Separation between quantum Lovász number and entanglement-assisted zero-error classical capacity

Transcription

1 Separation between quantum Lovász number and entanglement-assisted zero-error classical capacity Xin Wang QCIS, University of Technology Sydney (UTS) Joint work with Runyao Duan (UTS), arxiv: AQIS 2016, Taipei

2 Introduction

3 Capacity of a communication channel 1

4 Capacity of a communication channel The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point. Claude Elwood Shannon, 1948 Approximately (small-error) [Shannon-1948]: m, m m Exactly (zero-error) [Shannon-1956]: m, m = m 1

5 Zero-error capacity (Shannon capacity) of a classical channel Classical channel: N (y x), e.g., 2

6 Zero-error capacity (Shannon capacity) of a classical channel Classical channel: N (y x), e.g., One-shot zero-error capacity α(n ): the maximum number of messages can be sent via N without error, depends on G. Asymptotic zero-error capacity: C 0(N ) = sup n 1 1 n log α ( N n). Example: α(c 5) = 2, α(c 2 5 ) = 5 [Shannon-1956], C 0(C 5) = log 5 [Lovász-1979]. If there is feedback assistance, the zero-error capacity depends only on the bipartite graph [Shannon-1956]. 2

7 Zero-error capacity of a quantum channel One-shot zero-error classical capacity: The maximum integer n such that {N (ρ 1),..., N (ρ n)} are orthogonal (perfectly distinguished). Quantum Channel: completely positive (CP) and trace-preserving (TP) linear map N from A to B. Choi-Kraus Representation: N (ρ) = k E kρe k, k E k E k = 1 Choi-Jamiołkowski matrix: J N = ij i j A N ( i j A ) = (id A N ) Φ AA Φ AA, with Φ AA = k k A k A 3

8 Zero-error capacity of a quantum channel One-shot zero-error classical capacity: The maximum integer n such that {N (ρ 1),..., N (ρ n)} are orthogonal (perfectly distinguished). Quantum Channel: completely positive (CP) and trace-preserving (TP) linear map N from A to B. Choi-Kraus Representation: N (ρ) = k E kρe k, k E k E k = 1 Choi-Jamiołkowski matrix: J N = ij i j A N ( i j A ) = (id A N ) Φ AA Φ AA, with Φ AA = k k A k A Non-commutative (confusability) graph [Duan-Severini-Winter-13]: S = span{e k E j } (S = S, 1 S). The zero-error capacity of N depends only on S. Non-commutative bipartite graph [Duan-Severini-Winter-16]: K = K(N ) = span{e k }. If there is feedback assistance, the zero-error capacity depends only on K. 3

9 Quantum Lovász number The zero-error capacity C 0 is not known to be computable. 4

10 Quantum Lovász number The zero-error capacity C 0 is not known to be computable. Nevertheless, [Lovász-1979] introduce the Lovász number to upper bound the zero-error capacity of a classical channel: 2 C 0(N ) ϑ(g) = max{ 1+T : T i,j = 0 if {i, j} E, T = T, 1+T 0}. 4

11 Quantum Lovász number The zero-error capacity C 0 is not known to be computable. Nevertheless, [Lovász-1979] introduce the Lovász number to upper bound the zero-error capacity of a classical channel: 2 C 0(N ) ϑ(g) = max{ 1+T : T i,j = 0 if {i, j} E, T = T, 1+T 0}. [Duan-Severini-Winter-2013] introduced the quantum Lovász number to upper bound the zero-error capacity of a quantum channel: 2 C 0(N ) ϑ(s) = max Φ (1 ρ + T ) Φ s.t. T S L(A ), Tr ρ = 1, 1 ρ + T 0, ρ 0. 4

12 Quantum zero-error information theory Zero-error information theory has grown into a vast field over the past decades. (See review article [Korner-Orlitsky-1998]). 5

