IN studying classical communication, Shannon developed

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1 Sueradditivity in trade-off caacities of quantum channels Elton Yechao Zhu, untao Zhuang, Min-Hsiu Hsieh, Senior Member, IEEE, Peter W Shor trade-off regions for the resources that are consumed or generated [2] [4] The most common, useful, quantum resource in communication settings is quantum entanglement Unlike classical shared romness, which does not increase a classical channel s caability to send more messages, reshared quantum entanglement will generally increase the throughut of a quantum channel for sending classical messages or quantum messages or both [2], [5] [9] It thus makes sense to consider the trade-off caacity regions among these three useful resources: entanglement, classical communication, quantum communication, this was done in Ref [4] The result in Ref [4] further shows that a coding strategy that eloits the channel coding of these three resources as a whole erforms better than strategies that do not take advantage of channel coding On the other h, single-lettered channel caacity formulas in the classical regime generally become intractable regularized caacity formulas in the quantum regime [0] [4] In other words, evaluation of these caacity quantities requires otimizing channel inuts over an arbitrary finite number of uses of a given channel This largely blocks our understing of how quantum channels behave An etreme eamle shows the eistence of two quantum channels that cannot be used to send a quantum message individually but will have a ositive channel caacity when both are used simultaneously [5] However, there are also several eamles showing that when additional resources are used to assist, the corresonding assisted caacity will also become additive The classical caacity over quantum channels is generally sueradditive; however, when assisted by a sufficient amount of entanglement, the entanglement-assisted caacity becomes additive [6], [6] The quantum caacity also ehibits similar roerties When assisted by either entanglement [2], [3] or an unbounded symmetric side channel [7], its assisted quantum caacity becomes additive This sueradditive roerty of quantum channel caacities has accordingly attracted significant attention Hastings [8] roved that the classical caacity over quantum channels is not additive, a result built uon earlier develoments by Hayden- Winter [9] Shor [20] Recently, three of us showed a rather erleing result [2]: when assisted by an insufficient amount of entanglement, a channel s classical caacity could be sueradditive regardless of whether the unassisted classical caacity is additive or not Further, the additive roerty of the entanglement-assisted classical caacity shows a form of hase transition Even if the channel is additive when assisted by a sufficient amount of entanglement or no entanglement at all, it can still be sueradditive when assisted with an insufarxiv: v2 [quant-h] 6 Aug 207 Abstract In this article, we investigate the additivity henomenon in the dynamic caacity of a quantum channel for trading classical communication, quantum communication entanglement Understing such additivity roerty is imortant if we want to otimally use a quantum channel for general communication urose However, in a lot of cases, the channel one will be using only has an additive single or double resource caacity, it is largely unknown if this could lead to an sueradditive double or trile resource caacity For eamle, if a channel has an additive classical quantum caacity, can the classical-quantum caacity be sueradditive? In this work, we answer such questions affirmatively We give roof-of-rincile requirements for these channels to eist In most cases, we can rovide an elicit construction of these quantum channels The eistence of these sueradditive henomena is surrising in contrast to the result that the additivity of both classical-entanglement classical-quantum caacity regions imly the additivity of the trile caacity region Inde Terms Additivity; uantum hannel aacity; Tradeoff aacity Regions; uantum Shannon theory I INTRODUTION IN studying classical communication, Shannon develoed owerful robabilistic tools that connect the theoretic throughut of a channel to an entroic quantity defined on a single use of the channel [] Shannon s noiseless channel coding theorem involves a rom coding strategy to rove achievability entroic inequalities that show otimality, ie, the converse This methodology has now become stard in roving finite or asymtotic otimal resource conversions in information theory uantum Shannon information starts by mimicking classical information theory: tyical sets can be generalized to tyical subsaces to rove achievability while various entroic inequalities, such as the quantum data rocessing inequality, can be used to rove the converse However, the differences between quantum classical Shannon information are also significant On one h, additional resources available in the quantum domain diversify the allowable caacities, resulting in EY Zhu is with the enter of Theoretical Physics Deartment of Physics, Massachusetts Institute of Technology, ambridge, MA, 0239 USA eltonzhu@mitedu Zhuang is with the Research Laboratory of Electronics Deartment of Physics, Massachusetts Institute of Technology, ambridge, MA, 0239 USA quntao@mitedu Min-Hsiu Hsieh is with the entre of uantum Software Information UTS SI, University of Technology Sydney, Australia Min- HsiuHsieh@utseduau PW Shor is with the Deartment of Mathematics, Massachusetts Institute of Technology, ambridge, MA, 0239 USA shor@mathmitedu

