Single-letter one-way distillable entanglement for non-degradable states

1 Single-letter one-way distillable entanglement for non-de gradable states Rabsan Galib Ahmed 1 , 2 , Graeme Smith 1 , 2 and Peixue W u 1 , 2 1 Institute for Quantum Computing, University of W aterloo, 200 University A venue W est, W aterloo, ON N2L 3G1, Canada 2 Department of Applied Mathematics, University of W aterloo, 200 University A venue W est, W aterloo, ON N2L 3G1, Canada Abstract The one-way distillable entanglement is a central operational measure of bipartite entanglement, quantifying the optimal rate at which maximally entangled pairs can be extracted by one-way LOCC. Despite its importance, it is notoriously hard to compute, since it is defined by a regularized optimization over many copies and adaptive one-way protocols. At present, single-letter formulas are only known for (conjugate) degradable and PPT states. More generally , it has remained unclear when one-way distillable entanglement can still be additive beyond degradability and PPT settings, and ho w such additivity relates to additivity questions of quantum capacity of channels. In this paper, we address this gap by identifying three explicit families of non-degradable and non-PPT states whose one-way distillable entanglement is nevertheless single-letter . First, we introduce two weakened degradability-type conditions—regularized less- noisy and informationally degradable—and prov e that each guarantees additivity and hence a single-letter formula. Second, we show a stability result for orthogonally flagged mixtures: when one component has orthogonal support on Alice’ s system and zero one-way distillable entanglement, the mixture remains single-letter , e ven though degradability is typically lost under such mixing. Finally , we propose a generalized spin-alignment principle for entropy minimization in tensor-product settings, which we establish in sev eral key cases, including a complete Rényi-2 result. As an application, we obtain additivity results for generalized direct-sum channels and their corresponding Choi states. C O N T E N T S I Introduction 1 II Preliminary 4 II-A Bipartite quantum states and entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 II-B Distillable entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 II-C Degradable states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 II-D Connection to the quantum channel capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 III General principle of non-degradable states with single-letter one-way distillable entanglement 7 III-A States with weaker notion of degradability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 III-B Non-degradable states with useless component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 III-C Spin alignment phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 III-C1 n “ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 III-C2 Rényi-2 entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 IV Explicit examples of non-degradable states with single-letter one-way distillable entanglement 12 IV -A Mixture of degradable and anti-degradable states with orthogonal support . . . . . . . . . . . . . . . . . . . . . 12 IV -B Flagged mixture of de gradable and antidegradable states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 IV -C Generalized direct sum channels and their corresponding states . . . . . . . . . . . . . . . . . . . . . . . . . . 13 V Conclusion and Outlook 17 References 18 Appendix 18 A Explicit upper bound on each alignment term for Rényi-2 entropy . . . . . . . . . . . . . . . . . . . . . . . . . 18 B Generalized direct sum completely positive maps and states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 I . I N T RO D U C T I O N Distilling maximally entangled states from many copies of a noisy bipartite state via local operations and classical communication (LOCC) [ 2 ], [ 5 ], [ 4 ] is a foundational problem in quantum information science, enabling key protocols such as dense coding and teleportation [ 6 ], [ 3 ]. When the allowed classical communication is restricted to be one-way (say A Ñ B ), the optimal asymptotic rate at which ebits can be extracted is quantified by the one-way distillable entanglement D Ñ p ρ AB q . Beyond its operational meaning 2 as an entanglement measure, D Ñ is tightly connected to communication problems: through channel–state duality , additivity and single-letter phenomena for one-way distillation mirror (but are not identical to) additivity questions for coherent information and quantum capacity . A major obstruction is that D Ñ is generally non-additive [ 12 ], [ 16 ], and therefore is defined through a regularization [ 10 ]: D Ñ p ρ AB q “ lim n Ñ8 1 n D p 1 q Ñ p ρ b n AB q . Determining when such regularizations collapse to single-letter formulas is a central theme in quantum information theory . For (conjugate) degradable and anti-degradable states, one-way distillable entanglement admits a single-letter expression giv en by coherent information [ 14 ]. Like wise, PPT states have vanishing two-w ay (and hence one-way) distillable entanglement [ 11 ]. In contrast, explicit families of states that are non-degr adable , non-PPT , and yet satisfy D Ñ p ρ AB q “ D p 1 q Ñ p ρ AB q hav e remained une xplored. Motiv ated by recent progress on additi vity for non-degradable channels in quantum capacity theory [ 15 ], [ 17 ], we fill this gap by introducing three mechanisms that produce such states: (i) states satisfying weaker notions of de gradability , (ii) orthogonally mixtur es with a “useless” component , and (iii) states governed by a spin-alignment entr opy minimization principle . A channel-capacity viewpoint and a subtle mismatch. Giv en a bipartite state ρ AB , a natural perspecti ve suggested by the Choi correspondence is to compare additivity phenomena for one-way distillation with those for one-shot channel capacity . Concretely , let Φ A Ñ B ρ be the completely positiv e map induced by ρ AB via the Choi–Jamiołkowski isomorphism (equiv alently , ρ AB is proportional to the Choi operator of Φ ρ ). One may ask whether additivity of the maximal coherent information Q p 1 q p Φ ρ q should be equiv alent to additivity of D p 1 q Ñ p ρ AB q . While these two questions coincide in the (conjugate) degradable setting, in general they are a priori differ ent : the optimizations defining the two quantities impose different operational constraints. Recall [ 10 ] that D p 1 q Ñ p ρ AB q “ max ! I p A 1 y B M q ω A 1 BM : ω A 1 B M “ ÿ m p K m b I B q ρ AB p K : m b I B q b | m yx m | , ÿ m K : m K m “ I A ) . A natural relaxation is to dr op the completeness constraint ř m K : m K m “ I A , i.e., allo w trace non-preserving operations and renormalize. For any K : A Ñ A 1 , define the postselected state ω A 1 B p K q “ p K b I B q ρ AB p K b I B q : T r “ p K : K b I B q ρ AB ‰ . Then one obtains the simplified expression p D p 1 q Ñ p ρ AB q “ max K I p A 1 y B q ω A 1 B p K q , (1) which upper bounds D p 1 q Ñ p ρ AB q by construction. This relaxation admits an exact channel interpretation. Let ρ N AB denote the normalized Choi state of a channel N A Ñ B . Using the Choi–Jamiołko wski isomorphism, p D p 1 q Ñ becomes precisely the one-shot quantum capacity: Q p 1 q p N q “ p D p 1 q Ñ ` ρ N AB ˘ , where Q p 1 q p N q is the maximal coherent information of N . Consequently , additivity of p D p 1 q Ñ is equiv alent (via the Choi correspondence) to additi vity of Q p 1 q for the associated channels. In contrast, additivity for D p 1 q Ñ is intrinsically more delicate, since the optimization is restricted to complete one-way instruments ř m K : m K m “ I A rather than arbitrary postselected filters. Our three mechanisms provide structured settings in which this com- pleteness constraint can still be controlled, yielding explicit non-degradable, non-PPT families with single-letter one-way distillable entanglement. W e present three structural mechanisms that yield explicit families of non-de gradable , non-PPT states with single-letter one-way distillable entanglement. For each mechanism we first state the guiding principle, and then giv e a representativ e example. a) Mechanism 1: additivity fr om weaker de gradability (information dominance): Let ρ AB hav e a purification | ϕ y AB E . Standard degradability requires that E can be simulated from B by a channel; we instead impose only an information or dering saying that, under all relevant pre-processings on Alice, Bob is at least as informati ve as the purifying system. T wo con venient formulations are: ‚ (Regularized less noisy .) For ev ery n ě 1 and ev ery quantum instrument T A n Ñ A 1 M , letting ω A 1 B n E n “ T p ϕ b n AB E q , I p M ; B n q ω ě I p M ; E n q ω . (2) ‚ (Inf ormationally degradable.) For e very channel N A Ñ A 1 and ω A 1 B E “ N p ϕ AB E q , I p A 1 ; B q ω ě I p A 1 ; E q ω . (3) 3 These conditions are strictly weaker than degradability but are still strong enough to force single-letter behavior , by the same telescoping/chain-rule strategy that underlies many additi vity proofs in capacity theory (cf. [ 17 ]). Consequence (informal). If ( 2 ) holds for all n , then D Ñ p ρ AB q “ D p 1 q Ñ p ρ AB q “ I p A y B q ρ , and D p 1 q Ñ is additive for many copies of such states, i.e., it exhibits weak additivity under tensor products. Furthermore informationally degradable states satisfy additivity for D p 1 q Ñ under tensor products i.e., it exhibits strong additi vity under tensor products. Example (flagged mixtur es of amplitude damping channels). Let AD γ be the qubit amplitude damping channel with Kraus operators E 0 “ | 0 y x 0 | ` a 1 ´ γ | 1 y x 1 | , E 1 “ ? γ | 0 y x 1 | , γ P r 0 , 1 s . Fix parameters γ 0 , γ 1 P r 0 , 1 s and a mixing probability p P r 0 , 1 s , and define the flagged channel N A Ñ B F p ρ q : “ p AD γ 0 p ρ q b | 0 y x 0 | F ` p 1 ´ p q AD γ 1 p ρ q b | 1 y x 1 | F , where the classical flag F is av ailable to Bob . Its (normalized) Choi state ρ N RB F : “ p id R b N qp| Φ ` y x Φ ` |q , with | Φ ` y “ p| 00 y ` | 11 yq{ ? 