Additivity violation of the regularized Minimum Output Entropy
aa r X i v : . [ m a t h . OA ] J un ADDITIVITY VIOLATION OF THE REGULARIZED MINIMUMOUTPUT ENTROPY
BENOˆIT COLLINS AND SANG-GYUN YOUNA
BSTRACT . The problem of additivity of the Minimum Output Entropyis of fundamental importance in Quantum Information Theory (QIT). Itwas solved by Hastings [Has09] in the one-shot case, by exhibiting apair of random quantum channels. However, the initial motivation wasarguably to understand regularized quantities and there was so far noway to solve additivity questions in the regularized case. The purpose ofthis paper is to give a solution to this problem. Specifically, we exhibita pair of quantum channels which unearths additivity violation of theregularized minimum output entropy. Unlike previously known resultsin the one-shot case, our construction is non-random, infinite dimen-sional and in the commuting-operator setup. The commuting-operatorsetup is equivalent to the tensor-product setup in the finite dimensionalcase for this problem, but their difference in infinite dimensional settinghas attracted substantial attention and legitimacy recently in QIT withthe celebrated resolutions of Tsirelson’s and Connes embedding problem[JNV +
1. I
NTRODUCTION
A crucial problem in quantum information theory is the problem of addi-tivity of Minimum Output Entropy (MOE), which asks whether it is possi-ble to find two quantum channels Φ , Φ such that H min (Φ ⊗ Φ ) < H min (Φ ) + H min (Φ ) . This problem was stated by King-Ruskai in [KR01] as a natural questionin the study of quantum channels. Shor proved in 2004 [Sho04b, Sho04a]that a positive answer to the above question is equivalent to super-additivityof the Holevo capacity, i.e. there exist quantum channels Φ , Φ such that χ (Φ ⊗ Φ ) > χ (Φ ) + χ (Φ ) . Heuristically, super-additivity of the Holevo capacity implies that entan-glement inputs can be used to increase the transmission rate of classicalinformation. We refer to Section 2.1 for the definitions of the MOE H min and the Holevo capacity χ . This question attracted lots of attention, and it ii BENOˆIT COLLINS AND SANG-GYUN YOUN was eventually solved by Hastings in 2009 [Has09], with preliminary sub-stantial contributions by Hayden, Winter, Werner, see in particular [HW08].Subsequently, the mathematical aspects of the proof have been clarified invarious directions by [ASW11, FKM10, BaH10, BCN16, Col18, CFZ15].All previously known examples of additivity violation of MOE rely onsubtle random constructions. In particular, to date, no deterministic con-struction of additivity violation has ever been given. For attempts and par-tial results in the direction of non-random techniques we refer to [WH02,GHP10, BCLY20], etc.Note that the above results do not imply anything about the problem ofthe additivity of the regularized MOE (see Definition 2.3 for details). In-deed, additivity violation is not known to pertain when the MOE is regular-ized. More precisely, the additivity question for the regularized MOE askswhether it is possible to find two quantum channels Φ , Φ such that H min (Φ ⊗ Φ ) < H min (Φ ) + H min (Φ ) . where H min stands for the regularized MOE. This question was raised in[Fuk14] and the affirmative answer to this implies superadditivity of classi-cal capacity.Very few results are known about regularized entropic quantities – see forexample [Kin02] or [BCLY20] for partial results. In this paper, we focus onthe additivity question of the regularized minimum output entropy, and thetensor product channel will be understood as a composition of two quantumchannels whose systems of Kraus operators are commuting (see Section 2.2for details).In (quantum) information theory, one key paradigm is to allow repeateduses of a given quantum channel. To do this, we have to analyze a phys-ical system by separated subsystems. In view of quantum strategies fornon-local games, there are two natural models to describe separated subsys-tems. One is the tensor-product model and the other is commuting-operatormodel . This latter approach is the object of intense research, see for ex-ample [PT15, DP16, CLS17, Slo20, CCLP18], culminating with the recentresolution [JNV +
20] in the negative of the celebrated Connes Embeddingproblem whose origin dates back to [Con76]. In our case, commuting sys-tems of Kraus operators correspond to a commuting-operator model. Werefer to Section 2.2 for details on this.The main result of this paper is an explicit construction of a pair of quan-tum channels Φ and Φ which have commuting systems of Kraus operatorsand satisfy additivity violation of the regularized MOE. Specifically, ourmain result can be stated as follows: DDITIVITY VIOLATION OF THE REGULARIZED MINIMUM OUTPUT ENTROPY iii
Theorem 1.1.
