asymptotic security proof for discrete- modulated cvqkd … · p bob parameter estimation q p q p...

Uniform Continuity of Conditional Entropy W can be determined from parameter estimation, and characterizes the weight outside the finite subspace. Using a generalization of the result in [5], we have: 1 - W F (P, Q)= ) |f (P ) - f (Q)| ✏ N (W ) < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 p d y L y e 3 X X V j y e b c h O k 2 h + S C H u g = " > A A A C O H i c b V D J S g N B F O y J W 4 x b 1 K O X x i B M w I Q Z i S i e g o J 4 k g T M A p k Q e j p v k i Y 9 i 9 0 9 Q h j y L 3 6 H H + B V r 9 6 8 B a 9 + g Z 3 l o I k P H h T 1 q i h e u R F n U l n W h 5 F a W V 1 b 3 0 h v Z r a 2 d 3 b 3 s v s H d R n G g k K N h j w U T Z d I 4 C y A m m K K Q z M S Q H y X Q 8 M d 3 E z u j S c Q k o X B g x p G 0 P Z J L 2 A e o 0 R p q p O 9 s g s N 7 H B 4 x L d m 5 b S a x w 7 z d S 5 I 7 B B X J p 5 Z y R c 8 s 5 o f T T U O R J J x 7 b s 3 G / l O N m c V r e n g Z W D P Q Q 7 N p 9 L J j p 1 u S G M f A k U 5 k b J l W 5 F q J 0 Q o R j m M M k 4 s I S J 0 Q H r Q 0 j A g P s h 2 M v 1 x h E 8 0 0 8 V e K P Q G C k / Z 3 4 6 E + F I O f V c r f a L 6 c v E 2 I f + 9 R f 2 h Z F Q u x C v v s p 2 w I I o V B H S W 7 s U c q x B P W s R d J o A q P t S A U M H 0 A 5 j 2 i S B U 6 a 4 z u h l 7 s Y d l U D 8 r 2 q X i e b W U K 1 / P O 0 q j I 3 S M T G S j C 1 R G d 6 i C a o i i Z / S K 3 t C 7 8 W J 8 G m P j a y Z N G X P P I f o z x v c P 1 w y q p A = = < / l a t e x i t > Main Theorem Asymptotic Security Proof for Discrete - Modulated CVQKD without Cutoff Assumption Twesh Upadhyaya | Thomas van Himbeeck | Jie Lin | Norbert Lütkenhaus Introduction Goal is to compute tight key rates for DMCVQKD with a small number of modulated states. Want to harness the existing numerics framework for finite- dimensional optimizations [1][2]. Previous work assumes the state is finite-dimensional; the cutoff assumption. This gives numerically stable results but is not a rigorous security proof [3]. Our Contribution We establish the asymptotic security of DMCVQKD with a small number of modulated states. In particular, we do not use a cutoff assumption. We develop a framework to provide tight, reliable key rates for other protocols with infinite-dimensional Hilbert spaces. Key Rate Formula Under collective attacks in the asymptotic limit, given by Devetak-Winter formula [4]: Evaluated on worst-case state compatible with statistics from parameter estimation; after post-processing. Reformulate: If ɸ is the