Discriminating idempotent quantum channels

Discriminating idemp oten t quan tum c hannels Satvik Singh 1 and Bjarne Bergh 2 1 Dep artment of Mathematics, T e chnic al University of Munich, Gar ching, Germany 1 Munich Center for Quantum Scienc e and T e chnolo gy (MCQST), Munich, Germany 2 Dep artment of Applie d Mathematics and The or etic al Physics, University of Cambridge, Unite d Kingdom Abstract W e study binary discrimination of idemp oten t quantum c hannels. When the tw o c hannels share a common full-rank in v ariant state, w e show that a simple image inclu- sion condition completely determines the asymptotic b eha vior: when it holds, a broad family of channel divergences collapse to a closed-form, single-letter expression, regu- larization is unnecessary , and all error exp onen ts (Stein/Chernoﬀ/strong-con verse) are explicitly computable with no adaptiv e adv antage. Crucially , this yields the strong con- v erse prop ert y for this channel family , which is an imp ortan t op en problem for general c hannels. When the inclusion fails, asymmetric exp onen ts b ecome inﬁnite, implying p erfect asymptotic discrimination. W e apply the results to GNS-symmetric c hannels, sho wing discrimination rates for large num b er of self iterations con v erge exp onen tially fast to those of the corresp onding idemp oten t p eripheral pro jections. If the tw o chan- nels do not share a common inv ariant state, w e provide a single-letter conv erse b ound on the regularized sandwiched R ´ en yi cb-div ergence, which suﬃces to establish a strong con verse upp er b ound on the Stein exp onen ts. Con ten ts 1 In tro duction 2 2 Preliminaries 3 2.1 En tropies and Div ergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Quan tum Hyp othesis T esting . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Quan tum Channel Discrimination . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3.1 Adaptiv e and P arallel Strategies . . . . . . . . . . . . . . . . . . . . . 6 2.3.2 Asymmetric Error Exp onen ts . . . . . . . . . . . . . . . . . . . . . . 8 2.3.3 Strong Con v erse Exp onen ts . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.4 Symmetric Error Exp onen ts . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Multiplicativ e Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1 3 Main results 14 3.1 Noiseless vs Noisy: Identit y Against an Idemp oten t Channel . . . . . . . . . 15 3.2 Noisy vs Noisy: Two Idemp oten t Channels . . . . . . . . . . . . . . . . . . . 19 3.2.1 Common In v ariant State . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.2 F ully general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Pimsner-P opa Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 Application 32 4.1 GNS-symmetric c hannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 App endices 37 A Three-la yer decomp osition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 B Pro of of Eq. (3.116) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 C Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 D Auxilliary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1 In tro duction Quan tum c hannel discrimination—the task of iden tifying an unknown quan tum channel as one of tw o candidates given black-box access—is a central problem in quantum information theory [ 5 , 13 , 24 , 23 ], with applications spanning device veriﬁcation, quan tum communica- tion, and b enc hmarking of quantum hardware. Despite extensiv e study , fundamen tal asp ects of the problem remain p o orly understo o d. In particular, the regularised c hannel div ergence go verning the optimal asymptotic asymmetric error exp onent is generally in tractable to com- pute [ 14 ], and it is unknown whether the strong-conv erse prop erty holds for arbitrary chan- nel pairs [ 16 ]. Additionally , essentially nothing is kno wn in general regarding the optimal asymptotic symmetric error exp onents. In this w ork, w e resolve man y of these diﬃculties for the class of idemp otent quantum c hannels with full rank in v ariant states, i.e. channels P : L ( H ) → L ( H ) satisfying P ◦ P = P and suc h that P ( 1 H ) has full rank. This class includes conditional exp ectations on to unital ∗− subalgebras A ⊆ L ( H ), replacer (constan t) c hannels, pro jections onto decoherence-free subspaces and noiseless subsystems, and crucially , the asymptotic limits of quantum Mark ov semigroups satisfying suitable rev ersibility (or detailed balance) with resp ect to full-rank in v ariant states [ 21 ]. Our main results sho w that if t wo such c hannels P and Q satisfy a natural image inclusion condition im( Q ∗ ) ⊆ im( P ∗ ), and share a common in v ariant state τ = P ( τ ) = Q ( τ ), then (c.f. Theorems 3.2 , 3.6 and Corollaries 3.3 , 3.7 , 3.8 ): 1. all quantum c hannel div ergences of interest collapse to a single closed-form expression, 2. regularisation is unnecessary , 3. all asymptotic error exp onents are explicitly computable 4. the strong conv erse prop ert y holds, and 5. adaptiv e discrimination strategies oﬀer no adv an tage ov er parallel ones. 2 Additionally , whenever this inclusion condition is not satisﬁed, all the asymmetric error exp onen ts are inﬁnite (c.f. Lemma 3.4 ). When the t wo channels do not share a common in v ariant state, suc h a drastic collapse of c hannel div ergences do es not o ccur. Nev ertheless, w e still pro vide a single-letter con verse b ound on the regularized sandwiched R´ en yi cb- div ergence (c.f. Theorem 3.10 ), which suﬃces to establish a strong con verse upp er b ound on the Stein exp onents (c.f. Corollary 3.12 ). Our pap er is structured as follows. Section 2 starts with an introduction of all the entropic quan tities we use, as well as a summary of quan tum hypothesis testing and previous results on quan tum channel discrimination. Section 3 con tains our main theorems with their proofs. Section 4 presents an application of our results to the class of GNS symmetric channels. 2 Preliminaries W e denote quantum systems and their asso ciated complex Hilb ert spaces 1 b y capital letters A, B , C , with corresponding dimensions d A , d B and d C , resp ectiv ely . The space of linear op erators acting on a Hilb ert space H is denoted b y L ( H ) and the con vex set of quantum states (i.e. p ositive semi-deﬁnite operators in L ( H ) with unit trace) is denoted b y D ( H ). F or a unit vector | ψ ⟩ ∈ H , we denote the pure state | ψ ⟩⟨ ψ | ∈ D ( H ) b y ψ . The identit y op erator in L ( H ) is denoted by 1 H . The trace-norm, Schatten α − norm for α ∈ (1 , ∞ ), and the op erator norm on L ( H ) are denoted b y ∥·∥ 1 , ∥·∥ α , and ∥·∥ ∞ , resp ectively [ 4 ]. A quantum channel Φ : L ( A ) → L ( B ) is a linear, completely p ositiv e, and trace- preserving map. The adjoint Φ ∗ : L ( B ) → L ( A ) of a quantum channel Φ : L ( A ) → L ( B ) is a linear, unital, and completely p ositiv e map deﬁned through the following relation: ∀ X ∈ L ( A ) , ∀ Y ∈ L ( B ) : T r( Y Φ( X )) = T r(Φ ∗ ( Y ) X ) . (2.1) 2.1 En tropies and Div ergences Deﬁnition 2.1 L et ρ ∈ D ( H ) b e a state and σ ∈ L ( H ) b e a p ositive semi-deﬁnite op er ator. • The quan tum (Umegaki) relative entrop y b etwe en ρ and σ is deﬁne d as [ 46 ] D ( ρ ∥ σ ) := ( T r ρ (log ρ − log σ ) if supp ρ ⊆ supp σ + ∞ otherwise • The α - Petz R´ en yi relative entrop y b etwe en ρ and σ with α ∈ (0 , 1) ∪ (1 , ∞ ) is deﬁne d as [ 36 , 37 ] D α ( ρ ∥ σ ) :=      1 α − 1 log T r [ ρ α σ 1 − α ] ( if α ∈ (0 , 1) and ρσ  = 0 or α ∈ (1 , ∞ ) and supp ρ ⊆ supp σ + ∞ otherwise 1 All Hilb ert space are assumed to be ﬁnite-dimensional throughout this work. 3 • The α - sandwic hed R´ en yi relativ e entrop y b etwe en ρ and σ with α ∈ (0 , 1) ∪ (1 , ∞ ) is deﬁne d as [ 33 , 52 ] e D α ( ρ ∥ σ ) :=      1 α − 1 log T r h σ 1 − α 2 α ρσ 1 − α 2 α  α i ( if α ∈ (0 , 1) and ρσ  = 0 or α ∈ (1 , ∞ ) and supp ρ ⊆ supp σ + ∞ otherwise • The max-relativ e entrop y b etwe en ρ and σ is deﬁne d as [ 8 ] D max ( ρ ∥ σ ) := ( log   σ − 1 / 2 ρσ − 1 / 2   ∞ if supp ρ ⊆ supp σ + ∞ otherwise • The min-relativ e entrop y b etwe en ρ and σ is deﬁne d as [ 9 ] D min ( ρ ∥ σ ) := ( − log T r(Π ρ σ ) if ρσ  = 0 + ∞ otherwise , wher e Π ρ denotes the ortho gonal pr oje ction onto supp ort of ρ . These div ergences are ordered as follows: for 0 < α ≤ β < 1 and 1 < α ′ ≤ β ′ < ∞ , D min ( ·∥· ) ≤ D α ( ·∥· ) ≤ D β ( ·∥· ) ≤ D ( ·∥· ) ≤ e D α ′ ( ·∥· ) ≤ e D β ′ ( ·∥· ) ≤ D max ( ·∥· ) . (2.2) Moreo ver, lim α → 0 D α ( ·∥· ) = D min ( ·∥· ) , lim α → 1 D α ( ·∥· ) = D ( ·∥· ) = lim α → 1 e D α ( ·∥· ), and lim α →∞ e D α ( ·∥· ) = D max ( ·∥· ) (see e.g. [ 29 ]). F or any div ergence D from Deﬁnition 2.1 , and tw o c hannels Φ , Ψ : L ( A ) → L ( B ), w e also deﬁne the asso ciated c hannel div ergence as D (Φ ∥ Ψ) : = sup ν ∈D ( A ) D (Φ( ν ) ∥ Ψ( ν )) . (2.3) Using this, w e can deﬁne the completely b ounded (or stabilized) and regularized channel div ergences as D cb (Φ ∥ Ψ) : = D (id A ′ ⊗ Φ ∥ id A ′ ⊗ Ψ) (2.4) D cb , reg (Ψ ∥ Φ) : = lim n →∞ 1 n D cb (Φ ⊗ n ∥ Ψ ⊗ n ) = lim n →∞ 1 n D ((id A ′ n ⊗ Φ ⊗ n ) ∥ (id A ′ n ⊗ Ψ ⊗ n )) , (2.5) where A ′ ∼ = A and id A ′ denotes the identit y c hannel on L ( A ′ ). 