Classical communication over classical channels using non-classical correlation. Will Matthews University of Waterloo

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1 Classical communication over classical channels using non-classical correlation. Will Matthews University of Waterloo 1

2 Classical data over classical channels Q F E n Y X G ˆQ Finite input alphabet A, output alphabet B. 2

3 Classical data over classical channels Q F E n Y X G ˆQ Finite input alphabet A, output alphabet B. Channel input X in A n, output Y in B n (random variables). 2

4 Classical data over classical channels Q F E n Y X G ˆQ Finite input alphabet A, output alphabet B. Channel input X in A n, output Y in B n (random variables). Pr(Y = y X = x, E) = E n (y x). Discrete Memoryless Channel: E n (y x) = n i=1 E(y i x i ). 2

5 Classical data over classical channels Q F E n Y X G ˆQ Finite input alphabet A, output alphabet B. Channel input X in A n, output Y in B n (random variables). Pr(Y = y X = x, E) = E n (y x). Discrete Memoryless Channel: E n (y x) = n i=1 E(y i x i ). Message Q, decoded message ˆQ in [M] := {1,..., M} 2

6 Classical data over classical channels Q F E n Y X G ˆQ Finite input alphabet A, output alphabet B. Channel input X in A n, output Y in B n (random variables). Pr(Y = y X = x, E) = E n (y x). Discrete Memoryless Channel: E n (y x) = n i=1 E(y i x i ). Message Q, decoded message ˆQ in [M] := {1,..., M} Classical code: Encoder Pr(X = x Q = q) = F (x q); 2

7 Classical data over classical channels Q F E n Y X G ˆQ Finite input alphabet A, output alphabet B. Channel input X in A n, output Y in B n (random variables). Pr(Y = y X = x, E) = E n (y x). Discrete Memoryless Channel: E n (y x) = n i=1 E(y i x i ). Message Q, decoded message ˆQ in [M] := {1,..., M} Classical code: Encoder Pr(X = x Q = q) = F (x q); Decoder Pr( ˆQ = ˆq Y = y) = G(ˆq y). 2

8 Classical data over classical channels Q F E n Y X G ˆQ Finite input alphabet A, output alphabet B. Channel input X in A n, output Y in B n (random variables). Pr(Y = y X = x, E) = E n (y x). Discrete Memoryless Channel: E n (y x) = n i=1 E(y i x i ). Message Q, decoded message ˆQ in [M] := {1,..., M} Classical code: Encoder Pr(X = x Q = q) = F (x q); Decoder Pr( ˆQ = ˆq Y = y) = G(ˆq y). M ɛ (E n ) :=max M such that code with Pr(Q ˆQ E n, F, G) ɛ (hard to compute exactly). 2

9 Classical data over classical channels Q E n Y X ˆQ Correlated Encoder/Decoder: Pr(X = x, ˆQ = ˆq Q = q, Y = y) = Z(x, ˆq q, y) 3

10 Classical data over classical channels Q X E n Y ˆQ Correlated Encoder/Decoder: Pr(X = x, ˆQ = ˆq Q = q, Y = y) = Z(x, ˆq q, y) Entanglement assisted: Z(x, ˆq q, y) = Tr F (q) x G (y) ˆq ρ AB 3

11 Classical data over classical channels Q E n Y X ˆQ Correlated Encoder/Decoder: Pr(X = x, ˆQ = ˆq Q = q, Y = y) = Z(x, ˆq q, y) Entanglement assisted: Z(x, ˆq q, y) = Tr F (q) x G (y) ˆq ρ AB Non-signalling assisted: Z(x, ˆq q, y) is non-signalling. 3

12 Classical data over classical channels Q E n Y X ˆQ Correlated Encoder/Decoder: Pr(X = x, ˆQ = ˆq Q = q, Y = y) = Z(x, ˆq q, y) Entanglement assisted: Z(x, ˆq q, y) = Tr F (q) x G (y) ˆq ρ AB Non-signalling assisted: Z(x, ˆq q, y) is non-signalling. M E ɛ (E n ) :=max M such that entanglement assisted code with Pr(Q ˆQ E n, Z) ɛ. 3

