Quantum Bilinear Optimisation
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1 Quantum Bilinear Optimisation ariv: Mario Berta (IQIM Caltech), Omar Fawzi (ENS Lyon), Volkher Scholz (Ghent University) March 7th, 2016 Louisiana State University
2 Quantum Bilinear Optimisation ariv: Mario Berta (IQIM Caltech), Omar Fawzi (ENS Lyon), Volkher Scholz (Ghent University) March 7th, 2016 Louisiana State University Goal: understand the power of quantum assistance and quantum adversaries for operational setups
3 Example: Classical noisy channel coding (vs.) Entanglement-assisted channel coding
4 Classical noisy channel coding (I) Alice Bob i 2 [k] e(x i) x2 W!Y (y x) d(i y) i2[k] Encoder Decoder Given noisy channel W!Y mapping to Y with transition probability: W!Y (y x) 8(x, y) 2 Y The goal is to send k different messages using W while minimising the error probability for decoding: p succ (W, k) := maximize (e,d) subject to 1 W!Y (y x)e(x i)d(i y) bilinear optimisation k x,y,i e(x i) =1 8i 2 [k], d(i y) =1 8y 2 Y x i 0 apple e(x i) apple 1 8(x, i) 2 [k], 0 apple d(i y) apple 1 8(i, y) 2 [k] Y.
5 Classical noisy channel coding (II) p succ (W, k) := maximize (e,d) subject to 1 W!Y (y x)e(x i)d(i y) k e W d x,y,i e(x i) =1 8i 2 [k], d(i y) =1 8y 2 Y x i 0 apple e(x i) apple 1 8(x, i) 2 [k], 0 apple d(i y) apple 1 8(i, y) 2 [k] Y. compared to Shannon s asymptotic independent and identical distributed (iid) channel capacity: Definition: C(W ):=sup n R o lim p succ W n, b2 Rn c =1 n!1 W Answer: C(W ) = max P I( : Y ) mutual information e (n) W d (n) W Encoder Decoder
6 Classical noisy channel coding (II) p succ (W, k) := maximize (e,d) subject to 1 W!Y (y x)e(x i)d(i y) k e W d x,y,i e(x i) =1 8i 2 [k], d(i y) =1 8y 2 Y x i 0 apple e(x i) apple 1 8(x, i) 2 [k], 0 apple d(i y) apple 1 8(i, y) 2 [k] Y. compared to Shannon s asymptotic independent and identical distributed (iid) channel capacity: Definition: C(W ):=sup n R o lim p succ W n, b2 Rn c =1 n!1 W Answer: C(W ) = max P I( : Y ) mutual information e (n) W d (n) W Encoder Decoder
7 Example: Classical noisy channel coding (vs.) Entanglement-assisted channel coding
8 Entanglement-assisted channel coding (I) Alice Bob i 2 [k] E(x i) x2 W!Y (y x) D(i y) i2[k] Encoder i 2H H Decoder p succ(w, k) := maximize (H,,E,D) subject to 1 quantum bilinear optimisation W!Y (y x)h E(x i) D(i y) i k x,y,i E(x i) =1 H 8i 2 [k], D(i y) =1 H 8y 2 Y x i 0 apple E(x i) apple 1 H 8(x, i) 2 [k], 0 apple D(i y) apple 1 H 8(i, y) 2 [k] Y.