4 2.2 Quan tum Hyp othesis T esting Hyp othesis testing deals with the question of asserting the truth of one of multiple p ossible h yp otheses given some ob ject or data. In the simplest case, the giv en ob ject is a quantum state and the task is to identify which of tw o (fully sp eciﬁed) options it is. Generically , if there are t wo h yp otheses, there are t wo wa ys of making an error (mistaking the ﬁrst for the second, or mistaking the second for the ﬁrst). Given a decision strategy , we will call the corresp onding probabilities of making suc h an error the type-I and type-I I error probabilities. In the simple case where the task is to identify a given state as either ρ ∈ D ( H ) or σ ∈ D ( H ), these t w o probabilities are given by α : = P [w e claim the state is σ | state is actually ρ ] (t yp e-I error), (2.6) β : = P [w e claim the state is ρ | state is actually σ ] (t yp e-I I error). (2.7) The most general wa y to arrive at such a decision in the case of distinguishing ρ and σ is b y p erforming a binary (i.e., a tw o-outcome) PO VM measuremen t, whic h is fully sp eciﬁed b y one of its elements 0 ≤ M ≤ 1 H , and we use the con ven tion that an outcome corresp onding to the measurement M (resp. 1 H − M ) leads to the inference that the state is ρ (resp. σ ). In that case the corresp onding error probabilities are given by α = T r(( 1 H − M ) ρ ) = 1 − T r( M ρ ) (2.8) β = T r( M σ ) . (2.9) W e can then optimize ov er M to ﬁnd the optimal measurement in a certain sense. In the so-called asymmetric setting of hypothesis testing the aim is to minimize the t yp e-I I error probabilit y giv en the constrain t that the type-I error is b elow a chosen threshold ε ∈ [0 , 1]: min M : 0 ≤ M ≤ 1 H { T r( M σ ) | 1 − T r( M ρ ) ≤ ε } = min M : 0 ≤ M ≤ 1 H T r( M ρ ) ≥ 1 − ε T r( M σ ) . (2.10) The negativ e logarithm of this is called the hyp othesis testing r elative entr opy [ 47 ]: D ε H ( ρ ∥ σ ) : = − log   min M : 0 ≤ M ≤ 1 H T r( M ρ ) ≥ 1 − ε T r( M σ )   = max M : 0 ≤ M ≤ 1 H T r( M ρ ) ≥ 1 − ε − log(T r( M σ )) . (2.11) It has b een giv en this name, since it shares some prop erties with the quan tum relative en- trop y , in particular, it satisﬁes the data-pro cessing inequality [ 47 ]. F amously , the hypothesis testing relativ e en trop y satisﬁes Stein’s Lemma [ 27 , 35 ]: ∀ ϵ ∈ (0 , 1) : lim n →∞ 1 n D ε H ( ρ ⊗ n ∥ σ ⊗ n ) = D ( ρ ∥ σ ) , (2.12) meaning that for an y ﬁxed type-I error constraint ϵ , the type-I I error probability deca ys exp onen tially in the n umber of samples n , and the optimal error exp onent is giv en b y the 5 quan tum relativ e entrop y D ( ρ ∥ σ ). Additionally , the follo wing one-shot b ounds hold for all α ∈ (0 , 1), α ′ > 1 and ε ∈ (0 , 1) (see e.g. [ 29 ]): D α ( ρ ∥ σ ) + α α − 1 log  1 ε  ≤ D ε H ( ρ ∥ σ ) ≤ e D α ′ ( ρ ∥ σ ) + α ′ α ′ − 1 log  1 1 − ε  . (2.13) In the symmetric setting of hypothesis testing, w e assume that b oth hypotheses can o ccur with equal (prior) probability and then try to minimize the exp ected decision error probabilit y (one can show that the asymptotic decay rate of this error (with n ) is actually indep enden t of the exact form of the prior as long as the prior inv olves both hypotheses with non-zero probability and stays constant in n ). F or state discrimination, this corresp onds to the follo wing expression: p err ( ρ, σ ) : = 1 2 inf 0 ≤ M ≤ 1 H ( α + β ) = 1 2 inf 0 ≤ M ≤ 1 H  T r(( 1 H − M ) ρ ) + T r( M σ )  (2.14) = 1 2  1 − 1 2 ∥ ρ − σ ∥ 1  (2.15) where the last equality is known as the Holevo-Helstr¨ om theorem [ 26 , 28 ]. This error proba- bilit y also decays exp onen tially in n when one considers m ultiple copies of ρ and σ , and the optimal deca y rate is given by the quantum Chernoﬀ divergence [ 1 , 34 ]: lim n →∞ − 1 n log p err ( ρ, σ ) = sup α ∈ (0 , 1) (1 − α ) D α ( ρ ∥ σ ) = : ξ ( ρ ∥ σ ) . (2.16) 2.3 Quan tum Channel Discrimination Quan tum Channel Discrimination is the natural generalization of quantum state discrimina- tion to the setting where the ob jects to b e iden tiﬁed are quantum channels. Concretely , the task of binary quantum channel discrimination is as follows: Given an unknown quantum c hannel as a blac k b o x and the side information that it is one of tw o p ossible channels, de- termine the channel’s identit y [ 5 , 13 , 24 , 23 ]. This is again a fundamental building blo c k in quan tum information theory and has v arious applications in testing and v eriﬁcation of quan- tum devices and communication links. Nonetheless, man y fundamen tal asp ects of quantum c hannel discrimination are still not fully understo o d. 2.3.1 Adaptiv e and Parallel Strategies Quan tum channel discrimination is substan tially more complex than quantum state discrim- ination due to the fact that one has the additional freedom of c ho osing input states, and if m ultiple uses of the channel are allow ed, this choice can in volv e adaptivity and/or en tangle- men t. Characterizing the optimal discrimination strategy and c hoice of input state(s) can th us b e very tricky . F or quan tum channel discrimination, one often considers tw o classes of discrimination strategies: general adaptive strategies, or strategies without adaptivity – the so-called p ar al lel strategies. 6 • In a parallel strategy the input state to all the channels is ﬁxed at the b eginning and do es not (adaptively) dep end on channel outputs, ho wev er it can still inv olve states that are arbitrarily entangled. A parallel strategy is depicted in Figure 1 , and it is sp eciﬁed by a join t input state ν RA n ∈ D ( R A n ). On the output side, we are then left with the state (id R ⊗ Φ ⊗ n )( ν RA n ) or (id R ⊗ Ψ ⊗ n )( ν RA n ) dep ending on whether the giv en c hannel is Φ or Ψ. • In an adaptive strategy , the inputs are allow ed to dep end on previous c hannel inputs and also a (in principle un b ounded) reference system which can b e though of as a ‘memory’ for the adaptive computation. See Figure 2 for a depiction of an adaptive strategy . W e call an adaptive strategy Θ that uses the unknown channel n times an adaptive n -str ate gy , and lab el with Θ[Φ] (resp. Θ[Ψ]) the joint state of the output and memory registers after the n th use of the channel Φ (resp. Ψ) (i.e. the joint state of all registers b efore the measurement M in Figure 2 ). It is easy to see from the ﬁgure that every adaptive n -strategy can b e decomp osed as Θ[Φ] = (id R ⊗ Φ)(Λ n (Θ n − 1 [Φ])), where Θ n − 1 is an adaptive ( n − 1)-strategy , and Λ n b eing the last input-preparation c hannel of the strategy . 1 2 3 . . . A 1 A 2 A 3 R Φ | Ψ M Φ | Ψ ν A n R Φ | Ψ Figure 1: Illustration of a general parallel strategy with n uses of the black-box c hannel, and a join t binary POVM measurement { M , 1 − M } at the end. The strategy is fully sp eciﬁed b y a join t input state ν . It is not hard to see that every parallel strategy can be expressed as an adaptive strategy , ho wev er the reverse is not true. 7 . . . . . . Φ | Ψ Λ 2 Φ | Ψ Λ 3 Φ | Ψ Λ n Φ | Ψ M Θ[ Φ ] | Θ[ Ψ ] Figure 2: Illustration of a general adaptive strategy Θ with n uses of the black-box channel, and a joint binary POVM measuremen t { M , 1 − M } at the end. The top row uses the giv en black-boxes while the b ottom row depicts the memory system R . The strategy is fully sp eciﬁed by an input state ν and in termediate input-preparation c hannels Λ 2 , . . . , Λ n . 2.3.2 Asymmetric Error Exp onen ts The asymmetric error exp onen t is the only asymptotic error exp onent for quantum c hannel discrimination for which one has b een able to prov e an entropic expression (although still in volving a regularization limit). Concretely , for a parallel strategy as depicted in Figure 1 , the n -shot type-I I error exp onent is characterized by the expression 1 n D ε H ((id R ⊗ Φ ⊗ n )( ν RA n ) ∥ (id R ⊗ Ψ ⊗ n )( ν RA n )) , (2.17) and for the optimal type-I I parallel error exp onent one would optimize ov er all input states: e P (Φ , Ψ , n, ε ) := sup ν RA n 1 n D ε H ((id R ⊗ Φ ⊗ n )( ν RA n ) ∥ (id R ⊗ Ψ ⊗ n )( ν RA n )) . (2.18) One can show that the optimal asymptotic ( n → ∞ ) t yp e-I I parallel error exp onent as the t yp e-I error go es to zero ( ϵ → 0) is precisely the regularized cb-c hannel div ergence [ 48 ]: lim ϵ → 0 lim n →∞ e P (Φ , Ψ , n, ε ) = lim ε → 0 lim n →∞ sup ν RA n 1 n D ε H ((id R ⊗ Φ ⊗ n )( ν RA n ) ∥ (id R ⊗ Ψ ⊗ n )( ν RA n )) (2.19) = lim n →∞ sup ν RA n 1 n D ((id R ⊗ Φ ⊗ n )( ν RA n ) ∥ (id R ⊗ Ψ ⊗ n )( ν RA n )) (2.20) = D cb , reg (Φ ∥ Ψ) . (2.21) While the righ t-hand side is an en tropic expression only inv olving the t w o channels, there is no general w ay to compute it (in fact it is not even kno wn whether it is theoretically computable), and explicit v alues are only known for very small classes of channels suc h as replacer c hannels, for which one can sho w that the regularization limit is not necessary [ 50 ]. In terestingly , even though we ha ve no go o d wa y to compute this expression, it is p ossible to show that it is equal to the corresp onding error exp onent for adaptive discrimination strategies deﬁned as: e A (Φ , Ψ , n, ε ) := sup Θ adaptiv e n -strategy 1 n D ε H (Θ(Φ) ∥ Θ(Ψ)) , (2.22) 8 where for a giv en adaptiv e n -strategy Θ, Θ[ · ] denotes the join t state of all registers before the measuremen t M in Figure 2 (see e.g. [ 2 ] for an explicit expression of e A as an optimization problem inv olving the preparation maps and the input state that speciﬁes Θ). More precisely , the c hain rule for the quantum relative entrop y [ 14 ] implies that lim ε → 0 lim n →∞ e A (Φ , Ψ , n, ε ) = D cb , reg (Φ ∥ Ψ) = lim ϵ → 0 lim n →∞ e P (Φ , Ψ , n, ε ) . (2.23) Therefore, adaptiv e strategies do not oﬀer an asymptotically b etter asymmetric error exp o- nen t than parallel strategies. Besides these asymptotic expressions, the b ounds from ( 2.13 ) can b e turned in to the following one-shot c hannel discrimination b ounds that hold for all α ∈ (0 , 1), α ′ ∈ (1 , ∞ ), and ε ∈ (0 , 1) [ 48 , 50 , 16 ]: e P (Φ , Ψ , n, ε ) ≥ 1 n D cb α (Φ ⊗ n ∥ Ψ ⊗ n ) + α n ( α − 1) log  1 ε  , (2.24) e A (Φ , Ψ , n, ε ) ≤ e D cb , reg α (Φ ∥ Ψ) + α ′ n ( α ′ − 1) log  1 1 − ε  . (2.25) In particular, we get the following b ounds: D cb α (Φ ∥ Ψ) + α n ( α − 1) log  1 ε  ≤ e P (Φ , Ψ , n, ε ) ≤ e A (Φ , Ψ , n, ε ) ≤ e D cb , reg α ′ (Φ ∥ Ψ) + α ′ n ( α ′ − 1) log  1 1 − ε  . (2.26) 2.3.3 Strong Conv erse Exp onents While the Channel Stein’s Lemma ( 2.23 ) gives the optimal ac hiev able error exp onent for the c hannel discrimination task, it has curren tly only b een pro ven with a w eak conv erse, meaning that there is no discrimination strategy that ac hieves a b etter asymptotic t yp e-I I error exp onen t with asymptotically v anishing type-I error. Man y other (similar) information theoretic tasks also satisfy the strong-conv erse prop erty , which (for this discrimination prob- lem) means that the optimal asymptotic type-I I error exp onen t do es not increase even if one allo ws a non-v anishing t yp e-I error probability strictly b ounded aw a y from 1. Concretely , this w ould mean that one can replace the limit ε → 0 with a limit ε → 1 in ( 2.21 ) and the righ t-hand side stays unc hanged (in fact, this w ould imply that one need not take a limit in ε at all and the expression is the same for all ε ∈ (0 , 1)). The works [ 50 , 16 ] hav e established that the optimal asymptotic t yp e-I I error exp onent (for b oth adaptive and parallel strategies) with the a type-I error probability just b ounded a wa y from 1 is b ounded from ab o ve by lim ε → 1 lim sup n →∞ e A/P (Φ , Ψ , n, ε ) (2.27) ≤ lim α → 1 + e D cb , reg α (Φ ∥ Ψ) = lim α → 1 lim n →∞ sup ν