2 2 ficient amount of entanglement This henomenon indicates that quantum channels behave fundamentally differently from classical channels, our understing of it is still quite limited This aer is insired by, aims to etend Ref [2] Will additivity of single or double resource caacities always lead to additivity of a general resource trade-off region? We will study sueradditi vity in a general framework that considers the three most common resources of: entanglement, noiseless classical communication quantum communication Our results show that i additivity of single resource caacities of a quantum channel does not generally imly additivity of double resource caacities, ecet for the known result [2] that an additive quantum caacity yields an additive entanglementassisted quantum caacity region see Table I; ii additive double resource caacities does not generally imly an additive trile resource caacity, ecet for the known case [8] that additive classical-entanglement classical-quantum caacity regions yield an additive trile dynamic caacity see Table II These results again demonstrate how comle a quantum channel can be, further investigation is required The aer is structured as follows Section II introduces the various definitions, notations revious results on the trile resource quantum Shannon theory Section III summarizes the various sueradditivity results that we establish in the aer Section IV establishes the switch channel that we use for all our constructions, how this reduces the trile resource trade-off formula Section V gives a detailed construction of all the ossible sueradditivity henomena II PRELIMINARIES In this section, we give definitions of basic entroic quantities used in the aer We also describe the dynamic caacity theorem Secial cases of this include the various single double resource caacities Finally, we define the elementary channels that will be used in our elicit constructions A biartite quantum state AB is a ositive semi-definite matri in Hilbert sace H A H B with trace one We define the von Neumann entroy, conherent information quantum mutual information of AB, resectively, as follows: SAB = Tr [ AB log AB ], IA B = SB SAB, IA; B = SA + IA B, where SA is the von Neumann entroy of the reduced state A = Tr B [ AB ] For an ensemble {, AB } X, let X AB = X AB, X where { } forms a fied orthonormal comutational basis in Hilbert Sace H X We need the following information quantities as well: IA BX = IA B, IA; B X = IA; B, 2 IAX; B = IX; B + IA; B X, 3 where IA BX IA; B X in Eqs 2 are the conditional coherent information the conditional mutual information, resectively IX; B in Eq 3 is the Holevo information of XB = Tr A [ X AB ] A quantum channel N is a comletely ositive tracereserving ma With it, we can transmit either classical or quantum information or both with ossible entanglement assistance between the sender the receiver [8] More generally, the authors in Ref [4] roved the following caacity theorem that involves a noisy quantum channel N the three resources mentioned above; namely, classical communication, quantum communication quantum entanglement E Theorem trade-off [4]: The dynamic caacity region N of a quantum channel N is equal to the following eression: N = k= k N k, where the overbar indicates the closure of a set The region N is equal to the union of the state-deendent regions, N: N, N The state-deendent region, E, such that, N is the set of all rates + 2 IAX; B, 4 + E IA BX, E IX; B + IA BX 6 The above entroic quantities are with resect to a classicalquantum state cq state X AB, where X AB X N A B φ AA, 7 the states φ AA are ure We say that the dynamic caacity of a channel N is additive if N = N 8 The dynamic caacity region N in Theorem allows us to recover known caacity theorems by choosing certain,, E in Eqs 4-6 as follows: the classical caacity N when choosing = E = 0 [0], []; the quantum caacity N when choosing = E = 0 [2] [4]; the classical quantum caacity N when choosing E = 0 trade-off [22]; the entanglement assisted classical caacity E N when choosing = 0 E trade-off [7], [6]; the entanglement assisted quantum caacity E N when choosing = 0 E trade-off [2], [3]; Additivity of these secial cases follows similarly from Eq 8 We note that the dynamic caacity region is concave, as a conve combination of any two oints in the region can be

3 3 achieved by a time-sharing strategy, ie, using the channel for a fraction of uses to achieve one oint, using it for the other fraction to achieve the second oint Below we will briefly describe a few channels which we will reeatedly use Definition 2: A Hadamard channel is a quantum channel whose comlementary channel is entanglement breaking Suose Ψ A B is a Hadamard channel, with the comlementary channel Ψ c A E Then there is a degrading ma D B E such that Ψ c A E = D B E Ψ A B Moreover, D can be decomosed as D B E = D 2 Y E D B Y, where Y is a classical variable A Hadamard channel has an additive quantum dynamic caacity region, when tensored with an arbitrary quantum channel [23] Eamles of Hadamard channels include the qubit dehasing channel, N cloning channels, the Unruh channel We ll define the qubit dehasing channel below, but refrain from giving definitions of other Hadamard channels, since their eact forms are not needed for understing this work We refer the interested readers to Ref [23] for more details roerties of these channels Definition 3: The qubit dehasing channel Ψ dh, with dehasing robability, is defined as Ψ dh ρ = ρ + Z ρz Definition 4: The qubit deolarizing channel Ψ do, with deolarizing robability, is defined as Ψ do ρ = ρ + I 2 The qubit deolarizing channel is known to have an additive classical caacity [24], but a sueradditive quantum caacity [25] Definition 5: A rom orthogonal channel Ψ ro is defined as D Ψ ro ρ = P i O i ρo i, i= where O i are chosen from the orthogonal grou the robabilities P i are roughly equal For D N, with N the inut dimension, such a channel will have a subadditive minimum outut entroy with high robability [8] Definition 6: onsider an arbitrary channel Ψ B Aend a register R to the inut, with a set of orthonormal bases { j } R = B 2 We define its unitally etended channel [20], [26] Φ R B as Φ R B ρ R = XjΨ B j ρ R j R Xj, 9 j where {X j : j {,, R }} are the Heisenberg-Weyl oerators The unital etension of a rom orthogonal channel will have a sueradditive classical caacity with high robability [20] Imly the additivity of Additive caacities E E N [2] N [25] N [25] N SecV- N SecV- Y [3],,E N SecV-B N SecV-D Y[3] TABLE I SUMMARY OF RESULTS FOR DOUBLE RESOURES N STANDS FOR DOES NOT IMPLY ADDITIVITY, WHILE Y MEANS IMPLIES ADDITIVITY Imly the additivity of Additive caacities E N SecV- N SecV-G E N SecV-E E, E,E N SecV-F E, Y[8] TABLE II SUMMARY OF RESULTS FOR TRIPLE RESOURES N STANDS FOR DOES NOT IMPLY ADDITIVITY, WHILE Y MEANS IMPLIES ADDITIVITY III SUMMARY OF RESULTS We summarize all of our results here We will denote the single caacity region by a single letter, eg for N We will also use short notation for double trile tradeoff regions, eg E for E N for N We will use the arrow notation, with meaning additivity of the left-h side caacity imlies additivity of the right-h side caacity, meaning additivity of the left-h side caacity does not imly additivity of the right-h side caacity A Double resources see table I E: a E [2]: There eists a quantum channel N, such that its classical caacity is additive, but its E trade-off caacity region is sueradditive We will give a simlified construction in Sec V-A b, E: There eists a quantum channel N, such that its classical quantum caacities are both additive, but its E trade-off caacity region is sueradditive, ie,, a quantum channel N st but N = N N = N E N E N An elicit construction of N is given in Sec V-B 2 E [3]: For all quantum channels N, if its quantum caacity is additive, then its E trade-off caacity region is always additive