There exist systems of operators { E i } mi =1 and { F j } nj =1 in B ( H ) such that (1) E i F j = F j E i for all ≤ i ≤ m and ≤ j ≤ n , (2) m X i =1 E ∗ i E i = Id H = n X j =1 F ∗ j F j , (3) Φ , Φ : T ( H ) → T ( H ) are quantum channels given by Φ ( ρ ) = m X i =1 E i ρE ∗ i and Φ ( ρ ) = n X j =1 F j ρF ∗ j , (4) H min (Φ ◦ Φ ) < H min (Φ ) + H min (Φ ) . Note that the above discussion for the regularized MOE makes sensesince the given channels are generated by finitely many Kraus operators,and given commuting systems will be chosen as an infinite dimensionalanalogue of i.i.d. Haar distributed unitary matrices, which will be explainedin Section 2.2 and Theorem 4.1 in details. One of the biggest benefit fromthis shift in perspective is that the regularized minimum output entropy be-comes computable, whereas for random unitary channels, computing suchregularized quantities still seems to remain totally out of reach at this point.One of the key ingredients is to extend the
Haagerup inequality [Haa79]to products of free groups (Proposition 3.2). This result itself is a crucialfact. Indeed, the Haagerup inequality has numerous applications in operatoralgebras, non-commutative harmonic analysis and geometric group theory[Bo˙z81, DCH85, Jol89, Laf00, Laf02].This paper is organized as follows. After this introduction, Section 2gathers some preliminaries about entropic quantities, quantum channelsand the infinite dimensional framework. Section 3 contains the proof ofa Haagerup-type inequality for products of free groups as well as estimatesfor the regularized Minimum Output Entropy of our main family of quan-tum channels. Section 4 explains how we can obtain additivity violationof the regularized MOE in the commuting operator setup, and Section 5contains concluding remarks.
Acknowledgements : BC. was supported by JSPS KAKENHI 17K18734and 17H04823. S-G. Youn was funded by Natural Sciences and Engineer-ing Research Council of Canada and by the National Research Founda-tion of Korea (NRF) grant funded by the Korea government (MSIT) (No.2020R1C1C1A01009681). S-G. Youn acknowledges the hospitality of Ky-oto University on the occasion of two visits during which this project wasinitiated and completed. Part of this work was also done during the con-ference MAQIT 2019, at which the authors acknowledge a fruitful working v BENOˆIT COLLINS AND SANG-GYUN YOUN environment. Finally, both authors would like to thank Mike Brannan, Ja-son Crann and Hun Hee Lee for inspiring discussions on this paper.2. P
RELIMINARIES
Minimum output entropy in infinite dimensional setting.
Let V : H A → H B ⊗ H E be an isometry. Then partial traces on H B and H E de-fine the following completely positive trace preserving maps (aka quantumchannels ) Φ : T ( H A ) → T ( H B ) , ρ (id ⊗ tr)( V ρV ∗ ) (2.1) Φ c : T ( H A ) → T ( H E ) , ρ (tr ⊗ id)( V ρV ∗ ) (2.2)where T ( H ) denotes the space of trace class operators on a Hilbert space H . The map Φ c is called the complementary channel of Φ . The tensorproduct channels Φ ⊗ k : T ( H ⊗ kA ) → T ( H ⊗ kB ) are defined in the obviousway. A (quantum) state in H is a positive element of T ( H ) of trace , andfor a state ρ , its R´enyi entropy for p ∈ (1 , ∞ ) is defined as H p ( ρ ) = 11 − p log ( tr ( ρ p )) . Its limit as p → + is called the von Neumann entropy , and if λ ( ρ ) ≥ λ ( ρ ) ≥ . . . are the eigenvalues of ρ (counted with multiplicity), then thevon Neumann entropy is H ( ρ ) = − X i λ i ( ρ ) log λ i ( ρ ) . The
Holevo capacity of a quantum channel is χ (Φ) = sup ( H (Φ( X i λ i ρ i ) − X i λ i H (Φ( ρ i )) ) , where the supremum is taken over all probability distributions ( p i ) i and allfamilies of states ( ρ i ) i . It describes the amount of classical information thatcan be carried through a single use of a quantum channel. If repeated usesof a given quantum channel is allowed, the ultimate transmission rate ofclassical information is described by C (Φ) = lim k →∞ k χ (Φ ⊗ k ) , which is called the classical capacity . DDITIVITY VIOLATION OF THE REGULARIZED MINIMUM OUTPUT ENTROPY v
For a quantum channel Φ , the Minimum Output Entropy (MOE) and the regularized MOE are defined as H min (Φ) = inf ξ H (Φ( | ξ ih ξ | )) and (2.3) H min (Φ) = lim k →∞ k H min (Φ ⊗ k ) . (2.4)respectively, where the infimum runs over all unit vectors ξ of H A . If Φ has finitely many Kraus operators { E , E , · · · , E N } satisfying Φ( ρ ) = N X i =1 E i ρE ∗ i (e.g. if H E is finite dimensional), then H min (Φ) + χ (Φ) ≤ log( N ) and H min (Φ) + C (Φ) ≤ log( N ) . Remark 2.1.