RA n 1 n e D α ((id R ⊗ Φ ⊗ n )( ν RA n ) ∥ (id R ⊗ Ψ ⊗ n )( ν RA n )) . (2.28) 9 If one can exchange the limits α → 1 and n → ∞ in the last expression, one would b e able to show that the right-hand side is equal to D cb , reg (Φ ∥ Ψ), thus establishing the strong con verse property for quan tum c hannel discrimination. Ho wev er, actually proving that these limits can b e exchanged has not b een p ossible yet. While this was claimed in [ 15 ], the argumen t turned out to b e incorrect, and this led to the discov ery of errors in the pro of of the generalized quantum Stein’s lemma [ 3 ]. While we no w hav e correct pro ofs of the generalized quan tum Stein’s lemma [ 25 , 30 ] the new approac hes use diﬀerent techn iques that do not seem to be directly applicable for the argument prop osed in [ 15 ] and so the question for the strong- con verse in c hannel discrimination remains op en. Note that the strong-con verse prop erty do es hold for channe ls for which one can sho w that the regularization is not necessary , i.e. for whic h e D cb α (Φ ⊗ n ∥ Ψ ⊗ n ) = n e D cb α (Φ ∥ Ψ). Another w ay to state the strong con verse prop ert y is that for any type-I I error exp o- nen t larger than the optimal achiev able exp onen t with v anishing type-I error (for channel discrimination this is giv en by D cb , reg (Φ ∥ Ψ)), the t yp e-I error has to approac h one in the asymptotic limit. One can then ask the question how fast this type-I error con verges to 1. This conv ergence is t ypically exp onential (in n ), and the corresp onding rate (or exp onent) is called the strong-conv erse exp onent. F ormally , for r > 0 one writes e sc A/P (Φ , Ψ , r ) := lim sup n →∞ inf A/P strategy  − 1 n log(1 − α n )   β n ≤ 2 − rn  , (2.29) e sc A/P (Φ , Ψ , r ) := lim inf n →∞ inf A/P strategy  − 1 n log(1 − α n )   β n ≤ 2 − rn  , (2.30) where A/P indicates the use of adaptiv e or parallel strategies, and α n / β n are the type-I/I I error probabilities of the chosen strategy with n channel uses. In [ 16 ], it w as shown that these exp onents are equal for adaptive and parallel strategies and are given by e sc A (Φ , Ψ , r ) = e sc P (Φ , Ψ , r ) = sup α> 1 α − 1 α ( r − e D cb , reg α  Φ ∥ Ψ)  , (2.31) where additionally 2 e sc A/P = e sc A/P = : e sc A/P . Note that the strong con v erse exp onen t is non-zero if and only if r > inf α> 1 e D cb , reg α (Φ ∥ Ψ), and it is not known whether inf α> 1 e D cb , reg α (Φ ∥ Ψ) ? = D cb , reg (Φ ∥ Ψ) , (2.32) whic h corresp onds precisely to the problem of establishing whether the (exp onential) strong con verse prop ert y holds in quan tum cha nnel discrimination [ 16 , 15 ]. 2.3.4 Symmetric Error Exp onen ts In the symmetric setting with equal (prior) probability , if one is allow ed to use the channel only once, we can easily show that the smallest exp ected error probability is given by 2 The pap ers [ 16 , 50 ] only consider e sc A/P , how ever it is easy to see that [ 50 , Prop. 20] also works for e sc A , whic h together with the results from [ 16 ] implies that the tw o limits are equal. 10 p err (Φ , Ψ) : = inf ν ∈D ( R A ) inf 0 ≤ M ≤ 1 H 1 2  T r(( 1 H − M )Φ( ν )) + T r( M Ψ( ν ))  (2.33) = 1 2  1 − 1 2 ∥ Φ − Ψ ∥ ⋄  , (2.34) where ∥·∥ ⋄ is the diamond norm [ 49 , Chapter 3]. Similarly , for multiple uses of the chan- nel one can again think ab out adaptiv e and parallel strategies, and arriv e at the following expressions for the error probabilities: p P err (Φ , Ψ , n ) : = p err (Φ ⊗ n , Ψ ⊗ n ) , (2.35) p A err (Φ , Ψ , n ) : = inf Θ adaptive n -strategy 1 2  1 − 1 2 ∥ Θ[Φ] − Θ[Ψ] ∥ 1  . (2.36) Since ev ery parallel strategy is an adaptive strategy it is easy to see that p A err (Φ , Ψ , n ) ≤ p P err (Φ ⊗ n , Ψ ⊗ n ) . (2.37) Moreo ver, in con trast to the asymmetric setting ( 2.23 ), there can b e a strict separation b et w een the optimal asymptotic parallel and adaptive error exp onen ts in the symmetric setting, i.e. there exist c hannels Φ , Ψ suc h that [ 23 , 41 ] lim sup n →∞ − 1 n log( p P err (Φ ⊗ n , Ψ ⊗ n )) < lim inf n →∞ − 1 n log( p A err (Φ , Ψ , n )) . (2.38) There are curren tly no matc hing upp er and lo w er b ounds for these symmetric error exp onen ts, although using the established results for symmetric state discrimination one can obtain the following t w o b ounds. Lemma 2.2 ([ 51 , Prop osition 21]) F or two quantum channels Φ , Ψ : L ( A ) → L ( B ) : lim sup n →∞ − 1 n log( p A err (Φ , Ψ , n )) ≤ D cb max (Φ ∥ Ψ) (2.39) Lemma 2.3 F or two quantum channels Φ , Ψ : L ( A ) → L ( B ) : lim inf n →∞ − 1 n log( p P err (Φ ⊗ n , Ψ ⊗ n )) ≥ sup α ∈ (0 , 1) (1 − α ) D reg , cb α (Φ ∥ Ψ) (2.40) = sup α ∈ (0 , 1) (1 − α ) lim n →∞ 1 n sup ν ∈D ( R A n ) D α ((id R ⊗ Φ ⊗ n )( ν ) ∥ (id R ⊗ Ψ ⊗ n )( ν )) (2.41) = ξ cb , reg (Φ ∥ Ψ) (2.42) wher e ξ cb , reg (Φ ∥ Ψ) is the r e gularize d cb-channel diver genc e asso ciate d to the Chernoﬀ diver- genc e ξ ( ρ ∥ σ ) : = sup α ∈ (0 , 1) (1 − α ) D α ( ρ ∥ σ ) . 11 Pro of. It is easy to see that the limit in n on the right-hand side is increasing. Hence, we can tak e the three suprema in any order, and let α , n and ν = ν AR n b e chosen such that sup α ∈ (0 , 1) (1 − α ) D cb , reg α (Φ ∥ Ψ) = (1 − α ) 1 n D α ((id R ⊗ Φ ⊗ n )( ν ) ∥ (id R ⊗ Ψ ⊗ n )( ν )) + δ (2.43) for some δ > 0 arbitrarily small. F rom here on in this argument w e will mak e the id R implicit, i.e. we will write Φ ⊗ n ( ν ) for (id R ⊗ Φ ⊗ n )( ν ). No w consider the parallel discrimination strategy that, given m copies of either Φ or Ψ divides the m copies into k = k ( m ) = ⌊ m/n ⌋ groups of n channels, inputs the state ν in to each group and just discards any remaining channels. W e are then left with k copies of either Φ ⊗ n ( ν ) or Ψ ⊗ n ( ν ), and by the Chernoﬀ b ound for quan tum state discrimination [ 1 ] we can discriminate these with error probability − log p err ((Φ ⊗ n ( ν )) ⊗ k , (Ψ ⊗ n ( ν )) ⊗ k ) ≥ k (1 − α ) D α (Φ ⊗ n ( ν ) ∥ Ψ ⊗ n ( ν )) + o ( k ) (2.44) = nk sup α ∈ (0 , 1) (1 − α ) D cb , reg α (Φ ∥ Ψ) − k nδ + o ( k ) (2.45) Since n is ﬁxed, it is easy to see that lim m →∞ ⌊ m/n ⌋ n m = 1 (2.46) and hence lim inf m →∞ 1 m − log( p P err (Φ ⊗ m , Ψ ⊗ m )) ≥ lim inf m →∞ 1 m − log p err ((Φ ⊗ n ( ν )) ⊗ k ( m ) , (Ψ ⊗ n ( ν )) ⊗ k ( m ) ) (2.47) = lim inf m →∞ ( nk ( m ) m sup α ∈ (0 , 1) (1 − α ) D cb , reg α (Φ ∥ Ψ) − δ nk ( m ) m + o ( k ) m ) (2.48) = sup α ∈ (0 , 1) (1 − α ) D cb , reg α (Φ ∥ Ψ) − δ (2.49) and δ w as arbitrary small. 2.4 Multiplicativ e Domain F or an y unital completely p ositiv e (UCP) linear map Φ : L ( H ) → L ( H ), the multiplic ative domain is deﬁned as M Φ :=  X ∈ L ( H ) | ∀ Y ∈ L ( H ) : Φ( X Y ) = Φ( X )Φ( Y ) Φ( Y X ) = Φ( Y )Φ( X )  . (2.50) Recall that an y UCP linear map Φ satisﬁes the Schwarz ine quality : Φ( X † X ) ≥ Φ( X † )Φ( X ) for all X ∈ L ( H ). Moreov er, Choi show ed that [ 6 ] M Φ = { X ∈ L ( H ) | Φ( X † X ) = Φ( X † )Φ( X ) , Φ( X X † ) = Φ( X )Φ( X † ) } . (2.51) 12 Belo w, we prov e the rank non-decreasing prop ert y of unital quantum channels in tw o simple lemmas, which also app eared in [ 39 ]. W e pro vide short pro ofs for completeness. Lemma 2.4 L et Φ : L ( H ) → L ( H ) b e a unital quantum channel. Then, ∀ X ≥ 0 : rank Φ( X ) ≥ rank X . (2.52) Pro of. Recall from Uhlmann’s theorem that for Hermitian matrices X , Y ∈ L ( H ), Φ( X ) = Y for some unital quantum channel Φ ⇐ ⇒ λ ( X ) ≻ λ ( Y ), where λ ( X ) , λ ( Y ) ∈ R dim H are the v ectors of eigenv alues (coun ted with m ultiplicity) of X , Y respectively , and ≻ denotes the ma jorization preorder [ 49 , see Chap. 4]. No w, let X , Y b e Hermitian matrices such that Y = Φ( X ). Let λ ( X ) = ( r 1 , r 2 , . . . , r n , 0 , . . . , 0) , (2.53) λ ( Y ) = ( s 1 , s 2 . . . , s m , 0 , . . . , 0) , (2.54) where n = rank( X ) , m = rank( Y ), and the en tries are arranged in a non-increasing order. W e wan t to pro v e that m ≥ n . Supp ose this is not the case and m < n . Then, since Φ is trace preserving and λ ( X ) ≻ λ ( Y ), w e get r 1 + . . . + r m + . . . + r n = s 1 + . . . + s m (2.55) r 1 + . . . + r m ≥ s 1 + . . . + s m , (2.56) whic h leads to a contradiction. Hence, m ≥ n . Lemma 2.5 L et Φ : L ( H ) → L ( H ) b e a unital quantum channel. Then, for an ortho gonal pr oje ction P = P † = P 2 , rank(Φ( P )) = rank( P ) ⇐ ⇒ P ∈ M Φ . (2.57) Pro of. Let P ∈ M Φ . Then, Φ( P ) = Φ( P P ) = Φ( P )Φ( P ), so that Φ( P ) is also a pro jection. Moreo ver, since Φ is trace-preserving, we obtain the desired assertion: rank(Φ( P )) = T rΦ( P ) = T r P = rank( P ) . (2.58) Con versely , supp ose rank(Φ( P )) = rank( P ). Let Q b e the orthogonal pro jection onto im Φ( P ). Since Φ is unital, w e get Φ( P ) ≤ ∥ Φ( 1 H ) ∥ ∞ Q = Q [ 40 ]. Since Φ is trace- preserving, we get T rΦ( P ) = T r P = rank P = rank(Φ( P )) = rank Q = T r Q , implying that Φ( P ) = Q is a pro jection. Hence, Φ( P ) = Φ( P )Φ( P ) ≤ Φ( P P ) = Φ( P ) , (2.59) whic h, according to Eq. ( 2.51 ), implies that P ∈ M Φ . 13 3 Main results W e start this section with a brief explanation of the (well-kno wn) algebraic structure of idemp oten t c hannels that is essential for phrasing our main results. Let P : L ( H ) → L ( H ) b e an idemp otent quan tum channel ( P = P 2 , where P 2 = P ◦ P ) suc h that P ( 1 H ) has full rank 3 . Then, the image of the adjoint P ∗ : L ( H ) → L ( H ) is a unital ∗− subalgebra 4 im( P ∗ ) ⊆ L ( H ) [ 31 ]. Since H is ﬁnite-dimensional, im( P ∗ ) ∼ = ⊕ K k =1 M d k ( C ) in the sense of ∗− algebra isomorphism [ 45 , Theorem 11.2], and is canonically represented on H . Hence, there exists an orthogonal decomp osition H = ⊕ K k =1 D k ⊗ C k with dim D k = d k suc h that [ 45 , Theorem 11.9] (see also App endix A ): im( P ∗ ) = M k ( L ( D k ) ⊗ 1 C k ) . (3.2) In fact, the channel acts like a conditional exp ectation [ 54 , Prop osition 1.5]: ∀ X ∈ L ( H ) : P ( X ) = X k V k  T r C k ( V † k X V k ) ⊗ δ k  V † k , (3.3) P ∗ ( X ) = X k V k  T r C k  ( V † k X V k )(1 D k ⊗ δ k )  ⊗ 1 C k  V † k , (3.4) where V k : D k ⊗ C k → H are the canonical inclusion isometries, T r C k is the partial trace o ver C k , and δ k ∈ D ( C k ) full rank states. Let δ := ⊕ k (1 D k ⊗ d C k δ k ). Clearly , δ has full rank. Moreo ver, one can c heck P ( X ) = √ δ P T r ( X ) √ δ (3.5) where P T r = P ∗ T r is the unique trace-preserving conditional exp ectation on to im( P ∗ ). Note that Eqs. ( 3.3 ), ( 3.4 ) can alternatively b e written in terms of a direct sum [ 18 ]: P = M k id D k ⊗ R δ k (3.6) P ∗ = M k id D k ⊗ R ∗ δ k , (3.7) where id D k is the identit y c hannel and R δ k ( · ) = T r( · ) δ k is a replacer channel. Remark 3.1 In terms of blo ck matric es, the dir e ct sum Φ ⊕ Ψ : L ( A ⊕ C ) → L ( B ⊕ D ) of two channels Φ : L ( A ) → L ( B ) , Ψ : L ( C ) → L ( D ) c an b e deﬁne d as ∀ X ∈ L ( A ) , ∀ Y ∈ L ( C ) : Φ ⊕ Ψ  X ∗ ∗ Y  :=  Φ( X ) 0 0 Ψ( Y )  . (3.8) 3 Note that this is equiv alent to saying that P admits a full-rank inv ariant state δ = P ( δ ). 