2 , is ρ N RB F “ p ρ p γ 0 q RB b | 0 y x 0 | F ` p 1 ´ p q ρ p γ 1 q RB b | 1 y x 1 | F , (4) where, in the computational basis t| 00 y , | 01 y , | 10 y , | 11 yu of RB , ρ p γ q RB “ 1 2 ¨ ˚ ˚ ˝ 1 0 0 ? 1 ´ γ 0 0 0 0 0 0 γ 0 ? 1 ´ γ 0 0 1 ´ γ ˛ ‹ ‹ ‚ . (5) This flagged family for some parameter regions provides explicit non-de gradable Choi states for which the weaker information- dominance property holds and hence D Ñ p ρ N RB F q is single-letter [ 17 ]. b) Mechanism 2: orthogonal flags with a “useless” component: Suppose Alice locally knows which component of a mixture she holds, i.e. the components have orthogonal support on A . Then one-way distillation can be performed conditionally on that classical information. If, moreov er , one component is “useless” for one-way distillation (e.g. anti-degradable / separable, hence zero one-way distillable entanglement), the mixture inherits a single-letter formula from the “useful” component. Operationally , orthogonality on A turns the problem into a direct-sum decomposition. Prototype statement (inf ormal). Let ρ 0 , ρ 1 be bipartite states with ρ A 0 K ρ A 1 and assume the product additi vity bound D p 1 q Ñ p  n i “ 1 ρ w i q ď ř n i “ 1 D p 1 q Ñ p ρ w i q for all words w n . Then for an y p P r 0 , 1 s , D Ñ ` pρ 0 ` p 1 ´ p q ρ 1 ˘ “ D p 1 q Ñ ` pρ 0 ` p 1 ´ p q ρ 1 ˘ “ p D Ñ p ρ 0 q ` p 1 ´ p q D Ñ p ρ 1 q . Example (a 3 ˆ 3 state with a separable “junk” bloc k). Consider , for s P r 0 , 1 s , the state on C 3 b C 3 ρ AB p s q : “ 1 3 ´ | 0 y x 0 | A b “ s | 0 y x 0 | B ` p 1 ´ s q | 2 y x 2 | B ‰ ` | 0 y x 1 | A b ? s | 0 y x 2 | B ` | 1 y x 0 | A b ? s | 2 y x 0 | B ` | 1 y x 1 | A b | 2 y x 2 | B ¯ ` 1 3 | 22 y x 22 | AB . Here the last term 1 3 | 22 y x 22 | is separable and supported on an A -subspace orthogonal to the support of the first block. The remaining “useful” component is a Choi-type amplitude-damping structure on span t| 0 y , | 1 yu A . By the above principle, ρ AB p s q has single-letter one-way distillable entanglement, and one obtains D Ñ p ρ AB p s qq “ D p 1 q Ñ p ρ AB p s qq “ 2 3 max " h ´ s 2 ¯ ´ h ˆ 1 ` s 2 ˙ , 0 * . c) Mechanism 3: spin alignment and gener alized dir ect-sum structur e: Certain block-structured channels/states reduce multi-copy optimizations (entropy minimization or coherent information maximization) to a classical mixture over “which block is used” [ 20 ]. The key step is a spin-alignment entropy minimization rule: when outputs are mixtures of the form ρ b σ 1 and σ 0 b ρ , entropy is minimized by aligning the free input ρ with maximal-eigenv alue directions of the fixed states σ 0 , σ 1 . This forces optimal n -copy inputs to be product-aligned across sites, turning a priori noncommutativ e problems into tractable classical ones. Spin alignment (inf ormal). Fix σ 0 P D p B 0 q and σ 1 P D p B 1 q and define N 0 p ρ q “ ρ b σ 1 , N 1 p ρ q “ σ 0 b ρ . Given a distribution t p  x u on t 0 , 1 u n , consider min t ρ  x u S ˜ ÿ  x p  x N  x p ρ  x q ¸ . The conjectured (and partially proved) alignment rule says the optimum is attained by product inputs ρ  x “  i τ p i q x i with each τ p i q 0 (resp. τ p i q 1 ) a projector onto a maximal-eigenspace of σ 0 (resp. σ 1 ). W e establish the n “ 1 von Neumann case, and the full n -copy statement for Rényi- 2 entropy . 4 Example (a gener alized dir ect-sum state with single-letter D Ñ ). Let A, B » C d 0 ` d 1 and define ρ AB “ 1 2 d 0 d 1 d 0 ´ 1 ÿ i “ 0 d 1 ´ 1 ÿ j “ 0 ´ | i y| j y ` | j ` d 0 y| i ` d 1 y ¯´ x i |x j | ` x j ` d 0 |x i ` d 1 | ¯ . This state is a canonical “two-block coherent coupling” (a Choi-type generalized direct-sum structure). Using the alignment principle to control the rele v ant entropy minimizations, we show that D Ñ p ρ AB q “ D p 1 q Ñ p ρ AB q , providing an explicit non-degradable, non-PPT family where one-way distillation is single-letter . T ogether, these three mechanisms—information dominance without full degradability , orthogonal flags with a useless component, and spin-alignment-driven block reduction—yield broad, explicit sources of additivity and single-letter behavior for D Ñ beyond the degradable/PPT regimes. Open questions and future work. An important open problem is to prove the spin alignment conjecture with von-Neumann entropy for arbitrary tensor powers. Although partial progress has been made [ 1 ], the general case remains unsolved. Futhermore, understanding the comparative additivity of D p 1 q Ñ and Q p 1 q remains to be explored. Finally , it remains to be understood whether the techniques de veloped here extend to two-way distillation or to other quantum resource theories. The rest of the paper is structured as follows. In Section II , we provide the necessary preliminary background for our results. In Section III , we hav e identified the general principles that allo w us to show single-letter one-way distillable entanglement for non-degradable states. Finally in Section IV , we provide explicit examples for such states. Additionally we show that a large class of generalised direct sum channels exhibit single-letter quantum capacity complying with one of our general principles. I I . P R E L I M I NA RY A. Bipartite quantum states and entanglement Consider two quantum systems with corresponding Hilbert spaces, H A and H B abbreviated as A (for Alice) and B (for Bob). The states of this composite system, called the bipartite states, are described by elements of D p H A b H B q , where D p H q is the set of all positi ve semi-definite operators on the Hilbert space H with unit trace. The set of bipartite states, D p H A b H B q is further classified into two categories: separable states and entangled states . W e call ϱ AB P D p H A b H B q a separable state if there exists a probability vector t p i u and states ρ i P D p H A q , σ i P D p H B q such that ϱ AB “ ÿ i p i ρ i b σ i (6) The set of all separable states are often denoted as Sep p A : B q . The states which cannot be written in this form are called entangled states . These are elements of D p H A b H B q z Sep p A : B q . B. Distillable entanglement Let ρ AB be a bipartite state on a finite-dimensional Hilbert space H A b H B . W e denote by LOCC A Ñ B the set of protocols implementable by local operations and classical communication from A to B only . For an integer M , let Φ M : “ 1 ? M ř M i “ 1 | i y | i y be a maximally entangled state of Schmidt rank M on C M b C M . From here onward all the logarithms are taken to be base 2 . a) Achievable rate.: A number R ě 0 (in ebits per copy) is said to be one-way achie vable for ρ AB if there exist a sequence of integers M n and a sequence of one-way LOCC protocols Λ n P LOCC A Ñ B such that σ n : “ Λ n ` ρ b n AB ˘ P D ` p C M n b C M n q ˘ satisfies lim n Ñ8 › › σ n ´ Φ M n › › 1 “ 0 and lim inf n Ñ8 1 n log M n ě R. b) One-way distillable entanglement.: The one-way distillable entanglement of ρ AB is D Ñ p ρ AB q : “ sup ␣ R ˇ ˇ R is one-way achie v able for ρ AB ( . (7) c) Equivalent fidelity formulation.: Equiv alently , for ε P p 0 , 1 q and n P N define the ε -error, n -blocklength one-way distillable entanglement by D p n,ε q Ñ p ρ AB q : “ 1 n max Λ n P LOCC A Ñ B t log M : F ` Λ n p ρ b n AB q , Φ M ˘ ě 1 ´ ε ) , where F p¨ , ¨q is the Uhlmann fidelity . Then D Ñ p ρ AB q “ lim ε Ñ 0 lim inf n Ñ8 D p n,ε q Ñ p ρ AB q . (8) 5 d) Regularization expr ession: Devetak and W inter [ 10 ] prov ed the following regularized expression for the one-way distillable entanglement D Ñ p ρ AB q “ lim n Ñ8 1 n D p 1 q Ñ ` ρ b n AB ˘ , (9) where D p 1 q Ñ p ρ AB q : “ max T ÿ m λ m I p A 1 y B q ρ m “ max T I p A 1 y B M q T p ρ AB q , ρ m : “ 1 λ m T m p ρ AB q , λ m : “ T r r T m p ρ AB qs , (10) where the instrument T “ ř m T m b | m y x m | has a single Kraus operator T m p¨q “ K m p¨q K : m , K m : A Ñ A 1 . W e can obtain an alternativ e expression in terms of an isometric extension V : A Ñ A 1 M N