entanglement-breaking the proof is the analysis in [WGK+04, Sections II and when ξ 1 [Hol08], and by Remark1, the entanglement III], where it is shown that every bosonic Gaussian state cost is equal≥ to zero for these values. can be decomposed as a Gaussian mixture of local dis- Kraus representations for the channels in (106)–(108) placements acting on a fixed Gaussian pure state and are available in [ISS11], which can be helpful for further that such a decomposition is optimal for the Gaussian en- understanding their action on input quantum states. tanglement of formation [WGK+04, Proposition 1]. The Due to the entanglement-breaking regions discussed Gaussian mixture of local displacements can be under- ω above, we are left with a limited range of single-mode stood as an LOCC channel A0B0 , and let ψA0B0 denote bosonic Gaussian channels to consider, which is delin- the aforementioned fixed GaussianG pure state such that ω eated by the white strip in Figure 1 of [GGPCH14]. A0B0 (ψA0B0 ) = ωA0B0 . G ω ω Since ψA0B0 is Gaussian, the marginal state ψB0 is 0 Gaussian, and thus it has finite entropy H(B )ψω , as well D. Upper bound on the entanglement cost of as finite entropy variance, i.e., teleportation-simulable channels with bosonic 0 ω ω 0 2 Gaussian resource states V (B )ψω Tr ψB0 [ log2 ψB0 H(B )ψω ] < , ≡ { − − } ∞(111) In this section, I determine an upper bound on the the latter statement following from the Williamson de- entanglement cost of any channel A→B that is telepor- composition [Wil36] for Gaussian states as well as the tation simulable with associated resourceN state given by formula for the entropy variance of a bosonic thermal a bosonic Gaussian state. Related bosonic teleportation state [WRG16]. For δ > 0, recall that the entropy-typical δ ω channels have been considered previously [BK98, BST02, projector ΠB0n [OP93, Sch95] of the state ψB0 is defined TBS02, GIC02, WPGG07, NFC09], in the case that the as the projection onto LOCC channel associated to A→B is a Gaussian LOCC −1 n 0 span ξ n : n log (p n (z )) H(B ) ω δ , channel. Proposition3 belowN states that the entangle- z 2 Z ψ {| i − − ≤ (112)} ment cost of these channels is bounded from above by ω where a countable spectral decomposition of ψ 0 is given the Gaussian entanglement of formation [WGK+04] of B by the underlying bosonic Gaussian resource state, and as such, this proposition represents a counterpart to Propo- ω X ψ 0 = pZ (z) ξz ξz , (113) sition2. Before stating it, let us note that the Gaussian B | ih | g z entanglement of formation EF (A; B)ρ of a bipartite state + ρAB [WGK 04] is given by the same formula as in (7), and x with the exception that the pure states ψAB in the en- n semble decomposition are required to be Gaussian. Note ξz ξz1 ξzn , (114) | ni ≡ | i ⊗ · · · ⊗ | i that continuous probability measures are allowed for the pZn (z ) pZ (z1) pZ (zn). (115) decomposition (for an explicit definition, see [WGK+04, ≡ ··· δ Section III]). Let us note here that the first part of the The entropy-typical projector ΠB0n projects onto a finite- ω ⊗n proof outlines a procedure for the formation of n approx- dimensional subspace of [ψB0 ] , and satisfies the condi- δ ω ⊗n imate copies of a bipartite state, and even though this tions ΠB0n , [ψB0 ] = 0 and kind of protocol has been implicit in prior literature, I 0 have included explicit steps for clarity. After proving −n[H(B )ψω +δ] δ δ ω ⊗n δ 2 Π 0n Π 0n [ψ 0 ] Π 0n B ≤ B B B Proposition2, I discuss its application to thermal, am- 0 −n[H(B )ψω −δ] δ plifier, and additive-noise bosonic Gaussian channels. 2 Π 0n . (116) ≤ B 0 δ n[H(B )ψω +δ] Proposition 3 Let A→B be a channel that is teleporta- It then follows that Tr ΠB0n 2 . Further- N { } ≤ δ tion simulable as defined in (28), where the resource state more, consider that the entropy-typical projector ΠB0n 0 0 ω ⊗n ωA B is a bosonic Gaussian state composed of k modes for the state [ψB0 ] satisfies for system A0 and ` modes for system B0, with k, ` 1. ≥ δ ω ⊗n δ ω ⊗n Then the entanglement cost of is never larger than Tr IA0n Π 0n [ψ 0 0 ] = Tr Π 0n [ψ 0 ] A→B { ⊗ B A B } { B B } the Gaussian entanglement ofN formation of the bosonic 0 V (B )ψω Gaussian resource state ω 0 0 : 1 , (117) A B ≥ − δ2n g 0 0 EC ( ) EF (A ; B )ω. (110) with the inequality following from the definition of the N ≤ entropy-typical projector and an application of the Cheb- Proof. The main idea of the proof is to first form n ap- shev inequality. By the gentle measurement lemma proximate copies of the bosonic Gaussian resource state [Win99, ON07] (see [Wil17, Lemma 9.4.1] for the version ωA0B0 , by using entanglement and LOCC as related to employed here), we conclude that the approach from [BBPS96], and then after that, simu- r 0 late n uses of the channel A→B by employing the struc- 1 ω ⊗n ω V (B )ψω N [ψ 0 0 ] ψ 0n 0n , (118) ture of the channel A→B from (28). Indispensable to A B eA B 2 N 2 − 1 ≤ δ n 16 where 1. Upper