4 This means that im( P ∗ ) ⊆ L ( H ) is a vector subspace such that 1 H ∈ im( P ∗ ) and ∀ X, Y ∈ im( P ∗ ) : X † , X Y ∈ im( P ∗ ) . (3.1) 14 The rigid algebraic structure of idemp otent c hannels makes them amenable to study . The communication capacities of such channels were recently c haracterized in [ 43 ] (see also [ 42 ], [ 17 ], [ 44 ]), and the problem of simulating such channels from one another w as recen tly studied in [ 10 ]. F or detailed pro ofs of the structure of idemp otent channels stated in this section, see [ 54 , Chapter 6], [ 42 , Theorem 2.7.3] or [ 10 , Prop osition 1]. 3.1 Noiseless vs Noisy: Iden tit y Against an Idemp oten t Channel In this section, w e completely c haracterize the optimal error-exp onen ts for discriminating b et w een the noiseless identit y c hannel and an arbitrary idemp oten t c hannel. Theorem 3.2 L et Q : L ( H ) → L ( H ) b e an idemp otent channel with Q ( 1 H ) of ful l r ank, i.e., Q = ⊕ L l =1 id A l ⊗ R ω l , wher e H = ⊕ L l =1 A l ⊗ B l and ω l ∈ D ( B l ) ar e states with ful l r ank. Then, D min (id H ∥Q ) = D (id H ∥Q ) = D max (id H ∥Q ) = log X l T r min( d A l ,d B l ) ( ω − 1 l ) , (3.9) wher e for k ∈ N , T r k ( X ) = λ 1 ( X ) + . . . + λ k ( X ) denotes the sum of the lar gest k eigenvalues of a Hermitian matrix X . Conse quently, D cb min (id H ∥Q ) = D cb (id H ∥Q ) = D cb max (id H ∥Q ) = log X l T r( ω − 1 l ) , (3.10) and al l the cb-diver genc es ar e additive: ∀ n ∈ N : D cb (id ⊗ n H ∥Q ⊗ n ) = nD cb (id H ∥Q ) . (3.11) Pro of. Let us ﬁrst tackle the conv erse b ound. Consider the follo wing calculation for the case of a single summand D max (id AB ∥ id A ⊗ R ω B ) = sup ρ D max ( ρ AB ∥ ρ A ⊗ ω B ) (3.12) = sup ψ D max ( ψ AB ∥ ψ A ⊗ ω B ) (3.13) = sup ψ log    ( ψ − 1 / 2 A ⊗ ω − 1 / 2 B ) | ψ ⟩⟨ ψ | AB ( ψ − 1 / 2 A ⊗ ω − 1 / 2 B )    ∞ (3.14) ≤ sup Π=Π 2 =Π † T rΠ ≤ min( d A ,d B ) log T r(Π ω − 1 ) (3.15) = log T r min( d A ,d B ) ( ω − 1 ) , (3.16) where we restricted the optimization to pure states b ecause of the quasi-conv exit y of D max . T o obtain the inequality , w e used the fact that for an y | ψ ⟩ AB = P i √ µ i | α i ⟩ A | β i ⟩ B (Sc hmidt decomp osition), we hav e ψ − 1 / 2 A | ψ ⟩ AB = P i | α i ⟩ A | β i ⟩ B , so that    ( ψ − 1 / 2 A ⊗ ω − 1 / 2 B ) | ψ ⟩⟨ ψ | AB ( ψ − 1 / 2 A ⊗ ω − 1 / 2 B )    ∞ = X i ⟨ β i | ω − 1 | β i ⟩ = T r(Π ω − 1 ) , (3.17) 15 where Π = P i | β i ⟩⟨ β i | is an orthogonal pro jection with T rΠ ≤ SR( ψ ) ≤ min( d A , d B ). The ﬁnal equality in Eq. ( 3.16 ) follows from Ky F an’s maxim um principle [ 4 , Problem I.6.15 and Exercise I I.1.13]. F or the general case, note that for any pro jective measuremen t 5 ( P l ) l ⊆ L ( H ) and a p ositiv e semi-deﬁnite X ∈ L ( H ), the generalized pinching inequality shows that [ 53 ] X ≤ X l α l P l X P l (3.18) for any  α ∈ R L satisfying diag (  α ) ≥ J , where J ∈ M L ( R ) is the all-ones matrix. Deﬁne t l = T r min( d A l ,d B l ) ( ω − 1 l, 2 ), the sum T = P l t l and let α l = T /t l . This  α satisﬁes the required constrain t diag(  α ) ≥ J , since for any v ector | v ⟩ ∈ C L , ⟨ v | J | v ⟩ =      X l v l      2 =      X l v l √ t l √ t l      2 ≤ X l | v l | 2 t l X l t l = X l T | v l | 2 t l = ⟨ v | diag (  α ) | v ⟩ . (3.19) Hence, choosing P l = V l V † l to b e the orthogonal pro jection onto A l ⊗ B l ⊆ H with V l : A l ⊗ B l → H b eing the canonical inclusion isometry , generalized pinching shows ∀ ρ ∈ D ( H ) : ρ ≤ T X l 1 t l P l ρP l ≤ T X l V l  T r B l ( V † l ρV l ) ⊗ ω l  V † l = T Q ( ρ ) , (3.20) where the second inequality follows from the previous calculation (Eq. ( 3.16 )). Hence, D max (id ∥ ⊕ l id A l ⊗ R ω l ) ≤ log T = log X l T r min( d A l ,d B l ) ( ω − 1 l ) . (3.21) T o sho w ac hiev ability , consider the following calculation for the case of a single summand: sup ψ D ( ψ AB ∥ ψ A ⊗ ω B ) = sup ψ − T r( ψ AB log( ψ A ⊗ ω B )) (3.22) = sup ψ − T r( ψ AB (log ψ A ⊗ 1 B + 1 A ⊗ log ω B )) (3.23) = sup ψ − T r( ψ A log ψ A ) − T r( ψ B log ω B ) (3.24) ≥ max { µ i } i H ( { µ i } ) + X i µ i log( λ i ( ω − 1 )) (3.25) ( a ) = log X i λ i ( ω − 1 ) (3.26) = log T r min( d A ,d B ) ( ω − 1 ) (3.27) where the mi inequality follows by choosing | ψ ⟩ AB = P i √ µ i | α i ⟩ A | β i ⟩ B to b e of full Schmidt rank such that | β i ⟩ B are precisely the eigenv ectors of ω − 1 B asso ciated with the largest min( d A , d B ) 5 This means that eac h P l = P 2 l = P † l is an orthogonal pro jection and P l P l = 1 H . 16 eigen v alues, and the equality in ( a ) follo ws from maximizing a function of the form H ( { µ i } i )+ P i µ i c i o ver the simplex of probabilit y distributions { µ i } i : max µ i ≥ 0 , P i µ i =1 H ( { µ i } i ) + X i µ i c i ! = log X i 2 c i ! , (3.28) see App endix D . This pro ves that D (id AB ∥ id A ⊗ R ω B ) = D max (id AB ∥ id A ⊗ R ω B ) . (3.29) F or D min ( ·∥· ), w e can do a similar calculation: D min (id AB ∥ id A ⊗ R ω B ) = sup ρ D min ( ρ AB ∥ ρ A ⊗ ω B ) (3.30) ≥ sup ψ D min ( ψ AB ∥ ψ A ⊗ ω B ) (3.31) = sup ψ − log T r( ψ AB ( ψ A ⊗ ω B )) (3.32) ≥ max { µ i } i − log X i µ 2 i 1 λ i ( ω − 1 ) (3.33) ( a ) = − log 1 P i λ i ( ω − 1 ) (3.34) = log T r min( d A ,d B ) ( ω − 1 ) , (3.35) where the middle inequality follows by c ho osing | ψ ⟩ AB = P i √ µ i | α i ⟩ A | β i ⟩ B to b e of full Sc hmidt rank suc h that the Schmidt vectors | β i ⟩ B are precisely the eigenv ectors of ω − 1 B asso- ciated with the largest min( d A , d B ) eigen v alues, and the equalit y in ( a ) follows b y minimizing a function of the form P i µ 2 i c i for non-negativ e num b ers { c i } i o ver the simplex of probabilit y distributions { µ i } i (see App endix D ): min { µ i } i X i µ 2 i c i = 1 P i 1 /c i . (3.36) The full achiev ability result follo ws from a similar argument. Consider H = ⊕ l A l ⊗ B l and let | ψ ⟩ = ⊕ l √ p l P i √ µ i,l | α i,l ⟩ A l | β i,l ⟩ B l , where p l ≥ 0, P l p l = 1, and within eac h l − blo c k, the vectors | α i,l ⟩ ∈ A l and | β i,l ⟩ ∈ B l and the Schmidt co eﬃcients µ i,l are chosen as ab ov e. Let W l : A l ⊗ B l → H b e the canonical inclusion isometries, and Q l := id A l ⊗ R ω l . Then, 17 D min ( ψ ∥Q ( ψ )) = − log T r( ψ Q ( ψ )) (3.37) = − log X l T r( W † l ψ W l Q l ( W † l ψ W l )) (3.38) = − log X l p 2 l T r W † l ψ W l p l Q l W † l ψ W l p l !! (3.39) = − log X l p 2 l 1 T r min( d A l ,d B l ) ( ω − 1 l ) (3.40) = − log 1 P l T r min( d A l ,d B l ) ( ω − 1 l ) (3.41) = log X l T r min( d A l ,d B l ) ( ω − 1 l ) , (3.42) where the ﬁnal tw o equalities follow by choosing p l ∝ T r min( d A l ,d B l ) ( ω − 1 l ). The expressions for the cb − divergences follow easily: D cb (id H ∥Q ) = D (id H ⊗ id H ∥ id H ⊗ Q ) (3.43) = D (id H⊗H ∥ ⊕ l id H⊗ A l ⊗ R ω l ) (3.44) = log X l T r min( d H d A l ,d B l ) ( ω − 1 l ) (3.45) = log X l T r( ω − 1 l ) . (3.46) Finally , the additivity claim follows since the max − cb divergence is alwa ys additive [ 51 ]. Alternativ ely , we can directly chec k that the stated form ula is additiv e. Corollary 3.3 L et Q : L ( H ) → L ( H ) b e an idemp otent channel with Q ( 1 H ) of ful l r ank. Then, for any ϵ ∈ (0 , 1) and r > D cb (id H ∥Q ) : lim n →∞ e P (id H , Q , n, ϵ ) = lim n →∞ e A (id H , Q , n, ϵ ) = D cb (id H ∥Q ) (3.47) e sc P (id H , Q , r ) = e sc A (id H , Q , r ) = r − D cb (id H ∥Q ) (3.48) lim n →∞ − 1 n log( p P err (id H , Q , n )) = lim n →∞ − 1 n log( p A err (id H , Q , n )) = D cb (id H ∥Q ) . (3.49) Pro of. Besides just the asymptotic expressions, w e actually get the following sligh tly stronger n -shot b ounds from ( 2.26 ), for all α ∈ (0 , 1), α ′ > 1, and ϵ ∈ (0 , 1): D cb (id H ∥Q ) + α n ( α − 1) log  1 ϵ  ≤ e P (id H , Q , n, ϵ ) (3.50) ≤ e A (id H , Q , n, ϵ ) (3.51) ≤ D cb (id H ∥Q ) + α ′ n ( α ′ − 1) log  1 1 − ϵ  , (3.52) 18 where w e used Theorem 3.2 to deduce that D cb α (id H ∥Q ) = D cb (id H ∥Q ) = e D cb α ′ (id H ∥Q ) for all α ∈ (0 , 1), α ′ > 1, as well as D cb (id ⊗ n H ∥Q ⊗ n ) = nD cb (id H ∥Q ). The ﬁrst claim then follows b y taking the limit n → ∞ . The second claim follows from Eq. ( 2.31 ) and the additivity result from Theorem 3.2 : e sc A (id H , Q , r ) = e sc P (id H , Q , r ) = sup α> 1 α − 1 α ( r − e D cb , reg α  id H ∥Q )  (3.53) = r − D cb (id H ∥Q ) . (3.54) The ﬁnal claim follows from Lemma 2.2 and Lemma 2.3 : lim sup n →∞ − 1 n log( p A err (id H , Q , n )) ≤ D cb max (id H ∥Q ) = D cb (id H ∥Q ) , (3.55) lim inf n →∞ − 1 n log( p P err (id H , Q , n )) ≥ sup α ∈ (0 , 1) (1 − α ) D cb , reg α (id H ∥Q ) (3.56) = D cb (id H ∥Q ) , (3.57) where D cb max (id H ∥Q ) = D cb (id H ∥Q ) = D cb , reg α (id H ∥Q ) again follow from Theorem 3.2 . 3.2 Noisy vs Noisy: Tw o Idemp otent Channels In this section, w e attempt to characterize the optimal error-exp onents for discriminating b et w een tw o idemp otent quantum c hannels P , Q : L ( H ) → L ( H ). Throughout this section, w e assume that b oth P ( 1 H ) and Q ( 1 H ) ha v e full rank. W e b egin by noting a general lemma. Lemma 3.4 L et P , Q : L ( H ) → L ( H ) b e idemp otent channels such that b oth P ( 1 H ) , Q ( 1 H ) have ful l r ank. If im( Q ∗ ) ⊆ im( P ∗ ) , then D ( P ∥Q ) = + ∞ . Pro of. As noted already , there exist p ositiv e deﬁnite op erators δ ∈ im( P ) and ω ∈ im( Q ) suc h that (see Eq. ( 3.5 )) P ( X ) = √ δ P T r ( X ) √ δ , Q ( X ) = √ ω Q T r ( X ) √ ω , (3.58) where P T r , Q T r are the unique trace-preserving conditional exp ectations on to im( P ∗ ) , im( Q ∗ ) resp ectiv ely , see Eq. ( 3.5 ). Assume that im( Q ∗ ) ⊆ im( P ∗ ). Since any ∗− subalgebra in L ( H ) is generated b y its pro jections, there exists a pro jection Q ∈ im( Q ∗ ) suc h that Q ∈ im( P ∗ ). Moreo ver, since P T r , Q T r are unital quantum c hannels, Lemmas 2.4 , 2.5 show that rank( P ( Q )) = rank( P T r ( Q )) > rank Q = rank( Q T r ( Q )) = rank( Q ( Q )) . (3.59) Hence, b y c ho osing ρ = Q/ T r Q , we see D ( P ∥Q ) ≥ D ( P ( ρ ) ∥Q ( ρ )) = + ∞ . 