In Equation (2.3), taking the minimum over all states insteadof pure states does not modify the quantity thanks to operator convexity ofthe function x log( x ) , see e.g. [Seg60, NU61] . As in the finite dimensional setting, the following Schmidt decomposi-tion theorem tells us that Φ( | ξ ih ξ | ) and Φ c ( | ξ ih ξ | ) have the same eigenval-ues for each pure state | ξ ih ξ | ∈ T ( H ) . Proposition 2.2.
Let V : H A → H B ⊗ H E be an isometry and ξ ∈ H A bea unit vector. If we suppose that Φ( | ξ ih ξ | ) has the spectral decomposition X i λ i | e i ih e i | with λ i > , where ( e i ) i ∈ I is an orthonormal subset of H B ,then there exists an orthonormal subset ( f i ) i ∈ I of H E satisfying V | ξ i = X i p λ i | e i i ⊗ | f i i and Φ c ( | ξ ih ξ | ) = X i λ i | f i ih f i | . (2.5) In particular, H (Φ( | ξ ih ξ | )) = H (Φ c ( | ξ ih ξ | )) for each unit vector ξ ∈ H A .Proof. Since ( e i ) i is an orthonormal basis of H B , we can write V | ξ i as X i | e i i ⊗ | η i i for a family ( η i ) i ⊆ H E . Moreover, the given spectral de-composition of Φ( | ξ ih ξ | ) tells us that h η j | η i i = λ i δ i,j , which is equivalentto that ( f i ) i := (cid:16) λ − i η i (cid:17) i is an orthonormal set. Then we have V | ξ i = X i p λ i | e i i ⊗ | f i i and Φ c ( | ξ ih ξ | ) = X i λ i | f i ih f i | . (cid:3) i BENOˆIT COLLINS AND SANG-GYUN YOUN Commuting systems of Kraus operators.
Let H be a Hilbert spaceand ( a ij ) ( i,j ) ∈ I × J be a family of bounded operators in B ( H ) satisfying X i ∈ I a ∗ i,j a i,j = Id H for each j ∈ J . We assume that I is finite and J isarbitary. Let us define a family of quantum channels (Φ j ) j ∈ J by Φ j : T ( H ) → T ( H ) , X X i ∈ I a ij Xa ∗ ij . Their complement channels are given by Φ cj : T ( H ) → M | I | ( C ) , X X i,i ′ ∈ I tr( a ij Xa ∗ i ′ j ) | i ih i ′ | . We say that (Φ j ) j ∈ J is in commuting-operator setup if the given channels Φ j have commuting systems of Kraus operators in the sense that a ij a i ′ j ′ = a i ′ j ′ a ij for any i, i ′ ∈ I and j, j ′ ∈ J such that j = j ′ .An example of this is the tensor-product setup but it is not the onlyexample. A property is that Φ j and Φ j ′ commute and their products areagain quantum channels. If J is finite and Φ , · · · , Φ | J | are in commuting-operator setup, then it is natural to ask whether the following additivityproperty holds when F is one of H min , H min , χ, C : F Y j ∈ J Φ j ! = X j ∈ J F (Φ j ) . In particular, in the case | J | = 2 , the product channel Φ ◦ Φ is called a local map in the context of [CKLT19].Let us construct a non-trivial quantum channel within the commuting-operator setup from the view of abstract harmonic analysis and operatoralgebra. Let F ∞ be the free group whose generators are g , g , · · · and letus define unitary operators U i and V j on ℓ ( F ∞ ) by ( U i f )( x ) = f ( g − i x ) and ( V j f )( x ) = f ( xg j ) for any f ∈ ℓ ( F ∞ ) , x ∈ F ∞ and i, j ∈ N . Since U i V j = V j U i for all i, j ∈ N , we have the following quantum channels that have commutingsystems of Kraus operators. Φ N,l : T ( ℓ ( F ∞ )) → T ( ℓ ( F ∞ )) , ρ N N X i =1 U i ρU ∗ i andΦ N,r : T ( ℓ ( F ∞ )) → T ( ℓ ( F ∞ )) , ρ N N X j =1 V j ρV ∗ j . DDITIVITY VIOLATION OF THE REGULARIZED MINIMUM OUTPUT ENTROPY vii
