p qsr p t - carnegie mellon school of computer scienceyangke/thesis/prop-slide.pdf · u u u u u u u...
TRANSCRIPT
�����
��� ��� �
�� ��� ��� � ����� ��� �
���� ��� � ��� �� � �� ���� �
��
��� ���� �� �� �� ���� �
�� ! "#
$%�& '&()**'++��
,- ./0123 0
4567 .89 ./:8 ;<
= 87 -7 >43? /@ ;A @ ;B /.C
,>? ./< ,06B />/<AED,=FHG>/-7 .< /8I 5 JKB8 - /BC
LM NOPQSRP TP UUV O
W
�� � ���� ���� �� ��� �� �
LM NOPQSRP TP UUV O
X
�YZ ���[\����[][^� ��_�� [
��� �`
ab5 ..7 2B8 /5 >b5 ..3c 8 /5 >
,2/@7B >?156< ;B .7 /0c 7 .J7@8 2I @5 ..7 2B87? />J5 .0B8 /5 >
ab5 ..7 2B8 /5 >47@5 -7 .I
,2/@7B >?1568B d7B@8 /5 >85 .7@5 -7 .c 7 .J7@8@5 ..7 2B8 /5 >
LM NOPQSRP TP UUV O
e
�����Y� �
� ��`�� [[� �
0 1
0 1
10
1
0 1
11
0 1
0
noisy channel
Alice
Bob
time
a,2/@7<7 >?<6 /8<85 156
ab5 ..7 2B8 /5 >@5 ..3c 8 /5 >6I 8 ;7>5 /<I @ ;B >>7 2
LM NOPQSRP TP UUV O
f
Z� [_Z
� ��`�� [[� �
Alice
noisy channel
Bob
imperfect
entanglement
|0>
time
a,2/@7<7 >?<g 36 /8<85 156
ah>8B >i 27 07>8@5 ..3c 8 /5 >6I 8 ;7>5 /<I @ ;B >>7 2
LM NOPQSRP TP UUV O
j
k� ��� ��_�� [ml
n� ����o�[\
a@ 2B<< /@B 2 pp@5 ..7 2B8 /5 > q@5 ..7 2B8 /5 >
ag 3B >83 0pp@5 ..7 2B8 /5 > q7>8B >i 27 07>8
LM NOPQSRP TP UUV O
r
_��_�\ ��� ^� �� ��� ��_�� [
�Y� n� �`
ast� u� "+ 'u�v + t!+�# w
,?? />i .7?3 >?B >@I 67 J5 .78 ;7@5 ..3c 8 /5 >
ax�y ! t!+ 'u�v + t!+�# w
47@5 -7 ./>i @5 ..7 2B8 /5 >5 >2I B J87 .@5 ..3c 8 /5 >
LM NOPQSRP TP UUV O
z
�� n� [_�n�_��_�\ `
encode
decode
Alice
noisy channel
Bob
perfect correlation
aD>J5 .0B8 /5 >7>@5?7?67 J5 .78 ;7@5 ..3c 8 /5 >
ah..5 .b5 ..7@8 />i b5?7<A hbb<C
a{ 3B >83 0h..5 .b5 ..7@8 />i b5?7<A { hbb<C
LM NOPQSRP TP UUV O
|
��� �� ���Y_�[\�o��
noisy channelE
D~c
Alice
Bob
c
m
m
aA~}������C �hbb p������� ������� �� }� <3@ ;8 ;B8
�D= �AE�A~��C � �A~��CC ��
ab5?7� -7 .;7B? pA~}���C 6 /8<
a�5 /<7 �5 27 .B >@7 p�A � ��C���6 /8 �/c <
A 7 >@5? />iF ?7@5? />i @5 0c 27�/8I >5853 .J5@3<C
LM NOPQSRP TP UUV O
�
�� � ��_�n�_��_�\ `
}s bits
noisy channel
local operation
local operation
Alice
Bob
perfect correlation
ab5 ..7 2B8 /5 >.7c B /.7?B J87 .8 ;7@5 ..3c 8 /5 >
a,2/@7B >?1567�@ ;B >i 7�6 /8<85 .7@5 -7 .8 ;7@5 ..7 2B8 /5 >
,==G¡¢�D� �p>5 /<7 27<<@ 2B<< /@B 2@5 003 >/@B8 /5 >
9� ,Kp 0/>/0/£7�
A @5 0c 38B8 /5 >B 2@5 0c 27�/8I >5853 .J5@3<C
LM NOPQSRP TP UUV O
¤
� ��� ��_�� [
��_����_�� [}
s bitsnoisy channel
local operation
local operationB
ob
Alice
ab 2B<< /@B 2b5 ..7 2B8 /5 >�/<8 /22B8 /5 >¢.585@5 2A b �¢C
a{ 3B >83 0h>8B >i 27 07>8 �/<8 /22B8 /5 >¢.585@5 2A h�¢C