13 Capacity C ɛ (E) := lim n 1 n log M ɛ(e n ), C(E) := lim ɛ 0 C Ω ɛ (E) 4

14 Capacity C ɛ (E) := lim n 1 n log M ɛ(e n ), C(E) := lim ɛ 0 C Ω ɛ (E) In general, zero-error capacity C 0 (E) C(E). 4

15 Capacity C ɛ (E) := lim n 1 n log M ɛ(e n ), C(E) := lim ɛ 0 C Ω ɛ (E) In general, zero-error capacity C 0 (E) C(E). Bennett, Shor, Smolin, Thapliyal (2001): For all DMCs: C E (E) = C(E) (quant-ph/ ). 4

16 Capacity C ɛ (E) := lim n 1 n log M ɛ(e n ), C(E) := lim ɛ 0 C Ω ɛ (E) In general, zero-error capacity C 0 (E) C(E). Bennett, Shor, Smolin, Thapliyal (2001): For all DMCs: C E (E) = C(E) (quant-ph/ ). Entanglement doesn t change capacity. But.. 4

17 Capacity C ɛ (E) := lim n 1 n log M ɛ(e n ), C(E) := lim ɛ 0 C Ω ɛ (E) In general, zero-error capacity C 0 (E) C(E). Bennett, Shor, Smolin, Thapliyal (2001): For all DMCs: C E (E) = C(E) (quant-ph/ ). Entanglement doesn t change capacity. But.. Entanglement can increase rate for fixed n and ɛ / reduce error prob. for fixed rate and n Cubitt, Leung, W.M., Winter: ( and ) Demonstration by Resch group at IQC ( ) 4

18 Capacity C ɛ (E) := lim n 1 n log M ɛ(e n ), C(E) := lim ɛ 0 C Ω ɛ (E) In general, zero-error capacity C 0 (E) C(E). Bennett, Shor, Smolin, Thapliyal (2001): For all DMCs: C E (E) = C(E) (quant-ph/ ). Entanglement doesn t change capacity. But.. Entanglement can increase rate for fixed n and ɛ / reduce error prob. for fixed rate and n Cubitt, Leung, W.M., Winter: ( and ) Demonstration by Resch group at IQC ( ) Leung, Mancinska, W.M., Ozols, Roy ( ): Example of DMC with C(E) = C E 0 (E) > C 0(E). 4

19 Motivation: Beyond Capacity Converse and achievability bounds on the rate 1 n log M ɛ(e n ) when ɛ = 1/1000 and E is the BSC with Pr(bit flip) = Capacity 0.4 Converse Rate 0.3 Achievability yond Capaci Blocklength n log ( n ) 1 Polyanskiy, Poor, Verdú. IEEE Trans. Inf. T., 56,

20 Motivation: Beyond Capacity Asymptotics of rate as function of n for DMCs: 1 V (E) n log M ɛ(e n ) = C(E) Q 1 (ɛ) + O(log n)/n n By Polyanskiy, Poor, Verdú (PPV converse) converse and achievability results for classical codes which match up to O(log n)/n. 6

21 Motivation: Beyond Capacity Asymptotics of rate as function of n for DMCs: 1 V (E) n log M ɛ(e n ) = C(E) Q 1 (ɛ) + O(log n)/n n By Polyanskiy, Poor, Verdú (PPV converse) converse and achievability results for classical codes which match up to O(log n)/n. Use of entanglement can t change capacity but can give striking qualitative changes to error behaviour below capacity. 6