of the instrument, T , defined by V : “ ÿ m K m b | m y M b | m y N , (11) for a classical register N – M . Since ř m K : m K m “ I A , we hav e V : V “ I A , and T p ρ AB q “ T r N ` V ρ AB V : ˘ . For any isometry V A Ñ A 1 M N with the form ( 11 ), and let ρ AB be a bipartite state with purification | ϕ y AB E , define | ω y A 1 M N B E “ p V b I B b I E q | ϕ y AB E , (12) we hav e I p A 1 y B M q T p ρ AB q “ I p A 1 y B M q ω “ S p B M q ω ´ S p A 1 B M q ω “ S p B M q ω ´ S p E N q ω “ S p B M q ω ´ S p E M q ω , where the third equality follows from the fact that ω is pure on A 1 M N B E , so S p A 1 B M q ω “ S p E N q ω (complementary systems with a joint pure state hav e equal von Neumann entropy); the last equality follo ws from M Ø N symmetry in the construction of ω (in particular , the joint law of p E , M q and p E , N q is the same), hence S p E N q ω “ S p E M q ω . Therefore, we ha ve Lemma II.1. F or any bipartite state ρ AB with purification | ϕ y AB E , we have D p 1 q Ñ p ρ AB q “ max V I p A 1 y B M q ω “ max V r S p B M q ω ´ S p E M q ω s , | ω y A 1 M N B E “ p V b I B b I E q | ϕ y AB E , (13) wher e V A Ñ A 1 M N is any isometry with the form ( 11 ) . From the abov e lemma, it is immediate to see that if U A and V B are isometries, then D p 1 q Ñ ` p U A b V B q ρ p U : A b V : B q ˘ “ D p 1 q Ñ p ρ q , D Ñ ` p U A b V B q ρ p U : A b V : B q ˘ “ D Ñ p ρ q (14) for e very ρ P D p AB q . C. Degr adable states Definition II.2. Let ρ AB be a bipartite state with purification | ϕ y AB E . The state ρ AB is called: 1) degradable , if there is a quantum channel D B Ñ E such that p id b D B Ñ E qp ρ AB q “ ϕ AE . (15) 2) antidegradable , if there is a quantum channel D E Ñ B such that p id b D E Ñ B qp ϕ AE q “ ρ AB . (16) V ia data processing inequality , for the state giv en by | ω y A 1 M N B E “ p V b I B b I E q | ϕ y AB E , we have S p B M q ω ´ S p E M q ω ď S p B q ω ´ S p E q ω “ I p A y B q ρ . Then using Lemma II.1 , we find that D p 1 q Ñ p ρ q “ I p A y B q ρ for degradable state ρ . Similarly , D p 1 q Ñ p ρ q “ 0 for anti-degradable state ρ . This provides an alternativ e proof of the results obtained in [ 14 ]. Using the additi vity of coherent information under tensor product states, we immediately have D Ñ p ρ AB q “ # I p A y B q ρ , when ρ is de gradable ; 0 , when ρ is anti-de gradable . (17) 6 D. Connection to the quantum channel capacities T o study the additivity property of D p 1 q Ñ , we introduce an upper bound of it. Recall ( 10 ): D p 1 q Ñ p ρ AB q “ max # I p A 1 y B M q ω A 1 BM : ω A 1 B M “ ÿ m p K m b I B q ρ AB p K m b I B q : b | m yx m | , @t K A Ñ A 1 m u s.t. ÿ m K : m K m “ I A + . Therefore, a natural upper bound for D p 1 q Ñ is to withdraw the condition ř m K : m K m “ I A : Definition II.3. F or a bipartite state ρ AB , we intr oduce the quantity p D p 1 q Ñ p ρ AB q “ max $ & % I p A 1 y B M q ω A 1 BM : ω A 1 B M “ ř m p K m b I B q ρ AB p K m b I B q : b | m yx m | ř m T r ´ p K : m K m b I B q ρ AB ¯ , @t K m u , . - . Interestingly , the abov e quantity is actually the maximal coherent information of the channel with Choi operator gi ven by ρ AB : Proposition II.4. Let ρ AB be bipartite state and N be the corr esponding completely positive map fr om A to B . Then, Q p 1 q p N q “ p D p 1 q Ñ p ρ AB q . A first step is to simplify the above expression. Lemma II.5. p D p 1 q Ñ p ρ AB q “ max " I p A 1 y B q ω A 1 B : ω A 1 B “ p K b I B q ρ AB p K b I B q : T r pp K : K b I B q ρ AB q , @ K : A Ñ A 1 * . (18) Pr oof. It is obvious that p D p 1 q Ñ p ρ AB q ě max " I p A 1 y B q ω A 1 B : ω A 1 B “ p K b I B q ρ AB p K b I B q : T r pp K : K b I B q ρ AB q , @ K : A Ñ A 1 * . T o show the other direction, suppose t K m u are the optimal operators achieving p D p 1 q Ñ p ρ AB q . Denote p m “ T r ` p K : m K m b I B q ρ AB ˘ ř m T r ´ p K : m K m b I B q ρ AB ¯ , ρ m A 1 B “ p K m b I B q ρ AB p K m b I B q : T r ´ p K : m K m b I B q ρ AB ¯ Then the coherent information of ω A 1 B M “ ř m p m ρ m A 1 B b | m yx m | is calculated as I p A 1 y B M q ω A 1 BM “ ÿ m p m I p A 1 y B q ρ m A 1 B ď I p A 1 y B q ρ m 0 A 1 B , where m 0 is chosen such that I p A 1 y B q ρ m 0 A 1 B “ max m I p A 1 y B q ρ m A 1 B . Therefore, by choosing K as K m 0 , one has p D p 1 q Ñ p ρ AB q ď max " I p A 1 y B q ω A 1 B : ω A 1 B “ 1 T r pp K : K b I B q ρ AB q p K b I B q ρ AB p K b I B q : @ K : A Ñ A 1 * , which concludes the proof. For any bipartite state ρ AB , it naturally induces a completely positive map N A Ñ B via the follo wing N A Ñ B p σ q : “ d A T r A pp σ T b I B q ρ AB q , ρ AB “ d A p id b N A Ñ B qp| Φ yx Φ |q . (19) For any completely positi ve map N , one can define its maximal coherent information by Q p 1 q p N q : “ max " I p A 1 y B q ω A 1 B : ω A 1 B “ p id b N qp| ψ yx ψ | A 1 A q T r pp id b N qp| ψ yx ψ | A 1 A qq , @| ψ y A 1 A * . (20) Pr oof of Pr oposition II.4 . Recall that there is a one-to-one correspondence (up to normalization) between pure states | ψ y A 1 A on A 1 A and operators K : A Ñ A 1 such that | ψ y A 1 A “ p K b I A q| Φ y AA . Note that p K b I B q ρ AB p K b I B q : “ p K b I B qp id b N qp| Φ yx Φ |qp K b I B q : “ p id b N qp| ψ yx ψ | A 1 A qq , thus using the definitions ( 20 ) and ( 18 ), we conclude the proof. Lemma II.6. Let ρ AB be a bipartite state . Then we have, p D p 1 q Ñ p ρ AB q “ 0 ð ñ D p 1 q Ñ p ρ AB q “ 0 7 Pr oof. As p D p 1 q Ñ p ρ AB q ě D p 1 q Ñ p ρ AB q ě 0 , it trivially follows that p D p 1 q Ñ p ρ AB q “ 0 ù ñ D p 1 q Ñ p ρ AB q “ 0 . T o show the other way around, assume that p D p 1 q Ñ p ρ AB q ą 0 . Therefore, there exists K : A Ñ A 1 such that I p A 1 y B q ω A 1 B ą 0; ω A 1 B “ 1 T r pp K : K b I B q ρ AB q p K b I B q ρ AB p K b I B q : Now choose 0 ă c ď 1 such that E 0 : “ c p K : K q ď I A . W e further perform the following spectral decomposition I A ´ E 0 “ dim A ÿ k “ 1 E k Here E k for k ě 1 are rank-1 positiv e operators with 0 ď T r p E k q ď 1 . Firstly , we note that t E k u dim A k “ 0 forms a PO VM, therefore defines an instrument, T : A Ñ A 1 M where M – C dimA ` 1 . Secondly , for k ě 1 , define Kraus operators, K k : “ U ? E k “ | β y x α | for some vectors | α y P A, | β y P A 1 , then the state ω p k q A 1 B : “ | β y x β | A 1 b 1 x α | ρ A | α y T r A rp | α y x α | b I B q ρ AB s is a product state on A 1 B , i.e., ha ving I p A 1 y B q ω p k q A 1 B “ 0 . Therefore, we hav e I p A 1 y B M q T p ρ q “ c T r ` p K : K b I B q ρ AB ˘ I p A 1 y B q ω A 1 B ą 0 ù ñ D p 1 q Ñ p ρ AB q ą 0 (21) By contrapositi vity this is equi valent to D p 1 q Ñ p ρ AB q “ 0 ù ñ p D p 1 q Ñ p ρ AB q “ 0 I I I . G E N E R A L P R I N C I P L E O F N O N - D E G R A D A B L E S TA T E S W I T H S I N G L E - L E T T E R O N E - W AY D I S T I L L A B L E E N TA NG L E M E N T It is well-known that the one-way distillable entanglement of (anti-)degradable states has a single-letter expression (see [ 14 ]) and is equal to the coherent information. Moreov er , two-way (thus one-way) distillable entanglement of PPT states is kno wn to be zero. In this section, we introduce three distinct classes of non-degradable states that exhibit additivity in the one-way distillable entanglement. The first class relies on a weaker notion of degradable states. The second class concerns with states having an useless component. Thirdly , we propose a special class of generalised direct sum states which has a single-letter one-way distillable entanglement. A. States with weaker notion of de gradability Motiv ated from previous work [ 17 ], we propose the follo wing two notions of weaker degradability Definition III.1. W e say that the bipartite state ρ AB with purification | ϕ y AB E is ‚ less noisy at level n , if for any quantum instrument T A n Ñ A 1 M , for the state ω A 1 B n E n “ T A n Ñ A 1 M p ϕ b n AB E q , we have I p M ; B n q ě I p M ; E n q . (22) W e say ρ AB is r e gularized less noisy , if ( 22 ) holds for any n ě 1 . ‚ informationally degr adable, if for any quantum channel N A Ñ A 1 , for the state ω A 1 B E “ N A Ñ A 1 p ϕ AB E q , we have I p A 1 ; B q ě I p A 1 ; E q . (23) Proposition III.2. Let ρ AB be a bipartite state which is less noisy at level n ě 1 . Then we have D p 1 q Ñ p ρ b n AB q “ nD p 1 q Ñ p ρ AB q “ n I p A y B q . (24) Thus if ρ AB is r e gularized less noisy , we have D Ñ p ρ AB q “ D p 1 q Ñ p ρ AB q . Pr oof. Fix n ě 1 and suppose the optimal quantum instrument for D p 1 q Ñ p ρ b n AB q is given by T A n Ñ A 1 M . Denote V : A n Ñ A 1 M N is the isometry of T , defined as in ( 11 ). V ia Lemma II.1 and the optimality of the quantum instrument, D p 1 q Ñ p ρ b n AB q “ S p B n M q ω ´ S p E n M q ω , | ω y A 1 M N B n E n : “ V A n Ñ A 1 M N | ϕ y b n AB E . Note that I p M ; B n q ω ě I p M ; E n q ω is equi valent to S p B n M q ω ´ S p E n M q ω ď S p B n q ω ´ S p E n q ω which implies D p 1 q Ñ p ρ b n AB q ď S p B n q ω ´ S p E n q ω “ I p A n y B n q ρ b n “ n I p A y B q ρ . where the last two equalities follow from