LM NOPQSRP TP UUV O
XW
][^� ��_�� [
�� [������ [
,2/@7¥ /< ;7<858 .B >< 0/8�85 156� >5 /<7 27<< 2I
y t� u� "+ 'u�noisy channel
E
D~c
c
m
m
t�y ! t!+ 'u�
}s bits
noisy channel
local operation
local operation
�¦h>@5? />i p§ q�A~�C
�¦�.B >< 0/<< /5 >p§�©¨§
ª ¦�7@5? />i p� q«A ¨§C
�¦�.B >< 0/<< /5 >p�� ¨�
�¦�/<8 /22B8 /5 >p
A~�� ¨�C¬��A~����C
� -7 .;7B? q® §® �® �®
� -7 .;7B? q�
LM NOPQSRP TP UUV O
XX
}s bits
noisy channel
local operation
local operation
noisy channelE
D~c
c
m
m
Quantum
Error C
orrecting Code
Error C
orrecting Code
Entanglem
ent Distillation P
rotocol
Correlation D
istillation Protocol
classical
quantum
s
statusw
ell−studied, w
ell−understood
less studied, fewer results
overheadc
m
preven
tiverep
arative
| | − | |
LM NOPQSRP TP UUV O
Xe
}s bits
noisy channel
local operation
local operation
noisy channelE
D~c
c
m
m
Quantum
Error C
orrecting Code
Error C
orrecting Code
Entanglem
ent Distillation P
rotocol
Correlation D
istillation Protocol
classical
quantum
s
statusw
ell−studied, w
ell−understood
less studied, fewer results
overheadc
m
preven
tiverep
arative
| | − | |
LM NOPQSRP TP UUV O
Xf
LM NOPQSRP TP UUV O
Xj
��� �� ���Y_�� [��
���_¯
y t� u� "+ 'u�noisy channel
E
D~c
c
m
m
t�y ! t!+ 'u�
}s bits
noisy channel
local operation
local operation
>5 > �/>87 .B@8 /-7
/>87 .B@8 /-7
LM NOPQSRP TP UUV O
Xr
"An ounce of prevention is w
orth a pound of cure."
AE°±Dp�c 53 >? q�² 53 >@7<C
LM NOPQSRP TP UUV O
Xz
³´"µ ¶ "·�)¸st� u� "+ ') "'&¹) t+ %! s)¶ "º)¸(¶ t� »½¼
y t� u� "+ 'u�noisy channel
E
D~c
c
m
m
t�y ! t!+ 'u�
}s bits
noisy channel
local operation
local operation
� -7 .;7B? q® §® �® �®
� -7 .;7B? q�
<B 0727-7 25 J@5 ..3c 8 /5 >� �² ¾ 05 .77 :@ /7>8 ¿
LM NOPQSRP TP UUV O
X|
�_
�Y���� ���`
b5 ..7 2B8 /5 >? /<8 /22B8 /5 >/< ¦¦¦
�¦B<7 :@ /7>8B<7 ..5 .@5 ..7@8 /5 >
�¦Bcc 2/@B6 2785B ¥ /?7 ..B >i 75 JBcc 2/@B8 /5 ><
LM NOPQSRP TP UUV O
X�
][^� ��_�� [
�� [������ [
y t� u� "+ 'u�noisy channel
E
D~c
c
m
m
t�y ! t!+ 'u�
}s bits
noisy channel
local operation
local operation
� -7 .;7B? q® §® �® �®
� -7 .;7B? q�
$ÀÁA~}������C �2/>7B .hbbÂb �¢5 J5 -7 .;7B?� qA~}���C
$ÀÁA~}������C �<8B6 /2/£7 .{ hbbÂh�¢5 J5 -7 .;7B?� qA~}���C
LM NOPQSRP TP UUV O
X¤
��� ^
$ÀÁA~}������C �2/>7B .hbbÂA~}���C �6 /8b �¢
message
k
checksum
kn
−k
message
sxµµ Ã�¦,2/@7<7 >?<8 ;7A~}���C �6 /8@ ;7@ d �<3 0
�¦156?7@5?7<
LM NOPQSRP TP UUV O
eW
an ounce
"An ounce of prevention is w
orth a pound of cure."