4 4 3 : a [25]: There eists a quantum channel N, such that its classical caacity is additive, but its tradeoff caacity region is sueradditive The deolarizing channel has a sueradditive quantum caacity hence a sueradditive trade-off caacity, while its classical caacity is additive b : There eists a quantum channel N, such that its quantum caacity is additive, but its tradeoff caacity region is sueradditive, ie, a quantum channel N st but N = N N N A construction of this eamle quantum channel is given in Sec V- c, : Moreover, there eists a quantum channel N, such that its classical quantum caacities are additive, but its trade-off caacity region is sueradditive, ie, a quantum channel N st but N = N N = N, N N A construction of this eamle quantum channel is given in Sec V-D B Trile resources see table II E : There eists a quantum channel N such that its E trade-off caacity region is additive, but its dynamic caacity region is sueradditive, ie, a quantum channel N st but E N = E N N N An eamle is constructed in Sec V-E 2 E, : There eists a quantum channel N such that its quantum caacity its E trade-off caacity region are additive, but its dynamic caacity region is sueradditive, ie, a quantum channel N st but N = N E N = E N, N N An eamle is constructed in Sec V-F 3 : There eists a quantum channel N such that its trade-off caacity region is additive, but its dynamic caacity region is sueradditive, ie, a quantum channel N st but N = N N N An eamle is given in Sec V-G 4 E, [8]: If a quantum channel N has additive E trade-off caacity regions, then its dynamic caacity region is also additive This statement is first observed in Ref [8], an elicit argument can be found in Ref [23] IV FRAMEWORK This section resents technical tools that we require for demonstration of sueradditivity in trade-off caacities We first define the concet of switch channels Definition 7: A switch channel N M B between N B 0 N B with M being a -bit switch register is defined as N M B ρ M =N 0 B 0 ρ M 0 M + N B ρ M M In quantum information theory, switch channels were first used in Ref [7] to demonstrate the eistence of quantum channels such that the quantum caacity is nonzero, but for which re-shared entanglement does not imrove the classical caacity Subsequently, they are used in Ref [27] to show the sueradditivity of rivate information, with an alternative definition Recently, they are also used in Ref [2] to show the sueradditivity of the classical caacity with limited entanglement assistance One immediate difficulty is that, even if N 0 N are well-studied, the dynamic caacity region of N may not always have a simle eression in terms of those of N 0 N This is due to the fact that the switch register M can be in a suerosition state However, if N 0 N are unitally etended channels, then the dynamic caacity region of N does have a simle eression Lemma 8: onsider a switch channel N A B between NR B 0 N R B, with inut artition A = MR M being a switch register Here NR B 0 N R B are unital etensions of Ψ B 0 Ψ B resectively Then N = onv N 0, N, where onv denotes the conve hull of oints from the two sets If the quantum dynamic caacity region for N 0 Ψ is additive for any Ψ, then we also have N = onv N 0, N The rest of this section is devoted to the roof of this lemma Firstly, we note that switch channels unitally etended channels fall under a broader class of channels that we call artial classical-quantum channels artial cq channels

5 5 Definition 9: A channel Ψ R B is a artial cq channel if there eists a noiseless classical channel Π R R with orthonormal basis { j R }, such that Ψ R B = Ψ R B Π R R 0 If there is no register, then such channels are classicalquantum channels cq channels For artial cq channels, one can always assume inuts are cq states with resect to the inut artition R for the urose of evaluating caacities, as we show in Lemma 0 below Lemma 0: If Ψ A B is a artial cq channel with artition A = R, then the otimal trade-off surface of the -shot dynamic caacity region Ψ can be achieved with resect to cq states X AB = Ψ A B ρ X AA, where ρ X AA is of the form ρ X AA = X j j R φ j A Proof We will show that, for any inut state ϱ X AA = X φ AA, 2 with its outut state ς X AB Ψ A B ϱ X AA = X ςab, where ςab = Ψ A B φ AA, there eists a corresonding state ρ X AA, in the form of Eq, which can achieve the same rate, if not better In fact, the state ρ X AA can be obtained by alying Π R R on ϱ X AA eing its classical register X This can be achieved by the following quantum instrument T : R RX R, T ψ R := j ψ R j j j R j j XR so that j ρ X AA = T ϱ X AA = X j j R φ j A 3 where we abuse the notation X to denote X X R in Eq 3, j, j = Tr[ j j φ A ] Let X AB = Ψ A B ρ X AA Then where It follows that X AB = X j AB j AB = Ψ A B j j R φ j A ς AB = j j j AB 4 Since the dynamic caacity region is fully determined by the three entroic quantities IAX; B, IA BX IX; B in Eqs 4-6, it suffices to show that all three entroic quantities evaluated on ρ X AA are greater than those evaluated on ϱ X AA First consider IA BX IA BX = IA B j = j IA B j IA B ς = IA BX ς, 5 where the inequality is due to Eq 4 the conveity of coherent information with resect to inuts 2 Now consider IAX; B Similarly, IAX; B = SB + IB AX SB ς + IB AX ς = IAX; B ς, where the inequality is due to B = ς B Eq 5 3 Finally consider IX; B Writing as j, it can be shown IX; B IX; B ς using the data rocessing inequality when we aly the artial trace ma j j to XB Lemma : The otimal trade-off surface of the -shot quantum dynamic caacity region of a unitally etended channel can always be achieved with X AB such that SB = log B This etends similarly to higher shots Proof Suose Φ R B is unitally etended from Ψ B Since a unitally etended channel Φ R B is a artial cq channel, by Lemma 0, we can consider states of the form ϱ X AA = X j j R φ j A Let ς X AB = Φ R B ϱ X AA with A R Then ς X AB = X ς j j AB, where ς jk AB = Xkς j AB Xk ς j AB = Ψ B φ j A We can construct another state of the form in Eq : ρ X AA =, k, k, k X k k R φ j A,,k 6 where, k = / R, X AB = Φ R B ρ X AA : X AB =, k, k, k X jk AB,,k where jk AB = Xk j AB Xk j AB = Ψ B state X AB satisfies SB = S,k S = log B,, k jk B R k jk B φ j A The