Let J ∈ B ( ℓ ( F ∞ )) be a unitary given by ( J f )( x ) = f ( x − ) for any f ∈ ℓ ( F ∞ ) and x ∈ F ∞ . Then, since J U i J = V i and J = Id , theabove channels Φ N,l and Φ N,r are equivalent in the sense that Φ N,r ( ρ ) = J Φ N,l ( J ρJ ) J for any ρ ∈ T ( ℓ ( F ∞ )) . In particular, H min (Φ ⊗ kN,l ) = H min (Φ ⊗ kN,r ) for any k ∈ N .Also, in order to express k -fold tensor product quantum channels Φ ⊗ kN,l ,we will use the following notation U m = U m ⊗ U m ⊗ · · · ⊗ U m k ∈ B ( ℓ ( F k ∞ )) for any m = ( m , · · · , m k ) ∈ I k , where I = { , , · · · , N } .3. G ENERALIZED H AAGERUP INEQUALITY AND REGULARIZED
MOEIn this section, we prove that the
Haagerup inequality extends naturallyto r -products of free groups F r ∞ . Then we explain how this generalizationallows to prove lower bounds of the regularized minimum output entropies(MOE) for Φ N,l . Let us simply write Φ N,l as Φ N in this section.3.1. A generalized Haagerup inequality.
For x in the free group F ∞ , wecall | x | its reduced word length with respect to the canonical generators andtheir inverses. We consider products of free groups F r ∞ for any r ∈ N . Letus use the following notations E j = { x ∈ F ∞ : | x | = j } for any j ∈ N and E m = E m × E m × · · · × E m r ⊆ F r ∞ for any m = ( m , · · · , m r ) ∈ N r .We view F r ∞ as an orthonormal basis that generates the Hilbert space ℓ ( F r ∞ ) . As an algebraic vector space, F r ∞ spans C [ F r ∞ ] , on which we maydefine the convolution f ∗ g and the pointwise product f · g . For A ⊂ F r ∞ , χ A denotes the indicator function of A . First of all, we can generalize Lemma1.3 of [Haa79] as follows: Lemma 3.1.
Let l, m, k ∈ N r and let f, g be supported in E k and E l re-spectively. Then k ( f ∗ g ) · χ E m k ℓ ( F r ∞ ) ≤ k f k ℓ ( F r ∞ ) · k g k ℓ ( F r ∞ ) (3.1) if | k j − l j | ≤ m j ≤ k j + l j and k j + l j − m j is even for all ≤ j ≤ r .Otherwise, we have k ( f ∗ g ) · χ E m k ℓ ( F r ∞ ) = 0 .Proof. Let us suppose that | k j − l j | ≤ m j ≤ k j + l j and k j + l j − m j iseven for all ≤ j ≤ r . If not, it is not difficult to see that ( f ∗ g ) χ m = 0 .Also, it is enough to suppose that f, g are finitely supported since ( f, g ) ( f ∗ g ) · χ E m is bilinear. iii BENOˆIT COLLINS AND SANG-GYUN YOUN Let us use the induction argument with respect to r ∈ N . The first case r = 1 follows from [Haa79, Lemma 1.3] and let us suppose that (3.1) holdstrue for F r ∞ . Under the notation m = ( m , m ′ ) ∈ N r +10 , we have k ( f ∗ g ) χ E m k ℓ ( F r +1 ∞ ) = X s ∈ E