the additivity of v on Neumann entropy over the tensor products: S p B n q ρ b n “ n S p B q ρ and S p A n B n q ρ b n “ n S p AB q ρ , hence I p A n y B n q ρ b n “ S p B n q ´ S p A n B n q “ n I p A y B q ρ . W e conclude the proof by recalling that n I p A y B q ρ is a lo wer bound for D p 1 q Ñ p ρ b n AB q . Proposition III.3. Let ρ AB and σ r A r B be informationally de gradable. Then for all n, k ě 0 , D p 1 q Ñ ` ρ b n AB b σ b k r A r B ˘ “ n D p 1 q Ñ p ρ AB q ` k D p 1 q Ñ p σ r A r B q “ n I p A y B q ρ ` k I p r A y r B q σ . (25) 8 In particular , informationally degr adable states ar e single-letter for one-way distillable entanglement and ar e additive under tensor pr oducts. Pr oof. Let | ϕ y AB E , | r ϕ y r A r B r E be the purification states of ρ, σ respectiv ely . Fix n, k ě 0 ( 0 means empty here), suppose T A n r A k Ñ A 1 M is the optimal quantum instrument for D p 1 q Ñ ` ρ b n AB b σ b k r A r B ˘ . Denote V : A n r A k Ñ A 1 M N is the isometry of T , defined as in ( 11 ). V ia Lemma II.1 and the optimality of the quantum instrument, we hav e D p 1 q Ñ ` ρ b n AB b σ b k r A r B ˘ “ S p B n r B k M q ω ´ S p E n r E k M q ω , where | ω y A 1 M N B n r B k E n r E k : “ V A n r A k Ñ A 1 M N b I B n r B k b I E n r E k ´ | ϕ y b n AB E b | r ϕ y b k r A r B r E ¯ . (26) Now denote B n “ B 1 B 2 ¨ ¨ ¨ B n , r B k “ r B 1 r B 2 ¨ ¨ ¨ r B k , and similar for E n , r E k . Then via telescoping argument, S p B n r B k M q ω ´ S p E n r E k M q ω “ S p B n r B k M q ω ´ S p B n r E k M q ω ` S p B n r E k M q ω ´ S p E n r E k M q ω “ k ÿ j “ 1 ´ S p r B j r V j B n M q ω ´ S p r E j r V j B n M q ω ¯ ` n ÿ i “ 1 ´ S p B i V i r E k M q ω ´ S p E i V i r E k M q ω ¯ , where V i , r V j are n ´ 1 , k ´ 1 subsystems defined by V i “ $ ’ ’ & ’ ’ % B 2 ¨ ¨ ¨ B n , i “ 1 , E 1 ¨ ¨ ¨ E i ´ 1 B i ` 1 ¨ ¨ ¨ B n , 2 ď i ď n ´ 1 , E 1 ¨ ¨ ¨ E n ´ 1 , i “ n. , r V j “ $ ’ ’ & ’ ’ % r B 2 ¨ ¨ ¨ r B k , j “ 1 , r E 1 ¨ ¨ ¨ r E j ´ 1 r B j ` 1 ¨ ¨ ¨ r B k , 2 ď j ď k ´ 1 , r E 1 ¨ ¨ ¨ r E k ´ 1 , j “ k . Now we claim that S p r B j r V j B n M q ω ´ S p r E j r V j B n M q ω ď S p r B j q ω ´ S p r E j q ω , S p B i V i r E k M q ω ´ S p E i V i r E k M q ω ď S p B i q ω ´ S p E i q ω , @ i, j. (27) In fact, for each 1 ď j ď k , one can construct a quantum channel Ă N j r A j Ñ r V j B n M by Ă N j p ρ r A j q : “ T r p r V j B n M q c „ V A n r A k Ñ A 1 M N p | ϕ y x ϕ | b n AB E b ρ r A j b | r ϕ y x r ϕ | bp k ´ 1 q r A r B r E qp V A n r A k Ñ A 1 M N q : ȷ , where p r A j r V j B n M q c denotes the complement of r A j r V j B n M . Therefore, we ha ve Ă N j r A j Ñ r V j B n M p | r ϕ y x r ϕ | r A j r B j r E j q “ ω r B j r E j r V j B n M , where ω r B j r E j r V j B n M is the reduced density of ω defined in ( 26 ). Since | r ϕ y r A j r B j r E j is the purification of the informational degradable state σ r A r B , via ( 23 ), we have I p r V j B n M ; r B j q ě I p r V j B n M ; r E j q , concluding the claim that S p r B j r V j B n M q ω ´ S p r E j r V j B n M q ω ď S p r B j q ω ´ S p r E j q ω . Similarly , we ha ve S p B i V i r E k M q ω ´ S p E i V i r E k M q ω ď S p B i q ω ´ S p E i q ω . Therefore, one has S p B n r B k M q ω ´ S p E n r E k M q ω “ k ÿ j “ 1 ´ S p r B j r V j B n M q ω ´ S p r E j r V j B n M q ω ¯ ` n ÿ i “ 1 ´ S p B i V i r E k M q ω ´ S p E i V i r E k M q ω ¯ ď k p S p r B q σ ´ S p r E q σ q ` n p S p B q ρ ´ S p E q ρ q “ k I p r A y r B q σ ` n I p A y B q ρ . W e conclude the proof by recalling that D p 1 q Ñ ` ρ b n AB b σ b k r A r B ˘ ě k I p r A y r B q σ ` n I p A y B q ρ . 9 B. Non-degr adable states with useless component In this subsection we exhibit a family of generally non-degradable states whose one-way distillable entanglement is single-letter . Heuristically , these states are orthogonally flagged mixtures on Alice, so the additivity inherited from the components persists ev en though degradability may fail. The formal statement is: Theorem III.4. Let ρ 0 and ρ 1 be bipartite states on AB satisfying D p 1 q Ñ p n â i “ 1 ρ w i q ď n ÿ i “ 1 D p 1 q Ñ p ρ w i q “ p n ´ | w n |q D p 1 q Ñ p ρ 0 q ` | w n | D p 1 q Ñ p ρ 1 q (28) for all w n “ p w 1 , ¨ ¨ ¨ w n q P t 0 , 1 u n and n P N . Mor eover , assume ρ A 0 K ρ A 1 . Then for all p P r 0 , 1 s , we have D Ñ ` pρ 0 ` p 1 ´ p q ρ 1 ˘ “ D p 1 q Ñ ` pρ 0 ` p 1 ´ p q ρ 1 ˘ “ pD Ñ p ρ 0 q ` p 1 ´ p q D Ñ p ρ 1 q . T o prove the above theorem, we need the following lemmas: Lemma III.5. Let σ, τ be bipartite states on AB such that σ A K τ A , i.e., supp p σ A q K supp p τ A q . Then for p P r 0 , 1 s , we have D p 1 q Ñ p ρ q ě p D p 1 q Ñ p σ q ` p 1 ´ p q D p 1 q Ñ p τ q , D Ñ p ρ q ě p D Ñ p σ q ` p 1 ´ p q D Ñ p τ q . (29) Pr oof. Define A 0 : “ supp p σ A q , A 1 : “ supp p τ A q , we hav e supp p σ q Ď A 0 B , supp p τ q Ď A 1 B . Define an isometry V : A Ñ X b A 1 : V ˇ ˇ A 0 “ | 0 y X b id A 0 , V ˇ ˇ A 1 “ | 1 y X b id A 1 , where X “ span t| 0 y , | 1 yu and A 1 is a copy of A 0 ‘ A 1 . Let ω : “ p V b I B q ρ p V : b I B q “ p | 0 y x 0 | X b σ ` p 1 ´ p q | 1 y x 1 | X b τ . V ia ( 14 ), we hav e D p 1 q Ñ p ω q “ D p 1 q Ñ p ρ q , D Ñ p ω q “ D Ñ p ρ q . It remains to show that D p 1 q Ñ p ω q “ D p 1 q Ñ p p | 0 y x 0 | X b σ ` p 1 ´ p q | 1 y x 1 | X b τ q ě p D p 1 q Ñ p σ q ` p 1 ´ p q D p 1 q Ñ p τ q , (30) D Ñ p ω q “ D Ñ p p | 0 y x 0 | X b σ ` p 1 ´ p q | 1 y x 1 | X b τ q ě p D Ñ p σ q ` p 1 ´ p q D Ñ p τ q . (31) T o show ( 30 ), suppose T 0 : A Ñ M 0 A 1 0 , T 1 : A Ñ M 1 A 1 1 are the optimal quantum instruments for D p 1 q Ñ p σ q and D p 1 q Ñ p τ q , respectiv ely . Then one finds the quantum instrument T “ | 0 y x 0 | ¨ | 0 y x 0 | b T 0 ` | 1 y x 1 | ¨ | 1 y x 1 | b T 1 provides a lower bound of D p 1 q Ñ p p | 0 y x 0 | X b σ ` p 1 ´ p q | 1 y x 1 | X b τ q , which is p D p 1 q Ñ p σ q ` p 1 ´ p q D p 1 q Ñ p τ q . T o show ( 31 ), for each n , ω b n is block-diagonal in the computational basis of X n : ω b n “ ÿ x n Pt 0 , 1 u n p N 0 p x n q p 1 ´ p q N 1 p x n q | x n y x x n | X n b σ b N 0 p x n q b τ b N 1 p x n q , where N 0 p x n q and N 1 p x n q “ n ´ N 0 p x n q denote the type counts (numbers of 0’ s and 1’ s). Let ∆ be the completely dephasing channel on X n in this basis; then ∆ p ω b n q “ ω b n . Therefore prepending ∆ to any protocol does not change its action on ω b n . Hence, without loss of generality , any protocol may be assumed to measure X n in this basis at the beginning and obtain the classical string x n . Fix ε ą 0 . By the definition of D Ñ p σ q , D Ñ p τ q , there exists K ě 1 such that for all N ě K there are one-way protocols Π σ N , Π τ N on σ b N and τ b N producing p N , M σ N , ε q and p N , M τ N , ε q with 1 N log M σ N ě D Ñ p σ q ´ ε, 1 N log M τ N ě D Ñ p τ q ´ ε. (32) On input ω b n , measure X n to obtain x n and the conditional state σ b N 0 p x n q b τ b N 1 p x n q . If N 0 p x n q , N 1 p x n q ě K , run Π σ N 0 p x n q on the σ -positions and Π τ N 1 p x n q on the τ -positions; otherwise output a trivial product (zero ebits). For large enough n , n D p n,ε q Ñ p ω q ě ÿ x n Pt 0 , 1 u n ,N 0 p x n q ,N 1 p x n qě K p N 0 p x n q p 1 ´ p q N 1 p x n q ´ log M σ N 0 p x n q ` log M τ N 1 p x n q ¯ ě ÿ x n Pt 0 , 1 u n ,N 0 p x n q ,N 1 p x n qě K p N 0 p x n q p 1 ´ p q N 1 p x n q p N 0 p x n qp D Ñ p σ q ´ ε q ` N 1 p x n qp D Ñ p τ q ´ ε qq , where for the first inequality , we use the one-way protocols consisting of Π σ N 0 p x n q , Π τ N 1 p x n q depending on the measurement outcome on X n , and the second inequality follows from ( 32 ). Finally , using ÿ x n Pt 0 , 1 u n ,N 0 p x n q ,N 1 p x n qě K p N 0 p x n q p 1 ´ p q N 1 p x n q N 0 p x n q “ pn p 1 ´ o n p 1 qq , ÿ x n Pt 0 , 1 u n ,N 0 p x n q ,N 1 p x n qě K p N 0 p x n q p 1 ´ p q N 1 p x n q N 1 p x n q “ p 1 ´ p q n p 1 ´ o n p 1 qq , we hav e D p n,ε q Ñ p ω q ě p p 1 ´ o n p 1 qqp D Ñ p σ q ´ ε q ` p 1 ´ p qp 1 ´ o n p 1 qqp D Ñ p τ q ´ ε q 10 Let n Ñ 8 and then ε Ñ 0 , by definition ( 8 ) we hav e D Ñ p ρ q “ D Ñ p ω q ě p D Ñ p σ q ` p 1 ´ p q D Ñ p τ q . For the other direction, we use the follo wing lemma, which is Proposition 2.7 in [ 14 ]: Lemma III.6. Let ρ 0 and ρ 1 be bipartite states on AB satisfying D p 1 q Ñ p n â i “ 1 ρ w i q ď n ÿ i “ 1 D p 1 q Ñ p ρ w i q “ p n ´ | w n |q D p 1 q Ñ p ρ 0 q ` | w n | D p 1 q Ñ p ρ 1 q (33) for all w n “ p w 1 , ¨ ¨ ¨ w n q P t 0 , 1 u n and n P N . Then for all p P r 0 , 1 s , D Ñ ` pρ 0 ` p 1 ´ p q ρ 1 ˘ ď p D Ñ p ρ 0 q ` p 1 ´ p q D Ñ p ρ 1 q . W e are able to prove Theorem III.4 : Pr oof of Theorem III.4 . The first conclusion follows directly from Lemma III.5 and Lemma III.6 . For the second conclusion (addi- tivity), we hav e p D p 1 q Ñ p ρ 0 q ` p 1 ´ p q D p 1 q Ñ p ρ 1 q ď Lemma III.5 D p 1 q Ñ ` pρ 0 ` p 1 ´ p q ρ 1 ˘ ď D Ñ ` pρ 0 ` p 1 ´ p q ρ 1 ˘ ď Lemma III.6 p D Ñ p ρ 0 q ` p 1 ´ p q D Ñ p ρ 1 q “ p D p 1 q Ñ p ρ 0 q ` p 1 ´ p q D p 1 q Ñ p ρ 1 q . C. Spin alignment phenomenon In this subsection we introduce the spin-alignment phenomenon, generalized