bound for the entanglement cost of thermal, amplifier, and additive-noise bosonic Gaussian channels δ ω ⊗n δ ω IA0n ΠB0n [ψA0B0 ] IA0n ΠB0n ψ 0n 0n ⊗ ⊗ . eA B δ ω ⊗n I now discuss how to apply Proposition2 to single- ≡ Tr IA0n Π 0n [ψ 0 0 ] { ⊗ B A B } mode thermal, amplifier, and additive-noise channels. (119) Some recent papers [LSMGA17, KW17, TDR18] have 0n ω Observe that the system B of ψeA0nB0n is supported on shown how to simulate each of these channels by using 0n a finite-dimensional subspace of B . a bosonic Gaussian resource state along with variations ω of the continuous-variable quantum teleportation proto- Now, the idea of forming n approximate copies ψA0B0 is then the same as it is in [BBPS96]: Alice prepares the col [BK98]. Of these works, the one most relevant for us ω is the latest one [TDR18], because these authors proved state ψeA0nB0n locally, Alice and Bob require beforehand a maximally entangled state of Schmidt rank no larger that the entanglement of formation of the underlying re- 0 source state is equal to the entanglement of formation than 2n[H(B )ψω +δ], and then they perform quantum tele- that results from transmitting through the channel one portation [BBC+93] to teleport the B0n system to Bob. share of a two-mode squeezed vacuum state with arbi- ω At this point, they share exactly the state ψeA0nB0n , which trarily large squeezing strength. That is, let de- ω ⊗n A→B becomes less and less distinguishable from [ψA0B0 ] as note a single-mode thermal, amplifier, or additive-noiseN n grows large, due to (118). Now applying the Gaussian channel. Then one of the main results of [TDR18] is ⊗n LOCC channel ( A0B0 ) , the data processing inequal- that, associated to this channel, there is a bosonic Gaus- G ω ity to (118), and the fact that A0B0 (ψ 0 0 ) = ωA0B0 , we A B sian resource state ωA0B0 and a Gaussian LOCC channel conclude that G AA0B0→B such that G 0 0 EF (A ; B )ω = sup EF (R; B)σ(N ) (121) r 0 S 1 ⊗n ⊗n ω V (B )ψω NS ≥0 0 0 ωA0B0 ( A B ) (ψeA0nB0n ) . (120) 2 1 δ2n − G ≤ = lim EF (R; B)σ(NS ), (122) NS →∞ where 0 Thus, to see that H(B )ψω is an achievable rate for ⊗n NS forming ω 0 0 , fix ε (0, 1] and δ > 0. Then choose σ(NS) A→B(φRA), (123) A B ∈ ≡ N q V (B0) ω NS NS NS ψ φ φ φ RA, (124) n large enough so that δ2n ε. Apply the above RA ≤ ≡ | ih | s procedure, using LOCC and a maximally entangled state 1 ∞ N n 0 NS X S of Schmidt rank no larger than 2n[H(B )ψω +δ]. Then the φ RA n R n A, | i ≡ √N + 1 NS + 1 | i | i rate of entanglement consumption to produce n approx- S n=0 0 (125) imate copies of ωA0B0 satisfying (120) is H(B )ψω + δ. Since this is possible for ε (0, 1], δ > 0, and suffi- and for all input states ρA, ∈ 0 ciently large n, we conclude that H(B )ψω is an achiev- (ρ ) = 0 0 (ρ ω 0 0 ). (126) able rate for the formation of ωA0B0 . Now, since achieving A→B A AA B →B A A B ω N G ⊗ this rate is possible for any pure state ψA0B0 such that NS ω In the above, φRA is the two-mode squeezed vacuum state ωA0B0 = A0B0 (ψA0B0 ), we conclude that the infimum of 0 G [Ser17]. Note that the equality in (122) holds because one H(B ) ω with respect to all such pure states is an achiev- 0 ψ NS NS 0 can always produce φ from φ such that N NS, able rate. But this latter quantity is exactly the Gaus- RA RA S ≥ sian entanglement of formation according to [WGK+04, by using Gaussian LOCC and the local displacements Proposition 1]. involved in the Gaussian LOCC commute with the channel A→B [GECP03] (whether it be thermal, amplifier, The idea for simulating n uses of the channel A→B N N or additive-noise). Furthermore, the entanglement of for- is then the same as the idea used in the proof of Propo- mation does not increase under the action of an LOCC sition2. First form n approximate copies of ωA0B0 ac- channel. cording to the procedure described above. Then, when Thus, applying the above observations and Proposi- the ith call to the channel A→B is made, use the LOCC N tion3, it follows that there exist bosonic Gaussian re- channel AA0B0→B from the definition in (28) along with η,NB G,NB ξ source states ω 0 0 , ω 0 0 , and ω 0 0 associated to the the ith LA0 and B0 systems of the state approximating A B A B A B ⊗n respective thermal, amplifier, and additive-noise channels ωA0B0 to simulate it. By the same reasoning that led to in (106)–(108), such that the following inequalities hold (38), the distinguishability of the final states of any se- 0 0 η,N quential test is limited by the distinguishability of the EC ( η,NB ) EF (A ; B )ω B , (127) ⊗n L ≤ 0 0 state ω 0 0 from its approximation, which I argued in G,N A B EC ( G,NB ) EF (A ; B )ω B , (128) (120) can be made arbitrarily small with increasing n. A ≤ 0 0 EC ( ξ) EF (A ; B )ωξ . (129) Thus, the Gaussian entanglement of formation ωA0B0 is T ≤ an achievable rate for sequential channel simulation of Analytical formulas for the upper bounds on the right A→B. can be found in [TDR18, Eqs. (4)–(6)]. N 17

