Universal superposition codes: capacity regions of compound quantum broadcast channel with confidential messages Holger Boche 12 , Gisbert Janßen 2 , Sajad Saeedinaeeni 2 1 Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany. 2 Lehrstuhl f¨ ur Theoretische Informationstechnik, Technische Universit¨ at M¨ unchen, 80290 M¨ unchen, Germany. ISIT 2020 1
Overview 1 Preview of results 2 Channel uncertainty 3 Broadcast channel and communication protocols 4 Discussion on the results 2
Preview of results • We derive universal codes for transmission of broadcast and confidential messages over classical-quantum-quantum and fully quantum channels. 3
Preview of results • We derive universal codes for transmission of broadcast and confidential messages over classical-quantum-quantum and fully quantum channels. • These codes are robust to channel uncertainties considered in the compound model. To construct these codes we generalize random codes for transmission of public messages, to derive a universal superposition coding for the compound quantum broadcast channel. 3
Preview of results • We derive universal codes for transmission of broadcast and confidential messages over classical-quantum-quantum and fully quantum channels. • These codes are robust to channel uncertainties considered in the compound model. To construct these codes we generalize random codes for transmission of public messages, to derive a universal superposition coding for the compound quantum broadcast channel. • As an application, we give a multi-letter characterization of regions corresponding to capacity of the compound quantum broadcast channel for transmitting broadcast and confidential messages simultaneously.(Full version of the paper is available at Journal of Mathematical Physics 61 (4), 2020.) 3
• Compound cqq broadcast channel For finite alphabet X and Hilbert spaces H B , H E , let W := { W s } s ∈ S ⊂ CQ ( X , H B ⊗ H E ) be a set of cqq channels. The compound cqq broadcast channel generated by this set is given by family { W ⊗ n , s ∈ S } ∞ n =1 . In other words, using n instances of the s compound channel is equivalent to using n instances of one of the channels from the uncertainty set. Abbildung: Compound model 4
Some remarks on the importance of the compound model • In general, a strong converse cannot be established on the capacity of the compound channel under average decoding error criterion (even for finite uncertainty sets), and hence no second-order capacity theorem is possible. • Further, calculation of the so called ǫ -capacity of the compound channel under the average error criterion is still an open question. We note however, that determining a second order ǫ -capacity for the compound channel is not possible, due to the observation, that there are examples of the compound channel where the optimistic ǫ -capacity is strictly larger than its pessimistic one. • In 1967 Ahlswede posed the question of whether or not there exist simple recursive formulas for the ǫ -capacity of the compound channel. This question was answered negatively by Boche, Schaefer, Poor (see ISIT 2020). • The non-computability of ǫ -capacity of the compound channel on the set of computable channels and under average error criterion, is implied by non-continuity of this function in its error input. 5
• AVQC The arbitrarily varying quantum channel generated by a set J := {N s } s ∈ S of CPTP maps with input Hilbert space H A and output Hilbert space H B , is given by family of CPTP maps B ) , s l ∈ S l , l ∈ N } ∞ {N s l : L ( H ⊗ l A ) → L ( H ⊗ l l =1 , where N s l := N s 1 ⊗ . . . N s l ( s l ∈ S l ) . Abbildung: Arbitrarily varying quantum channel 6
• Fully quantum AVC Let N ∈ C ( H A ⊗ H J , H B ) be a quantum channel whose input space is a tensor product of a Hilbert space H A (the legitimate sender’s space) and a Hilbert space H J which is under control of a quantum jammer. The fully quantum AVC generated by N is given by the family N n,σ ( · ) := N ⊗ n ( · ⊗ σ ) : σ ∈ S ( H ⊗ n � � J ) , n ∈ N of CPTP maps. Abbildung: Arbitrarily varying model with fully quantum jammer 7