19 Remark 3.5 L emma 3.4 also gives a suﬃcient c ondition for e D α ( P ∥Q ) = ∞ when α > 1 , sinc e e D α ( P ∥Q ) ≥ D ( P ∥Q ) . However, a similar statement is not true for α < 1 . F or α < 1 , D α ( P ∥Q ) = ∞ if and only if ther e exists a state ρ such that P ( ρ ) and P ( σ ) ar e ortho gonal (to se e this note that the set of states is c omp act, and for α < 1 , D α is c ontinuous). However, for idemp otent channels P , Q : L ( H ) → L ( H ) with P ( 1 H ) , Q ( 1 H ) of ful l r ank, this never happ ens. T o se e this, note that The or em 3.2 implies that for any state ρ ∈ D ( H ) the supp ort pr oje ctions satisfy Π ρ ≤ Π P ( ρ ) and Π ρ ≤ Π Q ( ρ ) , and so P ( ρ ) and Q ( ρ ) c annot b e ortho gonal. In accordance with Lemma 3.4 , we no w restrict ourselves to discriminating b et ween idemp oten t c hannels P , Q : L ( H ) → L ( H ) with im( Q ∗ ) ⊆ im( P ∗ ) ⊆ L ( H ). Since H is ﬁnite-dimensional, these image algebras are ∗ -isomorphic to direct sums of matrix blo cks [ 45 , Theorem 11.2]: im( P ∗ ) ∼ = ⊕ K k =1 M d k ( C ) and im( Q ∗ ) ∼ = ⊕ L l =1 M a l ( C ), and there exist orthogonal direct sum decomp ositions of the underlying Hilb ert space (see App endix A ): H = ⊕ K k =1 ⊕ L l =1 A l ⊗ B k,l ⊗ C k (3.60) = ⊕ k ( ⊕ l A l ⊗ B k,l ) | {z } := D k ⊗ C k (3.61) = ⊕ l A l ⊗ ( ⊕ k B k,l ⊗ C k ) | {z } := E l (3.62) suc h that dim D k = d k , dim A l = a l , dim B k,l ≥ 0, dim C k ≥ 1 and im( P ∗ ) = M k L ( D k ) ⊗ 1 C k (3.63) im( Q ∗ ) = M l L ( A l ) ⊗ 1 E l . (3.64) Denote the orthogonal pro jection onto a single k , l summand A l ⊗ B k,l ⊗ C k ⊆ H by Π k,l , so that the minimal central pro jections of im( P ∗ ) and im( Q ∗ ) b ecome P k = X l Π k,l and Q l = X k Π k,l , (3.65) whic h pro ject on to D k ⊗ C k ⊆ H and A l ⊗ E l ⊆ H , resp ectively . Note that P k Q l = Q l P k = Π k,l and for some k , l , the pro jection Π k,l migh t b e zero, which corresp onds to the fact that dim B k,l = 0 for those k , l . W e can now write the actions of P , Q as direct sums with resp ect to the appropriate orthogonal decomp ositions (see Eq. ( 3.6 )): P = M k id D k ⊗ R δ k , Q = M l id A l ⊗ R ω l , (3.66) where δ k ∈ D ( C k ) and ω l ∈ D ( E l ) are full-rank states. Alternativ ely , P ( X ) = X k V k P k ( V † k X V k ) V † k , (3.67) Q ( X ) = X l W l Q l ( W † l X W l ) W † l , (3.68) 20 where V k : D k ⊗ C k → H and W l : A l ⊗ E l → H are the canonical isometries, and P k : L ( D k ⊗ C k ) → L ( D k ⊗ C k ) and Q l : L ( A l ⊗ E l ) → L ( A l ⊗ E l ) act as tensor pro ducts of iden tity and replacer: P k = id D k ⊗ R δ k and Q l = id A l ⊗ R ω l . Since im( Q ∗ ) ⊆ im( P ∗ ), P ∗ ◦ Q ∗ = Q ∗ and Q ◦ P = Q . (3.69) In general, ω l ∈ D ( E l ) = D ( ⊕ k B k,l ⊗ C k ) can hav e coherences across diﬀerent k -blo cks, and within each k -blo ck, it can b e entangled across the B k,l − C k cut, which mak es it trickier to compute the corresp onding c hannel divergences D ( P ∥Q ) (see Section 3.2.2 ). Hence, in the next section, w e ﬁrst work with the assumption that P and Q share a common inv ariant state, whic h forces ω l to b e k − blo ck diagonal and pro duct across B k,l − C k . 3.2.1 Common Inv ariant State With the setup of the previous section, we further assume that P , Q share a common inv arian t state τ = P ( τ ) = Q ( τ ). Since any such τ has to b e blo ck diagonal with resp ect to b oth the k and l blo ck decomp ositions (Eq. ( 3.66 )), the states ω l ∈ D ( E l ) = D ( ⊕ k B k,l ⊗ C k ) must b e k − blo ck diagonal and pro duct across the B k,l − C k cut: ω l = M k p kl τ k,l ⊗ δ k , (3.70) where for each l , p k,l ≥ 0 , P k p k,l = 1, and τ k,l ∈ D ( B k,l ) are full-rank states. Note that since ω l has full rank, p kl > 0 whenev er dim B k,l > 0. W e can no w state the main result of this section. Theorem 3.6 L et P , Q : L ( H ) → L ( H ) b e idemp otent channels with im( Q ∗ ) ⊆ im( P ∗ ) that shar e a c ommon ful l-r ank invariant state. Then, H de c omp oses as H = ⊕ K k =1 ⊕ L l =1 A l ⊗ B k,l ⊗ C k (3.71) = ⊕ k ( ⊕ l A l ⊗ B k,l ) | {z } := D k ⊗ C k (3.72) = ⊕ l A l ⊗ ( ⊕ k B k,l ⊗ C k ) | {z } := E l , (3.73) with P = M k id D k ⊗ R δ k , Q = M l id A l ⊗ R ω l , (3 .74) wher e δ k ∈ D ( C k ) and ω l ∈ D ( E l ) ar e ful l-r ank states such that ω l = M k p k,l τ k,l ⊗ δ k (3.75) 21 for some pr ob ability distributions ( p k,l ) k and ful l-r ank states τ k,l ∈ D ( B k,l ) . F urthermor e, D min ( P ∥Q ) = D ( P ∥Q ) = D max ( P ∥Q ) = max k log X l T r min( d A l ,d B k,l ) ( τ − 1 k,l ) p k,l ! (3.76) D cb min ( P ∥Q ) = D cb ( P ∥Q ) = D cb max ( P ∥Q ) = max k log X l T r( τ − 1 k,l ) p k,l ! . (3.77) Conse quently, the cb -diver genc es ar e al l additive: ∀ n ∈ N : D cb ( P ⊗ n ∥Q ⊗ n ) = nD cb ( P ∥Q ) . (3.78) Pro of. The pro of of the ﬁrst claim is describ ed in the discussion that precedes the theorem (see also App endix A ). F or the second claim, note that D max ( P ∥Q ) := sup ρ ∈D ( H ) D max ( P ( ρ ) ∥Q ( ρ )) (3.79) = sup ρ ∈D ( H ) D max ( P ( ρ ) ∥Q ◦ P ( ρ )) (3.80) = sup ρ ∈ im( P ) D max ( ρ ∥Q ( ρ )) (3.81) = sup ρ ∈ im( P ) pure D max  ρ   Q ( ρ )  , (3.82) where the equality Q ◦ P = Q follows from im( Q ∗ ) ⊆ im( P ∗ ) and the ﬁnal equalit y follows from the join t quasi-conv exit y of D max . Note that pure states in im( P ) = ⊕ k L ( D k ) ⊗ δ k are of the form ρ = 0 ⊕ . . . ⊕ ( ψ ⊗ δ k ) ⊕ . . . ⊕ 0, supp orted on D k ⊗ C k for some k and pure state ψ ∈ L ( D k ). Recall that D k = ⊕ l A l ⊗ B k,l . Moreov er, Q ( ρ ) = M l q l T r B k,l ( ψ l ) ⊗ ω l = M k ′ M l p k ′ l q l T r B k,l ( ψ l ) ⊗ τ k ′ ,l ⊗ δ k ′ (3.83) where ψ l := 1 q l V † l | k ψ V l | k ∈ D ( A l ⊗ B k,l ) and q l = T r( V † l | k ψ V l | k ), with V l | k : A l ⊗ B k,l → D k b eing the canonical inclusion isometries. Hence, sup ρ ∈ im( P ) pure D max ( ρ ∥Q ( ρ )) = max k sup ψ ∈D ( D k ) D max ( ψ ∥ ⊕ l p k,l q l T r B k,l ( ψ l ) ⊗ τ k,l ) + D max ( δ k ∥ δ k ) (3.84) = max k D max (id D k ∥ ⊕ l p k,l id A l ⊗ R τ k,l ) (3.85) = max k log X l T r min( d A l ,d B k,l ) ( τ − 1 k,l ) p k,l ! , (3.86) where the last equality follows from Theorem 3.2 . 22 F or the ac hiev ability claim, we can c ho ose the blo c k k ∗ that achiev es the maxim um in Eq. ( 3.86 ) and let ρ ∗ = 0 ⊕ . . . ⊕ ( ψ ∗ ⊗ δ k ∗ ) ⊕ . . . ⊕ 0 b e supp orted on D k ∗ ⊗ C k ∗ suc h that ψ ∗ ∈ D ( D k ∗ ) is constructed as in the pro of of Theorem 3.2 (see Eq. ( 3.37 )-( 3.42 )), so that D min ( ρ ∗ ∥Q ( ρ ∗ )) = log X l T r min( d A l ,d B k ∗ ,l ) ( τ − 1 k ∗ ,l ) p k ∗ ,l (3.87) = max k log X l T r min( d A l ,d B k,l ) ( τ − 1 k,l ) p k,l ! . (3.88) The expression for the cb − divergences follow similarly . Finally , the additivity claim follo ws since the max − cb div ergence is alw a ys additive [ 51 ]. Alternativ ely , we can directly chec k that the stated form ula is additiv e. Corollary 3.7 L et P , Q : L ( H ) → L ( H ) b e idemp otent channels that shar e a c ommon ful l-r ank invariant state. Then, for any ϵ ∈ (0 , 1) and r > D cb ( P ∥Q ) : lim n →∞ e P ( P , Q , n, ϵ ) = lim n →∞ e A ( P , Q , n, ϵ ) = D cb ( P ∥Q ) (3.89) e sc P ( P , Q , r ) = e sc A ( P , Q , r ) = r − D cb ( P ∥Q ) . (3.90) Pro of. If im( Q ∗ ) ⊆ im( P ∗ ), then Lemma 3.4 sho ws that D cb ( P ∥Q ) = + ∞ . Hence, lim inf n →∞ e P ( P , Q , n, ϵ ) ≥ + ∞ . Otherwise, if im( Q ∗ ) ⊆ im( P ∗ ), again as b efore, we pro v e not just the asymptotic expressions, but slightly stronger n -shot b ounds. W e get the follow- ing from ( 2.26 ) for all α ∈ (0 , 1), α ′ > 1, and ϵ ∈ (0 , 1): D cb ( P ∥Q ) + α n ( α − 1) log  1 ϵ  ≤ e P ( P , Q , n, ϵ ) (3.91) ≤ e A ( P , Q , n, ϵ ) (3.92) ≤ D cb ( P ∥Q ) + α ′ n ( α ′ − 1) log  1 1 − ϵ  , (3.93) and Eq. ( 3.89 ) follows by taking the limit n → ∞ . Note that here, we used Theorem 3.6 to infer that D cb α ( P ∥Q ) = D cb ( P ∥Q ) = e D cb α ′ ( P ∥Q ) for all α ∈ (0 , 1), α ′ > 1. Finally , Eq. ( 3.90 ) follo ws from Eq. ( 2.31 ) and the additivity result from Theorem 3.6 : e sc A ( P , Q , r ) = e sc P ( P , Q , r ) = sup α> 1 α − 1 α ( r − e D reg , cb α  P ∥Q )  (3.94) = r − D cb ( P ∥Q ) . (3.95) Corollary 3.8 L et P , Q : L ( H ) → L ( H ) b e idemp otent channels that shar e a c ommon ful l-r ank invariant state. If im( Q ∗ ) ⊆ im( P ∗ ) , then lim n →∞ − 1 n log( p P err ( P , Q , n )) = lim n →∞ − 1 n log( p A err ( P , Q , n )) = D cb ( P ∥Q ) . (3.96) 23 Similarly, if im( P ∗ ) ⊆ im( Q ∗ ) , then lim n →∞ − 1 n log( p P err ( P , Q , n )) = lim n →∞ − 1 n log( p A err ( P , Q , n )) = D cb ( Q∥P ) . (3.97) Pro of. It suﬃces to prov e the case im( Q ∗ ) ⊆ im( P ∗ ). F rom Lemma 2.2 , we can write lim sup n →∞ − 1 n log( p A err ( P , Q , n )) ≤ D cb max ( P ∥Q ) = D cb ( P ∥Q ) , (3.98) where the equality follo ws from Theorem 3.6 . Similarly , using Lemma 2.3 , we get lim inf n →∞ − 1 n log( p P err ( P , Q , n )) ≥ sup α ∈ (0 , 1) (1 − α ) D reg , cb α ( P ∥Q ) (3.99) = D cb ( P ∥Q ) , (3.100) where the equality again follows from Theorem 3.6 . 