9 Entanglement-assisted channel coding (I) Alice Bob i 2 [k] E(x i) x2 W!Y (y x) D(i y) i2[k] Encoder i 2H H Decoder p succ(w, k) := maximize (H,,E,D) subject to 1 quantum bilinear optimisation W!Y (y x)h h E(x i) D(i y) i k x,y,i E(x i) =1 H 8i 2 [k], D(i y) =1 H 8y 2 Y x i 0 apple E(x i) apple 1 H 8(x, i) 2 [k], 0 apple D(i y) apple 1 H 8(i, y) 2 [k] Y. Scalar (commutative) versus matrix (non-commutative) variables: p succ (W, k) := maximize (e,d) 1 k x,y,i W!Y (y x)e(x i)d(i y)
10 Entanglement-assisted channel coding (I) Alice Bob i 2 [k] E(x i) x2 W!Y (y x) D(i y) i2[k] Encoder i 2H H Decoder p succ(w, k) := maximize (H,,E,D) subject to 1 quantum bilinear optimisation W!Y (y x)h h E(x i) D(i y) i k x,y,i E(x i) =1 H 8i 2 [k], D(i y) =1 H 8y 2 Y x i 0 apple E(x i) apple 1 H 8(x, i) 2 [k], 0 apple D(i y) apple 1 H 8(i, y) 2 [k] Y. Scalar (commutative) versus matrix (non-commutative) variables: p succ (W, k) := maximize (e,d) Unknown if p succ(w, k) is computable! 1 k x,y,i W!Y (y x)e(x i)d(i y)
11 Entanglement-assisted channel coding (II) Understand the possible separation: p succ (W, k) versus p succ(w, k)
12 Entanglement-assisted channel coding (II) Understand the possible separation: p succ (W, k) versus p succ(w, k) For the asymptotic iid capacity entanglement (quantum) assistance does not help: C(W )=C (W ) [Bennett et al., PRL (1999)]
13 Entanglement-assisted channel coding (II) Understand the possible separation: p succ (W, k) versus p succ(w, k) For the asymptotic iid capacity entanglement (quantum) assistance does not help: C(W )=C (W ) [Bennett et al., PRL (1999)]
14 Entanglement-assisted channel coding (II) Understand the possible separation: p succ (W, k) versus p succ(w, k) For the asymptotic iid capacity entanglement (quantum) assistance does not help: C(W )=C (W ) [Bennett et al., PRL (1999)] In general, there is a separation: 0 1 1/3 1/ /3 1/3 Z = 1/3 0 1/3 0 B 0 1/3 0 1/3 1/ /3A 0 1/3 1/ /2 p succ (Z, 2) = vs. p succ(z, 2) [Prevedel et al., PRL (2011)]
15 Entanglement-assisted channel coding (II) Understand the possible separation: p succ (W, k) versus p succ(w, k) For the asymptotic iid capacity entanglement (quantum) assistance does not help: C(W )=C (W ) [Bennett et al., PRL (1999)] In general, there is a separation: Z = 0 1 1/3 1/ /3 1/3 1/3 0 1/3 0 B 0 1/3 0 1/3 1/ /3A 0 1/3 1/ /2 p succ (Z, 2) = vs. p succ(z, 2) [Prevedel et al., PRL (2011)] > this is also optimal with two-dimensional assistance [Hemenway et al., PRA (2013)] [Williams and Bourdon, ariv: ]
16 Entanglement-assisted channel coding (II) Understand the possible separation: p succ (W, k) versus p succ(w, k) For the asymptotic iid capacity entanglement (quantum) assistance does not help: C(W )=C (W ) [Bennett et al., PRL (1999)] In general, there is a separation: Z = However: 0 1 1/3 1/ /3 1/3 1/3 0 1/3 0 B 0 1/3 0 1/3 1/ /3A 0 1/3 1/3 0 [0.902, 1] 3 p succ(z, 2) =? 2+2 1/2 p succ (Z, 2) = vs. p succ(z, 2) [Prevedel et al., PRL (2011)] > this is also optimal with two-dimensional assistance [Hemenway et al., PRA (2013)] [Williams and Bourdon, ariv: ]
17 Entanglement-assisted channel coding (II) Understand the possible separation: p succ (W, k) versus p succ(w, k) For the asymptotic iid capacity entanglement (quantum) assistance does not help: C(W )=C (W ) [Bennett et al., PRL (1999)] In general, there is a separation: Z = However: 0 1 1/3 1/ /3 1/3 1/3 0 1/3 0 B 0 1/3 0 1/3 1/ /3A 0 1/3 1/3 0 [0.902, 1] 3 p succ(z, 2) =? 2+2 1/2 p succ (Z, 2) = vs. p succ(z, 2) [Prevedel et al., PRL (2011)] > this is also optimal with two-dimensional assistance [Hemenway et al., PRA (2013)] [Williams and Bourdon, ariv: ] linear objective function with semidefinite constraints We give a converging hierarchy of semidefinite programming (sdp) relaxations: p succ (W, k) apple p succ(w, k) =sdp 1 (W, k) apple...apple sdp 1 (W, k) < efficiently computable!