m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t,u ∈ F r +1 ∞ : tu = s f ( t ) g ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.2) = X s ∈ E m X s ′ ∈ E m ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t ,u ∈ F ∞ : t u = s X t ′ ,u ′ ∈ F r ∞ : t ′ u ′ = s ′ f ( t , t ′ ) g ( u , u ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.3)First of all, if we suppose that m = k + l , then we have = X s ∈ E m X s ′ ∈ E m ′ X t ,u ∈ F ∞ : t u = s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t ′ ,u ′ ∈ F r ∞ : t ′ u ′ = s ′ f ( t , t ′ ) g ( u , u ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) since there is a unique choice of t ∈ E k and u ∈ E l satisfying t u = s .Let us define functions f t ( t ′ ) = f ( t , t ′ ) and g u ( u ′ ) = g ( u , u ′ ) on F r ∞ .Then the above is written as k ( f ∗ g ) χ E m k ℓ ( F r +1 ∞ ) = X s ∈ E m X t ,u ∈ F ∞ : t u = s (cid:13)(cid:13) ( f t ∗ g t ) χ E m ′ (cid:13)(cid:13) ℓ ( F r ∞ ) , (3.4)which is dominated by ≤ X s ∈ E m X t ,u ∈ F ∞ : t u = s k f t k ℓ ( F r ∞ ) k g u k ℓ ( F r ∞ ) (3.5) ≤ X t ∈ E k X u ∈ E l k f t k ℓ ( F r ∞ ) k g u k ℓ ( F r ∞ ) = k f k ℓ ( F r +1 ∞ ) k g k ℓ ( F r +1 ∞ ) . (3.6)Here, the first inequality comes from the induction hypothesis. Furthemore,the same idea applies whenever m j = k j + l j for some ≤ j ≤ r . Now, letus suppose that m j < k j + l j and put q j = k j + l j − m j for all ≤ j ≤ r .Also, denote by q = ( q , · · · , q r ) and define two functions F and G on F r +1 ∞ as follows: F ( x ) = ( (cid:16)P v ∈ E q | f ( xv ) | (cid:17) for any x ∈ E k − q otherwise (3.7) G ( y ) = ( (cid:16)P v ∈ E q | g ( v − y ) | (cid:17) for any y ∈ E l − q otherwise (3.8) DDITIVITY VIOLATION OF THE REGULARIZED MINIMUM OUTPUT ENTROPY ix
Note that F and G are supported in E k − q and E l − q respectively with k F k ℓ ( F r +1 ∞ ) = X x ∈ E k − q X v ∈ E q | f ( xv ) | = k f k ℓ ( F r +1 ∞ ) and (3.9) k G k ℓ ( F r +1 ∞ ) = X v ∈ E q X y ∈ E l − q | g ( v − y ) | = k g k ℓ ( F r +1 ∞ ) . (3.10)Then we can show that the convolution F ∗ G dominates | f ∗ g | on E m .Indeed, for any s ∈ E m , there exists a unique ( x, y ) ∈ E k − q × E l − q suchthat s = xy and we have | ( f ∗ g )( s ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t ∈ E k ,u ∈ E l : tu = s f ( t ) g ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X v ∈ E q f ( xv ) g ( v − y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X v ∈ E q | f ( xv ) | X v ∈ E q | g ( v − y ) | = F ( x ) G ( y ) = ( F ∗ G )( s ) Finally, since F and G are supported in E k − q and E l − q respectively with m = ( k − q ) + ( l − q ) , we can conclude that k ( f ∗ g ) χ E m k ℓ ( F r +1 ∞ ) ≤ k ( F ∗ G ) χ E m k ℓ ( F r +1 ∞ ) ≤ k F k ℓ ( F r +1 ∞ ) k G k ℓ ( F r +1 ∞ ) = k f k ℓ ( F r +1 ∞ ) k g k ℓ ( F r +1 ∞ ) . (cid:3) Then we can generalize the Haagerup inequality to products of free groups F r ∞ as follows: Proposition 3.2.
