Entanglement Transmission over Arbitrarily Varying Quantum Channels
aa r X i v : . [ qu a n t - ph ] A p r Entanglement Transmission over Arbitrarily VaryingQuantum Channels
Rudolf Ahlswede ∗ , Igor Bjelakovi´c † , Holger Boche † and Janis N¨otzel †∗ Working Group Information and Complexity, Universit¨at Bielefeld, GermanyEmail: [email protected] † Heinrich Hertz-Lehrstuhl f¨ur Informationstheorie und theoretische Informationstechnik, Technische Universit¨at Berlin, GermanyEmail: { holger.boche, igor.bjelakovic, janis.noetzel } @mk.tu-berlin.de Abstract —We derive a regularized formula for the commonrandomness assisted entanglement transmission capacity of fi-nite arbitrarily varying quantum channels (AVQC’s). For finiteAVQC’s with positive capacity for classical message transmissionwe show, by derandomization through classical forward commu-nication, that the random capacity for entanglement transmissionequals the deterministic capacity for entanglement transmission.This is a quantum version of the famous Ahlswede dichotomy.In the infinite case, we derive a similar result for certainclasses of AVQC’s. At last, we give two possible definitions ofsymmetrizability of an AVQC.
I. I
NTRODUCTION
We consider the task of entanglement transmission over anarbitrarily varying channel. This can be viewed as a three-party game in the following sense.The sender’s goal is to transmit one half of a maximallyentangled state to the receiver by some (large) number of usesof a quantum channel which is under the control of a thirdparty, called the adversary. The adversary is free to choosethe channel out of a set of memoryless, partly nonstationarychannels (cf. the beginning of section III). Only this given setis previously known to both sender and receiver.To make the situation even worse, the adversary knows theencoding-decoding procedure employed by sender and re-ceiver, so that they have to choose this procedure such thatit works well for all possible choices of channels that theadversary might come up with.Earlier results in comparable situations have been obtained byAhlswede [1],[2],[3] for classical arbitrarily varying channelsand Ahlswede and Blinovsky [4] in the case of classicalmessage transmission over an arbitrarily varying quantumchannel.In both cases we encounter a dichotomy: Either the capacityfor classical message transmission over the arbitrarily varying(quantum) channel is zero or it equals its common-randomnessassisted capacity. Also, for these models there exists the notionof symmetrizability . This is a necessary and sufficient single-letter condition for an arbitrarily varying (quantum) channelto have zero capacity for message transmission. Our workis based on ideas mainly taken from [1], [2] and our earlierresults for compound quantum channels [6].The paper is organized as follows: In Section II we fix thebasic notation. Section III introduces our channel model, in Section IV we summarize those of our results that lead to thequantum Ahlswede dichotomy. An outline of the strategy ofproof is given in Section V. Finally, in Section VI we addressthe question of symmetrizability.Details of the proofs given in this paper as well as theconverse part of the coding theorem can be picked up in theaccompanying paper [7].II. N
OTATION AND CONVENTIONS