from [ 15 ]. Let σ 0 P D p B 0 q and σ 1 P D p B 1 q be fixed states, and assume A i » B i for i “ 0 , 1 . Define the (partially erasing) channels N A 0 Ñ B 0 B 1 0 p ρ q : “ ρ b σ 1 , (34) N A 1 Ñ B 0 B 1 1 p ρ q : “ σ 0 b ρ. (35) Giv en n ě 1 and suppose  x “ p x 1 , . . . , x n q P t 0 , 1 u n is a n -bit string. Define the product channel N  x : “ n â i “ 1 N A p i q x i Ñ B p i q 0 B p i q 1 x i . (36) For each position i , the input system is A p i q x i and the output system is B p i q 0 B p i q 1 . For n -bit string  x , the input state is ρ  x P D ´  n i “ 1 A p i q x i ¯ . The spin alignment phenomenon is as follows: Conjecture III.7 (Spin alignment phenemenon) . F ix a pr obability distribution t p  x u  x Pt 0 , 1 u n , the minimization of the von Neumann entr opy min ρ  x S ¨ ˝ ÿ  x Pt 0 , 1 u n p  x N  x p ρ  x q ˛ ‚ (37) has an optimal c hoice of input states t ρ  x u for which ρ  x “ n â i “ 1 τ p i q x i , where τ p i q 0 “ | ψ max yx ψ max | A p i q 0 , τ p i q 1 “ | ϕ max yx ϕ max | A p i q 1 , wher e | ψ max y (resp. | ϕ max y ) is any unit vector in the maximal-eigenspace of σ 0 (r esp. σ 1 ), and |  x | 0 : “ |t i : x i “ 0 u| . Equivalently , each freely chosen spin is aligned with a maximal eigen vector of the corr esponding fixed state. Note that since the map p ρ  x q  x ÞÑ ř  x Pt 0 , 1 u n p  x N  x p ρ  x q imposes a con vex combination and S p¨q is concave, there exists an optimal choice with e very ρ  x pure. Moreov er , if σ 0 “ σ 1 , it can be recovered by the spin alignment conjecture proposed in [ 15 ]. Here we provide the rigorous justifications for se veral special cases. 11 1) n “ 1 : Given p 0 , p 1 ě 0 with p 0 ` p 1 “ 1 , consider the entropy minimization problem min ρ A 0 P D p A 0 q , ρ A 1 P D p A 1 q S ` p 0 N 0 p ρ A 0 q ` p 1 N 1 p ρ A 1 q ˘ . (38) Proposition III.8. The minimization pr oblem ( 38 ) has an optimal choice ρ A 0 “ | ψ max yx ψ max | , ρ A 1 “ | ϕ max yx ϕ max | . (39) T o prove the above result, we first briefly revie w the majorization of matrices. For a Hermitian matrix X , let λ p X q “ p λ Ó 1 p X q , . . . , λ Ó d p X qq be the vector of eigen values in non-increasing order . For two vectors x, y P R d , we say x is majorized by y , written x ă y , if k ÿ i “ 1 x Ó i ď k ÿ i “ 1 y Ó i p k “ 1 , . . . , d ´ 1 q , d ÿ i “ 1 x i “ d ÿ i “ 1 y i . A function f on probability vectors is Schur concave if x ă y implies f p x q ě f p y q . The Shannon entropy H p x q “ ´ ř i x i log x i is Schur conca ve; hence the von Neumann entropy S p ρ q “ H p λ p ρ qq is Schur concav e in the spectrum. For a positiv e semidefinite matrix X , define X Ó to be the diagonal matrix (in some fixed reference basis) whose diagonal entries are the eigen values of X written in non-increasing order . W e will use the following known tensor rearrangement majorization principle, proved in [ 1 ]: Lemma III.9 (T ensor rearrangement majorization) . Let B 1 , B 2 ľ 0 act on C d B and C 1 , C 2 ľ 0 act on C d C . Then λ ` B 1 b C 1 ` B 2 b C 2 ˘ ă λ ` B Ó 1 b C Ó 1 ` B Ó 2 b C Ó 2 ˘ . (40) Pr oof of Pr oposition III.8 . Fix arbitrary unit v ectors | ψ y and | ϕ y and consider Ω : “ Ω p| ψ yx ψ | , | ϕ yx ϕ |q “ p 0 | ψ yx ψ | b σ 1 ` p 1 σ 0 b | ϕ yx ϕ | . Apply Lemma III.9 with B 1 “ p 0 | ψ yx ψ | , B 2 “ p 1 σ 0 , C 1 “ σ 1 , C 2 “ | ϕ yx ϕ | . W e obtain the spectral majorization λ p Ω q ă λ p Ω Ó q , Ω Ó : “ p 0 p| ψ yx ψ |q Ó b σ Ó 1 ` p 1 σ Ó 0 b p| ϕ yx ϕ |q Ó . (41) Since Ω and Ω Ó are density operators, their spectra are probability vectors. Because the von Neumann entropy is Schur concav e in the spectrum, ( 41 ) implies S p Ω q “ H p λ p Ω qq ě H p λ p Ω Ó qq “ S p Ω Ó q . (42) By definition, σ Ó 0 and σ Ó 1 are diagonal with eigen values in non-increasing order; thus the first basis vectors correspond to the maximal eigen v alues of σ 0 and σ 1 , respecti vely . Therefore, in the eigenbases of σ 0 and σ 1 , we can re write Ω Ó as Ω Ó “ p 0 | ψ max yx ψ max | b σ 1 ` p 1 σ 0 b | ϕ max yx ϕ max | , i.e. Ω Ó “ Ω p| ψ max yx ψ max | , | ϕ max yx ϕ max |q up to unitary conjugation, and hence has the same entropy as that aligned-output state. Combining with ( 42 ), we conclude that for ev ery pure pair p| ψ y , | ϕ yq , S ` Ω p| ψ yx ψ | , | ϕ yx ϕ |q ˘ ě S ` Ω p| ψ max yx ψ max | , | ϕ max yx ϕ max |q ˘ . Since an optimizer exists among pure pairs, this shows that ( 38 ) admits an optimizer with ρ A 0 “ | ψ max yx ψ max | and ρ A 1 “ | ϕ max yx ϕ max | , as claimed. 2) Rényi-2 entropy: Recall that the Rényi-2 entropy is defined by S 2 p ρ q : “ ´ log T r ` ρ 2 ˘ . (43) Thus minimizing S 2 p ρ q is equiv alent to maximizing T r ` ρ 2 ˘ . Dealing with 2 -norm can greatly simplify the optimization. In fact, one can separate the mixture and optimize each term, while for von Neumann entropy , the non-linear nature obstructs this analysis. W e hav e the follo wing: Proposition III.10 (Spin alignment for Rényi-2 entropy) . Fix a pr obability distribution t p  x u  x Pt 0 , 1 u n , the minimization of the Rényi-2 entr opy min ρ  x S 2 ¨ ˝ ÿ  x Pt 0 , 1 u n p  x N  x p ρ  x q ˛ ‚ (44) has an optimal c hoice of input states t ρ  x u for which ρ  x “ n â i “ 1 τ p i q x i , where τ p i q 0 “ | ψ max yx ψ max | A p i q 0 , τ p i q 1 “ | ϕ max yx ϕ max | A p i q 1 , 12 and | ψ max y (r esp. | ϕ max y ) is any eigen vector of σ 0 (r esp. σ 1 ) corr esponding to its largest eigen value λ 0 (r esp. λ 1 ). Pr oof. Define the constants λ 0 : “ λ max p σ 0 q , λ 1 : “ λ max p σ 1 q , α 0 : “ T r ` σ 2 0 ˘ , α 1 : “ T r ` σ 2 1 ˘ . (45) For  x,  y P t 0 , 1 u n define the counts N ab p  x,  y q : “ ˇ ˇ t i P r n s : p x i , y i q “ p a, b qu ˇ ˇ , a, b P t 0 , 1 u . (46) W e show that the maximizer of T r “` ÿ  x Pt 0 , 1 u n p  x N  x p ρ  x q ˘ 2 ‰ “ ÿ  x, y p  x p  y T r “ N  x p ρ  x q N  y p ρ  y q ‰ . can be chosen as ρ  x “ n â i “ 1 τ p i q x i , where τ p i q 0 “ | ψ max yx ψ max | A p i q 0 , τ p i q 1 “ | ϕ max yx ϕ max | A p i q 1 . In fact, in Appendix A , we show that giv en t ρ  x u  x Pt 0 , 1 u n , for any  x,  y P t 0 , 1 u n , we ha ve T r “ N  x p ρ  x q N  y p ρ  y q ‰ ď α N 00 p  x, y q 1 α N 11 p  x, y q 0 p λ 0 λ 1 q N 01 p  x, y q` N 10 p  x, y q . (47) On the other hand, via direct calculation, T r “ N x i p τ p i q x i q N y i p τ p i q y i q ‰ “ $ ’ ’ ’ ’ & ’ ’ ’ ’ % T r ` σ 2 1 ˘ “ α 1 , p x i , y i q “ p 0 , 0 q , T r ` σ 2 0 ˘ “ α 0 , p x i , y i q “ p 1 , 1 q , T r p| ψ max yx ψ max | σ 0 q T r p σ 1 | ϕ max yx ϕ max |q “ λ 0 λ 1 , p x i , y i q “ p 0 , 1 q , T r p σ 0 | ψ max yx ψ max |q T r p| ϕ max yx ϕ max | σ 1 q “ λ 0 λ 1 , p x i , y i q “ p 1 , 0 q . which concludes the proof that ρ  x “  n i “ 1 τ p i q x i is an optimal choice of ( 44 ) In the next section, we will illustrate ho w to derive single-letter expression for quantum capacities and one-way distillable entanglement using Spin alignment phenomenon. I V . E X P L I C I T E X A M P L E S O F N O N - D E G R A DA B L E S TA T E S W I T H S I N G L E - L E T T E R O N E - W A Y D I S T I L L A B L E E N TA N G L E M E N T In this section, we provide several examples of non-degradable states with single-letter distillable entanglement using the framework proposed in the previous section. For the first class consisting of states with weaker degradability , one can choose the Choi state of quantum channels proposed in [ 17 ]. Therefore, we focus on the second and third class in the remaining section. A. Mixture of degr adable and anti-de gradable states with orthogonal support T o illustrate our general framew ork, we consider the state ρ AB p s q gi ven by 1 3 ´ | 0 y x 0 | A b “ s | 0 y x 0 | B ` p 1 ´ s q | 2 y x 2 | B ‰ ` | 0 y x 1 | A b ? s | 0 y x 2 | B ` | 1 y x 0 | A b ? s | 2 y x 0 | B ` | 1 y x 1 | A b | 2 y x 2 | B ` | 2 y x 2 | A b | 2 y x 2 | B ¯ . This is an example of a state with an useless component having orthogonal support on Alice’ s side to the useful component. The antidegradable state here is a separable state | 2 y x 2 | A b | 2 y x 2 | B (appears in the mixture with probability 1 { 3 ) and the useful component here is the Choi state of an amplitude damping channel from the Span t | 0 y , | 1 yu Ă H A to Span t | 1 y , | 2 yu Ă H B , parameterized by s . Therefore, from Theorem III.4 , we conclude that this state has a single-letter distillable entanglement, given by D Ñ p ρ AB p s qq “ D p 1 q Ñ p ρ AB p s qq “ 2 3 max " h ´ s 2 ¯ ´ h ˆ 1 ` s 2 ˙ , 0 * . (48) B. Flagged mixtur e of degr adable and antide gradable states A concrete and very explicit family comes from flag ged mixtures of amplitude damping channels (from [ 17 ]). Let AD γ be the qubit amplitude damping channel with Kraus operators E 0 “ | 0 y x 0 | ` a 1 ´ γ | 1 y x 1 | , E 1 “ ? γ | 0 y x 1 | , γ P r 0 , 1 s . Fix parameters γ 0 , γ 1 P r 0 , 1 s and a mixing probability p P r 0 , 1 s , and define the flagged channel N A Ñ B F p ρ q : “ p AD γ 0 p ρ q b | 0 y x 0 | F ` p 1 ´ p q AD γ 1 p ρ q b | 1 y x 1 | F , where the classical flag F is av ailable to Bob . Its (normalized) Choi state ρ N RB F : “ p id R b N qp| Φ ` y x Φ ` |q , with | Φ ` y “ p| 00 y ` | 11 yq{ ? 