6 6 where we ve used the qudit twirl formula [28] jk R B = Xk j R B Xk = B I B 7 k k One can verify that the dynamic caacity region with X AB is larger than that with ς X AB as follows: IA BX =, kia B j k,k = IA B ς j j = IA BX ς 8 IAX ; B = SB +, kib A j k IX ; B,k = log B + IB A ς j j IAX; B ς = SB, ksb j k 9,k = log B SB ς j j IX; B ς 20 The key roerty used in the above equations is, for any Heisenberg-Weyl oerator Xk, S B = SXk B Xk Proof of lemma 2 Following from Lemma Eq 6, we only need to consider states of the form,k ρ X AA = m m m M ρ m X AR 2 m=0 where m =,k, m, k ρ m X AR =, m, k, m, k, m, k X k k R φ m A, m with, m, k =, m, k for all k, k m {0, } The corresonding channel outut is where m X AB =,k mk AB X AB =, m, k m m X m AB 22 m=0 = XkΨm B, m, k, m, k X mk AB 23 φ m A Xk Then all three of the entroic quantities evaluated on X AB in Eq 22 can be decomosed to the corresonding ones evaluated on X m AB given in Eq 23: IA BX = =, m, kia B mk m=0,k m IA BX m m=0 Likewise, IAX; B = log B + =, m, kib A mk m=0,k m IAX; B m m=0 IX; B = m IX; B m m=0 This means if we consider inuts of the form 2, the trile rate for using N can always be eressed as a linear combination of the trile rates of N 0 N It is also clear that any linear combination is achievable by the time-sharing rincile Since using states of the form 2 is otimal, we have N = 0 = onv N 0 + N N 0, N Here, addition means Minkowski sum Similarly, we have N N =onv N 0 N 0, N 0 N, N N If the quantum dynamic caacity region is additive for N 0 Ψ, for any Ψ, then N 0 N = N 0 + N 24 In this case the -shot quantum dynamic caacity region for N N can be greatly simlified to N N = onv 2 N 0, N N Similarly, N k =onv, Each term uer bounded as N k, N 0 N k, N 0 k N 0 k N, N 0 m N k m, 0 m k, can be N 0 m N k m N k m =m N 0 + m N 0 + k m N konv N 0, N For two sets of osition vectors A B in Euclidean sace, their Minkowski sum A+ B is obtained by adding each vector in A to each vector in B, ie, A + B = {a + b a A, b B} [29]

7 7 Here the second line follows from the addivity of the dynamic caacity region of N 0 The third line follows from the definition of The fourth line follows from the definition of conve hull Thus N k can also be uer bounded as N k konv N = k N k k= N 0, N onv N 0, N =onv N 0, N The last equality follows because of the toology of the dynamic caacity region, as we show in Aendi D By a time-sharing rotocol, it is obvious that N onv N 0, N Hence N = onv N 0, N Note that unital etensions are not unique, we only used the unitarity of Heisenberg-Weyl oerators the twirl formula Eq 7 in roving the above lemmas Hence, as long as we have K unitaries {U k } Ud that satisfy the twirl formula U k AU K k = TrA I 25 d k for any d d matri A, one has a valid unital etension, lemmas 8 will hold 2 Moreover, unital etensions are reserved under tensor roduct of channels: if Φ is a unital etension of Ψ, Φ 2 is a unital etension of Ψ 2, then Φ Φ 2 is also a unital etension of Ψ Ψ 2 This follows from the fact that if {U j } Ud {V k } Ud 2 both satisfy Eq 25, then {U j V k } Ud d 2 also satisfies Eq 25 V EXPLIIT ONSTRUTION OF VARIOUS SUPERADDITIVITY PHENOMENA With the tools develoed in Sec IV, we can now elicitly construct channels that satisfy the sueradditivity roerties stated in Sec III All our constructions utilize the switch channel idea We always assume that N is a switch channel of two unitally etended channels N 0 N Further, we assume that U N 0 has an additive dynamic caacity region, when tensored with another arbitrary channel In this setting, we can use Lemma 8 its reduction to various single-resource two-resource caacities 2 Note that we do not even require N 0 N to have the same unital etension However, to ensure the inut dimensions of N 0 N are the same, their unital etensions must involve the same number of unitaries For this reason, we stick with the Heisenberg-Weyl oerators most of the time In each construction, we first state the roerties that N 0 N need to satisfy, in addition to Proerty U We then show how the desired sueradditivity of the switch channel N follows from these roerties In the end, we elicitly construct channels that satisfy the roerties we required Before we start, we first roose two families of unital etended channels that satisfy U Many of our elicit constructions of N 0 will be chosen from these cidates The first family comes from unital etensions of Hadamard channels The following lemma shows that the dynamic caacity of the unitally etended Hadamard channels is also additive Lemma 2: The dynamic caacity region is additive for Φ 0 any other channel Ψ, if Φ 0 is a unital etension of a Hadamard channel Ψ 0 The second family is unital etensions of classical channels Lemma 3: If Ψ 0 is a classical channel, then the dynamic caacity region is additive for Ψ 0 Ψ, for arbitrary Ψ The same holds for a unital etension of a classical channel The roofs of the above lemmas are left to the Aendices, as they are not essential in understing the construction A Additive, Sueradditive E Here we review the original argument in [2] recast it in the current framework We use P N when we view N as a function of the amount of entanglement assistance P, where N, P are oints on the E trade-off curve of N When P = 0, we return to the classical caacity N When P is maimal, we arrive at the classical caacity with unlimited entanglement assistance E N P N denotes the -shot case We require N 0 N to have the following roerties: A N 0 = N A2 N has a sueradditive E trade-off caacity region, ie,, E N E N, E N is strictly concave sueradditive at a boundary oint of the trade-off region with entanglement consumtion P A3 E N 0 E N in the sense the E trade-off caacity region of N 0 is strictly smaller than that of N when entanglement consumtion is at P In the P notation, roerty A2 means at P = P, P N > P N P N is strictly concave 3 in P at P = P Proerty A3 imlies that P N 0 < P N at P = P Note that these roerties are weaker than the ones required in Ref [2] These three roerties A-A3, together with U, will guarantee that i the classical caacity of N is additive; ii the E trade-off caacity region of N is sueradditive at entanglement consumtion rate P 3 Here by saying a function f is strictly concave at y, we mean f y > f v + f w for all v < y < w satisfying v + w = y, with 0,