CPTNI map representing the post-processing performed by Alice and Bob, then: R = min ⇢2S [H (X |E )] - H (X )+ I (A : B ) < l a t e x i t s h a 1 _ b a s e 6 4 = " x f t 7 M N y G O b / h 2 8 r b A 6 h J x 7 c a O W 4 = " > A A A C L n i c b V D L S g M x F M 3 U V 6 2 v q k s 3 w S K 0 i G V G K o o g 1 I p Q d / X R B 3 S G k k k z b W g m M y Q Z o Y z 9 D b / D D 3 C r n y C 4 E J f 6 G a a P h b Y e C D m c c y / 3 3 u O G j E p l m u 9 G Y m 5 + Y X E p u Z x a W V 1 b 3 0 h v b t V k E A l M q j h g g W i 4 S B J G O a k q q h h p h I I g 3 2 W k 7 v Y u h n 7 9 n g h J A 3 6 n + i F x f N T h 1 K M Y K S 2 1 0 u b N G b R 9 y l u x L b q B T b n t I 9 V 1 v f h 2 M G j C c r b x c J l z D v S f 2 7 / K n p + W c q 1 0 x s y b I 8 B Z Y k 1 I B k x Q a a W / 7 H a A I 5 9 w h R m S s m m Z o X J i J B T F j A x S d i R J i H A P d U h T U 4 5 8 I p 1 4 d N k A 7 m m l D b 1 A 6 M c V H K m / O 2 L k S 9 n 3 X V 0 5 3 F t O e 0 P x X y / s 9 i X F c m q 8 8 k 6 c m P I w U o T j 8 X Q v Y l A F c J g d b F N B s G J 9 T R A W V B 8 A c R c J h J V O O K W T s a Z z m C W 1 w 7 x V y B 9 d F z L F 0 i S j J N g B u y A L L H A M i q A M K q A K M H g E z + A F v B p P x p v x Y X y O S x P G p G c b / I H x / Q M / B K d M < / l a t e x i t > Future Work Extend to finite-size numerical framework for CVQKD. Encapsulate different models of imperfect detectors, e.g. trusted noise. The Institute for Quantum Computing is supported in part by Innovation, Science and Economic Development Canada. T.U. acknowledges the support of NSERC through the Alexander Graham Bell Canada Graduate Scholarship. This research has been supported by NSERC under the Discovery Grants Program, Grant No. 341495, and under the Collaborative Research and Development Program, Grant No. CRDP J 522308-17. Financial support for this work has been partially provided by Huawei Technologies Canada Co., Ltd. References [1] A. Winick, N. Lütkenhaus, and P. J. Coles, Quantum 2, 77 (2018). [2] P. J. Coles, Phys. Rev. A 85, 042103 (2012). [3] J. Lin, T. Upadhyaya, and N. Lütkenhaus, Phys. Rev. X 9, 041064 (2019). [4] I. Devetak and A. Winter, Proc. R. Soc. A 461, 207 (2005). [5] A. Winter. (2016) , Communications in Mathematical Physics, 347(1):291–313. f (⇢)= H (X |E ) φ(⇢) < l a t e x i t s h a 1 _ b a s e 6 4 = " U 2 T N g 1 D W f R e e N l K m f k H 3 u x G 3 x t Q = " > A A A C G X i c b V D L S s N A F J 3 4 r P U V d d n N Y B H a T U m k o h u h K E K X F e w D m l A m 0 0 k z d D I J M x M h x C 7 8 D j / A r X 6 C O 3 H r y