2 , is ρ N RB F “ p ρ p γ 0 q RB b | 0 y x 0 | F ` p 1 ´ p q ρ p γ 1 q RB b | 1 y x 1 | F , (49) 13 where, in the computational basis t| 00 y , | 01 y , | 10 y , | 11 yu of RB , ρ p γ q RB “ 1 2 ¨ ˚ ˚ ˝ 1 0 0 ? 1 ´ γ 0 0 0 0 0 0 γ 0 ? 1 ´ γ 0 0 1 ´ γ ˛ ‹ ‹ ‚ . (50) This flagged family for some parameter regions provides explicit non-de gradable Choi states for which the weaker information- dominance property holds and hence D Ñ p ρ N RB F q is single-letter , see [ 17 ]. C. Generalized dir ect sum channels and their corr esponding states In this subsection, motiv ated by [ 20 ], we present a class of quantum channels and states with single-letter expression for capacities and one-way distillable entanglement. W e first show that this class of channels can have single-letter quantum capacity using the spin alignment phenomenon. W e say a completely positiv e map Φ has a generalized (partially coherent) direct sum structure if Φ ˆ X 00 X 01 X 10 X 11 ˙ “ ˆ Φ 0 p X 00 q Φ 01 p X 01 q Φ 10 p X 10 q Φ 1 p X 11 q ˙ where Φ 0 , Φ 1 are completely positiv e. It is shown in [ 8 ] that Φ admits partially coherent direct sum structure if and only if the Kraus operators of Φ are block diagonal, i.e., Φ p X q “ ř k E k X E : k with E k “ ˆ E 0 k 0 0 E 1 k ˙ . (51) A key observation is that the optimal states achieving the maximal coherent information of Φ b n can be chosen to be block diagonal. Lemma IV .1. Suppose Φ is a gener alized dir ect sum channel with Kraus operators ˆ E 0 k 0 0 E 1 k ˙ . Then Q p 1 q p Φ b n q “ max ! I c p ρ n , Φ b n q : ρ n “ ÿ  x Pt 0 , 1 u n p  x |  x yx  x | b ρ  x , ρ  x P D d |  x | 0 d n ´|  x | 1 , ÿ  x Pt 0 , 1 u n p  x “ 1 ) . Pr oof. Using the direct sum structure of the Hilbert space, any ρ n P D p d 0 ` d 1 q n can be decomposed as ρ n “ ÿ  x, y Pt 0 , 1 u n |  x yx  y | b ρ  x, y , ρ  x, y P M d |  x | 0 d n ´|  x | 1 ˆ d |  y | 0 d n ´|  y | 1 . Define the pinching map (conditional expectation onto block-diagonal entries) P p ρ n q : “ ÿ  x Pt 0 , 1 u n p|  x yx  x | b I q ρ n p|  x yx  x | b I q “ ÿ  x |  x yx  x | b ρ  x, x . Step 1: the complementary output is unchanged by pinching. Since each Kraus operator of Φ is block diagonal, ev ery Kraus operator of Φ b n is also block diagonal with respect to the decomposition indexed by  x P t 0 , 1 u n . Hence, for  x ‰  y one has T r ` E  k p|  x yx  y | b ρ  x, y q E :  ℓ ˘ “ 0 , and therefore the complementary channel depends only on the diagonal blocks: p Φ c q b n p ρ n q “ p Φ c q b n p P p ρ n qq . Step 2: the main output entropy increases under pinching. Set ω : “ Φ b n p ρ n q and ω 1 : “ Φ b n p P p ρ n qq . By construction, ω 1 is obtained from ω by pinching with respect to the orthogonal projections t|  x yx  x | b I u  x on the output space. Equiv alently , ω 1 is a con ve x combination of unitary conjugations of ω (hence the spectrum of ω 1 majorizes that of ω ), and since the von Neumann entropy is Schur conca ve we get S p Φ b n p ρ n qq “ S p ω q ď S p ω 1 q “ S p Φ b n p P p ρ n qqq . This is exactly the majorization/pinching trick [ 7 ]. Combining Step 1 and Step 2, I c p ρ n , Φ b n q “ S p Φ b n p ρ n qq ´ S pp Φ c q b n p ρ n qq ď S p Φ b n p P p ρ n qqq ´ S pp Φ c q b n p P p ρ n qqq “ I c p P p ρ n q , Φ b n q , so an optimizer e xists among block-diagonal states of the claimed form. Now we show the following using the spin alignment phenomenon, which generalizes the result in [ 19 ] to different dimensions. Proposition IV .2. Assume Φ 0 p X q “ T r p X q I d 1 0 { d 1 0 and Φ 1 p X q “ T r p X q I d 1 1 { d 1 1 , and we choose the generalized dir ect sum channel as Φ ˆ X 00 X 01 X 10 X 11 ˙ “ ¨ ˝ T r p X 00 q I d 1 0 d 1 0 X T 01 X T 10 T r p X 11 q I d 1 1 d 1 1 ˛ ‚ . (52) 14 Then Q p Φ q “ Q p 1 q p Φ q . Pr oof. Step 0: enlarge dimensions without changing coherent information. When d 1 0 , d 1 1 do not match the sizes needed to place X T 01 as an of f-diagonal block, we work with an enlarged GDS channel r Φ : B p C d 0 ‘ C d 1 q Ñ B p C D 0 ‘ C D 1 q where D 0 “ max t d 1 0 , d 1 u , D 1 “ max t d 0 , d 1 1 u . (53) Define r Φ exactly as in your construction, i.e. it has the block form r Φ ˆ X 00 X 01 X 10 X 11 ˙ “ ¨ ˚ ˚ ˚ ˚ ˝ T r p X 00 q I d 1 0 d 1 0 0 X T 01 0 0 0 0 0 X T 10 0 T r p X 11 q I d 1 1 d 1 1 0 0 0 0 0 ˛ ‹ ‹ ‹ ‹ ‚ , so that Φ is obtained from r Φ by restricting to the supported output subspace (equiv alently , composing with an isometry and its adjoint). Therefore, for e very n and ev ery input state ρ n , S p Φ b n p ρ n qq “ S p r Φ b n p ρ n qq , S pp Φ c q b n p ρ n qq “ S pp r Φ c q b n p ρ n qq , and hence Q p 1 q p Φ b n q “ Q p 1 q p r Φ b n q . In particular , it suffices to prove the claim for r Φ . Step 1: identify the spin-alignment structure on the complementary channel. W ith Kraus operators as in your definition, r Φ c can be chosen so that r Φ c ˆ X 00 X 01 X 10 X 11 ˙ “ 1 d 1 0 P d 1 0 b p X 00 ` p X 11 b 1 d 1 1 P d 1 1 , (54) where p X 00 (resp. p X 11 ) is the embedding of X 00 (resp. X 11 ) into B p C D 1 q (resp. B p C D 0 q ), and P d 1 0 (resp. P d 1 1 ) is the projection onto C d 1 0 Ă C D 0 (resp. C d 1 1 Ă C D 1 ). In particular , for any block-diagonal input state ρ “ ˆ ρ A 0 0 0 ρ A 1 ˙ , p 0 : “ T r p ρ A 0 q , p 1 : “ T r p ρ A 1 q , the complementary output is exactly a two-branch mixture of the spin-alignment form: r Φ c p ρ q “ p 0 ´ 1 d 1 0 P d 1 0 ¯ b p ρ A 0 ` p 1 p ρ A 1 b ´ 1 d 1 1 P d 1 1 ¯ . (55) Step 2: r educe the n -copy optimization to the aligned tw o-dimensional subspace (Spin Alignment Conjectur e). By Lemma IV .1 , for e very n we may restrict to inputs of the form ρ n “ ÿ  x Pt 0 , 1 u n p  x |  x yx  x | b ρ  x . For our specific choice ( 52 ) (hence also for r Φ ), the main output r Φ b n p ρ n q depends on ρ  x only through the weights p  x , because each diagonal branch is completely depolarizing. Therefore, for fix ed t p  x u , maximizing I c p ρ n , r Φ b n q is equiv alent to minimizing the entropy of the complementary output p r Φ c q b n p ρ n q “ ÿ  x Pt 0 , 1 u n p  x p r Φ c q b n ` |  x yx  x | b ρ  x ˘ , and each summand has the spin-alignment structure inherited from ( 55 ) on e very tensor factor . In voking the (multi-copy) Spin Alignment Conjecture for the entropy-minimization problem associated with ( 55 ), we may choose an optimizer such that, for each  x , the state ρ  x is supported on the tensor product of one-dimensional subspaces spanned by a fixed unit vector in C d 0 and a fixed unit vector in C d 1 (i.e. the “aligned spins”). Equiv alently , there exists an optimizer supported on ´ span t | 0 y , | d 1 yu ¯ b n Ă p C d 0 ‘ C d 1 q b n , (56) where | 0 y denotes a unit vector in the first summand C d 0 and | d 1 y denotes a unit vector in the second summand C d 1 (after fixing an orthonormal basis). Consequently , for every n , Q p 1 q p r Φ b n q “ Q p 1 q p Ψ b n q , (57) where Ψ is the restriction of r Φ to the two-dimensional input subspace span t | 0 y , | d 1 yu . Step 3: the restricted channel Ψ is degradable. On span t | 0 y , | d 1 yu , both diagonal blocks X 00 and X 11 are scalars, hence by ( 54 ) the complementary output Ψ c p¨q depends only on the block traces (and ignores the off-diagonal entry). Let P 0 and P 1 be the orthogonal projections onto the two output summands of r Φ . Define a CPTP map D on the output space of Ψ by D p Y q : “ T r p P 0 Y q ´ 1 d 1 0 P d 1 0 ¯ b | 0 yx 0 | ` T r p P 1 Y q | d 1 yx d 1 | b ´ 1 d 1 1 P d 1 1 ¯ . (58) 15 Since T r p P 0 Ψ p ρ qq “ T r p ρ A 0 q and T r p P 1 Ψ p ρ qq “ T r p ρ A 1 q , and since Ψ c depends only on these two weights on the restricted subspace, one checks directly from ( 54 ) that Ψ c “ D ˝ Ψ , i.e. Ψ is de gradable. Therefore Q p Ψ q “ Q p 1 q p Ψ q . Step 4: conclude single-letter capacity for Φ . Using ( 57 ) and degradability of Ψ , Q p Φ q “ lim n Ñ8 1 n Q p 1 q p Φ b n q “ lim n Ñ8 1 n Q p 1 q p r Φ b n q “ lim n Ñ8 1 n Q p 1 q p Ψ b n q “ Q p 1 q p Ψ q “ Q p 1 q p r Φ q “ Q p 1 q p Φ q , which prov es Q p Φ q “ Q p 1 q p Φ q . Now we provide the proof of the single-letterization of one-way distillable entanglement for a class of generalised direct sum (GDS) states as an example case of the Spin alignment phenomena. Proposition IV .3. Let A, B – C d 0 ` d 1 and define ρ AB “ 1 2 d 0 d 1 d 0 ´ 1 ÿ i “ 0 d 1 ´ 1 ÿ j “ 0 p | i y | j y ` | j ` d 0 y | i ` d 1 yq px i | x j | ` x j ` d 0 | x i ` d 1 | q . Then ρ AB has single-letter one-way distillable entanglement: D Ñ p ρ AB q “ D p 1 q Ñ p ρ AB q . Pr oof. W e prov e that for e very n P N , D p 1 q Ñ p ρ b n AB q “ n D p 1 q Ñ p ρ AB q , which implies the claim by regularization. Step 0: a canonical instrument. For 0 ď i ă d 0 and 0 ď j ă d 1 define the (orthonormal) v ectors | ψ ij y “ 1 ? 