Let n = ( n , · · · , n r ) ∈ N r and f be supported on E n ⊆ F r ∞ . Then k L f k ≤ ( n + 1) · · · ( n r + 1) k f k ℓ ( F r ∞ ) , where L f is the convolution operator on ℓ ( F r ∞ ) given by g f ∗ g .Proof. By density arguments, we may assume that f is finitely supportedand it is enough to consider finitely supported functions to evaluate the normof the associated convolution operator L f . Let g ∈ ℓ ( F r ∞ ) be finitely sup-ported and define g k = g · χ E k for each k ∈ N r . Then g = X k ∈ N r g · χ E k andwe have h := f ∗ g = X k ∈ N r f ∗ g k . BENOˆIT COLLINS AND SANG-GYUN YOUN
Then, by Lemma 3.1, we have the following estimate for h m = h · χ E m with m = ( m , · · · , m r ) ∈ N r as follows: k h m k ℓ ( F r ∞ ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ∈ N r ( f ∗ g k ) · χ E m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ ( F r ∞ ) ≤ X k ∈ N r k ( f ∗ g k ) · χ E m k ℓ ( F r ∞ ) ≤ X k ∈ N r n j + k j − m j : even | n j − k j |≤ m j ≤ n j + k j k f k ℓ ( F r ∞ ) k g k k ℓ ( F r ∞ ) =: A Writing k j = m j + n j − l j for all ≤ j ≤ r , we obtain A = k f k ℓ ( F r ∞ ) X l , ··· ,l r ≤ l j ≤ min { m j ,n j } k g m + n − l k ℓ ( F r ∞ ) ≤ k f k ℓ ( F r ∞ ) p (1 + n ) · · · (1 + n r ) X l , ··· ,l r ≤ l j ≤ min { m j ,n j } k g m + n − l k ℓ ( F r ∞ ) . Therefore, k h k ℓ ( F r ∞ ) = X m ∈ N r k h m k ℓ ( F r ∞ ) ≤ (1 + n ) · · · (1 + n r ) k f k ℓ ( F r ∞ ) X m ∈ N r X l , ··· ,l r ≤ l j ≤ min { m j ,n j } k g m + n − l k ℓ ( F r ∞ ) = (1 + n ) · · · (1 + n r ) k f k ℓ ( F r ∞ ) X l , ··· ,l r ≤ l j ≤ n j X m , ··· ,m r l j ≤ m j < ∞ k g m + n − l k ℓ ( F r ∞ ) = (1 + n ) · · · (1 + n r ) k f k ℓ ( F r ∞ ) X l , ··· ,l r ≤ l j ≤ n j X k , ··· ,k r n j − l j ≤ k j < ∞ k g k k ℓ ( F r ∞ ) ≤ (1 + n ) · · · (1 + n r ) k f k ℓ ( F r ∞ ) X l , ··· ,l r ≤ l j ≤ n j X k ∈ N r k g k k ℓ ( F r ∞ ) = (1 + n ) · · · (1 + n r ) k f k ℓ ( F r ∞ ) k g k ℓ ( F r ∞ ) , which gives us k L f k ≤ (1 + n ) · · · (1 + n r ) k f k ℓ ( F r ∞ ) . (cid:3) DDITIVITY VIOLATION OF THE REGULARIZED MINIMUM OUTPUT ENTROPY xi
A norm estimate.
In this subsection, we investigate the operator normof the following elements X v,w ∈ E k a v,w ( U v ) ∗ U w with a vw ∈ C , where E = { g , g , · · · } is the set of generators of F ∞ and E k = E ×· · · × E ⊆ E (1 , ··· , . Indeed, in Section 3.3, this estimate will be needed toevaluate the regularized MOE. Our estimate is as follows: Theorem 3.3.
For any a = ( a vw ) v,w ∈ E k ∈ M N k ( C ) such that tr( a ) = 0 ,we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X v,w ∈ E k a vw ( U v ) ∗ U w (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ N k p (1 + 9 N − ) k − k a k . Proof.