n E. Lower bound on the entanglement cost of R is equal to nNS. The second inequality follows from bosonic Gaussian channels the LOCC monotonicity of the entanglement of formation, here thinking of the person who possesses systems In this section, I establish a lower bound on the non- RAn to be in the same laboratory as the one possessing asymptotic entanglement cost of thermal, amplifier, or the systems Ai, while the person who possesses the Bi additive-noise bosonic Gaussian channels. After that, I systems is in a different laboratory. The first equality NS ⊗n show how this bound implies a lower bound on the en- follows from the fact that (φRA) is in tensor product with Φ , so that by a local channel, one may remove tanglement cost. Finally, by proving that the state re- A0B0 NS ⊗n sulting from sending one share of a two-mode squeezed (φRA) or append it for free. The final equality follows vacuum through a pure-loss or pure-amplifier channel has because the entanglement of formation of the maximally entanglement cost equal to entanglement of formation, I entangled state is equal to the logarithm of its Schmidt establish the exact entanglement cost of these channels rank. by combining with the results from the previous section. A direct consequence of Proposition4 is the following lower bound on the entanglement cost of the thermal, Proposition 4 Let A→B be a thermal, amplifier, or amplifier, and additive-noise channels: additive-noise channel,N as defined in (106)–(108). Let 0 n, M N, ε [0, 1/2), ε (√2ε, 1], δ = Proposition 5 Let A→B be a thermal, amplifier, 0 ∈ 0 ∈ ∈ or additive-noise channel,N as defined in (106)–(108). ε √2ε / [1 + ε ], and NS [0, ). Then the follow- − ∈ ∞ (p) ing bound holds for any (n, M, ε) sequential or parallel Then the entanglement costs EC ( ) and E ( ) are N C N channel simulation code for A→B: bounded from below by the entanglement cost of the state N NS A→B(φRA), where the two-mode squeezed vacuum state NNS 1 1 n n 0 NS /δ φ has arbitrarily large squeezing strength: log M EF (R ; B ) ⊗n (ε + 2δ) H(φ ) RA n 2 ≥ n ω − R (p) 1 0 0 EC ( ) EC ( ) (133) [2 (1 + ε ) g2(ε ) + 2h2(δ)] , (130) N ≥ N − n NS sup EC ( A→B(φRA)) (134) ≥ NS ≥0 N where ω (φNS ) and 1 log M is understood RB A→B RA n 2 NS as the non-asymptotic≡ N entanglement cost of the protocol. = lim EC ( A→B(φRA)). (135) NS →∞ N Proof. The reasoning here is very similar to that given in Proof. The first inequality follows from definitions, as the proof of Proposition1, but we can instead make use argued previously in (22). To arrive at the second in- of the continuity bound for the entanglement of forma- equality, in Proposition4, set ε0 = √4 2ε, and take the tion of energy-constrained states [Shi16, Proposition 5]. limit as n and then as ε 0. Employing the fact To begin, suppose that there exists an (n, M, ε) proto- → ∞ NS /ξ → that limξ→0 ξH(H(φR )) = 0 [Shi06, Proposition 1] col for sequential channel simulation. Then by previous and applying definitions, we find for all NS 0 that reasoning (also see Figure2), it can be thought of as a ≥ parallel channel simulation protocol, such that the cri- (p) EC ( ) EC ( ) (136) NS ⊗n N ≥ N terion in (2) holds. Let us take (φRA) to be a test 1 NS ⊗n ⊗n NS ⊗n lim E ([ (φ )] ) (137) input state, leading to ω = [ A→B(φ )] when the F A→B RA RB N RA ≥ n→∞ n N actual channels are applied and σR1···RnB1···Bn when the NS = EC ( A→B(φ )). (138) simulation is applied. Set N RA Since the above bound holds for all NS 0, we con- 0 0 NS /δ ≥ f(n, ε, ε ,NS) n (ε + 2δ) H(φR ) clude the bound in the statement of the proposition. The ≡ 0 0 equality in (135) follows for the same reason as given for + 2 (1 + ε ) g2(ε ) + 2h2(δ). (131) the equality in (122), and due to the fact that entan- Then we have that glement cost is non-increasing with respect to an LOCC channel by definition. n n EF (R ; B )ω⊗n n n 0 EF (R ; B )σ + f(n, ε, ε ,NS) ≤ F. Additivity of entanglement of formation for n n 0 EF (R A A0; B0)ψ⊗Φ + f(n, ε, ε ,NS) pure-loss and pure-amplifier channels ≤ 0 = EF (A0; B0)Φ + f(n, ε, ε ,NS) 0 The bound in Proposition5 is really only a formal = log M + f(n, ε, ε ,NS). (132) 2 statement, as it is not clear how to evaluate the lower The first inequality follows from the condition in (18), as bound explicitly. If it would however be possible to prove well as from the continuity bound for entanglement of for- that mation from [Shi16, Proposition 5], noting that the total 1 NS ⊗n ? NS EF ([ A→B(φ )] ) = EF ( A→B(φ )) (139) photon number of the reduced (thermal) state on systems n N RA N RA 18 for all integer n 1 and all NS 0, then we could 2. The determination of and method of proof for the ≥ ≥ conclude the following classically-conditioned entropy H(A E)ρ of an ar- | bitrary two-mode Gaussian state ρAE with covari- ? + NS ance matrix in certain standard forms [PSB 14]. EC ( ) lim EF ( A→B(φRA)), (140) N ≥ NS →∞ N (As remarked below, there is in fact a significant strengthening of the main result of [PSB+14], implying that this lower bound coincides with the up- which relies on item 3 below.) per bound from (127)–(129), due to the recent result of [TDR18] recalled in (121)–(122). 3. The multi-mode bosonic minimum output entropy In Proposition6 below, I prove that the additivity rela- theorem from [GHGP15] and [GHM15, Theorem 1] tion in (139) indeed holds whenever the channel A→B is (see the related work in [MGH14, GGPCH14] also), N a pure-loss channel η or pure-amplifier channel G. The which implies that the following identity holds for linchpin of the proofL is the multi-mode bosonic minimumA a phase-insensitive, single-mode bosonic Gaussian output entropy theorem from [GHGP15] and [GHM15, channel and for all integer n 1: Theorem 1]. G ≥ inf H( ⊗n(ρ(n))) = H( ⊗n([ 0 0 ]⊗n)) ρ(n) G G | ih | Proposition 6 For A→B a pure-loss channel η with transmissivity η (0,N1) or a pure-amplifier channelL = nH( ( 0 0 )), (148) G G | ih | with gain G > 1∈, the following additivity relation holdsA where the optimization is with respect to an arbi- for all integer n 1 and NS [0, ): ≥ ∈ ∞ trary n-mode input state ρ(n) and 0 0 denotes the bosonic vacuum state. | ih | 1 NS ⊗n NS EF ([ A→B(φ )] ) = EF ( A→B(φ )) (141) n N RA N RA g NS Indeed, these three key ingredients, with the third be- = E ( A→B(φ )), (142) F N RA ing the linchpin, lead to the statement of the proposition after making a few observations. Consider that a purifi- where φNS is the two-mode squeezed vacuum state from RA cation of the state ρ = (id )(φNS ) is given (125) and Eg denotes the Gaussian entanglement of for- AB R→A η RA F by ⊗L NS mation. Thus, the entanglement cost of A→B(φ ) is N RA equal to its entanglement of formation: η NS ψABE = (idR→A )(φ 0 0 E), (149) ⊗BAE→BE RA ⊗ | ih | NS NS EC ( A→B(φRA)) = EF ( A→B(φRA)). (143) where η represents the unitary for a beamsplitter N N BAE→BE interaction [Ser17] and 0 0 E again denotes the vacuum Proof. The proof of this proposition relies on three key | ih | state. Tracing over the system B gives the state ψAE = prior results: NS (idR→A 1−η)(φ ), where 1−η is a pure-loss channel ⊗L RA L of transmissivity 1 η. The state ψAE is well known to 1. The main result of [KW04] is that the entanglement have its covariance− matrix in standard form [Ser17] (see of formation EF (A; B)ψ is equal to the classically- discussion surrounding [PSB+14, Eq. (5)]) as conditioned entropy H(A E)ψ for a tripartite pure | state ψABE: a 0 c 0  0 a 0 c   (150) EF (A; B)ψ = H(A E)ψ, (144) c 0 b −0  | 0 c 0 b where − X and is also known as a two-mode squeezed thermal state H(A E)ψ = inf pX (x)H(A)σx , (145) + x [Ser17]. As such, the main result of [PSB 14] applies, | {Λ }x E x and we can conclude that heterodyne detection is the optimal measurement in (145), which in turn implies from with the optimization taken with respect to a pos- x (144) that the entanglement of formation of ρAB is equal itive operator-valued measure Λ x and { E} to the Gaussian entanglement of formation. x However, what we require is that the same results hold pX (x) Tr ΛEψE , (146) ⊗n ≡ { } for the multi-copy state ψAE. Inspecting Eqs. (9)–(14) x 1 x of [PSB+14], it is clear that the same steps hold, except σA TrE (IA ΛE)ψAE . (147) ≡ pX (x) { ⊗ } that we replace Eq. (12) therein with (148). Thus, it follows that n individual heterodyne detections on each The sum in (145) can be replaced with an integral E mode of ψ⊗n is the optimal measurement, so that for continuous-outcome measurements. The equal- AE ity in (144) can be understood as being a conse- 1 n n H(A E ) ⊗n = H(A E)ψ. (151) quence of the quantum steering effect [Sch35]. n | ψ | 19