13 Quantum zero-error information theory Zero-error information theory has grown into a vast field over the past decades. (See review article [Korner-Orlitsky-1998]). More capacities in quantum zero-error information theory Quantum assistance Shared Entanglement (E) Quantum No-signalling Correlations (QNSC) Feedback (F) Zero-error quantum or private capacity [Leung-Yu-2015, next talk!] 5

14 Quantum zero-error information theory Zero-error information theory has grown into a vast field over the past decades. (See review article [Korner-Orlitsky-1998]). More capacities in quantum zero-error information theory Quantum assistance Shared Entanglement (E) Quantum No-signalling Correlations (QNSC) Feedback (F) Zero-error quantum or private capacity [Leung-Yu-2015, next talk!] Entanglement-assisted zero-error classical capacity Make use of (pre-shared) entanglement for communication One-shot entanglement-assisted zero-error capacity α(n ): the maximum number n such that {ρ 1,..., ρ n} are orthogonal. 1 Asymptotics: C 0E (N ) = sup log α(n n n ) n C 0E depends on the non-commutative graph S [Duan-Severini-Winter-13]. 5

15 Separation between quantum Lovász number and entanglement-assisted zero-error classical capacity

16 Problem: C 0E (N ) = log ϑ(n )? Two remarkable phenomena in zero-error information theory Entanglement can improve the zero-error capacity Classical channel (no such advantage for the the normal capacity) One-shot: [PRL, Cubitt-Leung-Matthews-Winter-2010] Asymptotic: [CMP, Leung-Mancinska-Matthews-Ozols-Roy-2012] Quantum channel (Superdense Coding) 6

17 Problem: C 0E (N ) = log ϑ(n )? Two remarkable phenomena in zero-error information theory Entanglement can improve the zero-error capacity Classical channel (no such advantage for the the normal capacity) One-shot: [PRL, Cubitt-Leung-Matthews-Winter-2010] Asymptotic: [CMP, Leung-Mancinska-Matthews-Ozols-Roy-2012] Quantum channel (Superdense Coding) C 0E is upper bounded by the quantum Lovász number C 0E (N ) log ϑ(n ). Classical channel: [PRA, Beigi-2010] Quantum channel: [IT, Duan-Severini-Winter-2013, arxiv-2010] 6

18 Problem: C 0E (N ) = log ϑ(n )? Two remarkable phenomena in zero-error information theory Entanglement can improve the zero-error capacity Classical channel (no such advantage for the the normal capacity) One-shot: [PRL, Cubitt-Leung-Matthews-Winter-2010] Asymptotic: [CMP, Leung-Mancinska-Matthews-Ozols-Roy-2012] Quantum channel (Superdense Coding) C 0E is upper bounded by the quantum Lovász number C 0E (N ) log ϑ(n ). Classical channel: [PRA, Beigi-2010] Quantum channel: [IT, Duan-Severini-Winter-2013, arxiv-2010] A pressing problem in zero-error information theory C 0E (N ) = log ϑ(n )? (mentioned in the above articles) Open since If this is true, then C 0E is additive. 6

19 Difficulties of the problem and our approach Difficulties Entanglement assistance is difficult to characterize Understanding about C 0E is limited 7

20 Difficulties of the problem and our approach Difficulties Entanglement assistance is difficult to characterize Understanding about C 0E is limited Our approach Construct a class of qutrit-to-qutrit channel N α Consider the zero-error communication properties of N α with the assistance of quantum no-signalling correlations (QNSC), which is a broader class of resources than entanglement. Compare C 0,NS to quantum Lovász number 7

21 Construction of a class of qutrit-to-qutrit channel N α The class of channels we use is N α(ρ) = E αρe α + D αρd α (0 < α π/4) with E α = sin α , D α = cos α

22 Construction of a class of qutrit-to-qutrit channel N α The class of channels we use is N α(ρ) = E αρe α + D αρd α (0 < α π/4) with E α = sin α , D α = cos α The non-commutative graph of N α is S = span{f 1, F 2, F 3, F 4} with F 1 = cos 2 α 1 1, F 2 = sin 2 α , F 3 = 0 2 and F 4 = 2 0. The Choi-Jamiołkowski matrix of N α is given by J α = (1 + sin 2 α) u u + (1 + cos 2 α) v v, where sin α u = 1 + sin 2 α cos α 21, v = 1 + sin 2 α 1 + cos 2 α cos 2 α 01. 8