i / w N 5 y 2 W W j r g Q u H c + 7 l c I 8 X M y q V Z X 0 Z K 6 t r 6 x u b h a 3 i 9 s 7 u 3 r 5 5 c N i R U S I w a e O I R a L n I U k Y 5 a S t q G K k F w u C Q o + R r j e + n v r d e y I k j f i d S m P i h m j E q U 8 x U l o a m C W / 4 o g g q l 4 2 K 7 2 H m + o g c + K A z q X J w C x b N W s G u E z s n J R B j t b A / H a G E U 5 C w h V m S M q + b c X K z Z B Q F D M y K T q J J D H C Y z Q i f U 0 5 C o l 0 s 9 k T E 3 i i l S H 0 I 6 G H K z h T f 1 9 k K J Q y D T 2 9 G S I V y E V v K v 7 r x U E q K Z Y L 8 c q / c D P K 4 0 Q R j u f p f s K g i u C 0 J j i k g m D F U k 0 Q F l Q / A H G A B M J K l 1 n U z d i L P S y T z m n N r t f O b u v l x l X e U Q G U w D G o A B u c g w Z o g h Z o A w w e w T N 4 A a / G k / F m v B s f 8 9 U V I 7 8 5 A n 9 g f P 4 A l C u f 7 g = = < / l a t e x i t > Protocol Quantum Phase Classical Phase • Error Correction: Reverse Reconciliation • Privacy Amplification Alice QPSK |↵i < l a t e x i t s h a 1 _ b a s e 6 4 = " V h P J 5 I Z p 0 C 0 2 4 l S 6 s E a s F X f 7 v i I = " > A A A C C H i c b V B L S g N B E O 2 J v x h / U Z d u G o P g K s x I R J d B N y 4 j m A 9 k h t D T 6 U m a 9 P Q 0 3 T X C M O Q C H s C t H s G d u P U W n s B r 2 E l m o Y k P C h 7 v V V F V L 1 S C G 3 D d L 6 e 0 t r 6 x u V X e r u z s 7 u 0 f V A + P O i Z J N W V t m o h E 9 0 J i m O C S t Y G D Y D 2 l G Y l D w b r h 5 H b m d x + Z N j y R D 5 A p F s R k J H n E K Q E r + f 6 E Q e 4 T o c Z k O q j W 3 L o 7 B 1 4 l X k F q q E B r U P 3 2 h w l N Y y a B C m J M 3 3 M V B D n R w K l g 0 4 q f G q Y I n Z A R 6 1 s q S c x M k M 9 v n u I z q w x x l G h b E v B c / T 2 R k 9 i Y L A 5 t Z 0 x g b J a 9 m f i v p 8 a Z 4 d Q s r Y f o O s i 5 V C k w S R f b o 1 R g S P A s F T z k m l E Q m S W E a m 4 f w H R M N K F g s 6 v Y Z L z l H F Z J 5 6 L u N e q X 9 4 1 a 8 6 b I q I x O 0 C k 6 R x 6 6 Q k 1 0 h 1 q o j S h S 6 B m 9 o F f n y X l z 3 p 2 P R W v J K W a O 0 R 8 4 n z / Q b 5 r s < / l a t e x i t > |i↵i < l a t e x i t s h a 1 _ b a s e 6 4 = " B G 5 C 3 O / / d 1 2 o M d r 8 b q b P R 6 n j s S 8 = " > A A A C C X i c b V B L S g N B E O 2 J v x h / U Z d u G o P g K s y I o s u g G 5 c R z A c y Q + j p 1 G S a 9 P S 0 3 T 1 C G H I C D + B W j + B O 3 H o K T + A 1 7 C S z 0 M Q H B Y / 3 q q i q F 0 r O t H H d L 6 e 0 s r q 2 v l H e r G x t 7 + z u V f c P 2 j r N F I U W T X m q u i H R w J m A l m G G Q 1 c q I E n I o R O O b q Z + 5 x G U Z q m 4 N 2 M J Q U K G g k W M E m O l w B + B y Z l P u I z J p F + t u X V 