By applying (144) (as applied to the states ρ⊗n and Λxn that is optimal for the left-hand side. Fur- AB En x1,...,xn ⊗n } ψAE), we conclude that thermore, by the relation in (144), for any purification ψABE of the state ρAE mentioned above, we conclude 1 n n that E (A ; B ) ⊗n = E (A; B) . (152) n F ρ F ρ 1 n n Furthermore, since the optimal measurement is given EF (A ; B )ψ⊗n = EF (A; B)ψ, (158) n by heterodyne detection, performing it on mode E of ψABE induces a Gaussian ensemble of pure states for all integer n 1, thus giving a whole host of two- x pX (x), ψ , which is the optimal decomposition of ≥ { AB} mode Gaussian states for which their entanglement cost ψAB = ρAB, and thus we conclude that EF (A; B)ρ = g is equal to their entanglement of formation: EF (A; B)ρ = E (A; B)ρ. g F EC (ρAB) = EF (A; B)ρ. As far as I am aware, these are A similar analysis applies for the quantum-limited the first examples of two-mode Gaussian states for which amplifier channel. I give the argument for complete- the additivity relation in (158) has been explicitly shown. ness. Consider that a purification of the state σAB = NS (idR→A G)(φ ) is given by ⊗A RA Remark 3 One might wonder whether the same method G NS of proof as given in Proposition6 could be used to es- ϕABE = (idR→A AE→BE)(φ 0 0 E), (153) ⊗S RA ⊗ | ih | tablish the equalities in (141)–(142) for general thermal, where G represents the unitary for a two-mode amplifier, and additive-noise channels. At the moment, SAE→BE squeezer [Ser17] and 0 0 E again denotes the vacuum it is not clear how to do so. The issue is that the state | ih | NS state. Tracing over the system B gives the state ϕAE = (idR η,NB )(φRA) for NB > 0 is a faithful state, mean- NS ⊗L (idR→A eG)(φ ), where eG denotes the channel con- ing that it is positive definite and thus has two symplec- ⊗A RA A jugate to the quantum-limited amplifier. The state ϕAE tic eigenvalues > 1. This means that any purification has its covariance matrix in the form (see Mathemat- of it requires at least four modes [HW01, Section III- ica files included with the arXiv posting or alternatively D]. Then tracing over the B system leaves a three-mode [Ser17, Appendix D.4]) state, of which we should be measuring two of them, and so it is not clear how to apply the methods of [PSB+14] a 0 c 0 to such a state. The same issues apply to the states 0 a 0 c (id )(φNS ) for N > 0 and (id )(φNS ) for   , (154) R G,NB RA B R ξ RA c 0 b 0 ξ > 0⊗A, which are the states resulting from⊗T the amplifier 0 c 0 b and additive-noise channels, respectively. and so the same proof approach to get (151) can be used to conclude that G. Entanglement cost of pure-loss and 1 n n pure-amplifier channels H(A E ) ⊗n = H(A E)ϕ. (155) n | ϕ | Indeed, this additionally follows from the discussion after Based on the results in the previous sections, we con- [PSB+14, Eqs. (17)–(19)]. As such, we conclude in the clude the following theorem, which gives simple formulas same way that for the entanglement cost of two fundamental bosonic Gaussian channels: 1 n n g E (A ; B ) ⊗n = E (A; B) = E (A; B) . (156) n F σ F σ F σ Theorem 2 For a pure-loss channel η with transmis- L The final statement about entanglement cost in (143) sivity η (0, 1) or a pure-amplifier channel G with gain follows from the fact that it is equal to the regularized G > 1, the∈ following formulas characterizeA the entangle- entanglement of formation. ment costs of these channels:

Remark 2 As can be seen from the proof above, the (p) h2(1 η) multi-mode minimum output entropy theorem recalled in EC ( η) = E ( η) = − , (159) L C L 1 η (148) provides a significant strengthening of the results − from [PSB+14]. Indeed, for ρ any two-mode Gaus- (p) g2(G 1) AE EC ( G) = E ( G) = − , (160) sian state considered in [PSB+14], the following equality A C A G 1 − holds where h2( ) is the binary entropy defined in (66) and g2( ) 1 · · n n is the bosonic entropy function defined in (25). H(A E ) ⊗n = H(A E)ρ, (157) n | ρ | x implying that the measurement ΛE x optimal for the Proof. Recalling the discussion in SectionVD1, for a right-hand side leads to a measurement{ } Λx1 pure-loss and pure-amplifier channel, there exist respec- { E1 ⊗ · · · ⊗ 20

η G Rate tive resource states ωA0B0 and ωA0B0 such that

0 0 EC ( η) EF (A ; B )ωη (161) L ≤ = lim E (R; B) η , (162) 6 F σ (NS ) Ent. Cost NS →∞ 0 0 Dist. Ent. EC ( G) EF (A ; B ) G (163) A ≤ ω G 4 = lim EF (R; B)σ (NS ), (164) NS →∞ where 2

η NS σ (NS)RB (idR η)(φ ), (165) ≡ ⊗L RA G NS σ (NS)RB (idR G)(φRA), (166) η ≡ ⊗A 0.2 0.4 0.6 0.8 1.0 with the equalities in (162) and (164) being one of the main results of [TDR18]. Furthermore, explicit formu- h2(1−η) 0 0 0 0 FIG. 6. Plot of the entanglement cost EC (Lη) = and las for EF (A ; B )ωη and EF (A ; B )ωG have been given 1−η in [TDR18, Eqs. (4)–(6)], and evaluating these formulas the distillable entanglement ED(Lη) = − log2(1 − η) of the leads to the expressions in (159)–(160) (supplementary pure-loss channel Lη as a function of the transmissivity η ∈ Mathematica ﬁles that automate these calculations are [0, 1], with the shaded area demonstrating the gap between them. The units for rate on the vertical axis are ebits per available with the arXiv posting of this paper). channel use, and η on the horizontal axis is dimensionless. On the other hand, Propositions5 and6 imply that

(p) EC ( η) EC ( η) (167) produces a channel of gain 1/η. Finally, we have the L ≥ L η lim EC (σ (NS)RB) (168) Taylor expansions: ≥ NS →∞

= lim EF (R; B)ση (N ), (169) h2(1 η) η 2 N →∞ S − = (1 ln(η)) + O(η ), (176) S 1 η ln 2 − (p) − EC ( G) E ( G) (170) g (G 1) 1 + ln(G) A ≥ C A 2 2 G − = + O(1/G ), (177) lim EC (σ (NS)RB) (171) G 1 G ln 2 ≥ NS →∞ −

G which are relevant in the low-transmissivity and high- = lim EF (R; B)σ (NS ). (172) NS →∞ gain regimes. In [Pir17], simple formulas for the distillable entangle- Combining the inequalities above, we conclude the statement of these channels were determined and given by ment of the theorem.