23 Quantum no-signalling correlations (QNSC) Two-input and two-output quantum channels (bipartite CPTP linear map) Π : L(A i ) L(B i ) L(A o) L(B o) No-signalling correlations [Beckman-Gottesman-Nielsen-Preskill-01, Eggeling-Schlingemann-Werner-02, Piani-Horodecki et al.-06], A and B cannot use the channel to communicate classical information. The structure of QNSC is more mathematically tractable than the entanglement assistance. 9

24 QNSC-assisted zero-error capacity and simulation cost One-shot QNSC-assisted zero-error capacity is given by [Duan-Winter-2016]: Υ(N ) = max Tr S A s.t. 0 U AB S A 1 B, Tr A U AB = 1 B, Tr J AB (S A 1 B U AB ) = 0. 1 Asymptotic: C 0,NS (N ) = sup log Υ ( n 1 N n) n (1) 10

25 QNSC-assisted zero-error capacity and simulation cost One-shot QNSC-assisted zero-error capacity is given by [Duan-Winter-2016]: Υ(N ) = max Tr S A s.t. 0 U AB S A 1 B, (1) Tr A U AB = 1 B, Tr J AB (S A 1 B U AB ) = 0. 1 Asymptotic: C 0,NS (N ) = sup log Υ ( n 1 N n) n One-shot QNSC-assisted zero-error simulation cost [Duan-Winter-2016]: Σ(N ) = 2 H min(a B) J = min Tr T B, s.t. J AB 1 A T B, where H min (A B) J is the conditional min-entropy [Koenig-Renner-Schaffner-09]. log Υ(N ) C 0,NS (N ) C E (N ) S 0,NS (N ) = log Σ(N ) 10

26 QNSC-assisted zero-error capacity of N α For the channel N α (0 < α π/4), C 0,NS (N α) 2 Construct a feasible solution to the prime SDP of Υ(N α): R A =2(cos 2 α sin 2 α 2 2 ) U AB = cos 2 α sin sin α( ) + cos α( ). C 0,NS (N α) log Υ(N α) log Tr R A = 2 11

27 QNSC-assisted zero-error capacity of N α For the channel N α (0 < α π/4), C 0,NS (N α) 2 Construct a feasible solution to the prime SDP of Υ(N α): R A =2(cos 2 α sin 2 α 2 2 ) U AB = cos 2 α sin sin α( ) + cos α( ). C 0,NS (N α) log Υ(N α) log Tr R A = 2 Similarly, we can prove that S 0,NS (N α) = H min (A B) JAB = log Σ(N ) = 2. Recall C 0,NS (N ) C E (N ) S 0,NS (N ) Proposition: C 0,NS (N α) = C E (N α) = S 0,NS (N α) = 2. 11

28 Quantum Lovász number of N α Proposition: For the channel N α (0 < α π/4), ϑ(n α) = 2 + cos 2 α + cos 2 α > 4. 12

29 Quantum Lovász number of N α Proposition: For the channel N α (0 < α π/4), Outline of proof: ϑ(n α) = 2 + cos 2 α + cos 2 α > 4. Recall the primal SDP of ϑ(n α): ϑ(s) = max Φ (1 ρ + T ) Φ s.t. T S L(A ), 1 ρ + T 0, Tr ρ = 1, ρ 0. Suppose that ρ = cos2 α 1+cos 2 α cos 2 α 1 1 and T = T 1 T 2 + R with 1 T 1 = 1 + cos 2 α ( cos 2 α sin2 α cos 2 α 2 2 ), T 2 = cos 4 α , R = {ρ, T } is a feasible solution to primal SDP of ϑ(n α). Hence, ϑ(n α) Tr[ Φ Φ (1 ρ + T )] = 2 + cos 2 α + 1/cos 2 α. Similarly, we can use the dual SDP to show ϑ(n α) 2 + cos 2 α + 1/cos 2 α. 12