22 Motivation: Beyond Capacity Asymptotics of rate as function of n for DMCs: 1 V (E) n log M ɛ(e n ) = C(E) Q 1 (ɛ) + O(log n)/n n By Polyanskiy, Poor, Verdú (PPV converse) converse and achievability results for classical codes which match up to O(log n)/n. Use of entanglement can t change capacity but can give striking qualitative changes to error behaviour below capacity. Goal: Looking beyond zero-error, quantify the extent to which entanglement improves coding. 6

23 Motivation: Beyond Capacity Asymptotics of rate as function of n for DMCs: 1 V (E) n log M ɛ(e n ) = C(E) Q 1 (ɛ) + O(log n)/n n By Polyanskiy, Poor, Verdú (PPV converse) converse and achievability results for classical codes which match up to O(log n)/n. Use of entanglement can t change capacity but can give striking qualitative changes to error behaviour below capacity. Goal: Looking beyond zero-error, quantify the extent to which entanglement improves coding. E.g. what does the asymptotic expansion of rate as a function of n look like after the capacity term with entanglement assistance? 6

24 PPV converse Q F Y E X G ˆQ 7

25 PPV converse Q F Y E X G ˆQ P XY : Pr(X = a) = 1 M q F (x q) =: p x, Pr(Y = y X = x) = E(y x). 7

26 PPV converse Q F X R Y G ˆQ P XY : Pr(X = a) = 1 M q F (x q) =: p x, Pr(Y = y X = x) = E(y x). P X R Y : Pr(X = a) = p x, Pr(Y = y X = x) = r y. 7

27 PPV converse Q F X R Y G ˆQ P XY : Pr(X = a) = 1 M q F (x q) =: p x, Pr(Y = y X = x) = E(y x). P X R Y : Pr(X = a) = p x, Pr(Y = y X = x) = r y. Test T for P XY : Specifies Pr(pass T X = x, Y = y). 7

28 PPV converse Q F X R Y G ˆQ P XY : Pr(X = a) = 1 M q F (x q) =: p x, Pr(Y = y X = x) = E(y x). P X R Y : Pr(X = a) = p x, Pr(Y = y X = x) = r y. Test T for P XY : Specifies Pr(pass T X = x, Y = y). β ɛ (P XY, P X R Y ): Minimum Pr(pass T P X R Y ) for all T with Pr(pass T P XY ) 1 ɛ. 7

29 PPV converse Given (M, ɛ) code for E with input distribution P X, construct test T for P XY (vs. P X R Y ): Pr(pass T X = x, Y = y) = T xy := q [M] G(q y) Pr(Q = q X = x, F ). 8

30 PPV converse Given (M, ɛ) code for E with input distribution P X, construct test T for P XY (vs. P X R Y ): Pr(pass T X = x, Y = y) = T xy := q [M] G(q y) Pr(Q = q X = x, F ). For P XY : Pr(X = x, Y = y) = p x E(y x), the probability of passing the test is T xy p x E(y x) = G(q y)e(y x)p x Pr(Q = q X = x, F ) x,y,q x,y = x,y,q G(q y)e(y x) Pr(Q = q X = x F ) = 1 M G(q y)e(y x)f (x q) x,y,q =1 ɛ. 8

31 PPV converse Given (M, ɛ) code for E with input distribution P X, construct test T for P XY (vs. P X R Y ): Pr(pass T X = x, Y = y) = T xy := q [M] G(q y) Pr(Q = q X = x, F ). For P X R Y : Pr(X = x, Y = y) = p x r y, the probability of passing the test is T xy p x r y = G(q y)r y p x Pr(Q = q X = x, F ) x,y,q x,y = x,y,q G(q y)r y Pr(Q = q X = x F ) = 1 M = 1 M G(q y)r y F (x q) x,y,q G(q y)r y y,q =1/M. 9

32 PPV converse If there exists a code of size M with distribution P X for channel input X, and error prob. ɛ, then for all distributions R Y on Y : β ɛ (P XY, P X R Y ) 1 M. 10