ED( η) = log2(1 η), (178) It is interesting to consider various limits of the formu- L − − ED( G) = log2(1 1/G). (179) las in (159)–(160): A − − Thus, the prior results and the formulas in Theorem2 h2(1 η) g2(G 1) demonstrate that the resource theory of entanglement for lim − = lim − = , (173) η→1 1 η G→1 G 1 ∞ these channels is irreversible. That is, if one started from − − a pure-loss channel of transmissivity η and distilled ebits h2(1 η) g2(G 1) lim − = lim − = 0. (174) from it at the ideal rate of log (1 η), and then subse- η→0 1 η G→∞ G 1 − 2 − − − quently wanted to use these ebits to simulate a pure-loss We expect these to hold because the channels approach channel with the same transmissivity, this is not possible, the ideal channel in the limits η, G 1, which we because the rate at which ebits are distilled is not suffi- previously argued has infinite entanglement→ cost, while cient to simulate the channel again. The same statement they both approach the completely depolarizing (useless) applies to the pure-amplifier channel. Figures6 and7 channel in the no-transmission limit η 0 and infinite- compare the formulas for entanglement cost and distill- amplification limit G . Furthermore,→ these formulas able entanglement of these channels, demonstrating that obey the symmetry → ∞ there is a noticeable gap between them. I note here that the differences are given by

h2(1 η) g2(1/η 1) − = − , (175) η log2 η 1 η 1/η 1 EC ( η) ED( η) = − , (180) − − L − L 1 η − which is consistent with the idea that the transformation log2 G EC ( G) ED( G) = , (181) η 1/η takes a channel of transmissivity η [0, 1] and A − A G 1 → ∈ − 21