Now we are summing over ( N k ) elements and we have to split thesum according to whether there are simplification or not. To do this, for anysubset K of { , , · · · , k } , we define E K = (cid:8) ( v, w ) ∈ E k × E k : v i = w i for all i ∈ K (cid:9) . Note that E = ⊔ K ⊂{ , ··· ,k } E K . Then, by the triangle inequality, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X v,w ∈ E a vw ( U v ) ∗ U w (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X K ⊂{ , ··· ,k } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X ( v,w ) ∈ E K a vw ( U v ) ∗ U w (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k X s =0 X K ⊂{ , ··· ,k }| K | = s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X ( v,w ) ∈ E K a vw ( U v ) ∗ U w (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Note that | K | = k implies X ( v,w ) ∈ E K a vw ( U v ) ∗ U w = X v ∈ E k a vv = 0 . Fromnow on, let us suppose that | K | = s < k . Then E K can be identifiedwith the set (cid:8) ( z, x, y ) ∈ E s × E k − s × E k − s : x j = y j ∀ ≤ j ≤ k − s (cid:9) foreach K . Under this notation, ( a vw ) can be written as ( a K,zx,y ) and we have X K ⊂{ , ··· ,k }| K | = s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X ( v,w ) ∈ E K a vw ( U v ) ∗ U w (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X K ⊂{ , ··· ,k }| K | = s X z ∈ E s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X ( x,y ) ∈ E k − s × E k − s x j = y j a K,zx,y ( U x ) ∗ U y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . ii BENOˆIT COLLINS AND SANG-GYUN YOUN Moreover, Proposition 3.2 gives us ≤ X K ⊂{ , ··· ,k }| K | = s X z ∈ E s k − s X ( x,y ) ∈ E k − s × E k − s x j = y j | a K,zx,y | and, since we are summing N s (cid:0) ks (cid:1) elements, the Cauchy-Schwarz inequalitytells us that ≤ k − s N s (cid:18) ks (cid:19) X K ⊂{ , ··· ,k }| K | = s X z ∈ E s X ( x,y ) ∈ E k − s × E k − s x j = y j | a K,zx,y | = 3 k − s N s (cid:18) ks (cid:19) X K ⊂{ , ··· ,k }| K | = s X ( u,v ) ∈ E K | a vw | . Here, the Haagerup constant k − s appears due to s cancellations. Apply-ing the Cauchy-Schwartz inequality once more, we obtain that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X v,w ∈ E a vw ( U v ) ∗ U w (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k − X s =0 X K ⊂{ , ··· ,k }| K | = s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X ( v,w ) ∈ E K a vw ( U v ) ∗ U w (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k − X s =0 k − s N s (cid:18) ks (cid:19) X K ⊂{ , ··· ,k }| K | = s X ( u,v ) ∈ E K | a vw | ≤ k − X s =0 k − s N s (cid:18) ks (cid:19)! X K ⊂{ , ··· ,k } X ( u,v ) ∈ E K | a vw | = p ( N + 9) k − N k k a k = N k p (1 + 9 N − ) k − k a k . (cid:3) DDITIVITY VIOLATION OF THE REGULARIZED MINIMUM OUTPUT ENTROPY xiii
The regularized minimum output entropy of Φ N . Theorem 3.3 en-ables us to show that for any density matrix S (cid:13)(cid:13)(cid:13)(cid:13) (Φ cN ) ⊗ k ( S ) − N k Id N ⊗ k (cid:13)(cid:13)(cid:13)(cid:13) (3.11)is sufficiently small with respect to the Hilbert-Schmidt norm. This gener-alizes [Col18, Theorem 3.1]. Specifically, we prove the following theorem: Theorem 3.4.
For each k ∈ N , we have sup S (cid:13)(cid:13)(cid:13)(cid:13) (Φ cN ) ⊗ k ( S ) − N k Id ⊗ kN (cid:13)(cid:13)(cid:13)(cid:13) ≤ p (1 + 9 N − ) k − N k , (3.12) where S runs over all density matrices in T ( ℓ ( F k ∞ )) .Proof. Let X = ( x i,j ) i,j ∈ I k = (Φ cN ) ⊗ k ( S ) − N k Id ⊗ kN . Since tr( X ) = 0 , wehave tr( X ) = tr((Φ cN ) ⊗ k ( S ) X ) = tr( S (cid:0) (Φ cN ) ⊗ k (cid:1) ∗ ( X )) where (cid:0) (Φ cN ) ⊗ k (cid:1) ∗ denotes the adjoint map of (Φ cN ) ⊗ k . Moreover, (cid:0) (Φ cN ) ⊗ k (cid:1) ∗ ( X ) = 1 N k X i,j ∈ I k i = j x i,j U ∗ i U j where I = { , , · · · , N } . Hence, we have tr( S ((Φ cN ) ⊗ k ) ∗ ( X )) ≤ N k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X i,j ∈ I k i = j x i,j U ∗ i U j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . According to Theorem 3.3, we get k X k = tr( X ) ≤ N k p (1 + 9 N − ) k − k X k N k , (3.13)as claimed. (cid:3) This allows us to estimate the regularized minimum output entropies of Φ N as follows: Theorem 3.5.
For any k ∈ N we have H min (Φ ⊗ kN ) ≥ k log( N ) − p (1 + 9 N − ) k − . In particular, we have the following estimate for the regularized MOE H min (Φ N ) ≥ log( N ) − log(1 + 9 N ) ≥ log( N ) − N . iv BENOˆIT COLLINS AND SANG-GYUN YOUN
Proof.