All Hilbert spaces are assumed to have finite dimension andare over the field C . S ( H ) is the set of states, i.e. positive semi-definite operators with trace acting on the Hilbert space H . If F ⊂ H is a subspace of H then we write π F for the maximallymixed state on F , i.e. π F = p F tr( p F ) where p F stands for theprojection onto F . For a finite set A , P ( A ) denotes the set ofprobability distributions on A .The set of completely positive trace preserving (CPTP) mapsbetween the operator spaces B ( H ) and B ( K ) is denoted by C ( H , K ) . C ↓ ( H , K ) stands for the set of completely positivetrace decreasing maps between B ( H ) and B ( K ) .We use the base two logarithm which is denoted by log . Thevon Neumann entropy of a state ρ ∈ S ( H ) is given by S ( ρ ) := − tr ( ρ log ρ ) . The coherent information for
N ∈ C ( H , K ) and ρ ∈ S ( H ) isdefined by I c ( ρ, N ) := S ( N ( ρ )) − S (( id B ( H ) ⊗ N )( | ψ ih ψ | )) , where ψ ∈ H ⊗ H is an arbitrary purification of the state ρ . Following the usual conventions we let S e ( ρ, N ) := S (( id B ( H ) ⊗ N )( | ψ ih ψ | )) denote the entropy exchange.As a measure of entanglement preservation we use entangle-ment fidelity. For ρ ∈ S ( H ) and N ∈ C ↓ ( H , K ) it is givenby F e ( ρ, N ) := h ψ, ( id B ( H ) ⊗ N )( | ψ ih ψ | ) ψ i , with ψ ∈ H ⊗ H being an arbitrary purification of the state ρ .We use the diamond norm || · || ♦ as a measure of closeness inthe set of quantum channels, which is given by ||N || ♦ := sup n ∈ N max a ∈B ( C n ⊗H ) , || a || =1 || ( id n ⊗ N )( a ) || , (1)here id n : B ( C n ) → B ( C n ) is the identity channel, and N : B ( H ) → B ( K ) is any linear map, not necessarily com-pletely positive. The merits of || · || ♦ are due to the followingfacts (cf. [12]). First, ||N || ♦ = 1 for all N ∈ C ( H , K ) .Thus, C ( H , K ) ⊂ S ♦ , where S ♦ denotes the unit sphereof the normed space ( B ( B ( H ) , B ( K )) , || · || ♦ ) . Moreover, ||N ⊗ N || ♦ = ||N || ♦ ||N || ♦ for arbitrary linear maps N , N : B ( H ) → B ( K ) . Finally, the supremum in (1) needsonly be taken over n that range over { , , . . . , dim H} . We further use the diamond norm to define the function D ♦ ( · , · ) on { ( I , I ′ ) : I , I ′ ⊂ C ( H , K ) } , which is for I , I ′ ⊂ C ( H , K ) given by D ♦ ( I , I ′ ) :=max { sup N ∈ I inf N ′ ∈ I ′ ||N − N ′ || ♦ , sup N ′ ∈ I ′ inf N ∈ I ||N − N ′ || ♦ } . For compact sets, this is basically the Hausdorff distanceinduced by the diamond norm.For arbitrary I , I ′ ⊂ C ( H , K ) , D ♦ ( I , I ′ ) ≤ ǫ implies that forevery N ∈ I ( N ′ ∈ I ′ ) there exists N ′ ∈ I ′ ( N ∈ I ) suchthat ||N − N ′ || ♦ ≤ ǫ . In this way D ♦ gives a measure ofdistance between sets of channels.For an arbitrary set S , S l := { ( s , . . . , s l ) : s i ∈ S ∀ i ∈{ , . . . , l }} . We write s l for the elements of S l .For I ⊂ C ( H , K ) we denote its convex hull by conv ( I ) , anotation which is adapted from [14].III. C ODES FOR ENTANGLEMENT AND MESSAGETRANSMISSION
An arbitrarily varying quantum channel (AVQC) generatedby a set I = {N s } s ∈ S of CPTP maps with input Hilbert space H and output Hilbert space K is the family of CPTP maps {N s l : B ( H ) ⊗ l → B ( K ) ⊗ l } l ∈ N ,s l ∈ S l , where N s l := N s ⊗ . . . ⊗ N s l ( s l ∈ S l ) . In order to relieve ourselves of the burden of complicatednotation we will simply write I = {N s } s ∈ S for the AVQC.Even in the case of a finite set I = {N s } s ∈ S , showing theexistence of reliable codes for the AVQC I is a non-trivialtask, since for each block length l ∈ N we have to dealwith | I | l memoryless partly non-stationary quantum channelssimultaneously.Let I = {N s } s ∈ S be an AVQC.An ( l, k l ) − random entanglement transmission code for I is aprobability measure µ l on ( C ( F l , H ) × C ( K , F ′ l ) , σ l ) , where dim F l = k l , F l ⊂ F ′ l and the sigma-algebra σ l is chosen suchthat F e ( π F l , ( · ) ◦ N s l ◦ ( · )) is measurable w.r.t. σ l for every s l ∈ S l . Moreover, we assume that σ l contains all singletonsets. An example of such a sigma-algebra σ l is given by theproduct of sigma-algebras of Borel sets induced on C ( F l , H ) and C ( K , F ′ l ) by the standard topologies of the ambient spaces. Definition 1:
A non-negative number R is said to be anachievable entanglement transmission rate for I with randomcodes if there is a sequence of ( l, k l ) − random entanglementtransmission codes such that 1) lim inf l →∞ l log k l ≥ R and2) lim l →∞ inf s l ∈ S l R F e ( π F l , R l ◦ N s l ◦ P l ) dµ l ( P l , R l ) = 1 .The random capacity A r ( I ) for entanglement transmissionover I is defined by A r ( I ) := sup { R : R is an achievable entanglement trans-mission rate for I with random codes } . We are now in a position to introduce deterministic codes:An ( l, k l ) − code for entanglement transmission over I is an ( l, k l ) − random code for I with µ l ( { ( P l , R l ) } ) = 1 for someencoder-decoder pair ( P l , R l ) and µ l ( A ) = 0 for any A ∈ σ l with ( P l , R l ) / ∈ A . We will refer to such measures as pointmeasures in what follows. Definition 2:
A non-negative number R is a deterministi-cally achievable rate for entanglement transmission over I ifit is achievable in the sense of Definition 1 for random codeswith point measures µ l .The deterministic capacity A d ( I ) for entanglement transmis-sion over the AVQC I is given by A d ( I ) := sup { R : R is a deterministically achievable ratefor entanglement transmission over I } . Finally, we shall need the notion of the classical deterministiccapacity C det ( I ) of the AVQC I = {N s } s ∈ S with average error criterion. An ( l, M l ) -(deterministic) code for messagetransmission is a family of pairs C l = ( ρ i , D i ) M l i =1 where ρ , . . . , ρ M l ∈ S ( H ⊗ l ) , and positive semi-definite operators D , . . . , D M l ∈ B ( K ⊗ l ) satisfying P M l i =1 D i = K ⊗ l . Theunderlying error criterion we shall use is the worst-caseaverage probability of error of the code C l which is givenby ¯ P e,l ( I ) := sup s l ∈ S l ¯ P e ( C l , s l ) , (2)where for s l ∈ S l we set P e ( C l , s l ) := 1 M l M l X i =1 (1 − tr( N s l ( ρ i ) D i )) . The achievable rates and the classical deterministic capacity C det ( I ) of I , with respect to the error criterion given in (2),are then defined in the usual way (see e.g. [4]).IV. M AIN RESULTS
The compound quantum channel generated by conv ( I ) (cf.[6] for the relevant definitions) shall play the crucial role inour derivation of the coding results stated below.Our main result, a quantum version of Ahlswede’s dichotomyfor finite AVQCs, goes as follows: Theorem 3:
Let I = {N s } s ∈ S be a finite AVQC.1) The random capacity for entanglement transmission over I is given by A r ( I ) = lim l →∞ l max ρ ∈S ( H ⊗ l ) inf N ∈ conv ( I ) I c ( ρ, N ⊗ l ) . (3) This explains our requirement on σ l to contain all singleton sets. ) Either C det ( I ) = 0 or else A d ( I ) = A r ( I ) . Remark. It is clear from convexity of entanglement fidelity inthe input state that A d ( I ) ≤ C det ( I ) , so that C det ( I ) = 0 implies A d ( I ) = 0 . Therefore, Theorem 3 determines A d ( I ) ,in principle, up to required regularization on the right-handside of (3) and the question of when C det ( I ) = 0 happens. Wederive a non-single-letter necessary and sufficient conditionfor the latter in Section VI. In the case that S is infinite, we have the following statement: Theorem 4:
Let I = {N s } s ∈ S be any AVQC and ∂ C thetopological boundary of C ( H , K ) . If D ♦ (˜ I , ∂ C ) > , then A r ( I ) = lim l →∞ l max ρ ∈S ( H ⊗ l ) inf N ∈ conv ( I ) I c ( ρ, N ⊗ l ) . Remark. The condition D ♦ (˜ I , ∂ C ) > in Theorem 4 stemsfrom our strategy of approximation of an infinite AVQCthrough a sequence of finite AVQC’s. We hope to be able todrop this artificial constraint in the final version of the paper. V. O
UTLINE OF THE PROOF
This section is split into three parts. First, we demonstratethe existence of asymptotically optimal sequences of randomcodes (in the sense of (3)). We use Ahlswede’s robustificationtechnique originally presented in [2] in the form presented in[3] and our results on compound quantum channels [6] in orderto get a sequence of finitely supported probability measures µ l on the set of encoding and decoding maps. Second, weshow that the support of each µ l can be taken as a set withcardinality l .Third, we show that C d ( I ) > implies that we can deran-domize our code without any asymptotic loss of capacity, sothat A d ( I ) = A r ( I ) holds.Fourth, we briefly sketch how approximation of conv ( I ) byconvex polytopes leads to Theorem 4. A. Finite AVQC
Let l ∈ N and let P l denote the set of permutationsacting on { , . . . , l } . Suppose we are given a finite set S .Then each permutation P ∈ P l induces an action on S l by P : S l → S l , P ( s l ) i := s P ( i ) . By T ( l, S ) , we denote theset of types on S induced by the elements of S l , i.e. the setof empirical distributions on S generated by sequences in S l .Now Ahlswede’s robustification can be stated as follows. Theorem 5 (Robustification technique, cf. [3]):
If a func-tion f : S l → [0 , satisfies X s l ∈ S l f ( s l ) q ( s ) · . . . · q ( s l ) ≥ − γ (4)for all q ∈ T ( l, S ) and some γ ∈ [0 , , then l ! X P ∈ P l f ( P ( s l )) ≥ − ( l + 1) | S | · γ ∀ s l ∈ S l . (5)As another ingredient for the arguments to follow we need an achievability result for the compound channel conv ( I ) . We setfor k ∈ N conv ( I ) ⊗ k := {N ⊗ kq } q ∈ P ( S ) . Lemma 6:
Let k ∈ N . Suppose that max ρ ∈S ( H ⊗ k ) inf N ∈ conv ( I ) ⊗ k I c ( ρ, N ) > holds. Then for each sufficiently small η > there is a se-quence of ( l, k l ) -codes ( P l , R l ) l ∈ N such that for all l ≥ l ( η ) the inequalities F e ( π F l , R l ◦ N ⊗ l ◦ P l ) ≥ − − lc ∀N ∈ conv ( I ) , (6) l log dim F l ≥ k max ρ ∈S ( H ⊗ k ) inf N ∈ conv ( I ) ⊗ k I c ( ρ, N ) − η, (7)hold with a constant c = c ( k, dim H , dim K , conv ( I ) , η ) > . Proof:
The proof follows from an application of thecompound BSST Lemma and Lemma 9 in [6]. These twostatements show the existence of well behaved codes for thechannels N ⊗ m · kq , where m depends on conv ( I ) , k and η . Forfixed k , all we have to do is convert these codes to codes forthe channels N q .In the next step we will combine the robustification techniqueand Lemma 6 to prove the existence of good random codesfor the AVQC I = {N s } s ∈ S .Recall that there is a canonical action of P l on B ( H ) ⊗ l givenby P H ( a ⊗ . . . ⊗ a l ) := a P − (1) ⊗ . . . ⊗ a P − ( n ) . It is easyto see that P H ( a ) = U P aU ∗ P , ( a ∈ B ( H ) ⊗ l ) with the unitaryoperator U P : H ⊗ l → H ⊗ l defined by U P ( x ⊗ . . . ⊗ x l ) = x P − (1) ⊗ . . . ⊗ x P − ( l ) . Theorem 7 (Conversion of compound codes):