Rate ulable channel for which, by pre- and post-processing, one can seize the underlying resource state ν as 2.5 post pre ( A→B(κ )) = νE, (183) Ent. Cost FRB→E N RA 2.0 Dist. Ent. where κpre is a free state and post is a free channel, RA FRB→E 1.5 extending Definition2. The general notion of channel simulation, as presented 1.0 in SectionIIB, can be considered in any resource theory also. Again, the main idea is really to replace “LOCC 0.5 channel” with “free channel” and “maximally entangled state” with “resourceful state” in the protocol depicted in G Figure1, and the goal is to determine the minimum rate 2 4 6 8 10 at which resourcefulness is needed in order to simulate n uses of a given channel. If the channels are resource- seizable as discussed above, then the theory should sig- FIG. 7. Plot of the entanglement cost E (A ) = g2(G−1) C G G−1 nificantly simplify, as has occurred in this paper for the and the distillable entanglement E (A ) = − log (1 − 1/G) D G 2 entanglement theory of channels (see Theorem1). Fur- of the pure-amplifier channel AG as a function of the gain G ≥ 1, with the shaded area demonstrating the gap between thermore, along the lines of the discussion in SectionIIC them. The units for rate on the vertical axis are ebits per (and related to [CG18, Section III-D-5]), suppose that channel use, and G on the horizontal axis is dimensionless. a channel A→B can be realized from another channel N pre A0→B0 via a preprocessing free channel A→A0M and M post F a postprocessing free channel B0M→B as implying that these differences are strictly greater than F zero for all the relevant channel parameter values η post pre A→B = 0 A0→B0 0 . (184) (0, 1) and G > 1. ∈ N FB M→B ◦ M ◦ FA→A M Then for the same reasons given there, the simulation cost of should never exceed the simulation cost of . VI. EXTENSION TO OTHER RESOURCE Finally,N let us note that some discussions about chan-M THEORIES nel simulation for the resource theory of coherence have appeared in the last paragraph of [BDGDMW17], as well Let us now consider how to extend many of the con- as the last paragraphs of [DFW+18]. It is clear from the cepts in this paper to other resource theories (see [CG18] findings of the present paper that identifying interest- for a review of quantum resource theories). In fact, this ing resource-seizable channels could be a useful first step can be accomplished on a simple conceptual level by re- for understanding interconversion costs of simulating one placing “LOCC channel” with “free channel,” “separable channel from another in the resource theory of coherence. state” with “free state,” and (roughly) “maximally en- It could also be helpful in further understanding chan- tangled state” with resource state throughout the paper. nel simulation in the resource theory of thermodynamics To be precise, let F denote the set of free states for a [FBB18]. given resource theory, and let be a free channel, which takes a free state to a free state.F In [KW18, Section 7], a general notion of distillation of a resource from n uses of VII. CONCLUSION a channel was given (see Figure 4 therein). In particular, one interleaves n uses of a given channel by free channels, In summary, this paper has provided a new definition and the goal is to distill resource from the n channels. As for the entanglement cost of a channel, in terms of the a generalization of a teleportation-simulable channel with most general strategy that a discriminator could use to an associated resource state, the notion of a ν-freely sim- distinguish n uses of the channel from its simulation. I ulable channel was introduced as a channel that can established an upper bound on the entanglement cost of a be simulated as N teleportation-simulable channel in terms of the entangle- sim ment cost of the underlying resource state, and I proved A→B(ρA) = AE→B(ρA νE), (182) N F ⊗ that the bound is saturated in the case that the chan- where sim is a free channel and ν is some resource state. nel is resource-seizable (Definition2). I then established The implicationsF of this for distillation protocols was dis- single-letter formulas for the entanglement cost of era- cussed in [KW18, Section 7], which is merely that the sure, dephasing, three-dimensional Werner–Holevo chan- rate at which resource can be distilled is limited by the nels, and epolarizing channels (complements of depolar- resourcefulness of the underlying resource state ν. izing channels), by leveraging existing results about the Going forward, we can also consider a resource-seizable entanglement cost of their Choi states. I finally consid- channel in a general resource theory to be a ν-freely sim- ered single-mode bosonic Gaussian channels, establishing 22 bounds on the entanglement cost of the thermal, am- totic (parallel) channel simulation was reduced to simu- plifier, and additive-noise channels, while giving simple lating the channel on a single state, called the universal formulas for the entanglement cost of pure-loss and pure- de Finetti state. For the asymptotic theory of (sequen- amplifier channels. By relating to prior work on the dis- tial) entanglement cost of channels, could there be a sin- tillable entanglement of these channels, it became clear gle universal adaptive channel discrimination protocol to that the resource theory of entanglement for quantum consider, such that simulating the channel well for such channels is irreversible. a protocol would imply that it has been simulated well Going forward from here, there are many directions to for all protocols? pursue. The discrimination protocols considered in Sec- For the task of entanglement cost, one could modify tionIIB do not impose any realistic energy constraint the set of free channels to be either those that completely on the states that can be used in discriminating the ac- preserve the positivity of partial transpose [Rai99, Rai01] tual n uses of the channel from the simulation. We could or are k-extendible in the sense of [KDWW18]. Could certainly do so by imposing that the average energy of we find simpler lower bounds on entanglement cost of all the states input to the actual channel or its simu- channels in this way? The semi-definite programming lation should be less than a threshold, and the result quantity from [WD17] could be helpful here also. Af- is to demand only that the energy-constrained strategy ter the above question was posed in the arXiv posting of norm (defined naturally as an extension of both the strat- the present paper, the exact entanglement cost has been egy norm [CDP08a, CDP09b, Gut12] and the energy- solved in [WW18] for the case of exact channel simula- constrained diamond norm [Shi18, Win17]) is less than tion, with the set of free channels taken to be those that ε (0, 1). To be specific, let HA be a (positive semi- completely preserve the positivity of partial transpose. definite)∈ Hamiltonian acting on the input of the channel Another way to think about quantum channel sim- and let E [0, ) be an energy constraint. Then, A→B ulation is to allow the entanglement to be free but demandingN that the∈ supremum∞ in (18) is taken over all count the cost of classical communication. This was strategies such that the approach taken for the reverse Shannon theorem + n n [BDH 14, BCR11], and these works also considered only 1 X 1 X Tr HAρA , Tr HAτA E, (185) parallel channel simulation. How are the results there n { i } n { i } ≤ i=1 i=1 affected if the goal is sequential channel simulation instead? Is the previous answer from [BDH+14, BCR11], the resulting quantity is an energy-constrained strategy the mutual information of the channel, robust under this norm. With an energy constraint in place, one would change? How do prior results on simulation of quan- expect that less entanglement is required to simulate the tum measurements [Win04, WHBH12, BRW14] hold up channel than if there is no constraint at all, and the re- under this change? A comprehensive summary of results sulting entanglement cost would depend on the given on parallel simulation of quantum channels, including the energy constraint. For example, Proposition4 leads quantum reverse Shannon theorem, measurement simu- to a lower bound on entanglement cost for an energy- lation, and entanglement cost, is available in [Ber13]. constrained sequential simulation, but it remains open Finally, is there an example of a channel for which its to determine if there is a matching upper bound. sequential entanglement cost is strictly greater than its Similar to how measures like squashed entanglement parallel entanglement cost? The examples discussed here [CW04] and relative entropy of entanglement [VP98] al- are those for which either there are equalities or no con- low for obtaining converse bounds or fundamental lim- clusion could be drawn. Evidence from quantum chan- itations on the distillation rates of quantum states or nel discrimination [HHLW10] and related evidence from channels, simply by making a clever choice of a squash- [DW17] suggests the possibility. One concrete example ing channel or separable state, it would be useful to have to examine in this context is the channel presented in a measure like this for bounding entanglement cost from [CMH17, Appendix A], given that it is not implementable below. That is, it would be desirable for the measure to from its image, as discussed there. involve a supremum over a given set of test states or channels rather than an infimum as is the case for squashed entanglement and relative entropy of entanglement. For example, it would be useful to be able to bound the entanglement cost of thermal, amplifier, and additive-noise ACKNOWLEDGMENTS channels from below, in order to determine how tight are the upper bounds in (127)–(129). Progress on this front I thank Gerardo Adesso, Eneet Kaur, Debbie Leung, is available in [WD17], but more results in this area would Alexander Müller-Hermes,Kaushik Seshadreesan, Xin be beneficial. Wang, John Watrous, and Andreas Winter for helpful One of the main tools used in the analysis of the (par- discussions, especially Andreas Winter for inspiring dis- allel) entanglement cost of channels from [BBCW13] is a cussions from April 2017 regarding a general notion of de-Finetti-style approach, consisting of the post-selection channel simulation. I acknowledge support from the Na- technique [CKR09]. In particular, the problem of asymp- tional Science Foundation under grant no. 1350397. 23

Appendix A: Relation between resource-seizable which is understood to be implemented via LOCC by channels and those that are implementable from measuring Alice’s system XA, communicating the out- their image come x to Bob, who then prepares the state φx based BM on the outcome. We ﬁnd that Deﬁnition2 introduced the notion of a resource-