8 8 ombining roerty A with U yields statement i: N = ma { N 0, N } = N 0 { = ma N 0, N } = N, where Lemma 8 is used in the first equality Proerty A3 ensures that E N = onv E N 0, E N Since = E N 26 E N = onv E N 0, E N, there eists P 0, P 0 [0, ] such that P 0 + P = P P N = P 0 N 0 + P N Statement ii follows after considering three different cases = 0 N = N < P N = P N, P P where the inequality follows from the sueradditivity art of roerty A2 The second equality follows from Eq < < P N = N 0 + N P0 N + P N N = P N < P P 0 P where the first inequality follows from Proerty A3 The second inequality follows from the strict concavity art of roerty A2 The last equality follows from Eq 26 3 = Then N = P P N 0 < P N = P N Here the first equality follows from additivity of the E trade-off caacity region for N 0 The inequality follows from roerty A3 The last equality follows from Eq 26 Elicit onstruction of N: We quote the following roerty about concave functions [30]: A concave function uy is continuous, differentiable from the left from the right The derivative is decreasing, ie, for < y we have u u + u y u y+ We use ± to denote the right left derivatives when needed We first construct N hoose Ψ ro to be a rom orthogonal channel with a subadditive minimum outut entroy, Ψ ro has inut dimension N This is unitally etended to Φ ro Due to Lemma 0, the useful entanglement assistance is at most logn Thus we restrict to 0 P logn Let ɛ = Φ ro Φro > 0 27 Since P Φro Φro + P, 28 E Φ ro Φ ro + logn ɛ This imlies d P Φ ro /dp cannot always be Thus there eists P [0, logn such that d P Φ ro /dp =, d P Φ ro /dp <, Net we discuss different cases of P 0 P P P > P P > 0 Then P Φ ro is strictly concave at P Furthermore, P Φ ro P Φro ɛ since P Φ ro = Φ ro + P but P Φro Φro + P Thus N = Φ ro satisfies A2 2 P = 0 Let N = Φ ro Φ dh, where Φ dh is the unital etension of the qubit dehasing channel Since d P Φ ro /dp 0+ <, choose > 0 small such that d P Φ dh /dp > d P Φ ro /dp 0+ This is ossible, as P Φ dh = P Ψ dh d P Ψ dh /dp as 0 This ensures that when 0 < P, P N = Φ ro + P Φ dh, 29 where we ve also used Lemma 2 For Φ dh, it can be shown that P Φ dh is strictly concave in P when < /2 see Aendi Hence P N is also strictly concave with resect to P, for 0 < P Also, when P < ɛ, P N > Φ ro + Φ dh > Φro + Φ dh + P P N Here the first inequality comes from Eq 29 P Φ dh > Φ dh when P > 0 The second inequality comes from our assumtion P < ɛ Eq 27 The last inequality comes from Eq 28 This ensures that P N is sueradditive Thus when 0 < P < min{, ɛ}, P N is strictly concave sueradditive For N 0, as long as it is a unital etension of a classical channel with N 0 = N, it will automatically satisfy roerty A3 B Additive, Sueradditive E In Section V-A, we constructed a channel N with an additive classical caacity, but a sueradditive E trade-off caacity region It s unclear if our construction N has an additive quantum caacity To etend the argument, we need to make some modifications to the original construction In addition to roerties A-A3, the channels N 0 N need to satisfy B N 0 N

9 9 This ensures that the quantum caacity of N is also additive: N = ma { N 0, N } = N 0 { = ma N 0, N } = N Elicit onstruction of N: We take the channels N 0 N that were constructed in Sec V-A, comare their quantum caacities Since N 0 = 0, we can only have N 0 N If N 0 = N, then B is automatically satisfied Hence we will focus on the case where N 0 < N In this case, we call these two channels Φ 0 Φ resectively We will construct two new channels N 0 N that satisfy roerties A-A3 B We will use the qubit dehasing channel N cloning channel To make the argument work, we will modify them in the following manner For the N cloning channel Ψ N, we always tensor an aroriate classical channel, such that the resulting channel has its classical caacity equal to, the outut dimension is the same as the inut dimension We denote the resulting channel Ψ N For the dehasing channel, we will tensor a comlete deolarizing channel, so that its inut outut dimensions match those of Ψ N Since tensoring a comlete deolarizing channel does not modify the dynamic caacity region of the qubit dehasing channel, we will continue using Ψ dh to denote it Based on results in Ref [23], we can obtain the trade-off caacities of the qubit dehasing channel Ψ dh modified N cloning channel Ψ N We observe that for = 02 N = 5, their trade-off caacities satisfy the following roerties see Fig E Ψ dh Ψ dh > Ψ N E Ψ N, 30 in the sense that Ψ N achieves a strictly better classical communication rate than Ψ dh, if we have any non-zero amount of entanglement assistance In the P notation, it means P Ψ dh < P Ψ N for all P > 0 Since unital etensions do not change the E tradeoff caacity regions of these two channels see Aendi, the above roerties hold if we relace Ψ dh Ψ N by their unital etensions Φ dh Φ N resectively Since Φ dh > Φ N, let n be large enough so that n Φ dh + Φ 0 n Φ N + Φ uantum ommunication Rate Ψ N, N = 5 Ψ dh, = lassical ommunication Rate a lassical ommunication Rate Ψ N, N = 5 Ψ dh, = Entanglement onsumtion Rate Fig omarison of trade-off curves between qubit dehasing channel Ψ dh modified N cloning channel Ψ N, when = 02 N = 5 a trade-off b E trade-off Define N 0 = N = Φ dh n Φ 0 Φ N n Φ Our choice of n ensures that N 0 N We also need to ensure our newly constructed N 0 N still satisfy roerties A-A3 As Φ dh = Φ N = we immediately have Φ 0 = Φ, N 0 = N roerty A is satisfied The E trade-off curve of Ψ N is strictly concave for N [23], hence roerty A2 is also satisfied for N Proerty A3 is satisfied due to Eq 30 Additive, Sueradditive We require N 0 N to have the following roerties: N 0 N 2 N > N 3 N 0 < N These roerties -3 will allow us to show that i N = N; ii N N Statement i follows from roerty U that N 0 has an additive quantum caacity: N = ma { N 0, N } = N 0 { = ma N 0, N } = N Proerties 2 3 together ensure N = ma { N 0, N } = N { > ma N 0, N } = N, b