Let I = {N s } s ∈ S be a finite AVQC. For each k ∈ N and anysufficiently small η > there is a sequence of ( l, k l ) -codes ( P l , R l ) l ∈ N , P l ∈ C ( F l , H ⊗ l ) , R l ∈ C ( K ⊗ l , F ′ l ) , for thecompound channel built up from conv ( I ) satisfying l log dim F l ≥ k max ρ ∈S ( H ⊗ k ) inf N ∈ conv ( I ) ⊗ k I c ( ρ, N ) − η (8)and, for all sufficiently large l ∈ N and s l ∈ S l , X P ∈ P l l ! F e ( π F l , R l ◦ P − K ◦N s l ◦ P H ◦P l ) ≥ − ( l +1) | S | · − lc , (9)with a positive number c = c ( k, dim H , dim K , conv ( I ) , η ) . Proof:
We let ( R l , P l ) be as in Theorem 6. Setting f ( s l ) := F e ( π F l , R l ◦ N s l ◦ P l ) and applying Theorem 5proves the theorem.For l ∈ N , define a discretely supported probability measure µ l by µ l := 1 l ! X P ∈ P l δ ( P H ◦P l , R l ◦ P − K ) , where δ ( P H ◦P l , R l ◦ P − K ) denotes the probability measure thatputs measure on the point ( P H ◦ P l , R l ◦ P − K ) , we obtainfor each k ∈ N a sequence of ( l, k l ) -random codes achieving k max ρ ∈S ( H ⊗ k ) inf N ∈ conv ( I ) ⊗ l I c ( ρ, N ) . his leads to the following corollary to Theorem 7. Corollary 8:
For any finite AVQC I = {N s } s ∈ S we have A r ( I ) ≥ lim l →∞ l max ρ ∈S ( H ⊗ l ) inf N ∈ conv ( I ) I c ( ρ, N ⊗ l ) . B. Derandomization
In this section we will prove the second claim made inTheorem 3 by following Ahlswede’s elimination technique.The proof is based on the following lemma, which shows thatnot much of common randomness is needed to achieve A r ( I ) . Lemma 9 (Random Code Reduction):
Let I = {N s } s ∈ S bea finite AVQC, l ∈ N , and µ l an ( l, k l ) -random code for theAVQC I with min s l ∈ S l Z F e ( π F l , R l ◦ N s l ◦ P l ) dµ l ( P l , R l ) ≥ − − la (10)for some positive constant a ∈ R .Let ε ∈ (0 , . Then for all sufficiently large l ∈ N thereexist l codes { ( P li , R li ) : i = 1 , . . . , l } ⊂ C ( F l , H ⊗ l ) ×C ( K ⊗ l , F ′ l ) such that l l X i =1 F e ( π F l , R li ◦ N s l ◦ P li ) > − ε ∀ s n ∈ S n . (11) Proof:
We define random variables (Λ i , Ω i ) , i = 1 , . . . , l with values in C ( F l , H ⊗ l ) × C ( K ⊗ l , F ′ l ) which are i.i.d. ac-cording to µ ⊗ l l . Using Markov’s inequality and the inequality γt ≤ (1 − t )2 γ · + t γ ≤ t γ , t ∈ [0 , , γ > as wellas the union bound we get P (cid:16) l P l i =1 F e ( π F l , Λ i ◦ N s l ◦ Ω i ) > − ε ∀ s l ∈ S l (cid:17) ≥ −| S | l · − l ε . For large enough l the above probability ispositive. This shows the existence of the required realizationof (Λ i , Ω i ) l i =1 . Proof: (Of the second claim in Theorem 3) . As shownabove, in order to achieve A r ( I ) we need only randomcodes with discrete support on subexponentially many points.Whenever C d ( I ) > and A r ( I ) > the sender can transmitclassical information at rate zero over the AVQC in order toderandomize the code without any asymptotic reduction in thecapacity for entanglement transmission. C. Infinite AVQC’s
Let I = {N s } s ∈ S with | S | = ∞ . We consider the set ˜ I := conv ( I ) ||·|| ♦ - the closure of conv ( I ) w.r.t. || · || ♦ . Supposethat D ♦ (˜ I , ∂ C ) =: a > . (12)Our goal is to find an outer approximation of ˜ I in Hausdorffmetric (cf. Section II) by polytopes contained entirely in the set C ( H , K ) . To this end, we need the following result of convexanalysis (cf. Theorem 3.1.6, p. 109, in [14]). Theorem 10:
Let A be a non-empty compact convex setin R d and let ε > . Then there exist polytopes P, Q in R d such that P ⊆ A ⊆ Q and D ( A, P ) ≤ ε , D ( A, Q ) ≤ ε , where D ( · , · ) denotes the Hausdorff distance induced by theeuclidean norm on R d .We note that the presence of R d and the euclidean norm inTheorem 10 is not essential at all. The theorem holds as wellfor any finite dimensional normed space with correspondingHausdorff distance induced by the given norm. Proof: (Of Theorem 4.)