2 ´ | i y | j y ` | j ` d 0 y | i ` d 1 y ¯ , so that ρ AB “ 1 d 0 d 1 ř i,j | ψ ij y x ψ ij | is the maximally mixed state on the span of t | ψ ij yu i,j . Define an instrument T : A Ñ AM with Kraus operators K ij “ 1 ? d 1 | i y x i | ` 1 ? d 0 | d 0 ` j y x d 0 ` j | , 0 ď i ă d 0 , 0 ď j ă d 1 . A direct computation gi ves ř i,j K : ij K ij “ I A , hence T is trace-preserving and T b n is a v alid candidate for D p 1 q Ñ p ρ b n AB q . Step 1: reduction to block-diagonal PO VM elements (pinching). Decompose A “ C d 0 ‘ C d 1 and define projectors Π 0 “ d 0 ´ 1 ÿ i “ 0 | i y x i | , Π 1 “ d 1 ´ 1 ÿ j “ 0 | j ` d 0 y x j ` d 0 | . For  x P t 0 , 1 u n , set Π  x “ Π x 1 b ¨ ¨ ¨ b Π x n . Let T 1 : A b n Ñ A b n M be any trace-preserving instrument with Kraus operators t L m u m and PO VM elements E m “ L : m L m . Consider the pinched PO VM elements r E m : “ ÿ  x Pt 0 , 1 u n Π  x E m Π  x . Since ř m E m “ I , we still hav e ř m r E m “ I , so t r E m u m comes from a trace-preserving instrument r T 1 . The GDS structure of ρ AB implies that, conditioned on outcome m , the joint state of B b n and of a purification en vironment depends only on the diagonal blocks Π  x E m Π  x and is insensitiv e to the off-diagonal blocks Π  x E m Π  y with  x ‰  y . Moreov er , on the Bob system the replacement E m ÞÑ r E m corresponds to pinching the post-measurement state in the direct-sum decomposition index ed by  x , which increases the entropy by the Ky Fan majorization trick [ 7 ]. Since I p A b n y B b n M q “ S p B b n M q ´ S p E b n M q for a purification, and the environment term is unchanged under the abov e replacement, we obtain I p A b n y B b n M q r T 1 p ρ b n q ě I p A b n y B b n M q T 1 p ρ b n q . It therefore suf fices to optimize ov er block-diagonal instruments. Step 2: spin alignment within each block. Fix a block-diagonal instrument (still denoted T 1 ) with PO VM elements E m “ ř  x Π  x E m Π  x . For each m and  x define the weights α m, x : “ T r p Π  x E m Π  x q ě 0 . 16 For our state ρ b n AB , the marginal on B b n M depends on E m only through the collection t α m, x u  x (intuitiv ely , within a fixed block  x the state is maximally mixed on the corresponding support, so only the total weight matters for Bob’ s entropy). Thus, keeping t α m, x u fixed, to increase coherent information it is enough to minimize the entropy contribution coming from the en vironment. By the (multi-copy) spin alignment property applied to the entropy-minimization problem induced by ρ b n AB , the environment entropy is minimized when each positive block Π  x E m Π  x is replaced by a rank-one projector with the same weight. Concretely , define positive elements E 1 m “ ÿ  x Pt 0 , 1 u n α m, x n â i “ 1 | t p x i qy x t p x i q | , where t p 0 q “ 0 and t p 1 q “ d 0 . Let T 2 denote the corresponding (possibly non–trace-preserving) instrument (as in Definition II.3 ). Then, by construction, I p A b n y B b n M q T 2 p ρ b n q ě I p A b n y B b n M q T 1 p ρ b n q . Step 3: reduction to a degradable tensor-po wer state. Define σ 0 “ p d 0 d 1 qp K 00 b I q ρ AB p K : 00 b I q , σ “ σ b n 0 . One verifies (see the appendix) that σ 0 is a degradable state, hence so is σ . Moreover , from the explicit form of E 1 m , one can construct an instrument T 3 acting on σ such that the classical-quantum branch structure matches that of T 2 p ρ b n q and I p A b n y B b n M q T 2 p ρ b n q “ I p A b n y B b n M q T 3 p σ q . Since σ is degradable, coherent information is maximized by the tri vial instrument [ 14 ]. Therefore I p A b n y B b n q σ ě I p A b n y B b n M q T 3 p σ q . Combining inequalities gi ves the upper bound I p A b n y B b n q σ ě D p 1 q Ñ p ρ b n AB q . (59) Step 4: a matching lower bound from T b n . For all x i P p 0 , . . . , d 0 ´ 1 q and y i P p 0 , . . . , d 1 ´ 1 q , the normalized branch states p d 0 d 1 q n n â i “ 1 p K x i y i b I q ρ AB p K : x i y i b I q are related to σ by local unitaries on A b n and B b n (they simply relabel the basis vectors within each direct-sum block). Hence they hav e the same coherent information as σ , and these branches are precisely the outcomes of T b n . Therefore I p A b n y B b n q σ “ I p A b n y B b n M q T b n p ρ b n q ď D p 1 q Ñ p ρ b n AB q . (60) Combining the bounds yields D p 1 q Ñ p ρ b n AB q “ I p A b n y B b n q σ . (61) Since σ “ σ b n 0 , I p A b n y B b n q σ “ n I p A y B q σ 0 . Therefore, D p 1 q Ñ p ρ b n AB q “ n D p 1 q Ñ p ρ AB q . T aking the regularized limit yields D Ñ p ρ AB q “ lim n Ñ8 1 n D p 1 q Ñ p ρ b n AB q “ D p 1 q Ñ p ρ AB q . (62) One can notice the structural similarity in the proofs of the Propositions IV .2 and IV .3 till step 2. Howe ver , in step 3 and step 4 the proofs significantly dif fer . As demonstrated below , this dif ference explains why it is harder to pro ve additivity for D p 1 q Ñ than that for Q p 1 q . 17 Channel ( Φ ) State ( ρ ) Block diagonal input states are optimal for Q p 1 q p Φ b n q Rank-1 in each block is optimal: still a feasible solution Φ b n restricted to the optimum subspace: Ψ b n 6 Q p 1 q p Φ b n q “ Q p 1 q p Ψ b n q Ψ turns out to be degradable. ù ñ Q p 1 q p Φ b n q “ Q p 1 q p Ψ b n q “ n Q p 1 q p Ψ q “ n Q p 1 q p Φ q Block diagonal PO VM elements are optimal for D p 1 q Ñ p ρ b n q Rank-1 in each block yields larger coherent information. Howe ver , not a feasible solution A degradable state σ b n 0 is constructed such that D p 1 q Ñ p σ b n 0 q ě D p 1 q Ñ p ρ b n q There exists an instrument for ρ b n that achieves D p 1 q Ñ p σ b n 0 q ù ñ D p 1 q Ñ p ρ b n q “ D p 1 q Ñ p σ b n 0 q “ n D p 1 q Ñ p σ 0 q “ n D p 1 q Ñ p ρ q Remark IV .4. Unlike the GDS channel setting, for general GDS states with differ ent bloc k dimensions on Bob (e.g . B “ C d 1 0 ‘ C d 1 1 with p d 1 0 , d 1 1 q ‰ p d 0 , d 1 q ), the spin alignment conjecture still suggests that one can restrict to aligned rank-one structur es inside each block. However , after this r estriction the differ ent measur ement branches need not be r elated by local isometries on AB , so the shortcut used in ( 60 ) (identifying all branches with a single tensor-power de gradable state) can fail. This makes pr oving additivity of one-way distillable entanglement for arbitrary generalised dir ect sum states comparatively harder than the channel setting. V . C O N C L U S I O N A N D O U T L O O K W e identified new structural conditions under which one-way distillable entanglement admits a single-letter formula. In particular , we introduced the notions of less noisy at level n and informationally de gradable states, and sho wed that both classes are single- letter for D Ñ . Informationally degradable states are moreover additiv e under tensor products, i.e., they exhibit strong additivity . These results demonstrate that single-letter behavior extends strictly beyond the degradable regime. W e further proved a general single-letter theorem for mixtures with orthogonal support on Alice’ s side, introducing non-degradable states with “useless” components. This yields explicit examples where the distillable entanglement is simply the con vex combination of the components, despite the absence of degradability . A central conceptual ingredient is the spin-alignment phenomenon, which e xplains why entropy minimization for certain generalized direct-sum channels is achie ved by locally aligned inputs. W e established this ef fect for the single-copy von Neumann entropy and for arbitrary block length in the Rényi-2 case. Applying this principle, we showed that a class of generalized direct sum channels has single-letter quantum capacity , and that some particular case of the corresponding states have single-letter one-way distillable entanglement. Howe ver , we hav e found that ev en though numerical evidences for additivity of D p 1 q Ñ are present for certain states, proving additivity is dif ficult. Our results reveal ne w mechanisms enforcing additivity and single-letter formulas beyond degradability . An important open problem is to prove the spin alignment conjecture with von-Neumann entropy for arbitrary tensor powers. Understanding the comparative additivity of D p 1 q Ñ and Q p 1 q remains to be explored. 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The Theory of Quantum Information . Cambridge University Press (2018). [19] P . Wu and Y . W ang. “Quantum capacity amplification via privacy” . In 2025 IEEE International Symposium on Information Theory (ISIT) , pages 1–6, (2025). [20] Z. Wu and S.