18 Hierarchy of semidefinite programming (sdp) relaxations
19 First level semidefinite programming relaxation (I) Quantum bilinear program: p succ(w, k) := maximize (H,,E,D) subject to 1 W!Y (y x)h E(x i) D(i y) i k x,y,i E(x i) =1 H 8i 2 [k], D(i y) =1 H x i 8y 2 Y 0 apple E(x i) apple 1 H 8(x, i) 2 [k], 0 apple D(i y) apple 1 H 8(i, y) 2 [k] Y.
20 First level semidefinite programming relaxation (I) Quantum bilinear program: p succ(w, k) := maximize (H,,E,D) subject to 1 W!Y (y x)h h E(x i) D(i y) i appleh E(x i) D(i y) i k x,y,i with [E(x i),d(i y)] = 0 E(x i) =1 H 8i 2 [k], D(i y) =1 H 8y 2 Y x i 0 apple E(x i) apple 1 H 8(x, i) 2 [k], 0 apple D(i y) apple 1 H 8(i, y) 2 [k] Y.
21 First level semidefinite programming relaxation (I) Quantum bilinear program: p succ(w, k) := maximize (H,,E,D) subject to idea: relaxation of this bilinear form 1 W!Y (y x)h h E(x i) D(i y) i appleh E(x i) D(i y) i k x,y,i with [E(x i),d(i y)] = 0 E(x i) =1 H 8i 2 [k], D(i y) =1 H 8y 2 Y x i 0 apple E(x i) apple 1 H 8(x, i) 2 [k], 0 apple D(i y) apple 1 H 8(i, y) 2 [k] Y. motivated by: NPA hierarchy (Bell inequalities) [Lasserre, SIAM (2001)], [Parrilo, Math. Program. (2003)], [Navascues et al., PRL (2007)], [Doherty et al., IEEE CCC (2008)], [Navascues et al., NJP (2008)], [Pironio et al., SIAM (2010)]
22 First level semidefinite programming relaxation (I) Quantum bilinear program: p succ(w, k) := maximize (H,,E,D) subject to idea: relaxation of this bilinear form 1 W!Y (y x)h h E(x i) D(i y) i appleh E(x i) D(i y) i k x,y,i with [E(x i),d(i y)] = 0 E(x i) =1 H 8i 2 [k], D(i y) =1 H 8y 2 Y x i 0 apple E(x i) apple 1 H 8(x, i) 2 [k], 0 apple D(i y) apple 1 H 8(i, y) 2 [k] Y. First step: see as the part of the upper-right block of the Gram matrix = u,v h u v i uihv with u = E(x i) u =(i, x) D(j y) u =(j, y) = h E(x i) E(x 0 i 0 ) i h E(x 0 i 0 ) D(j 0 y 0 ) i for i=j h E(x i) D(j y) i h D(j y) D(j 0 y 0 ) i motivated by: NPA hierarchy (Bell inequalities) [Lasserre, SIAM (2001)], [Parrilo, Math. Program. (2003)], [Navascues et al., PRL (2007)], [Doherty et al., IEEE CCC (2008)], [Navascues et al., NJP (2008)], [Pironio et al., SIAM (2010)]
23 First level semidefinite programming relaxation (I) Quantum bilinear program: p succ(w, k) := maximize (H,,E,D) subject to idea: relaxation of this bilinear form 1 W!Y (y x)h h E(x i) D(i y) i appleh E(x i) D(i y) i k x,y,i with [E(x i),d(i y)] = 0 E(x i) =1 H 8i 2 [k], D(i y) =1 H 8y 2 Y x i 0 apple E(x i) apple 1 H 8(x, i) 2 [k], 0 apple D(i y) apple 1 H 8(i, y) 2 [k] Y. First step: see as the part of the upper-right block of the Gram matrix = u,v h u v i uihv with u = E(x i) u =(i, x) D(j y) u =(j, y) = h E(x i) E(x 0 i 0 ) i h E(x 0 i 0 ) D(j 0 y 0 ) i for i=j h E(x i) D(j y) i h D(j y) D(j 0 y 0 ) i Original constraints can be formulated as positivity conditions on : sdp 1 (W, k) motivated by: NPA hierarchy (Bell inequalities) [Lasserre, SIAM (2001)], [Parrilo, Math. Program. (2003)], [Navascues et al., PRL (2007)], [Doherty et al., IEEE CCC (2008)], [Navascues et al., NJP (2008)], [Pironio et al., SIAM (2010)]