33 PPV converse If there exists a code of size M with distribution P X for channel input X, and error prob. ɛ, then for all distributions R Y on Y : β ɛ (P XY, P X R Y ) 1 M. For all codes of size M and error prob. ɛ min max β ɛ (P XY, P X R Y ) 1 P X R Y M. 10

34 PPV converse If there exists a code of size M with distribution P X for channel input X, and error prob. ɛ, then for all distributions R Y on Y : β ɛ (P XY, P X R Y ) 1 M. For all codes of size M and error prob. ɛ PPV converse: min max β ɛ (P XY, P X R Y ) 1 P X R Y M. M ɛ (W ) Mɛ 1 (W ) := max min P X R Y β ɛ (P XY, P X R Y ) 10

35 Non-signalling assisted codes: Minimum error LP q [M] x A E y B ˆq [M] ɛ NS (M, E) :=1 max 1 M subject to Z(x, q q, y)e(y x) q [M] x A y B ˆq [M], y B : Z(x, ˆq q, y) = 1, Z 0 x,ˆq ˆq, y, q, q : Z(x, ˆq q, y) = Z(x, ˆq q, y) x A x A x, q, y, y : Z(x, ˆq q, y) = Z(x, ˆq q, y ) ˆq [M] ˆq [M] 11

36 Simplified LP Let π(q) denote the image of q [M] under permutation π S M. Z(x, ˆq q, y) = 1 Z(x, π(ˆq) π(q), y) M! π S M 12

37 Simplified LP Let π(q) denote the image of q [M] under permutation π S M. Z(x, ˆq q, y) = 1 Z(x, π(ˆq) π(q), y) M! π S M Z is still NS, yields same error prob. and it is symmetric under simultaneous permutation of q and ˆq. 12

38 Simplified LP Let π(q) denote the image of q [M] under permutation π S M. Z(x, ˆq q, y) = 1 Z(x, π(ˆq) π(q), y) M! π S M Z is still NS, yields same error prob. and it is symmetric under simultaneous permutation of q and ˆq. This symmetry is equivalent to Z having the form: { D xy if ˆq = q Z(x, ˆq q, y) = W xy if ˆq q. 12

39 Simplified LP Let π(q) denote the image of q [M] under permutation π S M. Z(x, ˆq q, y) = 1 Z(x, π(ˆq) π(q), y) M! π S M Z is still NS, yields same error prob. and it is symmetric under simultaneous permutation of q and ˆq. This symmetry is equivalent to Z having the form: { D xy if ˆq = q Z(x, ˆq q, y) = W xy if ˆq q. Substitute this form into the LP for the error... 12

40 Simplified LP ɛ NS (M, E) := 1 max 1 M Z(x, q q, y)e(y x) q [M] x A y B ˆq [M], y B : Z(x, ˆq q, y) = 1 x,ˆq Z 0 ˆq, y, q, q : Z(x, ˆq q, y) = Z(x, ˆq q, y) x A x A x, q, y, y : Z(x, ˆq q, y) = Z(x, ˆq q, y ) ˆq [M] ˆq [M] 13

41 Simplified LP ɛ NS (M, E) := 1 max x A D xy E(y x) y B ˆq [M], y B : Z(x, ˆq q, y) = 1 x,ˆq Z 0 ˆq, y, q, q : Z(x, ˆq q, y) = Z(x, ˆq q, y) x A x A x, q, y, y : Z(x, ˆq q, y) = Z(x, ˆq q, y ) ˆq [M] ˆq [M] 13

42 Simplified LP ɛ NS (M, E) := 1 max x A D xy E(y x) y B y B : D xy + (M 1)W xy = 1 x R 0, W 0 ˆq, y, q, q : Z(x, ˆq q, y) = Z(x, ˆq q, y) x A x A x, q, y, y : Z(x, ˆq q, y) = Z(x, ˆq q, y ) ˆq [M] ˆq [M] 13