We apply Theorem 10 to the space H ( H , K ) := B h ( B ( H ) , B ( K )) of hermiticity preserving linearmaps from B ( H ) into B ( K ) endowed with ||·|| ♦ and obtain foreach ε > a polytope ¯ Q ε with ˜ I ⊆ ¯ Q ε and D ♦ (˜ I , ¯ Q ε ) ≤ ε. Let E denote the affine hull of C ( H , K ) in H ( H , K ) and set Q ε := E ∩ ¯ Q ε . Then Q ε is a polytope and for all sufficientlysmall ε > ( ε ≤ a , say, is small enough for this purpose) wehave ˜ I ⊆ Q ε ⊂ C ( H , K ) by (12). More important, we alsohave D ♦ (˜ I , Q ε ) ≤ D ♦ (˜ I , ¯ Q ε ) ≤ ε. (13)Let I ε = {N , . . . , N K } be the extremal points of Q ε . Then I ε has the following properties: 1) conv ( I ) ⊂ ˜ I ⊂ Q ε = conv ( I ε ) , 2) D ♦ (˜ I , conv ( I ε )) ≤ ε for all sufficiently small ε > by (13).We can now apply all results from Section V-A to the finiteAVQC generated by I ε giving us to each sufficiently small η > and k ∈ N a sequence of ( l, k l ) -random codes ( P l , R l ) l ∈ N with P l ∈ C ( F l , H ⊗ l ) , R l ∈ C ( K ⊗ l , F ′ l ) , F e ( π F l , R l ◦N t l ◦P l ) ≥ − ( l +1) K · − lc ∀ t l ∈ { , . . . , K } l , (14)and l log k l ≥ k inf N ∈ conv ( I ε ) I c ( ρ, N ⊗ k ) − η , (15)for any ρ ∈ S ( H ⊗ k ) and all sufficiently large l ∈ N with a pos-itive constant c = c ( k, dim H , dim K , I ε , η ) . Since I ⊆ ˜ I ⊆ conv ( I ε ) we can find to any finite collection N ′ , . . . , N ′ l ∈ I probability distributions q , . . . , q l ∈ P ( { , . . . , K } ) with N ′ i = P Kj =1 q i ( j ) N j ( N j ∈ I ε , j ∈ { , . . . , K } ) . Thus,for any choice of N ′ , . . . , N ′ l ∈ I F e ( π F l , R l ◦ ( ⊗ li =1 N ′ i ) ◦ P l ) ≥ − ( l + 1) K · − lc , (16)by (14). On the other hand, Lemma 16 in [6] and D ♦ ( conv ( I ) , conv ( I ε )) ≤ D ♦ (˜ I , conv ( I ε )) ≤ ε shows that l log k l ≥ k inf N ∈ conv ( I ) I c ( ρ, N ⊗ k ) − η, (17)whenever ε is small enough. It should be noted that k and l inthe above equation tend to infinity when η goes to zero. Since η > was arbitrary, we are done.VI. S YMMETRIZABILITY
In this section we introduce a notion of symmetrizabilitywhich is a sufficient and necessary condition for C det ( I ) = 0 .Our approach is motivated by the corresponding concept forarbitrarily varying channels with classical input and quantumoutput (cq-AVC) given in [4]. In what follows we will restrictourselves to the case | S | < ∞ . efinition 11: Let S be a finite set and I = {N s } s ∈ S anAVQC.1) I is called l -symmetrizable, l ∈ N , if for each finiteset { ρ , . . . , ρ K } ⊂ S ( H ⊗ l ) , K ∈ N , there is amap p : { ρ , . . . , ρ K } → P ( S l ) such that for all i, j ∈ { , . . . , K } the following holds: X s l ∈ S l p ( ρ i )( s l ) N s l ( ρ j ) = X s l ∈ S l p ( ρ j )( s l ) N s l ( ρ i ) . (18)2) We call I symmetrizable if it is l -symmetrizable for all l ∈ N . Theorem 12:
Let I = {N s } s ∈ S , | S | < ∞ , be an AVQC.Then I is symmetrizable if and only if C det ( I ) = 0 . Proof:
The proof follows closely the corresponding argu-ments given in [11], [10], and [4].