Thanks to the fact that the von Neumann entropy is bigger thanR´enyi entropies of order α = 2 [MLDS + H ((Φ cN ) ⊗ k ( S )) ≥ α − α log (cid:0)(cid:13)(cid:13) (Φ cN ) ⊗ k ( S ) (cid:13)(cid:13) α (cid:1) ≥ − (cid:16) N − k (1 + p (1 + 9 N − ) k − (cid:17) = k log( N ) − (cid:16) p (1 + 9 N − ) k − (cid:17) for any density matrix S ∈ T ( ℓ ( F k ∞ )) by Theorem 3.4. In particular, wehave H min (Φ ⊗ kN ) ≥ k log( N ) − p (1 + 9 N − ) k − and the last conclusion follow from the following computation with L’Hˆopital’srule: lim k →∞ log(1 + p (1 + 9 N − ) k − k = lim k →∞ (1+ N ) k log(1+ N )2 √ (1+ N ) k − q (1 + N ) k − N ) . (cid:3)
4. A
DDITIVITY VIOLATION OF THE REGULARIZED
MOEIn this section, we choose two copies of Φ N as Φ N,l and Φ N,r . Indeed,these two quantum channels Φ N,l and Φ N,r are equivalent as explained inSection 2.2 and are in commuting-operator setup.Then we can obtain the following additivity violation of the regularizedMOE by generalizing Winter-Holevo-Hayden-Werner trick for Φ N,l ◦ Φ N,r : Theorem 4.1.
The regularized MOE is not additive: For any
N > e , wehave H min (Φ N,l ◦ Φ N,r ) < H min (Φ N,l ) + H min (Φ N,r ) . Proof.
Note that under notations from Subsection 2.2, (Φ N,l ◦ Φ N,r )( ρ ) = 1 N N X i,j =1 U i V j ρV ∗ j U ∗ i . Since | e ih e | is an invariant for U i V i , we have (Φ N,l ◦ Φ N,r )( | e ih e | ) = 1 N | e ih e | + 1 N X i,j : i = j | g i g − j ih g i g − j | , DDITIVITY VIOLATION OF THE REGULARIZED MINIMUM OUTPUT ENTROPY xv which implies H min (Φ N,l ◦ Φ N,r ) ≤ H min (Φ N,l ◦ Φ N,r ) ≤ H ((Φ N,l ◦ Φ N,r )( | e ih e | ))= log( N ) N + ( N − N ) · log( N ) N = 2 log( N ) − log( N ) N .
Moreover, Φ N,l and Φ N,r are copies of Φ N , so that we have N ) − log( N ) N < N ) − N ≤ H min (Φ N,l ) + H min (Φ N,r ) by Theorem 3.5 if N > e . (cid:3)
5. C
ONCLUDING REMARKS (1) Various versions of C ∗ -tensor products can be used to obtain commut-ing systems of operators. For example, let A, B be unital C ∗ -algebras andtake families of operators ( E i ) mi =1 ⊆ A and ( F j ) nj =1 ⊆ B . Also, supposethat A ⊗ max B ⊆ B ( K ) . Then { E i ⊗ B } mi =1 and { A ⊗ F j } nj =1 give us commuting systems of operators in B ( K ) . Moreover, if we supposethat C ∗ r ( F ∞ ) ⊗ max C ∗ r ( F ∞ ) ⊆ B ( K ) where C ∗ r ( F ∞ ) is the reduced group C ∗ -algebra of the free group F ∞ , then commuting systems { U i ⊗ Id } i and { Id ⊗ U j } j give another example of additivity violation in the commuting-operator setup.(2) Since Haagerup type inequalities exist for other groups (e.g hyper-bolic groups [dlH88]) or certain reduced free products of C ∗ -algebras [Bo˙z91],it is natural to expect that similar results should hold and will yield other ex-amples of additivity violation phenomena.(3) It is worthwhile to compare the main results of this paper and the casesof random unitary channels. On the side of random unitary channels, theregularized MOE is unknown, whereas our Theorem 3.5 gives us a strongestimate for the regularized MOE of Φ N .(4) One might wonder if we can evaluate the classical capacity of Φ cN whose output space is finite dimensional. Thanks to Theorem 3.5 and astandard argument, the classical capacity of Φ cN is upper bounded by C (Φ cN ) ≤ log( N ) − H min (Φ cN ) ≤ N .
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ENO ˆ IT C OLLINS , D
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APAN . E-mail address : [email protected] viii BENOˆIT COLLINS AND SANG-GYUN YOUN S ANG -G YUN Y OUN , D
EPARTMENT OF M ATHEMATICS E DUCATION , S
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