43 Simplified LP ɛ NS (M, E) := 1 max x A D xy E(y x) y B y B : D xy + (M 1)W xy = 1 x R 0, W 0 y B : D xy = W xy x A x A x, q, y, y : Z(x, ˆq q, y) = Z(x, ˆq q, y ) ˆq [M] ˆq [M] 13

44 Simplified LP ɛ NS (M, E) := 1 max x A D xy E(y x) y B y B : D xy + (M 1)W xy = 1 x R 0, W 0 y B : D xy = W xy x A x A x, y : D xy + (M 1)W xy = p x 13

45 Simplified LP ɛ NS (M, E) := 1 max x A D xy E(y x) y B subject to y B : D xy = 1/M(= γ) x A p x = 1, x A x A, y B : D xy p x, x A, y B : D xy 0. 14

46 LP for M NS ɛ Can rewrite to find the smallest value of 1/M for a given ɛ: Mɛ NS (E) 1 = min γ x A, y B : D xy p x y B : x A D xy γ E(y x)d xy 1 ɛ x A y B p x = 1 x A x A, y B : D xy 0, p x 0. 15

47 LP for M NS ɛ Can rewrite to find the smallest value of 1/M for a given ɛ: Mɛ NS (E) 1 = min γ x A, y B : D xy p x y B : x A D xy γ E(y x)d xy 1 ɛ x A y B p x = 1 x A x A, y B : D xy 0, p x 0. Also gives a converse for classical codes: M ɛ (E) Mɛ NS (E). 15

48 Equivalence to PPV converse The PPV converse is M ɛ (E) 1 = min p max r min T subject to E(y x)t xy p x 1 ɛ x A y B p x = 1, x y r y = 1 T xy p x r y x A y B x A, y B : p x 0, 0 T xy 1, r y 0 16

49 Equivalence to PPV converse The PPV converse is M ɛ (E) 1 = min p min T max r subject to E(y x)t xy p x 1 ɛ x A y B p x = 1, x y r y = 1 T xy p x r y x A y B x A, y B : p x 0, 0 T xy 1, r y 0 By von Neumann s minimax theorem. 16

50 Equivalence to PPV converse The PPV converse is M ɛ (E) 1 = min p min T max y B T xy p x x A subject to E(y x)t xy p x 1 ɛ x A y B p x = 1 x x A, y B : p x 0, 0 T xy 1 By convexity of optimisation over r. 16

51 Equivalence to PPV converse The PPV converse is Mɛ (E) 1 = min max D,p y B x A subject to E(y x)d xy 1 ɛ x A y B p x = 1 x D xy x A, y B : p x 0, 0 T xy 1 D xy := T xy p x 16

52 Equivalence to PPV converse The PPV converse is Mɛ (E) 1 = min max D,p y B x A subject to E(y x)d xy 1 ɛ x A y B p x = 1 x D xy x A, y B : p x 0, 0 D xy p x, D xy := T xy p x 16

53 Equivalence to PPV converse The PPV converse is M ɛ (E) 1 = min D,p γ subject to E(y x)d xy 1 ɛ x A y B p x = 1 x x A, y B : p x 0, 0 D xy p x, y B : γ x A D xy D xy := T xy p x 16

54 Equivalence to PPV converse The PPV converse is M ɛ (E) 1 = min D,p γ subject to E(y x)d xy 1 ɛ x A y B p x = 1 x x A, y B : p x 0, 0 D xy p x, y B : γ x A D xy Mɛ (E) = Mɛ NS (E) 16

55 Consequences All converse results which can be derived from PPV converse apply to non-signalling and entanglement assisted codes too (for discrete channels): 17

56 Consequences All converse results which can be derived from PPV converse apply to non-signalling and entanglement assisted codes too (for discrete channels): Non-signalling doesn t change capacity for completely general channels (using result of Verdú and Han). 17

57 Consequences All converse results which can be derived from PPV converse apply to non-signalling and entanglement assisted codes too (for discrete channels): Non-signalling doesn t change capacity for completely general channels (using result of Verdú and Han). Asymptotic expansion of rate unchanged by NS/entanglement assistance up to O(log n)/n. 17