Corollary 13:
If the AVQC I = {N } s ∈ S is symmetrizablethen A d ( I ) = 0 . Proof:
Note that A d ( I ) ≤ C det ( I ) and apply Theorem12.What is missing now is the reverse direction in Corollary 13:That an AVQC with A d ( I ) = 0 is symmetrizable. It is notknown yet whether this implication is true or not.The final issue in this section is a sufficient conditionfor A r ( I ) = 0 which is based on the notion of qc-symmetrizability. Let B + ( H ) ⊂ B ( H ) be the set of nonnega-tive operators. SetQC ( H , S ) := {{ T s } s ∈ S ⊂ B + ( H ) : X s ∈ S T s = H } . For a given finite set of quantum channels I = {N s } s ∈ S and T ∈ QC ( H , S ) we define a CPTP map M T, S : B ( H ) ⊗B ( H ) → B ( K ) by M T, S ( a ⊗ b ) := X s ∈ S tr ( T s a ) N s ( b ) . (19) Definition 14:
An arbitrarily varying quantum channel,generated by a finite set I = {N s } s ∈ S , is called qc-symmetrizable if there is T ∈ QC ( H , S ) such that for all a, b ∈ B ( H ) M T, S ( a ⊗ b ) = M T, S ( b ⊗ a ) (20)holds, where M T, S : B ( H ) ⊗ B ( H ) → B ( K ) is the CPTPmap defined in (19).The best illustration of the definition of qc-symmetrizabilityis given in the proof of our next theorem: Theorem 15:
If an arbitrarily varying quantum channel gen-erated by a finite set I = {N s } s ∈ S is qc-symmetrizable,then for any sequence of ( l, k l ) -random codes ( µ l ) l ∈ N with k l = dim F l ≥ for all l ∈ N we have inf s l ∈ S l Z F e ( π F l , R l ◦ N s l ◦ P l ) dµ l ( R l , P l ) ≤ , for all l ∈ N . Thus A r ( I ) = 0 , and consequently A d ( I ) = 0 . Remark: Our Definition 14 addresses the notion of qc-symmetrizability for block length l = 1 . In our accompanyingpaper [7] we show that the corresponding definition forarbitrary l is equivalent.Proof: Let l ∈ N . We fix σ ∈ S ( H ) and define E , E ∈C ( H , K ) by E ( a ) := M T, S ( σ ⊗ a ) , E ( a ) := M T, S ( a ⊗ σ ) = X s ∈ S tr ( E s a ) N s ( σ ) . (21)Setting E s l := E s ⊗ . . . ⊗ E s l , we can show that Z F e ( π F l , R l ◦ E ⊗ l ◦ P l ) dµ l ( R l , P l ) ≥ inf s l ∈ S l Z F e ( π F l , R l ◦ N s l ◦ P l ) dµ l ( R l , P l ) . (22)Now, by the assumed qc-symmetrizability, we get id F l ⊗ ( R l ◦ E ⊗ l ) = id F l ⊗ ( R l ◦ E ⊗ l ) , thus (23) F e ( π F l , R l ◦ E ⊗ l ◦ P l ) = F e ( π F l , R l ◦ E ⊗ l ◦ P l ) . (24)But E is entanglement breaking, implying that ( id F l ⊗ R l ◦ E ⊗ l ◦ P l )( | ψ l ih ψ l | ) (for a purification ψ l of π F l ) is separable.A standard result from entanglement theory implies that h ψ l , ( id F l ⊗ R l ◦ E ⊗ l ◦ P l )( | ψ l ih ψ l | ) ψ l i ≤ k l (25)holds, since ψ l is maximally entangled with Schmidt rank k l .Combining (22), (24), (25) and our assumption k l ≥ we get inf s l ∈ S l R F e ( π F l , R l ◦ N s l ◦ P l ) dµ l ( R l , P l ) ≤ .R EFERENCES[1] R. Ahlswede, “Elimination of Correlation in Random Codes for Arbi-trarily Varying Channels”,
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