LM NOPQSRP TP UUV O
eX
[_� [\ ��� [_
��_����_�� [
��_�
�
Ä 17>>788� �/Å />@7 >£5� = 05 2/>�ÇÆ55887 .< �ÈȲÉ
h>8B >i 27 07>8 �/<8 /22B8 /5 >/<c .5 -B6 2I 05 .7c 5 ¥7 .J3 28 ;B >{ hbb<
Ê>5 /<I @ ;B >>7 2� < ¦8 ¦
a�5{ hbb@B >¥5 .d
a138 h>8B >i 27 07>8 �/<8 /22B8 /5 >¢.585@5 2<@B >
LM NOPQSRP TP UUV O
ee
"In a corrupted quantum w
orld, prevention is useless, yet there is cure."
"An ounce of prevention is w
orth a pound of cure."
LM NOPQSRP TP UUV O
ef
ËÌÍÍÎ ÏÐÑ ÒÌÓÔÒÕÑ ÒÏÏÐÑ ÒÌÓÖÐÕÌÍÎ ×ØØ ÏÒÙÐÑ ÒÌÓÕ
,<<3 0c 8 /5 >< 0B?76I 7 ..5 .@5 ..7@8 /5 >ÛÚ
st� u� "+ 'u�7>@5? />i 03<8c .7@7?78 ;7>5 /<7
ÜÆ;B8 /J7>@5? />i /< /0c 5<< /6 27 ¿ÞÝ
ß) '&� *)º� à/?7>8 /@B 2/>?7c 7 >?7>8 >5 /<7� d>5 ¥ >>5 /<7 .B87
ÜÆ;B8 /J8 ;7>5 /<7 05?7 2/<? /á7 .7>8 ¿ÞÝ
âB -785i 37<<B >3cc 7 .653 >?5 >>5 /<7 .B87
LM NOPQSRP TP UUV O
ej
� [o�
��Y� [
0 1
10
1
,.7B 2 �8 /07� -7 ./ãB6 27.B >?5 0<53 .@7
a-7 ./ãB6 27 25887 .I
a/>J5 .0B8 /5 > �8 ;75 .78 /@B 22I <7@3 .7@ .I c 85i .Bc ;I Úd7I �7�@ ;B >i 7�
7>@ .I c 8 /5 >¦¦¦A B<<3 0/>i 653 >?7?<85 .Bi 7C
LM NOPQSRP TP UUV O
er
� ä_�Z ��o�� [o�
��Y� [
0 1
10
1
Alice (ow
ner)B
ob (verifier)
a¢5 />8B87 27<@5c 785Bc 3 2<B .
a¡7B<3 .78 ;7< /i >B 2� @5 >-7 .885 .B >?5 06 /8<
a47B 2 �8 /07-7 ./ãB6 27 pA B 205<8C 7-7 .I 5 >7@B ><778 ;7c 3 2<B .