3 B r x M v I L U U I F m v / r t D 1 K a J S A M 5 U T r n u d K E + R E G U Y 5 T C p + p k E S O i J D 6 F k q S A I 6 y G d H T / C J V Q Y 4 S p U t Y f B M / T 2 R k 0 T r c R L a z o S Y W C 9 6 U / F f T 8 Z j z a h e W G + i q y B n Q m Y G B J 1 v j z K O T Y q n s e A B U 0 A N H 1 t C q G L 2 A U x j o g g 1 N r y K T c Z b z G G Z t M / q 3 n n 9 4 u 6 8 1 r g u M i q j I 3 S M T p G H L l E D 3 a I m a i G K H t A z e k G v z p P z 5 r w 7 H / P W k l P M H K I / c D 5 / A K J I m 1 8 = < / l a t e x i t > |-i↵i < l a t e x i t s h a 1 _ b a s e 6 4 = " P y n L s 4 5 A e d I J n w R F 5 z B g 8 A P W P P 4 = " > A A A C C n i c b V D L S s N A F J 3 U V 6 2 v q k s 3 g 0 V w Y 0 m k o s u i G 5 c V 7 A O a W C b T S T N 0 M g k z N 0 I I / Q M / w K 1 + g j t x 6 0 / 4 B f 6 G 0 z Y L b T 1 w 4 X D O v d x 7 j 5 8 I r s G 2 v 6 z S y u r a + k Z 5 s 7 K 1 v b O 7 V 9 0 / 6 O g 4 V Z S 1 a S x i 1 f O J Z o J L 1 g Y O g v U S x U j k C 9 b 1 x z d T v / v I l O a x v I c s Y V 5 E R p I H n B I w 0 o M 7 Z p C f c Z e I J C S T Q b V m 1 + 0 Z 8 D J x C l J D B V q D 6 r c 7 j G k a M Q l U E K 3 7 j p 2 A l x M F n A o 2 q b i p Z g m h Y z J i f U M l i Z j 2 8 t n V E 3 x i l C E O Y m V K A p 6 p v y d y E m m d R b 7 p j A i E e t G b i v 9 6 S Z h p T v X C e g i u v J z L J A U m 6 X x 7 k A o M M Z 7 m g o d c M Q o i M 4 R Q x c 0 D m I Z E E Q o m v Y p J x l n M Y Z l 0 z u t O o 3 5 x 1 6 g 1 r 4 u M y u g I H a N T 5 K B L 1 E S 3 q I X a i C K F n t E L e r W e r D f r 3 f q Y t 5 a s Y u Y Q / Y H 1 + Q M S 0 J u W < / l a t e x i t > |-↵i < l a t e x i t s h a 1 _ b a s e 6 4 = " 2 Y Q D r 8 q Y B i 7 h Y W 4 K l + u I + P m n H a M = " > A A A C C X i c b V D L S s N A F J 3 U V 6 2 v q k s 3 g 0 V w Y 0 m k o s u i G 5 c V 7 A O a U C b T S T N 0 M o k z N 0 I I / Q I / w K 1 + g j t x 6 1 f 4 B f 6 G 0 z Y L b T 1 w 4 X D O v d x 7 j 5 8 I r s G 2 v 6 z S y u r a + k Z 5 s 7 K 1 v b O 7 V 9 0 / 6 O g 4 V Z S 1 a S x i 1 f O J Z o J L 1 g Y O g v U S x U j k C 9 b 1 x z d T v / v I l O a x v I c s Y V 5 E R p I H n B I w k u e O G e R n L h F J S C a D a s 2 u 2 z P g Z e I U p I Y K t A b V b 3 c Y 0 z R i E q g g W v c d O w E v J w o 4 F W x S c V P N E k L H Z M T 6 h k o S M e 3 l s 6 M n + M Q o Q x z E y p Q E P F N / T + Q k 0 j q L f N M Z E Q j 1 o j c V / / W S M N O c 6 o X 1 E F x 5 O Z d J C k z S + f Y g F R h i P I 0 F D 7 l i F E R m C K G K m w c w D Y k i F E x 4 F Z O M s 5 j D M u m c 1 5 1 G / e K u U W t e F x m V 0 R E 6 R q f I Q Z e o i W 