10 0 ie, the classical caacity of N is sueradditive; hence statement ii follows Elicit onstruction of N: Net we construct N 0 N that satisfy the above roerties Let Ψ ro be a rom orthogonal channel, such that its unital etension has a sueradditive classical caacity For convenience, we also assume Ψ ro has the inut dimension N = 2 n hoose for the qubit dehasing channel Ψ dh such that Ψ ro + Ψ dh = m for some integer m Define N = Φ ro Φ dh, where Φ ro is a unital etension of Ψ ro Φ dh is a unital etension of Ψ dh N has the roerty that its quantum caacity is N = m, whereas its classical caacity is sueradditive, greater than m Define N 0 = Φ I m Φ do n+ m, where Φ I is a unital etension of the noiseless qubit channel, Φ do is a unital etension of the comlete qubit deolarizing channel It s clear that N 0 has its classical quantum caacity as N 0 = N 0 = m, thus fulfiling the roerties 3 above D Additive, Sueradditive We require N 0 N to satisfy the following roerties: D N 0 N = N N 0 = N D2 N has a sueradditive trade-off caacity region, meaning N N N is strictly concave sueradditive at a boundary oint with classical communication rate D3 N 0 N in the sense the trade-off caacity region of N 0 is strictly smaller than that of N when classical communication rate is at With these roerties, we can show that i N = N; ii N = N; iii N N We ll focus on the trade-off curve Same as in Section V-A, we use a simlified notation N when we view N as a function of N In the -shot scenario, it is denoted by N We ll show there eists 0 such that N > N In the notation, roerty D2 means at =, N > N N is strictly concave in at = Proerty D3 imlies that N 0 < N at = Proerty D U that N 0 has an additive quantum caacity ensure that N = ma { N 0, N } = N { = ma N 0, N } = N N = ma { N 0, N } = N 0 = ma { N 0, N } = N, ie, N has an additive classical quantum caacity By roerty D3, we have N = onv N 0, N Since = N 3 N = onv N 0, N, there eists 0, [0, ] such that 0 + = N = 0 N 0 + N Now consider three different cases = 0 N = N < N = N, where the inequality follows from roerty D2 The second equality follows from Eq < < N = 0 N 0 + N 0 N + < N = N N Here the first inequality follows from the definition of roerty D3 The second inequality follows from the strict concavity art of roerty D2 The last equality follows from Eq 3 3 = Then N = N 0 < N = N Here the inequality follows from roerty D3 The last equality follows from Eq 3 Hence statement iii follows Elicit onstruction: Now we elicitly construct N 0 N hoose such that the qubit deolarizing channel Ψ do is known to have a sueradditive quantum caacity onsider its unital etension Φ do Note that the gradient d Φ do /d of the trade-off curve cannot always stay at 0 for the choice of Ψ do with a ositive quantum caacity It means there eists 0 < Φ do such that d d Φ do Φ do /d = 0, 0 32 /d < 0, + Φ do

11 > 0 In this case, we know concave at Also = 0 Φ do Φ do > 0 Φ do Φ do is strictly Φ do Here the equality follows from Eq 32 The first inequality follows because Ψ do has a sueradditive quantum caacity, as both remain unchanged after a unital etension, reduces to the quantum caacity at = 0 The second inequality follows as the rate of quantum communication along the trade-off curve must not eceed the quantum caacity hoose the noise arameter for the qubit dehasing channel Ψ dh aroriately such that Define Ψ dh = N = Φ do Ψ do Φ dh It s clear that N is a unitally etended channel of Ψ do Ψ dh has N = Ψ do Ψ dh = The trade-off curve is strictly concave sueradditive at The corresonding Ψ 0 is Ψ 0 = I Ψ do, ie, a noiseless channel tensor a comlete qubit deolarizing channel N 0 is a unital etension of Ψ 0 2 = 0 hoose close to /2 such that Let d Φ dh d Ψ dh N = Φ dh where 2 is chosen such that Φ do d > d Φ do Φ dh 2, 0+ N = Ψ dh Ψ do Ψ dh 2 = Ψ dh + Ψ do + Ψ dh 2 = By our choice of, Φ dh in for 0 < < Φ dh 2 Φ do is strictly concave is also strictly concave in Thus N is strictly concave in, for 0 < < In this case, the corresonding Ψ 0 is Ψ 0 = I Ψ do 2, ie, a noiseless channel tensor two coies of the comlete qubit deolarizing channel N 0 is a unital etension of Ψ 0 E Additive E, Sueradditive Here we construct a channel that has an additive E tradeoff caacity region, but a sueradditive quantum caacity, hence a sueradditive quantum dynamic caacity region Let Ψ 0 be a classical channel Ψ be the deolarizing channel Ψ do is chosen such that Ψ do has a sueradditive quantum caacity Also, we require Ψ 0 > E Ψ 33 Now consider the switch channel N, consisting of N 0 N, which are unital etensions of Ψ 0 Ψ It can be easily shown that unital etension does not change the classical caacity with umlimited entanglement assistance of the qubit deolarizing channel Thus Eq 33 imlies E N 0 E N E N 34 Hence E N = onv E N 0, E N = E N 0 = onv E N 0, E N = E N, ie, its E trade-off caacity region is additive Since N 0 = 0, it is clear that the quantum caacity of N is the same as that of N, which is sueradditive Note that N is a unitally etended channel This fact will be imlicitly used in Section V-F F Additive E, Sueradditive Previously in Section V-D, we give an eamle of a channel with an additive classical quantum caacity, but whose trade-off curve is sueradditive It is unclear if the channel has an additive E trade-off caacity region, because the E tradeoff caacity region of the deolarizing channel has not been shown to be additive This is itself an interesting question but we ll not elore it here We relace Ψ do in the original argument of Section V-D by the channel constructed in Section V-E It s clear that the rest of the argument is not changed N still has a sueradditive trade-off caacity region Now both N 0 N have an additive E trade-off caacity region It s clear that E N 0, E N E N = onv = onv E N 0, E N = E N, ie, the E trade-off caacity region of N is additive G Additive, Sueradditive Our construction in Section V-A has a sueradditive E trade-off caacity region But most likely its trade-off caacity region is also sueradditive This is because in Section V-A, N 0 is the unital etension of a classical channel, its trade-off caacity region is trivial Hence the trade-off caacity region of N is given by that of N, which is most likely sueradditive as well