24 First level semidefinite programming relaxation (II) First level relaxation: p succ (W, k) apple p succ(w, k) apple sdp 1 (W, k) sdp 1 (W, k) = maximize 1 k W!Y (y x) (i,x),(i,y) x,y,i subject to 2 Pos(1 + k + k Y ), ;,; =1 with; the empty symbol u,v 0 8u, v 2 [k] [ Y [k] [ {;} w,(i,x) = w,; 8i 2 [k],w 2 [k] [ Y [k] [ {;} x w,(i,y) = w,; 8y 2 Y,w 2 [k] [ Y [k] [ {;}. i
25 First level semidefinite programming relaxation (II) First level relaxation: p succ (W, k) apple p succ(w, k) apple sdp 1 (W, k) sdp 1 (W, k) = maximize 1 k x,y,i W!Y (y x) (i,x),(i,y) subject to 2 Pos(1 + k + k Y ), ;,; =1 with; the empty symbol u,v 0 8u, v 2 [k] [ Y [k] [ {;} w,(i,x) = w,; 8i 2 [k],w 2 [k] [ Y [k] [ {;} x w,(i,y) = w,; 8y 2 Y,w 2 [k] [ Y [k] [ {;}. i
26 First level semidefinite programming relaxation (II) First level relaxation: p succ (W, k) apple p succ(w, k) apple sdp 1 (W, k) sdp 1 (W, k) = maximize 1 k x,y,i W!Y (y x) (i,x),(i,y) subject to 2 Pos(1 + k + k Y ), ;,; =1 with; the empty symbol u,v 0 8u, v 2 [k] [ Y [k] [ {;} w,(i,x) = w,; 8i 2 [k],w 2 [k] [ Y [k] [ {;} x w,(i,y) = w,; 8y 2 Y,w 2 [k] [ Y [k] [ {;}. i Going back to our example: 0 1 1/3 1/ /3 1/3 Z = 1/3 0 1/3 0 B 0 1/3 0 1/3 1/ /3A 0 1/3 1/3 0 p succ (Z, 2) = p succ(z, 2) 2+2 1/ (known before, with twodimensional assistance)
27 First level semidefinite programming relaxation (II) First level relaxation: p succ (W, k) apple p succ(w, k) apple sdp 1 (W, k) sdp 1 (W, k) = maximize 1 k x,y,i W!Y (y x) (i,x),(i,y) subject to 2 Pos(1 + k + k Y ), ;,; =1 with; the empty symbol u,v 0 8u, v 2 [k] [ Y [k] [ {;} w,(i,x) = w,; 8i 2 [k],w 2 [k] [ Y [k] [ {;} x w,(i,y) = w,; 8y 2 Y,w 2 [k] [ Y [k] [ {;}. i Going back to our example: 0 1 1/3 1/ /3 1/3 Z = 1/3 0 1/3 0 B 0 1/3 0 1/3 1/ /3A 0 1/3 1/3 0 p succ (Z, 2) = p succ(z, 2) Relaxation: 2+2 1/ (known before, with twodimensional assistance) p succ(z, 2) apple sdp 1 (Z, 2) = p p 1 6 Four-dimensional assistance: p succ(z, 2)
28 First level semidefinite programming relaxation (II) First level relaxation: p succ (W, k) apple p succ(w, k) apple sdp 1 (W, k) sdp 1 (W, k) = maximize new condition > 1 k x,y,i W!Y (y x) (i,x),(i,y) subject to 2 Pos(1 + k + k Y ), ;,; =1 with; the empty symbol u,v 0 8u, v 2 [k] [ Y [k] [ {;} w,(i,x) = w,; 8i 2 [k],w 2 [k] [ Y [k] [ {;} x w,(i,y) = w,; 8y 2 Y,w 2 [k] [ Y [k] [ {;}. i Going back to our example: 0 1 1/3 1/ /3 1/3 Z = 1/3 0 1/3 0 B 0 1/3 0 1/3 1/ /3A 0 1/3 1/3 0 (NPA hierarchy and nonsignalling bounds are one) p succ (Z, 2) = p succ(z, 2) Relaxation: 2+2 1/ (known before, with twodimensional assistance) p succ(z, 2) apple sdp 1 (Z, 2) = p p 1 6 Four-dimensional assistance: p succ(z, 2)