3.2.2 F ully general case Recall that for idemp oten t c hannels P , Q : L ( H ) → L ( H ) with P ( 1 H ) , Q ( 1 H ) of full rank, the inclusion im( Q ∗ ) ⊆ im( P ∗ ) induces a three-lay er decomp osition of H : H = ⊕ k ⊕ l A l ⊗ B k,l ⊗ C k (3.101) = ⊕ k ( ⊕ l A l ⊗ B k,l ) | {z } := D k ⊗ C k (3.102) = ⊕ l A l ⊗ ( ⊕ k B k,l ⊗ C k ) | {z } := E l , (3.103) suc h that P = M k id D k ⊗ R δ k (3.104) Q = M l id A l ⊗ R ω l . (3.105) When P and Q do not share a common inv arian t state, the states ω l ma y carry coher- ences across k − blo c ks. Moreo v er, within eac h k − blo ck, the state need not factor across the B k,l − C k cut, prev enting the full collapse of div ergences seen in the previous section. Nev- ertheless, in this section, w e derive a general single-letter con v erse b ound on the regularized sandwic hed R´ enyi cb-divergence, whic h suﬃces to establish a strong conv erse b ound on the Stein exp onents. In order to tackle the general case, w e deﬁne the following divergences for each k , l : e D α ( k , l ) := e D α ( P | Π k,l ∥Q| Π k,l ) (3.106) = sup ν =Π k,l ν Π k,l e D α ( P ( ν ) ∥Q ( ν )) , (3.107) 24 where P | Π k,l and Q| Π k,l are the restrictions 6 of P and Q to the subspace Π k,l H = A l ⊗ B k,l ⊗ C k , and the sup is ov er all states ν ∈ D ( H ) satisfying ν = Π k,l ν Π k,l . Deﬁne canonical inclusion isometries W k | l : B k,l ⊗ C k → E l and W l : A l ⊗ E l → H , and the corresp onding embedding and compression maps ι k | l ( · ) := W k | l ( · ) W ∗ k | l , W ∗ l ( · ) = W † l ( · ) W l , (3.108) so that we can restrict the outputs of P | Π k,l , Q| Π k,l to the same co domain L ( A l ⊗ E l ): W ∗ l ◦ P | Π k,l = (id A l ⊗ ι k | l ) ◦ (id A l ⊗ id B k,l ⊗ R δ k ) , (3.109) W ∗ l ◦ Q| Π k,l = id A l ⊗ R ω l . (3.110) Hence, e D α ( k , l ) = e D α  (id A l ⊗ ι k | l ) ◦ (id A l ⊗ id B k,l ⊗ R δ k )    id A l ⊗ R ω l  . (3.111) Similarly , w e deﬁne the stabilized divergences e D cb α ( k , l ) := e D cb α ( P | Π k,l ∥Q| Π k,l ) . (3.112) W e note the follo wing lemma, which follo ws from the multiplicativit y of the completely b ounded α − norms [ 11 ] (see also [ 22 ] and [ 51 , Prop osition 41]). Lemma 3.9 L et R i b e r eplac er channels and N i b e arbitr ary channels for i = 1 , . . . N . Then, e D cb α ( ⊗ N i =1 N i ∥ ⊗ N i =1 R i ) = N X i =1 e D cb α ( N i ∥R i ) . (3.113) With this background, w e can prov e the main result of this section. Theorem 3.10 L et P , Q : L ( H ) → L ( H ) b e idemp otent channels such that b oth P ( 1 H ) and Q ( 1 H ) have ful l r ank and im( Q ∗ ) ⊆ im( P ∗ ) . Then, for α > 1 , e D α ( P ∥Q ) ≤ max k log X l 2 e D α ( k,l ) ! , (3.114) e D cb α ( P ∥Q ) ≤ max k log X l 2 e D cb α ( k,l ) ! , (3.115) wher e e D α ( k , l ) , e D cb α ( k , l ) ar e deﬁne d in Eqs. ( 3.106 ) , ( 3.112 ) . Conse quently, for al l n ∈ N : e D cb α ( P ⊗ n ∥Q ⊗ n ) ≤ n max k log X l 2 e D cb α ( k,l ) ! . (3.116) 6 F or any channel Φ : L ( A ) → L ( B ) and subspace C ⊆ A , the restriction Φ | C : L ( C ) → L ( B ) is deﬁned as Φ C ( · ) := Φ( V C ( · ) V † C ), where V C : C → A is the canonical inclusion isometry . 25 Mor e over, e quality holds in Eqs. ( 3.114 ) - ( 3.116 ) when the r eplac er states ω l ∈ D ( E l ) = D ( ⊕ k B k,l ⊗ C k ) in Eq. ( 3.105 ) ar e of the blo ck-diagonal form ω l = M k p k,l τ k,l ⊗ ν k , (3.117) for some pr ob ability distributions ( p k,l ) k and ful l-r ank states τ k,l ∈ D ( B k,l ) , ν k ∈ D ( C k ) . Pro of. W e b egin with the ﬁrst claim. Note that e D α ( P ∥Q ) := sup ρ ∈D ( H ) e D α ( P ( ρ ) ∥Q ( ρ )) (3.118) = sup ρ ∈D ( H ) e D α ( P ( ρ ) ∥Q ◦ P ( ρ )) (3.119) = sup ρ ∈ im( P ) e D α ( ρ ∥Q ( ρ )) (3.120) = sup ρ ∈ im( P ) pure e D α ( ρ ∥Q ( ρ )) , (3.121) where Q = Q ◦ P follows from im( Q ∗ ) ⊆ im( P ∗ ) and the ﬁnal equality follo ws from the joint quasi-con vexit y of e D α (see e.g. [ 29 , Chapter 7]). Note that pure states in im( P ) are of the form ρ = V k ( ψ ⊗ δ k ) V † k = 0 ⊕ . . . ⊕ ( ψ ⊗ δ k ) ⊕ . . . ⊕ 0, supp orted in D k ⊗ C k for some k and pure state ψ ∈ L ( D k ). Let σ = Q ( ρ ) = P l W l Q l ( W † l ρW l ) W † l := P l σ l . Recall that each D k = ⊕ l A l ⊗ B k,l . Let V l | k : A l ⊗ B k,l → D k b e the canonical inclusion isometries. A straightforw ard calculation shows σ l = W l Q l  W † l V k ( ψ ⊗ δ k ) V † k W l  W † l (3.122) = p l W l  T r B k,l ( ψ l ) ⊗ ω l  W † l , (3.123) where ψ l := 1 p l V † l | k ψ V l | k ∈ D ( A l ⊗ B k,l ) and p l = T r( V † l | k ψ V l | k ). Then, e D α ( ρ ∥ σ ) = 1 α − 1 log T r h σ 1 − α 2 α ρσ 1 − α 2 α  α i (3.124) = 1 α − 1 log T r h √ ρσ 1 − α α √ ρ  α i (3.125) = 1 α − 1 log T r h V k ( ψ ⊗ p δ k ) V † k σ 1 − α α V k ( ψ ⊗ p δ k ) V † k  α i (3.126) = 1 α − 1 log T r  p δ k T r D k  ( ψ ⊗ 1 C k ) V † k σ 1 − α α V k  p δ k  α  (3.127) = 1 α − 1 log T r  X l p δ k T r D k  ( ψ ⊗ 1 C k ) V † k σ 1 − α α l V k  p δ k  α  . (3.128) Deﬁne p ositive op erators X k,l := p δ k T r D k  ( ψ ⊗ 1 C k ) V † k σ 1 − α α l V k  p δ k ∈ L ( C k ) , (3.129) 26 so that e D α ( ρ ∥ σ ) = 1 α − 1 log T r "  X l X k,l  α # (3.130) ≤ 1 α − 1 log " X l (T r X α k,l ) 1 /α # α , (3.131) where the inequality follows from the con v exity of the Schatten norm ∥·∥ α . Recall that E l = ⊕ k B k,l ⊗ C k and let W k | l : B k,l ⊗ C k → E l are the canonical inclusion isometries. Then, b y using the form of σ l from Eq. ( 3.123 ), we see that V † k σ l V k = p l ( V l | k ⊗ 1 C k )  T r B k,l ( ψ l ) ⊗ W † k | l ω l W k | l  ( V † l | k ⊗ 1 C k ) (3.132) is only supp orted on a single k , l blo ck A l ⊗ B k,l ⊗ C k inside D k ⊗ C k . Hence, by retracing our steps, working exclusiv ely with op erators in L ( A l ⊗ B k,l ⊗ C k ), w e can chec k that T r X α k,l = p l T r  ( ψ l ⊗ p δ k )  (T r B k,l ψ l ) 1 − α α ⊗ W † k | l ω 1 − α α l W k | l  ( ψ l ⊗ p δ k )  α (3.133) ≤ p l 2 ( α − 1) e D α ( k,l ) , (3.134) where the inequality follo ws from c ho osing a particular input ψ l ⊗ δ k ∈ L ( A k ⊗ B k,l ⊗ C k ) in Eq. ( 3.106 ), ( 3.111 ). Thus, for the input ρ = V k ( ψ ⊗ δ k ) V † k supp orted in a single k − blo ck, w e obtain e D α ( ρ ∥Q ( ρ )) ≤ α α − 1 log " X l p 1 /α l 2 α − 1 α e D α ( k,l ) # (3.135) ≤ log " X l 2 e D α ( k,l ) # , (3.136) where the ﬁnal equality follo ws from the fact that for any set of non-negativ e num b ers { c l } l max p l ≥ 0 , P l p l =1 α α − 1 log X l p 1 α l c α − 1 α l = log X l c l , (3.137) see App endix D . Finally , Eq. ( 3.114 ) follows by taking a maxim um o ver the k − blo cks: e D α ( P ∥Q ) = sup ρ ∈ im( P ) pure e D α ( ρ ∥Q ( ρ )) (3.138) ≤ max k log " X l 2 e D α ( k,l ) # . (3.139) 27 F or the achiev abilit y claim, note that in Eq. ( 3.131 ), the Sc hatten norm conv exity b ound collapses in to an equality when the hypothesis of the theorem holds: if ω l = ⊕ k p k,l τ k,l ⊗ ν k , then for a ﬁxed k , all X k,l op erators are prop ortional to the same p ositive op erator X k,l ∝ p δ k ν 1 − α α k p δ k (3.140) indep enden t of l . Therefore, we can select the blo c k k ∗ that ac hieves the maximum in Eq. ( 3.139 ) and let ρ ∗ = 0 ⊕ . . . ⊕ ( ψ ∗ ⊗ δ k ∗ ) ⊕ . . . ⊕ 0 b e supp orted on D k ∗ ⊗ C k ∗ , with ψ ∗ ∈ D ( D k ∗ ) chosen in suc h a w a y that ψ ∗ l = V † l | k ψ ∗ V l | k /p ∗ l ac hieves 7 the b ound in Eq. ( 3.134 ), and p ∗ l = T r( V † l | k ψ ∗ V l | k ) ac hiev es the bound in Eq. ( 3.137 ). In the pro of ’s notation, this sho ws e D α ( ρ ∗ ∥Q ( ρ ∗ )) = 1 α − 1 log T r "  X l X k ∗ ,l  α # (3.141) = 1 α − 1 log " X l (T r X α k ∗ ,l ) 1 /α # α (3.142) = α α − 1 log " X l ( p ∗ l ) 1 /α 2 α − 1 α e D α ( k ∗ ,l ) # (3.143) = log " X l 2 e D α ( k ∗ ,l ) # (3.144) = max k log " X l 2 e D α ( k,l ) # . (3.145) The b ound for the cb − div ergence (Eq. ( 3.115 )) follo ws b y doing exactly the same calculation with an extra reference system attac hed (Eq. ( 2.4 )). Finally , Eq. ( 3.116 ) follo ws b ecause Lemma 3.9 shows that the RHS of Eq. ( 3.115 ) is additive, see App endix B . W e emphasize that the upp er b ound in Theorem 3.10 can b e strictly lo ose in general, since the Sc hatten norm conv exity bound used in the ab ov e pro of (Eq. ( 3.131 )) can b e strict. This can happ en even in the simplest case with a single k -blo ck, see App endix C . On the other hand, the stronger structural condition on the replacer states ω l = M k p k,l τ k,l ⊗ ν k (3.146) is suﬃcient to make the Sc hatten step an equality , since for each ﬁxed k it implies that all X k,l in Eq. ( 3.130 ) are prop ortional to each other. This condition, how ev er, migh t not b e necessary , since equality in the Schatten step only requires prop ortionalit y of the resulting op erators X k,l , and this may happ en in more general situations. 7 T o see that this bound is alwa ys attained, see e.g. [ 32 ] 28 Remark 3.11 When P , Q shar e a c ommon ful l r ank-invariant state, the blo ck-diagonal (Eq. ( 3.117 ) ) assumption on ω l r e quir e d in The or em 3.10 is automatic al ly satisﬁe d (Eq. ( 3.70 ) ), and the RHS of Eqs. ( 3.114 ) - ( 3.116 ) match the explicit expr essions derive d in The or em 3.6 . Henc e, in this setting, The or em 3.10 r e c overs the r esult of The or em 3.6 . Corollary 3.12 L et P , Q : L ( H ) → L ( H ) b e idemp otent channels such that b oth P ( 1 H ) , Q ( 1 H ) have ful l r ank and im( Q ∗ ) ⊆ im( P ∗ ) . Then, for any ϵ ∈ (0 , 1) : D cb ( P ∥Q ) ≤ D cb , reg ( P ∥Q ) ≤ lim inf n →∞ e P ( P , Q , n, ϵ ) (3.147) ≤ lim sup n →∞ e A ( P , Q , n, ϵ ) ≤ max k log X l 2 D cb ( k,l ) ! , (3.148) wher e D cb ( k , l ) is deﬁne d as in Eq. ( 3.112 ) . Mor e over, for r > max k  log P l 2 D cb ( k,l )  , e sc P ( P , Q , r ) = e sc A ( P , Q , r ) ≥ sup α> 1 α − 1 α r − max k log X l 2 e D cb α ( k,l ) !! . (3.149) F urthermor e, e quality holds in Eqs. ( 3.147 ) - ( 3.149 ) when the r eplac er states ω l ∈ D ( E l ) = D ( ⊕ k B k,l ⊗ C k ) in Eq. ( 3.105 ) ar e of the blo ck-diagonal form in Eq. ( 3.117 ) . Pro of. The b ound in Eq. ( 3.147 ) alwa ys holds [ 48 ]. T o prov e Eq. ( 3.148 ), we use the b ound from Eq. ( 2.26 ), which holds for all α > 1 , ϵ ∈ (0 , 1): e P ( P , Q , n, ϵ ) ≤ e A ( P , Q , n, ϵ ) (3.150) ≤ e D cb , reg α ( P ∥Q ) + α n ( α − 1) log  1 1 − ϵ  (3.151) ≤ max k log X l 2 e D cb α ( k,l ) ! + α n ( α − 1) log  1 1 − ϵ  , (3.152) where the last inequality follows from Theorem 3.10 . Hence, lim sup n →∞ e A ( P , Q , n, ϵ ) ≤ max k log X l 2 e D cb α ( k,l ) ! , (3.153) and the asserted claim follo ws by taking the limit α → 1 + , since e D cb α ( k , l ) → D cb ( k , l ) as α → 1 + [ 12 , Lemma 33]. The second claim follows from Eq. ( 2.31 ), since e sc P (Φ , Ψ , r ) = e sc A (Φ , Ψ , r ) = sup α> 1 α − 1 α ( r − e D cb , reg α  Φ ∥ Ψ)  (3.154) ≥ sup α> 1 α − 1 α r − max k log X l 2 e D cb α ( k,l ) !! , (3.155) 29 where the last inequality again follows from Theorem 3.10 . F or the ﬁnal assertion, note that when ω l are of the form in Eq. ( 3.117 ), then Theorem 3.10 sho ws that the sandwiched R´ en yi cb-divergences are additiv e for all α > 1: e D cb α ( P ∥Q ) = e D cb , reg α ( P ∥Q ) = max k log X l 2 e D cb α ( k,l ) ! , (3.156) whic h implies the same for the Umegaki div ergence b y taking α → 1 + limit: D cb ( P ∥Q ) = D cb , reg ( P ∥Q ) = max k log X l 2 D cb ( k,l ) ! . (3.157) This collapses all inequalities in Eqs. ( 3.147 )-( 3.149 ) into equalities. 