12 2 To achieve an additive trade-off caacity region, we have to substitute N 0 with a channel that has a non-trivial trade-off caacity region Recall that our construction in Section V-A requires N 0 N to have roerties A-A3 These three roerties ensure that N will have a sueradditive E trade-off caacity region, while its classical caacity is still additive In etending to a channel with an additive trade-off caacity region, the additional roerties we need are G N 0 N Proerty G U ensure the trade-off caacity region of N is additive, as N 0, N = N 0 N = onv = = N N 0 = onv N 0, N Unfortunately, we cannot find quantum channels N 0 N that satisfy all the roerties Hence we do not have an elicit construction in this case This is because there are very few channels that we underst their dynamic caacity regions This leaves us with a limited choice of cidates for N 0 However, in rincile there is no obstacle the construction will be readily available once we have a better understing of quantum channels VI ONLUSION Unlike revious studies on additivity of single resource channel caacity, our work aimed to underst how additivity of single or double resource caacity regions will effect additivity of a general resource trade-off caacity In contrast to the two known results in the literature; namely, i additivity of the quantum caacity imlies additivity of the entanglement-assisted quantum caacity region ii additivity of classical-quantum classical-entanglement caacity regions imlies additivity of the three resource caacity region, the additivity of all the remaining situations does not hold In this work, we identified all ossible occurrences where sueradditivity could occur in the trade-off quantum dynamic caacity Furthermore, we rovided an elicit construction of quantum channels for most instances Our main technical tool combines roerties of switch channels unital etension of known quantum channels An obvious oen question is an elicit construction of a quantum channel whose classical-quantum caacity region is additive, but its trile trade-off caacity is sueradditive Moreover, there are other trile resource trade-off caacity regions [4], [3] ould similar statements made in this work hold in these scenarios as well? AKNOWLEDGMENT EYZ PWS are suorted by the National Science Foundation under grant ontract Number F Z is suorted by the laude E Shannon Research Assistantshi MH is suorted by an AR Future Fellowshi under Grant FT PWS is suorted by the NSF through the ST for Science of Information under grant number F APPENDIX A PROOF OF LEMMA 2 Proof onsider Φ 0 Ψ, where Φ 0 is a unital R B 0 A B etension of a Hadamard channel Ψ 0, Ψ is an B arbitrary channel 0 The result follows if both the E trade-off caacity regions of Φ 0 are additive [4] To show that the trade-off caacity region is additive for Φ 0, it was shown in Ref [23] it suffices to rove that f λ Φ 0 Ψ = f λ Φ 0 + f λ Ψ 35 for any channel Ψ, where f λ N = ma IX; B + λia BX 36 ρ The state is the channel outut state with ρ being the inut state see, eg, Theorem In the following, we will only show that f λ Φ 0 Ψ f λ Φ 0 + f λ Ψ because the other direction is trivial from its definition Since Φ 0 Ψ : RA B 0 B is a artial cq channel, then by the same argument as that in Lemma 0, f λ Φ 0 Ψ can be achieved with inut states of the following form ρ XRA A = with outut states where R X AB 0 B = X j j R φ A A, R X j AB 0 B, 37 j AB 0 B = Φ 0 Ψ j j R φ A A Let U 0 U be the isometric etensions of B 0 E 0 A B E Ψ 0 Ψ, let ϱ X A A = X φ A A ω X AA B 0 E U 0 = 0 I ϱ X A A U 0 I ς X AB 0 B E 0 E U = 0 U ϱ X A A U 0 U Moreover, let θ XY AB E 0 E = D B 0 Y ς X AB 0 B E 0 E, where D 2 D Y E 0 B 0 Y = D B 0 E0 is a degrading ma for the Hadamard channel Ψ 0 For any state X AB 0 B in Eq 37, we have f λ Φ 0 Ψ =I X; B 0 B + λi A B 0 B X =S B 0 B [ + λ S B 0 B X λs =S B 0 B [ + λ S B 0 B X ς AB 0 B X ς λs AB 0 B X ς ] ],