0 0 seizable channel, and SectionVI discussed how this no- ( AM BBM →A B XA→BM A→B)(σAM AXA ) tion can play a role in an arbitrary resource theory. In D ◦ P ◦ N = ωA0B0 . (A7) [CMH17, Appendix A], a channel A→B was defined to N be implementable from its image if there exists a state We finally conclude that σA0A and an LOCC channel AA0B0→B such that the fol- L lowing equality holds for all input states ρA: A→B(ρA) N = AA0B0→B(ρA ωA0B0 ) (A8) A→B(ρA) = AA0B0→B(ρA A00→B0 (σA0A00 )), (A1) M ⊗ N L ⊗ N = AAM XAB¯→B(ρA A¯→B¯ (σAM AX¯ A )), (A9) where system A00 is isomorphic to system A and system L ⊗ N B0 is isomorphic to system B. An example of a channel where that is not implementable from its image was discussed at length in [CMH17, Appendix A]. ¯ LAAM XAB→B ≡ Here, I prove that a channel is resource-seizable in the 0 0 AA B →B A BB¯ →A0B0 XA→BM , (A10) resource theory of entanglement if and only if it is im- M ◦ D M M ◦ P plementable from its image. To see this, suppose that a so that the channel is implementable from its image by channel is implementable from its image. Then, given the inputting the state σAM AXA and postprocessing with the above structure in (A1), it is clear that is telepor- 0 0 0 0 A→B LOCC channel AA B →B AM BBM →A B XA→BM . tation simulable with associated resourceN state given by M ◦ D ◦ P ωA0B0 = A00→B0 (σA0A00 ). Thus, one can trivially seize N the resource state ωA0B0 by sending in the input state Appendix B: Relation between Choi state of a σA0A00 , which is clearly separable between Alice and Bob, complementary channel and maximally mixed state given that Bob’s “system” here is trivial. sent through isometric extension Now suppose that a teleportation-simulable channel is resource-seizable, as in Definition2. This means that The purpose of this appendix is to prove the equality in (98). Consider a d-dimensional depolarizing channel A→B(ρA) = AA0B0→B(ρA ωA0B0 ), (A2) N M ⊗ I where ω 0 0 is the resource state and 0 0 is ρ (1 p) ρ + p . (B1) A B AA B →B → − d an LOCC channel. Furthermore, sinceM it is resource- seizable, this means that there exists a separable As noted in [DFH06, Eq. (3.2)], a Kraus representation state ρAM ABM and a postprocessing LOCC channel for this channel is as follows: A BB →A0B0 such that D M M np p o 1 pI, p/d i j i,j . (B2) A BB →A0B0 ( A→B(ρA AB )) = ωA0B0 . (A3) − { | ih |} D M M N M M This is because To see that the channel is implementable from its image, consider that ρA AB has a decomposition as follows, hp i hp i M M 1 pI ρ 1 pI given that it is separable: − − X hp i hp i X x x + p/d i j ρ p/d j i pX (x)ψ φ , (A4) | ih | | ih | AM A BM i,j x ⊗ p X X x = (1 p) ρ + i i j ρ j (B3) for pX a probability distribution and ψA A x and − d | ih | h | | i x { M } i j φ x sets of pure states. Now define the input state { BM } I σAM AXA as = (1 p) ρ + p Tr ρ . (B4) − { }d X x σA AX pX (x)ψ x x X , (A5) M A AM A A Now consider a generic channel A→B with Kraus op- ≡ x ⊗ | ih | i N erators N i so that an isometric extension is given by P i { } and note that this is the state we can use for imple- i N i E. Send the maximally mixed state π = I/d ⊗ | i P i menting the channel’s image. Define the LOCC measure- through the isometric extension i N i E. This leads to the state ⊗| i prepare channel X →B as P A M 1 X x X i j† X →B ( ) x X ( ) x X φ , (A6) N N i j E. (B5) A M A A BM d ⊗ | ih | P · ≡ x h | · | i i,j 24

Furthermore, a complementary channel of the orig- the state resulting from sending in the maximally mixed inal channel, resulting from the isometric extension state to the isometric extension of the channel is the same P i i N i E , is then as the Choi state of the complementary channel. This is ⊗ | i the case for the depolarizing channel with the Kraus op- c X i j† ρ (ρ) = Tr N ρN i j E. (B6) erators in (B2). Since all complementary channels and → NA→E { }| ih | i,j isometric extensions of a channel are related by an isometry acting on the environment system, we are arrive at The Choi state for this complementary channel is given the same conclusion for any isometric extension and the by corresponding complementary channel to which it leads.

c 1 X i j† (ΦRA) = k l R Tr N k l AN i j E NA→E d | ih | ⊗ { | ih | }| ih | k,l,i,j Appendix C: Matlab code for computing Rains 1 X j† i = k l R l AN N k i j E relative entropy d | ih | ⊗ h | | i| ih | k,l,i,j

1 X j† i This appendix provides a brief listing of Matlab code = k l AN N k l R i j E d | ih | | ih | ⊗ | ih | that can be used to compute the Rains relative en- k,l,i,j tropy of a bipartite state ρAB [Rai01, ADMVW02]. 1 X j† i = T (N N ) i j E, (B7) The code requires the QuantInf package in order to d ⊗ | ih | i,j generate a random state [Cub], the CVX package for semi-deﬁnite programming optimization [GB14], and the where T (N j†N i) denotes the transpose of N j†N i. If it CVXQuad package [Faw] for relative entropy optimiza- holds that N iN j† = T (N j†N i), then we conclude that tion [FSP18, FF18].

Listing 1. Matlab code for calculating the Rains relative entropy of a random bipartite state ρAB . na = 2 ; nb = 2 ; rho = randRho(na nb); % Generate a random bipartite state rho ∗ cvx begin sdp variable tau(na nb , na nb) hermitian ; minimize ( quantum∗ rel∗ entr(rho, tau)/ log(2) ); tau >= 0 ; norm nuc(Tx(tau, 2, [ na nb ])) <= 1 ; cvx end

r a i n s r e l e n t = cvx optval ;

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