3.3 Pimsner-P opa Indices F or a unital ∗− subalgebra A = ⊕ l L ( A l ) ⊗ 1 B l ⊆ L ( H ) with conditional exp ectation (not necessarily trace-preserving) E A = ⊕ l id A l ⊗ R ω l , the Pimsp er-Pop a index is deﬁned as [ 38 , 19 ]: C ( E A ) := inf { c > 0 | ∀ ρ ∈ D ( H ) : ρ ≤ cE A ( ρ ) } , (3.158) C cb ( E A ) := sup n ∈ N C ( E A ⊗ id M n ( C ) ) . (3.159) A straigh tforw ard calculation sho ws that log C ( E A ) = D max (id ∥ E A ) (3.160) log C cb ( E A ) = D cb max (id ∥ E A ) . (3.161) When E A , T r is the unique trace-preserving conditional exp ectation on to A , i.e., ω l = 1 B l /d B l for all l , explicit formulae for these indices w ere deriv ed in [ 38 , Theorem 6.1]: C ( E A , T r ) = X l min( d A l , d B l ) d B l and C cb ( E A , T r ) = X l d 2 B l , (3.162) whic h arise as a sp ecial case of Theorem 3.2 . More generally , for an arbitrary conditional exp ectation E A = ⊕ l id A l ⊗ R ω l , Theorem 3.2 provides explicit expressions for the indices: C ( E A ) = X l T r min( d A l ,d B l ) ( ω − 1 l ) and C cb ( E A ) = X l T r( ω − 1 l ) . (3.163) Similarly , for tw o unital ∗− subalgebras A ⊆ B ⊆ L ( H ) with unique trace-preserving conditional exp ectations E A , T r and E B , T r , the Pimsp er-Pop a index is deﬁned as [ 38 ] C ( E A , T r , E B , T r ) := inf { c > 0 | ∀ ρ ∈ D ( H ) : E B , T r ( ρ ) ≤ cE A , T r ( ρ ) } , (3.164) C cb ( E A , T r , E B , T r ) := sup n ∈ N C ( E A , T r ⊗ id M n ( C ) , E B , T r ⊗ id M n ( C ) ) . (3.165) 30 As b efore, it is easy to see that log C ( E A , T r , E B , T r ) = D max ( E B , T r ∥ E A , T r ) (3.166) log C cb ( E A , T r , E B , T r ) = D cb max ( E B , T r ∥ E A , T r ) . (3.167) Clearly , b oth E A , T r and E B , T r share a common full-rank in v ariant state, namely the maximally mixed state in D ( H ). Hence, according to Theorem 3.6 (see also App endix A ), w e obtain a three-lay er decomp osition of H : H = ⊕ k ⊕ l A l ⊗ B k,l ⊗ C k (3.168) = ⊕ k ( ⊕ l A l ⊗ B k,l ) | {z } := D k ⊗ C k (3.169) = ⊕ l A l ⊗ ( ⊕ k B k,l ⊗ C k ) | {z } := E l , (3.170) suc h that B = M k L ( D k ) ⊗ 1 C k , (3.171) A = M l L ( A l ) ⊗ 1 E l . (3.172) Moreo ver, E B , T r = ⊕ k id D k ⊗ R δ k and E A , T r = ⊕ l id A l ⊗ R ω l with δ k = 1 C k d C k , and (3.173) ω l = 1 E l d E l = ⊕ k d B k,l d C k P k ′ d B k ′ ,l d C ′ k 1 B k,l d B k,l ⊗ 1 C k d C k . (3.174) Theorem 3.6 recov ers the form ula deriv ed in [ 38 , Theorem 6.1] for the indices log C ( E B , T r ∥ E A , T r ) = log max k X l min( d A l , d B k,l ) P k ′ d B k ′ ,l d C k ′ d C k ! (3.175) log C cb ( E B , T r ∥ E A , T r ) = log max k X l d B k,l P k ′ d B k ′ ,l d C k ′ d C k ! . (3.176) It also generalizes the result in [ 19 , Theorem 3.1], showing that log C ( E B , T r ∥ E A , T r ) = D max ( E B , T r ∥ E A , T r ) = D ( E B , T r ∥ E A , T r ) = D min ( E B , T r ∥ E A , T r ) . (3.177) In addition, Theorem 3.6 obtains explicit expressions for more general subalgebra inclusion indices that can b e deﬁned via arbitrary conditional exp ectations (not necessarily trace- preserving) in exactly the same wa y as in Eq. ( 3.164 )-( 3.165 ): C ( E A , E B ) := inf { c > 0 | ∀ ρ ∈ D ( H ) : E B ( ρ ) ≤ cE A ( ρ ) } (3.178) C cb ( E A , E B ) := sup n ∈ N C ( E A ⊗ id M n ( C ) , E B ⊗ id M n ( C ) ) . (3.179) 31 4 Application In this section, we apply our main results to GNS-symmetric Mark o vian dynamics. W e pro ve that discrimination rates for large iterates of a GNS-symmetric c hannel conv erge exp onen- tially fast to those of the corresp onding idemp otent p eripheral pro jections. W e b egin with some preliminaries on the asymptotic prop erties of quantum channels. Let Φ : L ( H ) → L ( H ) b e a quantum channel. Then, Φ admits a Jordan decomp osition Φ = X i λ i P i + N i with N i P i = P i N i = N i and P i P j = δ ij P i , (4.1) where the sum runs ov er the distinct eigen v alues λ i of Φ, P i are pro jectors whose rank equals the algebraic m ultiplicit y of λ i , and N i denote the corresp onding nilp otent op erators [ 54 , Chapter 6]. All the eigenv alues λ i of Φ satisfy | λ i | ≤ 1 and they are either real or come in complex conjugate pairs. Since Φ alwa ys admits a ﬁxed p oint, λ = 1 is alwa ys an eigenv alue of Φ. Moreov er, all λ i with | λ i | = 1 ha ve equal algebraic and geometric m ultiplicities, so that N i = 0 for all such eigenv alues. As l → ∞ , the image of Φ l := Φ ◦ Φ ◦ . . . ◦ Φ | {z } l times (4.2) con verges to the p eripher al space X (Φ) := span { X ∈ L ( H ) : ∃ θ ∈ R s.t. Φ( X ) = e iθ X } . Deﬁnition 4.1 L et Φ : L ( H ) → L ( H ) b e a quantum channel. The asymptotic p art of Φ and the pr oje ctor onto the p eripher al sp ac e X (Φ) , ar e r esp e ctively deﬁne d as fol lows: Φ ∞ := X i : | λ i | =1 λ i P i and P Φ = X i : | λ i | =1 P i . (4.3) Note that Φ ∞ = Φ ◦ P Φ = P Φ ◦ Φ and im( P Φ ) = X (Φ) . We denote the lar gest magnitude of a non-p eripher al eigenvalue of Φ by µ Φ := spr(Φ − Φ ∞ ) , so that 1 − µ Φ is the sp e ctr al gap. Giv en arbitrary channels Φ , Ψ : L ( H ) → L ( H ), the div ergence sequence D (Φ l ∥ Ψ l ) can dramatically oscillate. F or example, let Φ = id C 2 b e the qubit identit y c hannel and Ψ( · ) = X ( · ) X be the P auli- X channel. Then, for all k ∈ N : • D (Φ 2 k ∥ Ψ 2 k ) = 0, and • D (Φ 2 k +1 ∥ Ψ 2 k +1 ) = + ∞ . 4.1 GNS-symmetric c hannels Consider no w a channel Φ : L ( H ) → L ( H ) that satisﬁes GNS-detailed balance with resp ect to a full-rank inv arian t state δ = Φ( δ ): ∀ X , Y ∈ L ( H ) : T r(Φ † ( X ) Y δ ) = T r( X Φ † ( Y ) δ ) . (4.4) 32 Then, Φ satisﬁes (complete) strong data-pro cessing inequality (CSDPI) with resp ect to its p eripheral part, i.e. for all ρ ∈ D ( H ⊗ H ) and l ∈ N [ 21 , 20 ], D  (id H ⊗ Φ l )( ρ ) ∥ (id H ⊗ Φ l ◦ P Φ )( ρ )  ≤ s l Φ D ( ρ ∥ (id H ⊗ P Φ )( ρ )) (4.5) for some constant s Φ < 1. Hence, Pinsker inequality shows that for all l ∈ N :   Φ l − Φ l ∞   ⋄ ≤ q 2 s l Φ D cb (id H ∥P Φ ) − − − → t →∞ 0 (4.6) Moreo ver, GNS-symmetry forces spec(Φ) ⊆ [ − 1 , 1], so there is only a p ossibilit y of 2- cycles in the p eriphery . More precisely , one can show [ 21 ]: Φ 2 k ◦ P Φ = P Φ ◦ Φ 2 k = P Φ and   Φ 2 k − P Φ   ⋄ ≤ q 2 s 2 k Φ D cb (id H ∥P Φ ) − − − → k →∞ 0 . (4.7) Theorem 4.2 Supp ose Φ , Ψ : L ( H ) → L ( H ) ar e GNS-symmetric with r esp e ct to a c ommon ful l-r ank invariant state τ = Φ( τ ) = Ψ( τ ) and im( P ∗ Ψ ) ⊆ im( P ∗ Φ ) . Then, for 2 k > max  log D cb (id H ∥P Φ ) log 1 /µ Φ , log D cb (id H ∥P Ψ ) log 1 /µ Ψ  , (4.8) we have D cb ( P Φ ∥P Ψ ) + log  1 1 + ϵ 2 k Ψ  ≤ D cb min (Φ 2 k ∥ Ψ 2 k ) (4.9) ≤ D cb max (Φ 2 k ∥ Ψ 2 k ) (4.10) ≤ D cb ( P Φ ∥P Ψ ) + log  1 + ϵ 2 k Φ  + log  1 1 − ϵ 2 k Ψ  , (4.11) wher e ϵ 2 k ⋆ := µ 2 k ⋆ D cb (id H ∥P ⋆ ) − − − → k →∞ 0 for ⋆ ∈ { Φ , Ψ } . Pro of. F or the upp er b ound, w e use the chain rule (see [ 7 , Theorem I I I.1]): D cb max (Φ 2 k ∥ Ψ 2 k ) ≤ D cb max (Φ 2 k ∥P Φ ) + D cb max ( P Φ ∥P Ψ ) + D cb max ( P Ψ ∥ Ψ 2 k ) (4.12) ≤ log  1 + ϵ 2 k Φ  + log  1 1 − ϵ 2 k Ψ  + D cb ( P Φ ∥P Ψ ) , (4.13) where the second inequalit y follo ws from Theorem 3.6 and the cp-order mixing estimates [ 21 , Lemma A.1]: (1 − ϵ 2 k Φ ) P Φ ≤ cp Φ 2 k ≤ cp (1 + ϵ 2 k Φ ) P Φ (4.14) (1 − ϵ 2 k Ψ ) P Ψ ≤ cp Ψ 2 k ≤ cp (1 + ϵ 2 k Ψ ) P Ψ . (4.15) 33 F or the lo w er b ound, we note that, D cb min (Φ 2 k ∥ Ψ 2 k ) ≥ D cb min (Φ 2 k ∥P Ψ ) + log  1 1 + ϵ 2 k Ψ  (4.16) ≥ D cb min ( P Φ ∥P Ψ ) + log  1 1 + ϵ 2 k Ψ  , (4.17) = D cb ( P Φ ∥P Ψ ) + log  1 1 + ϵ 2 k Ψ  , (4.18) where the ﬁrst inequality follo ws from the mixing estimate Eq. ( 4.15 ) b ecause σ ≤ σ ′ implies D min ( ρ ∥ σ ) ≥ D min ( ρ ∥ σ ′ ), the second inequality follows from data-pro cessing: D cb min (Φ 2 k ∥P Ψ ) ≥ D cb min (Φ 2 k ◦ P Φ ∥P Ψ ◦ P Φ ) = D cb min ( P Φ ∥P Ψ ) , (4.19) where P Ψ ◦ P Φ = P Ψ follo ws from the image inclusion im( P ∗ Ψ ) ⊆ im( P ∗ Φ ), and the ﬁnal equalit y D cb min ( P Φ ∥P Ψ ) = D cb ( P Φ ∥P Ψ ) follo ws from Theorem 3.6 . Corollary 4.3 Supp ose Φ , Ψ : L ( H ) → L ( H ) ar e GNS-symmetric with r esp e ct to a c ommon ful l-r ank invariant state τ = Φ( τ ) = Ψ( τ ) and im( P ∗ Ψ ) ⊆ im( P ∗ Φ ) . Then, for 2 k > max  log D cb (id H ∥P Φ ) log 1 /µ Φ , log D cb (id H ∥P Ψ ) log 1 /µ Ψ  , (4.20) and any ϵ ∈ (0 , 1) , the fol lowing b ounds hold: D cb ( P Φ ∥P Ψ ) + log  1 1 + ϵ 2 k Ψ  ≤ lim inf n →∞ e P (Φ 2 k , Ψ 2 k , n, ϵ ) (4.21) ≤ lim sup n →∞ e A (Φ 2 k , Ψ 2 k , n, ϵ ) (4.22) ≤ D cb ( P Φ ∥P Ψ ) + log  1 + ϵ 2 k Φ  + log  1 1 − ϵ 2 k Ψ  , (4.23) wher e ϵ 2 k ⋆ := µ 2 k ⋆ D cb (id H ∥P ⋆ ) − − − → k →∞ 0 for ⋆ ∈ { Φ , Ψ } . F urthermor e, if r > D cb ( P Φ ∥P Ψ ) + log  1 1+ ϵ 2 k Ψ  , then e sc P (Φ 2 k , Ψ 2 k , r ) = e sc A (Φ 2 k , Ψ 2 k , r ) ≤ r − D cb ( P Φ ∥P Ψ ) − log  1 1 + ϵ 2 k Ψ  , (4.24) and if r > D cb ( P Φ ∥P Ψ ) + log  1 + ϵ 2 k Φ  + log  1 1 − ϵ 2 k Ψ  , then e sc P (Φ 2 k , Ψ 2 k , r ) = e sc A (Φ 2 k , Ψ 2 k , r ) ≥ r − D cb ( P Φ ∥P Ψ ) − log  1 + ϵ 2 k Φ  − log  1 1 − ϵ 2 k Ψ  . (4.25) 34 Pro of. F or the ﬁrst claim, we establish the following slightly stronger n -shot b ounds. F rom Eq. ( 2.26 ), the following holds for all ϵ ∈ (0 , 1) and n, k ∈ N : e P (Φ 2 k , Φ 2 k , n, ϵ ) ≥ D cb min (Φ 2 k ∥ Ψ 2 k ) ≥ D cb ( P Φ ∥P Ψ ) + log  1 1 + ϵ 2 k Ψ  , (4.26) where the last b ound follo ws from Theorem 4.2 . Similarly , using Eq. ( 2.26 ) again, e A (Φ 2 k , Ψ 2 k , n, ϵ ) ≤ D cb max (Φ 2 k ∥ Ψ 2 k ) + 1 n log  1 1 − ϵ  (4.27) ≤ D cb ( P Φ ∥P Ψ ) + log  1 + ϵ 2 k Φ  + log  1 1 − ϵ 2 k Ψ  + 1 n log  1 1 − ϵ  , (4.28) where the last b ound again follows from Theorem 4.2 . The desired claim then follows b y taking the limit n → ∞ . The b ounds on the strong con v erse exp onents follo w from Eq. ( 2.31 ): e sc P (Φ 2 k , Ψ 2 k , r ) = e sc A (Φ 2 k , Ψ 2 k , r ) (4.29) = sup α> 1 α − 1 α ( r − e D cb , reg α  Φ 2 k ∥ Ψ 2 k )  (4.30) ≥ r − D cb ( P Φ ∥P Ψ ) − log  1 + ϵ 2 k Φ  − log  1 1 − ϵ 2 k Ψ  , (4.31) where we used the fact that e D cb , reg α  Φ 2 k ∥ Ψ 2 k ) ≤ D cb max (Φ 2 k ∥ Ψ 2 k ), after which Theorem 4.2 is used to b ound D cb max as b efore. Similarly , e sc P (Φ 2 k , Ψ 2 k , r ) = e sc A (Φ 2 k , Ψ 2 k , r ) (4.32) = sup α> 1 α − 1 α ( r − e D cb , reg α  Φ 2 k ∥ Ψ 2 k )  (4.33) ≤ r − D cb ( P Φ ∥P Ψ ) − log  1 1 + ϵ 2 k Ψ  , (4.34) where we used the fact that e D cb , reg α  Φ 2 k ∥ Ψ 2 k ) ≥ D cb min (Φ 2 k ∥ Ψ 2 k ), after which Theorem 4.2 is used to b ound D cb min as b efore. Corollary 4.4 Supp ose Φ , Ψ : L ( H ) → L ( H ) ar e GNS-symmetric with r esp e ct to a c ommon ful l-r ank invariant state τ = Φ( τ ) = Ψ( τ ) and im( P ∗ Ψ ) ⊆ im( P ∗ Φ ) . Then, for 2 k > max  log D cb (id H ∥P Φ ) log 1 /µ Φ , log D cb (id H ∥P Ψ ) log 1 /µ Ψ  , (4.35) 35 the fol lowing holds true: D cb ( P Φ ∥P Ψ ) + log  1 1 + ϵ 2 k Ψ  ≤ lim inf n →∞ − 1 n log( p P err (Φ 2 k , Ψ 2 k , n )) (4.36) ≤ lim sup n →∞ − 1 n log( p A err (Φ 2 k , Ψ 2 k , n )) (4.37) ≤ D cb ( P Φ ∥P Ψ ) + log  1 + ϵ 2 k Φ  + log  1 1 − ϵ 2 k Ψ  , (4.38) wher e ϵ 2 k ⋆ := µ 2 k ⋆ D cb (id H ∥P ⋆ ) − − − → k →∞ 0 for ⋆ ∈ { Φ , Ψ } . Similarly, if im( P ∗ Φ ) ⊆ im( P ∗ Ψ ) , then D cb ( P Ψ ∥P Φ ) + log  1 1 + ϵ 2 k Φ  ≤ lim inf n →∞ − 1 n log( p P err (Φ 2 k , Ψ 2 k , n )) (4.39) ≤ lim sup n →∞ − 1 n log( p A err (Φ 2 k , Ψ 2 k , n )) (4.40) ≤ D cb ( P Ψ ∥P Φ ) + log  1 + ϵ 2 k Ψ  + log  1 1 − ϵ 2 k Φ  . (4.41) Pro of. It suﬃces to prov e the case im( P ∗ Ψ ) ⊆ im( P ∗ Φ ). F rom Lemma 2.2 , we can write lim sup n →∞ − 1 n log( p A err (Φ 2 k , Ψ 2 k , n )) ≤ D cb max (Φ 2 k ∥ Φ 2 k ) (4.42) ≤ D cb ( P Φ ∥P Ψ ) + log  1 + ϵ 2 k Φ  + log  1 1 − ϵ 2 k Ψ  , (4.43) where the last b ound follo ws from Theorem 4.2 . Similarly , using Lemma 2.3 , w e get lim inf n →∞ − 1 n log( p P err (Φ 2 k , Ψ 2 k , n )) ≥ sup α ∈ (0 , 1) (1 − α ) D cb , reg α (Φ 2 k ∥ Ψ 2 k ) (4.44) ≥ D cb min (Φ 2 k ∥ Ψ 2 k ) (4.45) ≥ D cb ( P Φ ∥P Ψ ) + log  1 1 + ϵ 2 k Ψ  (4.46) where the last b ound again follo ws from Theorem 4.2 . Ac kno wledgemen ts SS ackno wledges supp ort from the Deutsc he F orsc hungsgemeinsc haft (DF G, German Re- searc h F oundation) via TRR 352 – Pro ject-ID 470903074. BB ackno wledges supp ort from EPSR C UK under grant num b er EP/Y028732/1. 