-Q. Zhou. “Extreme Capacities in Generalized Dir ect Sum Channels” , (2025). A vailable online: https://arxiv .org/abs/2510.10711 . A P P E N D I X A. Explicit upper bound on each alignment term for Rényi-2 entr opy In this section, we present the details of the proof of ( 47 ). That is T r “ N  x p ρ  x q N  y p ρ  y q ‰ ď α N 00 p  x, y q 1 α N 11 p  x, y q 0 p λ 0 λ 1 q N 01 p  x, y q` N 10 p  x, y q , @  x,  y P t 0 , 1 u n , where N 0 p ρ q “ ρ b σ 1 and N 1 p ρ q “ σ 0 b ρ and the constants are given by λ 0 : “ λ max p σ 0 q , λ 1 : “ λ max p σ 1 q , α 0 : “ T r ` σ 2 0 ˘ , α 1 : “ T r ` σ 2 1 ˘ . (63) For  x,  y P t 0 , 1 u n , the counts are given by N ab p  x,  y q : “ ˇ ˇ t i P r n s : p x i , y i q “ p a, b qu ˇ ˇ , a, b P t 0 , 1 u . (64) Lemma A.1 (Single-site adjoint compositions) . W ith r espect to the Hilbert–Schmidt inner pr oduct, the adjoints satisfy N : 0 p X q “ T r B 1 “ p I b σ 1 q X ‰ , N : 1 p X q “ T r B 0 “ p σ 0 b I q X ‰ . Mor eover , for any positive operator X of compatible dimension, N : 0 N 0 p X q “ α 1 X, N : 1 N 1 p X q “ α 0 X, N : 0 N 1 p X q “ T r p σ 1 X q σ 0 , N : 1 N 0 p X q “ T r p σ 0 X q σ 1 . Pr oof of Lemma A.1 . The adjoint formulas follo w directly from T r “ p ρ b σ 1 q X ‰ “ T r “ ρ T r B 1 pp I b σ 1 q X q ‰ , T r “ p σ 0 b ρ q X ‰ “ T r “ ρ T r B 0 pp σ 0 b I q X q ‰ . The composition identities are then obtained by substitution. For example, N : 0 N 1 p X q “ N : 0 p σ 0 b X q “ T r B 1 “ p I b σ 1 qp σ 0 b X q ‰ “ T r p σ 1 X q σ 0 , and the remaining cases are analogous. Pr oof of ( 47 ) . Fix  x,  y . Using adjoints and Hölder’ s inequality , T r “ N  x p ρ  x q N  y p ρ  y q ‰ “ T r “ ρ  x p N :  x N  y qp ρ  y q ‰ ď } ρ  x } 1 › › p N :  x N  y qp ρ  y q › › 8 “ › › p N :  x N  y qp ρ  y q › › 8 . Since N  x “  i N x i , we ha ve N :  x N  y “ n â i “ 1 p N : x i N y i q . 19 By Lemma A.1 , each tensor factor is one of four completely positi ve maps: N : 0 N 0 p X q “ α 1 X, N : 1 N 1 p X q “ α 0 X, N : 0 N 1 p X q “ T r p σ 1 X q σ 0 , N : 1 N 0 p X q “ T r p σ 0 X q σ 1 . Let D : “ t i : x i ‰ y i u and E : “ t i : x i “ y i u . The equal sites contribute the scalar factor α N 00 p  x, y q 1 α N 11 p  x, y q 0 . On the mismatch sites D , the map is “measure-and-prepare”: it applies a weighted partial trace against σ 1 (resp. σ 0 ) and outputs σ 0 (resp. σ 1 ). Concretely , there exist positive operators M  x, y “ σ b N 01 p  x, y q 1 b σ b N 10 p  x, y q 0 , P  x, y “ σ b N 01 p  x, y q 0 b σ b N 10 p  x, y q 1 , such that for e very positive operator X on the input space of N  y , p N :  x N  y qp X q “ α N 00 1 α N 11 0 ` T r D “ p M  x, y b I E q X ‰˘ b P  x, y , (65) where T r D is the partial trace over the mismatch input registers and N ab “ N ab p  x,  y q . Now specialize to X “ ρ  y (a density operator). The operator R E : “ T r D “ p M  x, y b I E q ρ  y ‰ is positi ve, and hence } R E } 8 ď T r p R E q . Moreov er , T r p R E q “ T r “ p M  x, y b I E q ρ  y ‰ ď } M  x, y } 8 “ λ N 01 1 λ N 10 0 , since T r p K ρ q ď } K } 8 T r p ρ q for K ě 0 and T r p ρ  y q “ 1 . Also, } P  x, y } 8 “ λ N 01 0 λ N 10 1 . Combining with ( 65 ) and using } A b B } 8 “ } A } 8 } B } 8 giv es › › p N :  x N  y qp ρ  y q › › 8 ď α N 00 p  x, y q 1 α N 11 p  x, y q 0 p λ 0 λ 1 q N 01 p  x, y q` N 10 p  x, y q . (66) B. Generalized dir ect sum completely positive maps and states Definition A.2 (Generalized direct sum CP maps) . Let us consider Hilbert spaces, A – A 0 ‘ A 1 and B – B 0 ‘ B 1 . Moreover consider two CP maps, Φ i : B p A i q Ñ B p B i q , for i “ 0 , 1 , with particular Kraus r epr esentations, t F p i q k u . A gener alised direct sum (GDS) completely positive (CP) map, Φ : B p A q Ñ B p B q is defined by its Kraus repr esentation: t F k : “ F p 1 q k ‘ F p 2 q k u . Note that in the definition, we add zero operators in either of the set of Kraus operators to make them equally long. Furthermore, different ordering of the Kraus operators in the direct sum, can yield different generalised direct sum CP maps. As these maps are not necessarily trace-preserving, in general ř k F : k F k ‰ I A . The Choi-Jamiołko wski isomorphism [ 9 ], [ 13 ] implies that corresponding to the completely positiv e map, Φ : B p A q Ñ B p B q , there exists a positiv e semidefinite operator on the bipartite system AB , i.e., J Φ P Pos p A b B q . In fact, we can write [ 18 ] J Φ “ ÿ k v ec p F T k q v ec p F T k q : . (67) For non-zero maps, T r t J Φ u ‰ 0 . Therefore, we can define a corresponding GDS state ρ Φ P D p A b B q as ρ Φ “ 1 T r t J ϕ u J Φ . (68) Theorem A.3. Let A, B – C d 0 ` d 1 . The following bipartite state ρ AB “ 1 2 d 0 d 1 d 0 ´ 1 ÿ i “ 0 d 1 ´ 1 ÿ j “ 0 p | i y | j y ` | j ` d 0 y | i ` d 1 yq px i | x j | ` x j ` d 0 | x i ` d 1 | q (69) has single-letter one-way distillable entanglement, i.e., D Ñ p ρ AB q “ D p 1 q Ñ p ρ AB q . (70) Pr oof. First we define the instrument T : A Ñ AM , with Kraus operators K ij “ 1 ? d 1 | i y x i | ` 1 ? d 0 | d 0 ` j y x d 0 ` j | where i “ 0 , . . . , d 0 ´ 1 and j “ 0 , . . . , d 1 ´ 1 . W e hav e a total of d 0 d 1 number of Kraus operators. In the follo wing, we prove that T b n : A b n Ñ A b n M is the instrument that achieves the optimum for D p 1 q Ñ p ρ b n AB q for all n P N . Step 1 (Block diagonal PO VM elements ar e suf ficient): Consider any instrument T 1 : A b n Ñ A b n M with PO VM elements E m,n “ ÿ  x, y Pt 0 ,...,d 0 ` d 1 ´ 1 u n e m  x, y |  x y x  y | . 20 For all  x P t 0 , . . . , d 0 ` d 1 ´ 1 u n , define  s p  x q P t 0 , 1 u n such that s i “ # 0 , x i P t 0 , . . . , d 1 ´ 1 u 1 , x i P t d 1 , . . . , d 1 ` d 0 ´ 1 u . (71) The p  x,  x q -th diagonal element of ρ p m q B b n is proportional to T r Π  s p  x q E m,n Π  s p  x q . (72) Here Π  x “ Π x 1 b Π x 2 b ¨ ¨ ¨ b Π x n with Π 0 “ ř d 0 ´ 1 j “ 0 | j y x j | and Π 1 “ ř d 1 ´ 1 j “ 0 | j y x j | . The state of E b n is proportional to ÿ  x Pt 0 , 1 u n n â i “ 1 N x i p E  x q , (73) where E  x “ Π  x E T m,n Π  x . (74) Here we treat E  x to be a positiv e operator on the space  i C d x i . Furthermore, we define, N 0 : B p C d 0 q Ñ B p C d 1 b C d 0 q and N 1 : B p C d 1 q Ñ B p C d 1 b C d 0 q as: N 0 p ρ q “ I d 1 b ρ ; N 1 p σ q “ σ b I d 0 . (75) Note that the state of the en vironment does not depend on the the off-diagonal blocks of the PO VM elements, i.e., on Π  x E m,n Π  y for  x ‰  y . As these blocks appear only in the off-diagonal entries of B b n , we can choose PO VM elements E m,n such that Π  x E m,n Π  y “ δ  x, y Π  x E m,n Π  x without decreasing the coherent information. Step 2 (Spin-alignment phenomena): Furthermore, the spin alignment conjecture III.7 implies that the entropy of the en vironment is minimum when replace the PO VM elements E m,n by positi ve elements E 1 m,n “ ÿ  x Pt 0 , 1 u b n T r p Π  x E m,n Π  x q n â i “ 1 | t p x i qy x t p x i q | , (76) where t p 0 q “ 0 and t p 1 q “ d 0 . Let us denote the corresponding trace non-preserving instrument (which we hav e introduced in the definition II.3 ), by T 2 : A b n Ñ A b n M . Therefore, so far we hav e prov en that for any trace-preserving instrument, T 1 , we can find a (possibly trace non-preserving) instrument, T 2 such that I p A b n y B b n M q T 2 p ρ b n q ě I p A b n y B b n M q T 1 p ρ b n q . (77) Step 3 (Matching upper and lower bound on D p 1 q Ñ p ρ b n q ) : No w , we consider the state σ “ σ b n 0 “ p d 0 d 1 q n pp K 00 b I B q ρ AB p K : 00 b I B qq b n , (78) and the instrument T 3 : A b n Ñ A b n M giv en by the PO VM elements: F m “ ÿ  x Pt 0 , 1 u n 1 d N 0 p  x q 0 1 d N 1 p  x q 1 T r Π  x E m,n Π  x n â i “ 1 | t p x i qy x t p x i q | ; F “ I A b n ´ ÿ m F m . It immediately follo ws that I p A b n y B b n M q T 2 p ρ b n q “ I p A b n y B b n M q T 3 p σ q . (79) Now σ 0 is a degradable state, as can be seen as follo ws: σ AB 0 “ 1 2 ˆ 1 ? d 1 | 0 , 0 y ` 1 ? d 0 | d 0 , d 1 y ˙ ˆ 1 ? d 1 x 0 , 0 | ` 1 ? d 0 x d 0 , d 1 | ˙ ` 1 2 d 1 d 1 ´ 1 ÿ j “ 1 | 0 , j y x 0 , j | ` 1 2 d 0 d 0 ´ 1 ÿ i “ 1 | d 0 , i ` d 1 y x d 0 , i ` d 1 | (80) For any purification | σ 0 y AB E , by applying a fully amplitude damping channel from | d 1 y Ñ | 0 y on B alone, we obtain a state on AB that is equiv alent to the state σ AE 0 , upto local isometries on B . Therefore, σ “ σ b n 0 is degradable as well. W e kno w that for degradable states, the trivial instrument achiev es the optimal coherent information. W e get the following chain of inequality , I p A b n y B b n q σ ě I p A b n y B b n M q T 3 p σ q “ I p A b n y B b n M q T 2 p ρ b n q ě I p A b n y B b n M q T 1 p ρ b n q (81) As T 1 is any trace-preserving instrument, this inequality implies that I p A b n y B b n q σ ě D p 1 q Ñ p ρ b n AB q (82) 21 Now we note that for all x i P p 0 , . . . , d 0 ´ 1 q and y i P p 0 , . . . , d 1 ´ 1 q , the states p d 0 d 1 q n n â i “ 1 p K x i y i b I B q ρ AB p K : x i y i b I B q are related to σ via local unitaries on A b n and B b n , and there are p d 0 d 1 q n numbers of such states. Therefore, they all hav e the same coherent information as I p A b n y B b n q σ . Noting that  n i “ 1 K x i y i are nothing but the Kraus operators of the proposed optimum instrument T b n , we ha ve I p A b n y B b n q σ “ I p A b n y B b n M q T b n p ρ b n q ď D p 1 q Ñ p ρ b n AB q (83) where the last inequality follows from the definition of D p 1 q Ñ p ρ b n AB q . Step 4 (Additivity of D p 1 q Ñ p ρ b n AB ) ) : From 82 and 83 , we get D p 1 q Ñ p ρ b n AB q “ I p A b n y B b n q σ “ I p A b n y B b n q σ b n 0 “ nI p A y B q σ 0 “ nD p 1 q p ρ AB q . Therefore, we finally ha ve D Ñ p ρ AB q “ lim n Ñ8 1 n D p 1 q Ñ p ρ b n AB q “ D p 1 q p ρ AB q . (84)