5 R C 7 U R R Q / o G b 2 g V + v J e r P e r Y 9 5 a 8 k q Z g 7 R H 1 i f P 0 D I m y M = < / l a t e x i t > q p Bob Parameter Estimation q p q p Key Map Uncharacterized Quantum Channel Authenticated Classical Channel n rounds Heterodyne Measurement Results for Typical Simulation Lossy and noisy channel, with ξ=0.01 50 100 150 200 Distance (km) 10 -6 10 -4 10 -2 10 0 Key Rate N=10 N=20 Gaussian Cutoff Infinite and Finite Optimizations Require: ⇧ N S 1 ⇧ N ✓ S N < l a t e x i t s h a 1 _ b a s e 6 4 = " A + O q F w r 2 X g X C F 2 O c R 8 F M q b X i q 6 g = " > A A A C M n i c b V C 7 S s R A F J 3 4 X N f X q m B j M y i C 1 Z K I o q W s j Z W s 6 K p g l j C Z v X E H J 5 M 4 c y O E s D / i P 9 j 5 A R Y 2 + g d a C G J h 4 0 c 4 + y h 0 9 c D A 4 Z x z u X N P m E p h 0 H V f n J H R s f G J y d J U e X p m d m 6 + s r B 4 a p J M c 2 j w R C b 6 P G Q G p F D Q Q I E S z l M N L A 4 l n I V X + 1 3 / 7 A a 0 E Y k 6 w T y F Z s w u l Y g E Z 2 i l o L L l 1 0 V w S P 2 Y Y T u M i u N O 4 A s V Y U 4 H u s l C A w j X P x O H Q W X N r b o 9 0 L / E G 5 C 1 v e W j N 3 F X e 6 w H l U + / l f A s B o V c M m M u P D f F Z s E 0 C i 6 h U / Y z A y n j V + w S L i x V L A b T L H r X d e i 6 V V o 0 S r R 9 C m l P / T l R s N i Y P A 5 t s v t H M + x 1 x X + 9 t J 0 b w c 3 Q e o x 2 m 4 V Q a Y a g e H 9 7 l E m K C e 3 2 R 1 t C A 0 e Z W 8 K 4 F v Y A y t t M M 4 6 2 5 b J t x h v u 4 S 8 5 3 a x 6 W 9 X t I 1 t R j f R R I i t k l W w Q j + y Q P X J A 6 q R B O L k l D + S J P D v 3 z q v z 7 n z 0 o y P O Y G a J / I L z 9 Q 1 r N K 7 g < / l a t e x i t > W ≥ 1 - Tr(⇢⇧ N ) 8⇢ 2 S 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " A k A 9 Z 0 k j i d 9 f o E y x u u H S f k U K Z C Q = " > A A A C Q X i c b V D L S g M x F M 3 U d 3 1 V X b q 5 t A i K W G Z E 0 W X R j S u p a G 3 B l C G T Z t p g J j M m G W E Y + h f + h N / h B 7 i 1 X y D u x K 0 b 0 9 a F V g 8 E T s 6 5 l 3 v v C R L B t X H d g V O Y m p 6 Z n Z t f K C 4 u L a + s l t b W r 3 W c K s o a N B a x a g V E M 8 E l a x h u B G s l i p E o E K w Z 3 J 4 O / e Y 9 U 5 r H 8 s p k C W t H p C t 5 y C k x V v J L p 0 3 A X X Y H H u w B v l L b W P V i w H X u n + 8 A v k t J B 3 A Y K y I E j B 0 u A U f E 9 I I w v + z 7 9 h u a z C 9 V 3 K o 7 A v w l 3 j e p 1 M p 4 9 2 F Q y + p + 6 R V 3 Y p p G T B o q i N Y 3 n p u Y d k 6 U 4 V S w f h G n m i W E 3 p I u u 7 F U k o j p d j 4 6 t g 9 b V u m A X c o + a W C k / u z I S a R 1 F g W 2 c r i o n v S G 4 r 9 e 0 s s 0 p 3 p i v A m P 2 z m X S W q Y p O