30 C 0E (N ) log ϑ(n )!!! Theorem: For the channel N α (0 < α π/4), the quantum Lovász number is strictly larger than the entanglement-assisted zero-error capacity (or even with quantum no-signalling assistance), i.e., log ϑ(n α ) > C 0,NS (N α ) C 0E (N α ). C 0E (N α ) C 0,NS (N α ) = C E (N α ) = S 0,NS (N α ) = 2 log ϑ(n α ) = log(2 + cos 2 α + cos 2 α) > 2 13

31 Quantum fractional packing number and feedback-assisted or QNSC-assisted zero-error capacity For classical channel N [Shannon-1956, Cubitt-Leung-Matthews-Winter-2011], C 0F (N ) = C 0,NS (N ) = log α (N ), where α (N ) is the fractional packing number: α (N ) = max x v x s.t. x v x N (y x) 1 y, 0 v x 1 x. 14

32 Quantum fractional packing number and feedback-assisted or QNSC-assisted zero-error capacity For classical channel N [Shannon-1956, Cubitt-Leung-Matthews-Winter-2011], C 0F (N ) = C 0,NS (N ) = log α (N ), where α (N ) is the fractional packing number: α (N ) = max v x s.t. v x N (y x) 1 y, 0 v x 1 x. x x For classical-quantum channel N [Duan-Winter-2016], C 0,NS (N ) = log A(N ), where A(N ) is the quantum fractional packing number: A(N ) = max Tr R A s.t. 0 R A, Tr A P AB (R A 1 B ) 1 B, 14

33 Quantum fractional packing number and feedback-assisted or QNSC-assisted zero-error capacity For classical channel N [Shannon-1956, Cubitt-Leung-Matthews-Winter-2011], C 0F (N ) = C 0,NS (N ) = log α (N ), where α (N ) is the fractional packing number: α (N ) = max v x s.t. v x N (y x) 1 y, 0 v x 1 x. x x For classical-quantum channel N [Duan-Winter-2016], C 0,NS (N ) = log A(N ), where A(N ) is the quantum fractional packing number: A(N ) = max Tr R A s.t. 0 R A, Tr A P AB (R A 1 B ) 1 B, However, for quantum channel N α (0 < α π/4), we have that A(N α) = 2 + cos 2 α + cos 2 α and C 0F (N α) < log A(N α), C 0,NS (N α) < log A(N α). 14

34 Conclusion

35 Summary For the channel N α = E αρe α + D αρd α (0 < α π/4) with we have that E α = sin α , D α = cos α , C 0,NS (N α) = C E (N α) = S 0,NS (N α) = 2 (A kind of reversibility) ϑ(n α) = 2 + cos 2 α + cos 2 α > 4 A(K α) = 2 + cos 2 α + cos 2 α > 4 15

36 Summary For the channel N α = E αρe α + D αρd α (0 < α π/4) with we have that E α = sin α , D α = cos α , C 0,NS (N α) = C E (N α) = S 0,NS (N α) = 2 (A kind of reversibility) ϑ(n α) = 2 + cos 2 α + cos 2 α > 4 A(K α) = 2 + cos 2 α + cos 2 α > 4 A separation between quantum Lovász number and entanglement-assisted zero-error classical capacity: log ϑ(n α) > C 0,NS (N α) C 0E (N α). A gap between quantum fractional packing number and feedback-assisted or QNSC-assisted zero-error capacity: C 0F (N α) < log A(N α), C 0,NS (N α) < log A(N α). 15

37 Open Questions: It remains unknown whether Lovász number coincides with C 0E for every classical channel, i.e., C 0E (G) = log ϑ(g)? Other communication properties of N α? Characterize the channels which have reversibility in the sense that Additivity for C 0E (N ) or C 0,NS (N )? C 0,NS (N ) = S 0,NS (N ). 16

38 Thank you! Questions? 16