Channel uncertainty • Compound channel Let J := {N s } s ∈ S ⊂ C ( H A , H B ) be a set of CPTP maps. The compound quantum channel generated by J is given by family {N ⊗ n : N ∈ J } ∞ n =1 . In other words, using n instances of the compound channel is equivalent to using n instances of one of the channels from the uncertainty set. • AVQC The arbitrarily varying quantum channel generated by a set J := {N s } s ∈ S of CPTP maps with input Hilbert space H A and output Hilbert space H B , is given by family of CPTP maps B ) , s l ∈ S l , l ∈ N } ∞ {N s l : L ( H ⊗ l A ) → L ( H ⊗ l l =1 , where N s l := N s 1 ⊗ . . . N s l ( s l ∈ S l ) . • Fully quantum AVC Let N ∈ C ( H A ⊗ H J , H B ) be a quantum channel whose input space is a tensor product of a Hilbert space H A (the legitimate sender’s space) and a Hilbert space H J which is under control of a quantum jammer. The fully quantum AVC generated by N is given by the family N n,σ ( · ) := N ⊗ n ( · ⊗ σ ) : σ ∈ S ( H ⊗ n � � J ) , n ∈ N of CPTP maps. 8
Communication scenarios over broadcast channel Broadcasting common and confidential messages (BCC) : where the compound channel is used n ∈ N times by the sender Alice in control of the input of the channel, to send two types of messages ( m 0 , m c ) simultaneously over the channel. • m 0 ∈ [ M 0 ,n ] , called the common message, that has to be reliably decoded by receiver Bob in control of Hilbert space H B and Eve in control of Hilbert space H E . • m c ∈ [ M c,n ] , called the confidential message, that has to be decoded reliably by Bob while Eve, the wiretapper, is kept ignorant. Abbildung: Common and confidential messaging 9
Communication scenarios over broadcast channel Transmitting public and confidential messages : where along with the confidential message m c ∈ [ M 0 ,n ] and instead of the common message, Alice wishes to send a ”public”message m 1 ∈ [ M 1 ,n ] , that is reliably decoded by Bob while it may or may not be decoded by Eve. Abbildung: Public and confidential messaging 10
Formal definitions of the codes BCC codes : • An ( n, M 0 ,n , M c,n ) BCC code for W , is a family C = ( E ( ·| m ) , D B , m , D E , m 0 ) m ∈ M with M := [ M 0 , n ] × [ M c , n ] , stochastic encoder E : M → P ( X n ) , POVMs ( D B, m ) m ∈ M on H ⊗ n and B ( D E,m 0 ) m 0 ∈ [ M 0 ,n ] on H ⊗ n E . • e B ( C , W ⊗ n ) := 1 B , m W ⊗ n � � x ∈X n E ( x | m )( D c B ( x )) and | M | m ∈ M E , m 0 W ⊗ n • e E ( C , W ⊗ n ) := 1 x ∈X n E ( x | m )( D c � � E ( x )) , | M | m ∈ M where, W γ , γ ∈ { B, E } are the marginal channels of W . Moreover, the security condition will be achieved by upper-bounding I ( M c ; E | M 0 , σ s,n ) . (1) 11
TPC codes : • An ( n, M 1 ,n , M c,n ) TPC code for W , is a family C = ( E ( ·| m ) , D B , m ) m ∈ M with M := [ M 1 , n ] × [ M c , n ] , stochastic encoder E : M → P ( X n ) and a POVM ( D B, m ) m ∈ M on H ⊗ n B . • e B ( C , W ⊗ n ) := 1 B , m W ⊗ n � � x ∈X n E ( x | m )( D c B ( x )) . Moreover, | M | m ∈ M the security condition will be achieved by upper-bounding I ( M c ; E | M 1 , σ s,n ) , (2) where σ s,n is the code state defined for all s ∈ S and n ∈ N by 1 � � E ( x | m ) W ⊗ n | m >< m | ⊗ σ s,n := ( x ) . (3) s | M | m ∈ M x ∈X n 12
Achievable rates • A pair ( R 0 , R c ) of non-negative numbers is called an achievable BCC rate pair for W , if for each ǫ, δ > 0 , exists an n 0 ( ǫ, δ ) ∈ N , such that for all n > n 0 , we find an ( n, M 0 ,n , M c,n ) BCC code C = ( E ( ·| m ) , D B , m , D E , m 0 ) m ∈ M such that 1 n log M i,n ≥ R i − δ ( i ∈ { 0 , c } ), 1 2 sup s ∈ S e γ ( C , W ⊗ n ) ≤ ǫ ( γ ∈ { B, E } ), s 3 sup s ∈ S I ( M c ; E | M 0 , σ s,n ) ≤ ǫ , are simultaneously fulfilled. • A pair ( R 1 , R c ) of non-negative numbers is called an achievable TPC rate pair for W , if for each ǫ, δ > 0 , exists an n 0 ( ǫ, δ ) ∈ N , such that for all n > n 0 , we find an ( n, M 1 ,n , M c,n ) TPC code C = ( E ( ·| m ) , D B , m ) m ∈ M such that 1 n log M i,n ≥ R i − δ ( i ∈ { 1 , c } ), 1 2 sup s ∈ S e B ( C , W ⊗ n ) ≤ ǫ , s 3 sup s ∈ S I ( M c ; E | M 1 , σ s,n ) ≤ ǫ are simultaneously fulfilled. 13
Capacity regions Given finite alphabets U , Y and probability distribution p = p UY X ∈ P ( U × Y × X n ) , with the random variables U, Y, X distributed accordingly we define C (1) � ˆ ( R 0 , R c ) ∈ R + 0 × R + � � W , p, n := 0 : R 0 ≤ inf s ∈ S min { I ( U ; B, ω s ) , I ( U ; E, ω s ) } ∧ R c ≤ inf s ∈ S I ( Y ; B | U, ω s ) − sup I ( Y ; E | U, ω s ) � . s ∈ S Then we have � ∞ 1 �� � � ˆ C (1) � C BCC [ W ] = cl W , p, l , (4) l l =1 p where we have used 1 l A := { ( 1 l x 1 , 1 l x 2 ) : ( x 1 , x 2 ) ∈ A } . The second union is taken over all p UY X ∈ P ( U × Y × X l ) such that random variable U − Y − X form a Markov chain and alphabets U and Y are finite. 14
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