29 First level semidefinite programming relaxation (II) First level relaxation: p succ (W, k) apple p succ(w, k) apple sdp 1 (W, k) sdp 1 (W, k) = maximize new condition > 1 k x,y,i W!Y (y x) (i,x),(i,y) subject to 2 Pos(1 + k + k Y ), ;,; =1 with; the empty symbol u,v 0 8u, v 2 [k] [ Y [k] [ {;} w,(i,x) = w,; 8i 2 [k],w 2 [k] [ Y [k] [ {;} x w,(i,y) = w,; 8y 2 Y,w 2 [k] [ Y [k] [ {;}. i Going back to our example: 0 1 1/3 1/ /3 1/3 Z = 1/3 0 1/3 0 B 0 1/3 0 1/3 1/ /3A 0 1/3 1/3 0 (NPA hierarchy and nonsignalling bounds are one) p succ (Z, 2) = p succ(z, 2) Relaxation: 2+2 1/ (known before, with twodimensional assistance) p succ(z, 2) apple sdp 1 (Z, 2) = p p 1 6 Four-dimensional assistance: p succ(z, 2) > further work [Barman and Fawzi., ariv: ]
30 General Quantum Bilinear Optimisation and more applications
31 General Quantum Bilinear Optimisation p succ(w, k) := maximize (H,,E,D) subject to 1 W!Y (y x)h E(x i) D(i y) i k x,y,i E(x i) =1 H 8i 2 [k], D(i y) =1 H 8y 2 Y x i 0 apple E(x i) apple 1 H 8(x, i) 2 [k], 0 apple D(i y) apple 1 H 8(i, y) 2 [k] Y. and more applications
32 General Quantum Bilinear Optimisation p succ(w, k) := maximize (H,,E,D) subject to idea: relaxation of this bilinear form 1 W!Y (y x)h E(x i) D(i y) i appleh E(x i) D(i y) i k x,y,i with [E(x i),d(i y)] = 0 E(x i) =1 H 8i 2 [k], D(i y) =1 H 8y 2 Y x i 0 apple E(x i) apple 1 H 8(x, i) 2 [k], 0 apple D(i y) apple 1 H 8(i, y) 2 [k] Y. and more applications
33 Example: Randomness (vs.) Quantum-Proof Randomness
34 Quantum Cryptography (I) Privacy amplification: weak source of randomness 2 {0, 1} n Ext Z (up to 0) (with min-entropy k ) uniform random bits Z 2 {0, 1} m Example: two-universal hashing
35 Quantum Cryptography (I) Privacy amplification: weak source of randomness 2 {0, 1} n Ext (with min-entropy k ) What happens for quantum adversaries? weak source of randomness relative to E Ext E Z Z E uniform random bits Z 2 {0, 1} m (up to 0) uniform random bits relative to E (up to 0) Example: two-universal hashing
36 Quantum Cryptography (I) Privacy amplification: weak source of randomness 2 {0, 1} n Ext Z (up to 0) (with min-entropy k ) randomness extractor uniform random bits Z 2 {0, 1} m What happens for quantum adversaries? weak source of randomness relative to E Ext Z quantum-proof E randomness extractor E uniform random bits relative to E (up to 0) Example: two-universal hashing
37 Quantum Cryptography (I) Privacy amplification: weak source of randomness 2 {0, 1} n Ext Z (up to 0) (with min-entropy k ) randomness extractor uniform random bits Z 2 {0, 1} m What happens for quantum adversaries? weak source of randomness relative to E Ext Z quantum-proof E randomness extractor E uniform random bits relative to E (up to 0) Example: two-universal hashing Motivation: quantum key distribution, two-party cryptography, quantum-safe / quantum-proof / post-quantum
38 Quantum Cryptography (I) Privacy amplification: weak source of randomness 2 {0, 1} n Ext Z (up to 0) (with min-entropy k ) randomness extractor uniform random bits Z 2 {0, 1} m What happens for quantum adversaries? weak source of randomness relative to E Ext Z quantum-proof E randomness extractor E uniform random bits relative to E (up to 0) Example: two-universal hashing Motivation: quantum key distribution, two-party cryptography, quantum-safe / quantum-proof / post-quantum Classical versus quantum error (the ): C(Ext,k) versus Q(Ext,k) computable?