36 App endices A Three-la y er decomp osition Lemma 4.5 L et A ⊆ B ⊆ L ( H ) b e ﬁnite-dimensional unital ∗ -sub algebr as. Then, ther e exist ﬁnite-dimensional Hilb ert sp ac es A l , B k,l , C k , and a unitary U : H → M k,l A l ⊗ B k,l ⊗ C k (4.47) such that, with D k := L l A l ⊗ B k,l and E l := L k B k,l ⊗ C k , one has U B U † = M k L ( D k ) ⊗ 1 C k , U A U † = M l L ( A l ) ⊗ 1 E l . (4.48) Pro of. Since w e are in ﬁnite dimensions, b oth A and B are ∗ -isomorphic to direct sums of matrix algebras [ 45 , Theorem 11.2]: B ∼ = ⊕ K k =1 M d k ( C ) and A ∼ = ⊕ L l =1 M a l ( C ), b oth canonically represented on H . Hence, there exist Hilb ert spaces D k , C k with dim D k = d k and a unitary [ 45 , Theorem 11.9] U (1) : H → M k D k ⊗ C k (4.49) suc h that U (1) B ( U (1) ) † = M k L ( D k ) ⊗ 1 C k . (4.50) Since A ⊆ B , the algebra U (1) A ( U (1) ) † preserv es each summand D k ⊗ C k and acts trivially on C k , so for eac h k it determines a represen tation of A on D k . Applying [ 45 , Theorem 11.9] again, for each k there exist Hilb ert spaces A l , B k,l with dim A l = a l and a unitary U (2) k : D k → M l A l ⊗ B k,l (4.51) suc h that U (2) k  U (1) A ( U (1) ) † | D k  ( U (2) k ) † = M l L ( A l ) ⊗ 1 B k,l . (4.52) No w, set U :=  M k ( U (2) k ⊗ 1 C k )  U (1) . (4.53) Then U H = M k  M l A l ⊗ B k,l  ⊗ C k = M k,l A l ⊗ B k,l ⊗ C k , (4.54) and the asserted forms of U A U † and U B U † follo w immediately . The equiv alent descriptions via D k and E l are obtained by regrouping the same direct sum. 37 Remark 4.6 F or idemp otent channels P , Q : L ( H ) → L ( H ) with P ( 1 H ) , Q ( 1 H ) of ful l r ank, sinc e b oth im( Q ∗ ) and im( P ∗ ) ar e unital ∗ -sub algebr as in L ( H ) [ 31 ], the inclusion im( Q ∗ ) ⊆ im( P ∗ ) induc es a thr e e-layer de c omp osition of H as obtaine d in L emma 4.5 . After ﬁxing a unitary U as in the lemma, deﬁne e P := Ad U ◦P ◦ Ad U † , e Q := Ad U ◦Q ◦ Ad U † , (4.55) wher e Ad U ( · ) = U ( · ) U † . Sinc e al l channel diver genc es c onsider e d in this p ap er ar e invariant under unitary c onjugation, one may r eplac e ( P , Q ) by ( e P , e Q ) without changing any of the quantities of inter est. Thus, after this identiﬁc ation, we wil l always write H = M k,l A l ⊗ B k,l ⊗ C k = M k D k ⊗ C k = M l A l ⊗ E l (4.56) as liter al e qualities, and P = M k id D k ⊗ R δ k , Q = M l id A l ⊗ R ω l . (4 .57) In this r epr esentation, al l p artial tr ac es ar e or dinary tensor-factor p artial tr ac es, and V k : D k ⊗ C k → H W l : A l ⊗ E l → H (4.58) V l | k : A l ⊗ B k,l → D k W k | l : B k,l ⊗ C k → E l (4.59) ar e simply the c anonic al inclusion isometries asso ciate d with the dir e ct-sum de c omp ositions. B Pro of of Eq. ( 3.116 ) F or α > 1, deﬁne U α ( P , Q ) := max k log  X l 2 e D cb α ( k,l )  . (4.60) F or m ulti-indices k = ( k 1 , . . . , k n ) and l = ( l 1 , . . . , l n ) , set A l := n O i =1 A l i , B k , l := n O i =1 B k i ,l i , C k := n O i =1 C k i , E l := n O i =1 E l i , (4.61) so that H ⊗ n = M k , l A l ⊗ B k , l ⊗ C k . (4.62) Let W k | l := n O i =1 W k i | l i : B k , l ⊗ C k → E l , (4.63) W l := n O i =1 W l i : A l ⊗ E l → H ⊗ n (4.64) 38 and deﬁne the corresp onding embedding channel ι k | l ( · ) := W k | l ( · ) W † k | l . Then, the restricted tensor-p o w er c hannels on the subspace Π k , l H ⊗ n = A l ⊗ B k , l ⊗ C k are W ∗ l ◦ P ⊗ n   Π k , l = (id A l ⊗ ι k | l ) ◦ n O i =1  id A l i ⊗ id B k i ,l i ⊗ R δ k i  , (4.65) W ∗ l ◦ Q ⊗ n   Π k , l = n O i =1  id A l i ⊗ R ω l i  , (4.66) b oth viewed as channels into L ( A l ⊗ E l ). Denote e D cb α ( k , l ) := e D cb α  P ⊗ n   Π k , l    Q ⊗ n   Π k , l  . (4.67) Since tensoring both channels with the same iden tity channel does not change the cb div ergence, w e ma y remo ve the factors id A l i from b oth sides and obtain e D cb α ( k , l ) = e D cb α  ι k | l ◦ n O i =1 (id B k i ,l i ⊗ R δ k i )    n O i =1 R ω l i  . (4.68) No w, Lemma 3.9 shows e D cb α ( k , l ) = n X i =1 e D cb α  ι k i | l i ◦ (id B k i ,l i ⊗ R δ k i )   R ω l i  = n X i =1 e D cb α ( k i , l i ) . ( ∗ ) Therefore, U α ( P ⊗ n , Q ⊗ n ) = max k log  X l 2 e D cb α ( k , l )  = max k log  X l 1 ,...,l n 2 P n i =1 e D cb α ( k i ,l i )  b y ( ∗ ) = max k log  n Y i =1 X l 2 e D cb α ( k i ,l )  = max k n X i =1 log  X l 2 e D cb α ( k i ,l )  = n X i =1 max k i log  X l 2 e D cb α ( k i ,l )  = n max k log  X l 2 e D cb α ( k,l )  = n U α ( P , Q ) . 39 C Example Recall that for idemp oten t c hannels P , Q : L ( H ) → L ( H ) with P ( 1 H ) , Q ( 1 H ) of full rank, the inclusion im( Q ∗ ) ⊆ im( P ∗ ) induces a three-lay er decomp osition of H : H = ⊕ K k =1 ⊕ L l =1 A l ⊗ B k,l ⊗ C k (4.69) = ⊕ k ( ⊕ l A l ⊗ B k,l ) | {z } := D k ⊗ C k (4.70) = ⊕ l A l ⊗ ( ⊕ k B k,l ⊗ C k ) | {z } := E l , (4.71) suc h that P = M k id D k ⊗ R δ k (4.72) Q = M l id A l ⊗ R ω l . (4.73) No w, let K = 1 , L = 2 and A 1 = A 2 = C , B 1 , 1 = B 1 , 2 = C 2 , C 1 = C 2 , δ 1 = 1 2 1 C 1 . (4.74) Th us, H = ⊕ 2 l =1 B 1 ,l ⊗ C 1 (4.75) = ( B 1 , 1 ⊕ B 1 , 2 ) | {z } := D 1 ⊗ C 1 (4.76) = ( B 1 , 1 ⊗ C 1 ) | {z } := E 1 ⊕ ( B 1 , 2 ⊗ C 1 ) | {z } := E 2 . (4.77) Deﬁne states ω l ∈ D ( E l ) in the basis {| 0 ⟩ B | 0 ⟩ C , | 0 ⟩ B | 1 ⟩ C , | 1 ⟩ B | 0 ⟩ C , | 1 ⟩ B | 1 ⟩ C } as follows: ω 1 := 1 8 diag(1 , 2 , 3 , 2) , ω 2 := 1 8 diag(2 , 1 , 2 , 3) . (4.78) Neither ω 1 nor ω 2 is of pro duct form across the B 1 ,l – C 1 cut, since for a diagonal pro duct state diag( a, b, c, d ) one necessarily has ad = bc . T ak e α = 2, so that β = 1 − α α = − 1 2 . Then, ω − 1 / 2 1 = diag 2 √ 2 , 2 , 2 r 2 3 , 2 ! , ω − 1 / 2 2 = diag 2 , 2 √ 2 , 2 , 2 r 2 3 ! . (4.79) R estricte d blo ck diver genc es. Since A l = C , the restricted channels (Eqs. ( 3.109 ), ( 3.110 )) are W ∗ l ◦ P | Π 1 ,l = id B 1 ,l ⊗ R δ 1 , (4.80) W ∗ l ◦ Q| Π 1 ,l = R ω l . (4.81) 40 Let | v l ⟩ ∈ B 1 ,l b e a unit vector with q l := |⟨ 0 | v l ⟩| 2 . A direct calculation giv es ˆ X ( v 1 ) 1 , 1 := p δ 1 T r B 1 , 1  ( v 1 ⊗ 1 C 1 ) ω − 1 / 2 1  p δ 1 = √ 2  q 1 + 1 − q 1 √ 3  0 0 1 ! , (4.82) ˆ X ( v 2 ) 1 , 2 := p δ 1 T r B 1 , 2  ( v 2 ⊗ 1 C 1 ) ω − 1 / 2 2  p δ 1 = 1 0 0 √ 2  q 2 + 1 − q 2 √ 3  ! . (4.83) Hence, for e D 2 (1 , l ) deﬁned in Eq. ( 3.106 ), we get 2 e D 2 (1 , 1) = sup v 1 ∈D ( B 1 , 1 ) T r  ( ˆ X ( v 1 ) 1 , 1 ) 2  = max q 1 ∈ [0 , 1] 2  q 1 + 1 − q 1 √ 3  2 + 1 = 3 , (4.84) and similarly 2 e D 2 (1 , 2) = max q 2 ∈ [0 , 1] 2  q 2 + 1 − q 2 √ 3  2 + 1 = 3 , (4.85) with equalit y at q l = 1. Therefore, the righ t-hand side of Theorem 3.10 equals max k log  X l 2 e D 2 ( k,l )  = log(3 + 3) = log 6 . (4.86) The ful l channel diver genc e. No w, let | ψ ⟩ = √ p | ψ 1 ⟩ ⊕ p 1 − p | ψ 2 ⟩ ∈ B 1 , 1 ⊕ B 1 , 2 = D 1 , (4.87) where 0 ≤ p ≤ 1 and | ψ l ⟩ ∈ B 1 ,l are unit v ectors. Let q l := |⟨ 0 | ψ l ⟩| 2 . Then, the corresponding op erators X ( ψ ) 1 , 1 , X ( ψ ) 1 , 2 from Eq. ( 3.129 ) are X ( ψ ) 1 , 1 = √ p p δ 1 T r B 1 , 1  ( ψ 1 ⊗ 1 C 1 ) ω − 1 / 2 1  p δ 1 = √ 2 p  q 1 + 1 − q 1 √ 3  0 0 √ p ! , (4.88) X ( ψ ) 1 , 2 = p 1 − p p δ 1 T r B 1 , 2  ( ψ 2 ⊗ 1 C 1 ) ω − 1 / 2 2  p δ 1 = √ 1 − p 0 0 p 2(1 − p )  q 2 + 1 − q 2 √ 3  ! . Denoting t ( q ) := q + (1 − q ) / √ 3, w e get (see Eq. ( 3.130 )): 2 e D 2 ( P ∥Q ) = sup ψ ∈D ( D 1 ) T r( X ( ψ ) 1 , 1 + X ( ψ ) 1 , 2 ) 2 = max p ∈ [0 , 1] max q l ∈ [0 , 1]  p 2 p t ( q 1 ) + p 1 − p  2 +  √ p + p 2(1 − p ) t ( q 2 )  2 = max p ∈ [0 , 1]  p 2 p + p 1 − p  2 +  √ p + p 2(1 − p )  2 = 3 + 4 max p ∈ [0 , 1] p 2 p (1 − p ) = 3 + 2 √ 2 . 41 Both maxima are attained for p = 1 2 and q 1 = q 2 = 1, i.e. for | ψ ⟩ = 1 √ 2  | 0 ⟩ B 1 , 1 ⊕ | 0 ⟩ B 1 , 2  . (4.89) Hence, com bining the tw o computations, w e obtain e D 2 ( P ∥Q ) = log (3 + 2 √ 2) < log 6 = max k log  X l 2 e D 2 ( k,l )  . (4.90) D Auxilliary lemmas Lemma 4.7 F or any set of r e al numb ers { c i } i , the fol lowing holds true: max µ i ≥ 0 , P i µ i =1 H ( { µ i } i ) + X i µ i c i ! = log X i 2 c i ! . (4.91) Pro of. Let Z = P i 2 c i and q i = 2 c i / Z . Then, for any probability distribution { µ i } i : H ( { µ i } i ) + X i µ i c i = X i − µ i log µ i + X i µ i log( q i Z ) (4.92) = log Z − X i µ i log( µ i /q i ) (4.93) = log Z − D ( µ ∥ q ) , (4.94) where D ( µ ∥ q ) ≥ 0 denotes the classical KL-divergence b et ween µ and q . Hence, H ( { µ i } i ) + X i µ i c i ≤ log Z , (4.95) and equalit y holds when µ = q . Lemma 4.8 F or any set of non-ne gative numb ers { c i } i , the fol lowing holds true: min µ i ≥ 0 , P i µ i =1 X i µ 2 i c i = 1 P i 1 /c i . (4.96) Pro of. Using Cauc h y-Sch warz inequality , w e get for any probability distribution { µ i } i : 1 = X i µ i ! 2 = X i µ i √ c i √ c i ! 2 ≤ X i µ 2 i c i X i 1 c i , (4.97) and the minimum is achiev ed b y c ho osing µ i ∝ 1 /c i . 42 Lemma 4.9 F or any set of non-ne gative numb ers { c l } l and α > 1 , the fol lowing holds true: max p l ≥ 0 , P l p l =1 α α − 1 log X l p 1 α l c α − 1 α l ! = log X l c l ! . (4.98) Pro of. Using H¨ older’s inequalit y , we get for any probability distribution { p l } l : X l p 1 α l c α − 1 α l ≤ X l p l ! 1 α X l c l ! α − 1 α (4.99) = X l c l ! α − 1 α . (4.100) Hence, α α − 1 log X l p 1 α l c α − 1 α l ! ≤ log X l c l ! . (4.101) Moreo ver, equality is achiev ed when p l ∝ c l . References [1] Koenraad M. R. Audenaert, J. Calsamiglia, R. Mu ˜ noz-T apia, E. Bagan, Ll. Masanes, A. Acin, and F. V erstraete. Discriminating States: The Quantum Chernoﬀ Bound. 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