P p Y S r A x D C M E z p c M W p E Z g m h i t s D g P a I I t T Y 0 I s 2 G W 8 y h 7 / k e r / q H V Q P L 2 x E J 2 i M e b S J y m g b e e g I 1 d A Z q q M G o u g R P a M X N H C e n D f n 3 f k Y l x a c 7 5 4 N 9 A v O 5 x d Z R 7 J 1 < / l a t e x i t > min f (⇢) subject to: ⇢ 2 Pos(H 1 AB ) Tr(⇢)=1 ⇢ 2 S 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " g / / k b / E j T z K W P G 6 f / 3 J x b W N V M e 4 = " > A A A C k H i c b V H b b h M x E P U u t 5 I W C O U R q b J I K t K X a L c C U S E h 2 v J S 9 S m I p q 1 U h 5 X X 8 T a m v q z s W c T K 2 p / g 7 / i C / k I f 8 S Z 5 K A k j W T q e c 2 b G c 5 y X U j h I k j 9 R / O D h o 8 d P N p 5 2 N r e e P X / R f b l 9 7 k x l G R 8 z I 4 2 9 z K n j U m g + B g G S X 5 a W U 5 V L f p H f f G n 5 i 5 / c O m H 0 G d Q l n y h 6 r U U h G I W Q y r q / i T Z C T 7 m G P l F C 4 2 J A 7 M z s 9 Q n p 9 A n w X 2 C V d 1 X + g z P A Y D 4 2 O B C t A p M g J o r C L A h G x j U D P 7 8 x K v 1 J 8 z 2 w B d S Z P z p u m r 2 2 5 M w u G n 9 K 1 x r k h f / W Z I s K Q v q d r N t L h s k 8 8 D p I l 6 C H l j H K u r d k a l i l w h J M U u e u 0 q S E i a c W B J O 8 6 Z D K 8 Z K y G 3 r N r w L U V H E 3 8 X P v G r w b M l N c G B u O B j z P 3 q / w V D l X q z w o 2 8 e 6 V a 5 N / p c r Z 7 U T z K 2 M h + J g 4 o U u K + C a L a Y X l Q z W 4 v Z 3 8 F T Y 4 L S s A 6 D M i r A A Z j N q K Y P w h 6 0 z 6 a o P 6 + B 8 f 5 i + G 7 7 / u t 8 7 P F 5 6 t I F e o z d o g F L 0 A R 2 i E z R C Y 8 T Q X b Q T v Y 0 G 8 X Z 8 E H + O j x b S O F r W v E L / R H z 6 F y a I y B s = < / l a t e x i t > f (⇢ 1 )= < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 X x + L 2 S g K 4 p X G R h L X g o Q w u 9 C F z 8 = " > A A A C D X i c b V D L S s N A F J 3 4 r P X R q E s 3 g 0 W o m 5 J I R T d C 0 Y 3 L C v Y B T S y T 6 a Q Z O p k J M x M h h H 6 D H + B W P 8 G d u P U b / A J / w 2 m b h b Y e u H A 4 5 1 7 u v S d I G F X a c b 6 s l d W 1 9 Y 3 N 0 l Z 5 e 2 d 3 r 2 L v H 3 S U S C U m b S y Y k L 0 A K c I o J 2 1 N N S O 9 R B I U B 4 x 0 g / H N 1 O 8 + E q m o 4 P c 6 S 4 g f o x G n I c V I G 2 l g V 8 K a J y P x 4 F E e 6 u z 0 a m B X n b o z A 1 w m b k G q o E B r Y H 9 7 Q 4 H T m H C N G V K q 7 z q J 9 n M k N c W M T M p e q k i C 8 B i N S N 9 Q j m K i / H x 2 + A S e G G U I Q y F N c Q 1 n 6 u + J H M V K Z X F g O m O k I 7 X o T c V / v S T K F M V q Y b 0 O L / 2 c 8 i T V h O P 5 9 j B l U A s 4 j Q Y O q S R Y s 8 w Q h C U 1 D 0 A c I Y m w N g G W T T L u Y g 7 L p H N W d x v 1 8 7 t