39 Quantum Cryptography (II) For randomness extractors: classical versus quantum adversaries C(Ext,k) versus Q(Ext,k) computable? classical bilinear optimisation versus quantum bilinear optimisation scalar variables versus matrix variables
40 Quantum Cryptography (II) For randomness extractors: classical versus quantum adversaries C(Ext,k) versus Q(Ext,k) computable? classical bilinear optimisation versus quantum bilinear optimisation scalar variables versus matrix variables Converging hierarchy of semidefinite programming relaxations: C(Ext,k) apple Q(Ext,k)=sdp 1 (Ext,k) apple...apple sdp 1 (Ext,k) < efficiently computable! > upper bounding the power of quantum adversaries see ariv: for more (including references)
41 Conclusion / Outlook Understand quantum assistance (noisy channel coding) and quantum adversaries (randomness extractors) using optimisation methods Converging hierarchy of semidefinite programming relaxations: C apple Q =sdp 1 apple...apple sdp 1
42 Conclusion / Outlook Understand quantum assistance (noisy channel coding) and quantum adversaries (randomness extractors) using optimisation methods Converging hierarchy of semidefinite programming relaxations: C apple Q =sdp 1 apple...apple sdp 1 Improvement over NPA hierarchy for non-local games (Bell inequalities), first level also in [Sikora and Varvitsiotis, ariv: ] First outer hierarchy for optimisations over the completely positive semidefinite cone (symmetric matrices that admit a Gram representation by positive semidefinite matrices, [Laurent and Piovesan, ariv: ]) > quantum graph parameters
43 Conclusion / Outlook Understand quantum assistance (noisy channel coding) and quantum adversaries (randomness extractors) using optimisation methods Converging hierarchy of semidefinite programming relaxations: C apple Q =sdp 1 apple...apple sdp 1 Improvement over NPA hierarchy for non-local games (Bell inequalities), first level also in [Sikora and Varvitsiotis, ariv: ] First outer hierarchy for optimisations over the completely positive semidefinite cone (symmetric matrices that admit a Gram representation by positive semidefinite matrices, [Laurent and Piovesan, ariv: ]) > quantum graph parameters Study zero-error information theory Study full cryptographic protocols as optimisations Study SDP relaxation for quantum channel coding, work in progress (same authors)
44 Conclusion / Outlook Understand quantum assistance (noisy channel coding) and quantum adversaries (randomness extractors) using optimisation methods Converging hierarchy of semidefinite programming relaxations: C apple Q =sdp 1 apple...apple sdp 1 Improvement over NPA hierarchy for non-local games (Bell inequalities), first level also in [Sikora and Varvitsiotis, ariv: ] First outer hierarchy for optimisations over the completely positive semidefinite cone (symmetric matrices that admit a Gram representation by positive semidefinite matrices, [Laurent and Piovesan, ariv: ]) > quantum graph parameters Study zero-error information theory Study full cryptographic protocols as optimisations Study SDP relaxation for quantum channel coding, work in progress (same authors) Any other ideas what to quantise? Thanks!
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