58 Consequences All converse results which can be derived from PPV converse apply to non-signalling and entanglement assisted codes too (for discrete channels): Non-signalling doesn t change capacity for completely general channels (using result of Verdú and Han). Asymptotic expansion of rate unchanged by NS/entanglement assistance up to O(log n)/n. Linear program formulation of the PPV converse: 17

59 Consequences All converse results which can be derived from PPV converse apply to non-signalling and entanglement assisted codes too (for discrete channels): Non-signalling doesn t change capacity for completely general channels (using result of Verdú and Han). Asymptotic expansion of rate unchanged by NS/entanglement assistance up to O(log n)/n. Linear program formulation of the PPV converse: Dual formulation of LP gives a converse bound for any feasible point. 17

60 Consequences All converse results which can be derived from PPV converse apply to non-signalling and entanglement assisted codes too (for discrete channels): Non-signalling doesn t change capacity for completely general channels (using result of Verdú and Han). Asymptotic expansion of rate unchanged by NS/entanglement assistance up to O(log n)/n. Linear program formulation of the PPV converse: Dual formulation of LP gives a converse bound for any feasible point. For channels with permutation covariance (e.g. DMCs), LP can be simplified to size poly(n). 17

61 Dispersionless channels Information density: i(a; b) = log Pr(X=a,Y =b) Pr(X=a) Pr(Y =b). 18

62 Dispersionless channels Information density: i(a; b) = log Pr(X=a,Y =b) Pr(X=a) Pr(Y =b). Capacity C(E) is max. of expectation of information density over input distributions. 18

63 Dispersionless channels Information density: i(a; b) = log Pr(X=a,Y =b) Pr(X=a) Pr(Y =b). Capacity C(E) is max. of expectation of information density over input distributions. Dispersion V (E): Minimum variance of information density for capacity achieving input distributions. 18

64 Dispersionless channels Information density: i(a; b) = log Pr(X=a,Y =b) Pr(X=a) Pr(Y =b). Capacity C(E) is max. of expectation of information density over input distributions. Dispersion V (E): Minimum variance of information density for capacity achieving input distributions. 1 V (E) n log M ɛ(e n ) = C(E) Q 1 (ɛ) + O(log n)/n n 18

65 Dispersionless channels Information density: i(a; b) = log Pr(X=a,Y =b) Pr(X=a) Pr(Y =b). Capacity C(E) is max. of expectation of information density over input distributions. Dispersion V (E): Minimum variance of information density for capacity achieving input distributions. 1 V (E) n log M ɛ(e n ) = C(E) Q 1 (ɛ) + O(log n)/n n For DMCs C NS 0 (E) = C(E) V (E) = 0. 18

66 Future work Use dual LP formulation of converse to improve efficiently computable finite block length bounds for specific channels. 19

67 Future work Use dual LP formulation of converse to improve efficiently computable finite block length bounds for specific channels. Improved bounds for small block lengths by augmenting LP with certain families of Bell inequalities? 19

68 Future work Use dual LP formulation of converse to improve efficiently computable finite block length bounds for specific channels. Improved bounds for small block lengths by augmenting LP with certain families of Bell inequalities? Non-signalling converse generalises naturally to multi-terminal coding - investigate applications. 19

69 Future work Use dual LP formulation of converse to improve efficiently computable finite block length bounds for specific channels. Improved bounds for small block lengths by augmenting LP with certain families of Bell inequalities? Non-signalling converse generalises naturally to multi-terminal coding - investigate applications. 19

A Tight Upper Bound on the Second-Order Coding Rate of Parallel Gaussian Channels with Feedback

A Tight Upper Bound on the Second-Order Coding Rate of Parallel Gaussian Channels with Feedback Vincent Y. F. Tan (NUS) Joint work with Silas L. Fong (Toronto) 2017 Information Theory Workshop, Kaohsiung,