G t X l d Z F Q C R + A Y 1 I A L L k A T 3 I I W a A M M U v A M X s C r 9 W S 9 W e / W x 7 x 1 x S p m D s E f W J 8 / l 0 m b v g = = < / l a t e x i t > min f (⇢) subject to: ⇢ 2 Pos(H N AB ) 1 - W Tr(⇢) 1 ⇢ 2 S N < l a t e x i t s h a 1 _ b a s e 6 4 = " s L W 1 t u E z e 6 5 O k J g h V B x X b Q I 1 7 X A = " > A A A C l X i c b V H b b h M x E P V u u Z R w a Y A H H n i x S J C C R K P d C g S q V K k U h P p U g m i a S n V Y e R 1 v Y + r L 1 p 5 F R N Z + B 9 / G F / A V S H i z e Y C 0 I 1 k 6 n n N m 7 D m T l 1 I 4 S J J f U b x x 4 + a t 2 5 t 3 O n f v 3 X + w 1 X 3 4 6 M S Z y j I + Z k Y a e 5 p T x 6 X Q f A w C J D 8 t L a c q l 3 y S X 7 x v + M l 3 b p 0 w + h g W J Z 8 q e q 5 F I R i F k M q 6 P 4 k 2 Q s + 4 h j 5 R Q u N i Q O z c v O g T 0 u k T 4 D / A K u + q / B t n g M H s 1 r h l g g a T I C e K w j x I R s b V A 7 + 8 M S r 9 Y Z 3 5 d w f 1 1 6 O 6 7 Z R u T 4 j k l + T Y t u 2 b S 3 p N p 7 z w X + r s K D B Z t 5 c M k 2 X g q y B d g R 5 a x S j r / i Y z w y o V J m G S O n e W J i V M P b U g m O R 1 h 1 S O l 5 R d 0 H N + F q C m i r u p X x p Y 4 + c h M 8 O F s e F o w M v s v x W e K u c W K g / K 5 p t u n W u S 1 3 L l f O E E c 2 v P Q / F 2 6 o U u K + C a t a 8 X l Q z + 4 m Z F e C Z s s F s u A q D M i j A A Z n N q K Y O w y E 5 w J l 3 3 4 S o 4 2 R m m r 4 a v P + / 0 9 g 9 W H m 2 i p + g Z G q A U v U H 7 6 B C N 0 B g x 9 C f q R S + j 7 f h J v B d / i D + 2 0 j h a 1 T x G / 0 X 8 6 S + k J c h f < / l a t e x i t > f (⇢ N )= < l a t e x i t s h a 1 _ b a s e 6 4 = " P P R 7 C D o c 2 u f + Q / s Z o n r F N 3 i f e y Q = " > A A A C B n i c b V D L S s N A F L 2 p r 1 p f V Z d u g k W o m 5 J I R T d C 0 Y 0 r q W A f 2 M Y y m U 7 a o Z O Z M D M R Q u j e D 3 C r n + B O 3 P o b f o G / 4 b T N Q l s P X D i c c y / 3 3 u N H j C r t O F 9 W b m l 5 Z X U t v 1 7 Y 2 N z a 3 i n u 7 j W V i C U m D S y Y k G 0 f K c I o J w 1 N N S P t S B I U + o y 0 / N H V x G 8 9 E q m o 4 H c 6 i Y g X o g G n A c V I G + k + K H f l U D z c H F / 0 i i W n 4 k x h L x I 3 I y X I U O 8 V v 7 t 9 g e O Q c I 0 Z U q r j O p H 2 U i Q 1 x Y y M C 9 1 Y k Q j h E R q Q j q E c h U R 5 6 f T i s X 1 k l L 4 d C G m K a 3 u q / p 5 I U a h U E v q m M 0 R 6 q O a 9 i f i v F w 0 T R b G a W 6 + D c y + l P I o 1 4 X i 2 P Y i Z r Y U 9 y c T u U 0 m w Z o k h C E t q H r D x E E m E t U m u Y J J x 5 3 N Y J M 2 T i l u t n N 5 W S 7 X L L K M 8 H M A h l M G F M 6 j B N d S h A R g 4 P M M L v F p P 1 p v 1 b n 3 M W n N W N r M P f 2 B 9 / g D A