13 3 where the last equality follows from the same argument used in Eqs 8 20 Then subadditivity of the von Neumann entroy chain rule yield S B 0 + λ S B 0 X λs E 0 X ς ς ς +S B + λ S B B 0 X λs E E 0 X ς ς ς B 0 + λ S λs ς ς S +S B 0 X B + λ S B XY θ E 0 X ς θ λs E XY where the last inequality uses the fact that S B B 0 X ς S B Y X θ due to the eistence of D S E E 0 X ς S E Y X θ due to the eistence of D2 Finally, = I X; B 0 + λi AA B 0 X ω ω + I XY; B + λi AE 0 B XY θ θ f λ Φ 0 + f λ Ψ because SE 0 X ς = SAA B 0 X ω S E XY θ = S AB E 0 XY θ To rove that the E trade-off caacity region of the channel Φ 0 is additive is equivalent to showing that [23]: g λ Φ 0 Ψ = g λ Φ 0 + g λ Ψ, 38 where 0 λ <, g λ N = ma IAX; B λsa X 39 is of the form given in Eq 7 However this roof roceeds similarly; hence, we will omit it APPENDIX B PROOF OF LEMMA 3 Lemma 6: If Ψ 0 is a classical channel, then the dynamic caacity region is additive for Ψ 0 Ψ, for arbitrary Ψ Moreover, the dynamic caacity region of Ψ 0 is described by the following relation + 2 Ψ 0, + E 0, + + E Ψ 0, where Ψ 0 is the classical caacity of Ψ 0 The same holds for a unital etension of a classical channel Proof onsider the -shot dynamic caacity region of Ψ 0 A 0 B0 By Lemma 0, Ψ 0 can be achieved with resect to cq states X A 0 B 0 = Ψ0 ρ X A 0 A, where 0 ρx A 0 A 0 is of the form Thus ρ X A 0 A 0 = X A 0 B 0 = X j j A 0 φ j A 0 X j A 0 B 0, θ where j = Ψ 0 j j A 0 B 0 A 0 B 0 A 0 φ j A 0 is now a roduct state with resect to A 0 B 0 The three entroic quantities of interest can be simlied when evaluated with resect to X A 0 B0, as I A 0 X; B 0 B = S 0 S B 0 Ψ 0, j I A 0 B 0 X = I X; B 0 S Ψ 0 A 0 j 0, It s also clear that those inequalities can be achieved Thus Ψ 0 is described by + 2 Ψ 0, + E 0, + + E Ψ 0 Since the classical caacity of a classical channel is additive, the dynamic caacity region of Ψ 0 is additive is described by the same set of inequalities Net we show that the dynamic caacity region is additive for Ψ 0 Ψ, with Ψ arbitrary Since Ψ 0 Ψ is a artial cq channel, its A 0 B 0 A B -shot dynamic caacity region Ψ 0 Ψ can be achieved with resect to cq states X AB 0 B = Ψ0 A 0 B 0 Ψ ρx A B AA 0 A, where ρx AA 0 A is of the form ρ X AA 0 A = X j j A 0 φ j AA For ρ X AA 0 A of this form, X AB 0 B is of the form X AB 0 B = X j, AB 0 B with j AB 0 B = Ψ 0 A 0 B 0 Ψ A B j j A 0 φ j AA For such X AB 0 B, each of the three entroic quantities have simle uer bounds, AX; B 0 B X; I B 0 + I I I I A B 0 B X = I A B X, AX; B, X; B 0 B I X; B 0 + I X; B, where we ve used subadditivity of the von Neumann entroy Thus the -shot dynamic caacity region of Ψ 0 Ψ has a simle uer bound Ψ 0 Ψ Ψ 0 + Ψ It s trivial to etend it to the dynamic caacity region of Ψ 0 Ψ Ψ 0 Ψ Ψ 0 + Ψ

14 4 Since the other direction of inclusion is obvious, we have Ψ 0 Ψ = Ψ 0 + Ψ For unital etensions of a classical channel, we observe that, if the Heisenberg-Weyl oerators are defined on the stard basis for the outut of the channel, then the resulting channel is also a classical channel Hence the above result alies APPENDIX UNITAL EXTENSION OF THE UBIT DEPHASING HANNEL AND N LONING HANNEL Lemma 4: The E trade-off curve of the qubit dehasing channel N cloning channels are unchanged after a unital etension Proof onsider the qubit dehasing channel Ψ dh a N cloning channel Ψ N, their unital etensions Φ dh Φ N The statement of this lemma is equivalent to showing that for f λ Ψ = f λ Φ λ, g λ Ψ = g λ Φ 0 λ < Ψ, Φ = Ψ dh, Φ dh, Ψ N, Φ N In Lemma, we have argued that the -shot dynamic caacity region of a unitally etended channel can be achieved with inut of the form in Eq 6 Evaluating f λ Φ on such states, one obtains f λ Φ = log B + λ S B X λs AB X learly,ef N is bounded by the inut outut dimensions of N = log B +, k [λ SB j k λsab j k ] Then,k = log B + [λ SB j λsab j ] N = k,ef N k +,unit where log B + ma [λ S B λs AB ], 40 AB = Ψ B φ A 4 For such a AB = Ψ φ A that achieves Eq 40, one can construct ρ X AA = R k k X k k R φ A 42 This state will saturate the above inequality For Ψ dh Ψ N, it can be verified [23] that their f λ have the same form, ie, f λ Ψ = log B + ma [λ S B λs AB ], 43 with of the form given in Eq 4 The same argument also alies to g λ The trade-off curve of the qubit dehasing channel was comuted in Ref [22], the E trade-off curve was comuted in Ref [8] The E trade-off curves of the N cloning channel were given in Ref [23] Other than the secial cases = 0, /2 for the dehasing channel, N = for the N cloning channel, it can be verified that their E trade-off curves are strictly concave at every oint By Lemma 4, this roerty is true for their unital etensions Here we show that 4 APPENDIX D ONVEX HULL onv N 0, N =onv N 0, N We quote a few roerties about conve hull Minkowski addition that we will use [29]: i For two closed sets A B in R k, if A is bounded, then A + B is closed ii For two sets A B in R k, onva + B = onva + onvb iii The conve hull of a bounded set in R k is also bounded First, we note that, by Ref [4], all oints in the -shot dynamic caacity region can be achieved by the classically enhanced father rotocol, combined with unit rotocols, ie, N =,EF N +,unit, where,ef N = {IX; B, 2 IA; B X, 2 IA; E X } is the rate achieved using the classically enhanced father rotocol, is of the form in Eq 7,unit are all the rates achieved by the unit rotocols learly,unit is conve closed Define,EF N =,EF N Denote k= k= = k,ef N k +,unit A = k N k,,ef k= B =,unit Since A is bounded, B is closed, by i iii, A + B is also closed Since A + B A + B, A + B is closed, we have It is also obvious that hence Denote A + B A + B A + B A + B, A + B = A + B,EF N = k,ef N k k= 4 N 0 N are assumed to be finite-dimensional

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