v J k j < / l a t e x i t > f (⇢ N ) - ✏ N (W ) f (⇢ 1 ) < l a t e x i t s h a 1 _ b a s e 6 4 = " J 8 4 7 N J Y F d j 0 L 6 J p W V i P B N O H y l A E = " > A A A C K X i c b V D L S s N A F J 3 4 t r 6 i L t 0 M F q F F L I l U d C m 6 c S U K 9 g F N L J P p T T t 0 M o k z E y E E P 8 L v 8 A P c 6 i e 4 U 7 c u / A 2 n N Q u t H r h w O O d e D v c E C W d K O 8 6 b N T U 9 M z s 3 v 7 B Y W l p e W V 2 z 1 z e a K k 4 l h Q a N e S z b A V H A m Y C G Z p p D O 5 F A o o B D K x i e j v z W L U j F Y n G l s w T 8 i P Q F C x k l 2 k h d e z e s e H I Q X 5 9 X 9 z x I F O N G P K + 0 q t j j c I M L 0 2 M i 1 F m 1 a 5 e d m j M G / k v c g p R R g Y u u / e n 1 Y p p G I D T l R K m O 6 y T a z 4 n U j H K 4 K 3 m p g o T Q I e l D x 1 B B I l B + P n 7 q D u 8 Y p Y f D W J o R G o / V n x c 5 i Z T K o s B s R k Q P 1 K Q 3 E v / 1 k k G m G F U T 8 T o 8 8 n M m k l S D o N / p Y c q x j v G o N t x j E q j m m S G E S m Y e w H R A J K H a l F s y z b i T P f w l z f 2 a W 6 8 d X N b L x y d F R w t o C 2 2 j C n L R I T p G Z + g C N R B F 9 + g R P a F n 6 8 F 6 s V 6 t 9 + / V K a u 4 2 U S / Y H 1 8 A e u a p l c = < / l a t e x i t > ρ ∞ Π N ρ ∞ Π N ρ N H N H ∞ S N |f (ρ ∞ ) - f (Π N ρ ∞ Π N )| ≤ N (W ) f (Π N ρ ∞ Π N ) ≥ f (ρ N ) W R = I (A : B ) - χ(X : E ) < l a t e x i t s h a 1 _ b a s e 6 4 = " A 7 W 7 f 3 S E l D i I v v R + 8 r u T T K E r z y s = " > A A A C E X i c b V D L S s N A F J 3 4 r P U V 7 d J N s A j t w p J I R S k I t S L o r o p 9 Q B v K Z D p p h k 4 m Y W Y i h N C v 8 A P c 6 i e 4 E 7 d + g V / g b z h p s 9 D W A x c O 5 9 z L v f c 4 I S V C m u a X t r S 8 s r q 2 n t v I b 2 5 t 7 + z q e / t t E U Q c 4 R Y K a M C 7 D h S Y E o Z b k k i K u y H H 0 H c o 7 j j j q 9 T v P G I u S M A e Z B x i 2 4 c j R l y C o F T S Q C / c X 9 y W L m u N 8 n E f e a T U r V 2 X 8 w O 9 a F b M K Y x F Y m W k C D I 0 B / p 3 f x i g y M d M I g q F 6 F l m K O 0 E c k k Q x Z N 8 P x I 4 h G g M R 7 i n K I M + F n Y y P X 5 i H C l l a L g B V 8 W k M V V / T y T Q F y L 2 H d X p Q + m J e S 8 V / / V C L x Y E i b n 1 0 j 2 3 E 8 L C S G K G Z t v d i B o y M N J 4 j C H h G E k a K w I R J + o B A 3 m Q Q y R V i G k y 1 n w O i 6 R 9 U r G q l d O 7 a r H e y D L K g Q N w C E r A A m e g D m 5 A E 7 Q A A j F 4 B i / g V X v S 3 r R 3 7 W P W u q R l M w X w B 9 r n D y X W m z 4 = < / l a t e x i t >

asymptotic security proof for discrete- modulated cvqkd … · p bob parameter estimation q p q p...

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