LM NOPQSRP TP UUV O
ez
� ��`���Z ��� [_
0 1
0 1
10
1 0
11
10
10
Bob (verifier)
Alice (ow
ner)
¡7B<3 .7 07>87 ..5 .< Ú@5 ..3c 87?@5 ..7 2B8 /5 >
LM NOPQSRP TP UUV O
e|
ËÌÍÍÎ ÏÐÑ ÒÌÓåÎÙÌæÎ Íç�èÌÍåÐ ÓéÌêëÎÐÙÌÓÕ
0 1
0 1
10
1 0
11
10
10
Bob (verifier)
Alice (ow
ner)
ìµ ´íî85B@ ;/7-7A B 205<8C c 7 .J7@8@5 ..7 2B8 /5 >
LM NOPQSRP TP UUV O
e�
��� �� ��ïY_ðñ [ñ [òó ôõò ö÷ø
ùúûü ýþÿ���� �� úû� ý� ���û �� ü�� ���û ���ü �û �
ù �� �� �ü ����ü ��ü�� ��û�� ���ü�û ��
ù� ��� üû �û�������� ý ��� ��û �
� �ü�
�� ������� � !!" �
#$
%&''( )*+,& -.,/+ ,00*+,& -1&'2* -3&45(*6&-
0 1
11
0 1
0 0
10
11
01
ab
a=
bPr[ ]
1
Alice (ow
ner)B
ob (verifier)
�� ������� � !!" �
78
9: ;<=>?@:A= ;B@ CC= CA= CC@ADE= ;<=@F ;
GD:HHJIK
�� ������� � !!" �
7 L
MN öOP
Q
ò Oöõ
ùR S � ���� ���� � ÿT � �ü �û�U ��ü�V �ü ý� ��üû�üû ��
ùWû ��ü� �ü ÿ������ ���YX�� �� ��� �û �����ü�
ùZ RWW ý��üû� ���� � � ��û ��û ��û �����ü�
�� ������� � !!" �
7 #
óò PMó
[\OõM öO]óM ON P
^`_a[ò ôb
0 1
10
1
0 1
10
1
Alice
Bob
perfect entanglement
perfect correlation
c ú� ���üüedú����� ��f gV ú� ���üüh ijk �û� �l��m
ùþÿ����� ��� �� ��û �T � �ü�üû úû�� �� ��� ���û� ý� ��� ÿ�
ùkon��� ÿÿ�m � ����ü ÿ� � �ü� �� ÿ��T � �ü�
ùúûü ý���� ��pkon��� ÿÿ�m � ����ü ÿ��û ��� ÿ�ü��� �ü�
�� ������� � !!" �
77
óò PMó
[\OõM öO]óM ON P
^[ò ôôOq[b
Alice
Bob
Eve
Ucorrupted entanglem
ent
ùR�� ��ü� ��� ü��û ��T � �ü�� ��� ��üû �ü�ü ý� �
ù�û �� ü��� �ü� �� ÿ� �� �üp�û �� ü���û ��� ÿ�ü �û �
�� ������� � !!" �
7 r
ööN öN öö[sM ON PqN ö
t[ ÷
Z RWW��� ��� ��� �ü ��� ÿ���� � ��� �ü�û ���
�ü uuu
R�� ���� �� ��� ��� ÿ
�� ������� � !!" �
7v
Z � �ü �w��x��ü ���ü �û �y� ��û ��û ����ü �û ������ �z� �� �ü �û�� ÿ
�� ������� � !!" �
7{
|\[}�~ ö~M Ot[ ÷
� ���� ���
����� �
n��û ���ü �û ���� �� ���� �û �
Wû ��� ÿ�ü �û �� ��ü �ÿÿ�ü �û �����
� �� �� �ü��� ��û ��û ����ü �û �
k � ���� ��� �û ���� � ÿm
S� ��û �ú���û �
RWW��û � �ü� ÿ�
k �� ���ü� ��û ��û ����ü� ÿ�� �m
� üû ���� R S � ���
Z RWW�� �� ��� �� �� �ü
k �� �� ��� �û �����ü�m
Z � �ü �w��x��ü ���ü �û �Z RWW��û � �ü� ÿ�
k � �z� �� �ü�û ��� �û�� ÿ�m
�� ������� � !!" �
7 �
�
�
�
�� ������� � !!" �
7�
�~PM Oq\ OPQO�M O��~M ON P
�NMNsN��
}s bits
local operation
local operation
Alice
Bob
�� ������� � !!" �
7$
����� ��������� � ���� �� ¡¡������ �� �¢� £¤ �� ����
}s bits
local operation
local operation
Alice
Bob
"noise model"
communication
quality = CLOSENESS(output, "perfect")
�� ������� � !!" �
r8
quality
01
bounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
communicationmany
classicalquantum
�� ������� � !!" �
rL
U U U U U U U U
L L L L L Lbounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
LL L
UU
01
manyU U
quantum classical
communication
U =
L =
=
upper boundlow
er bound
=
independent result
L
my orignal result
�� ������� � !!" �
r#
[ �~M[a�] �Os~M ON P�
¥o¦� �§©¨ªª ¨« �� �ü ý�kon�m û�� �� �ÿ�ü�û �¬û � d��ü� ���ü ���Wû ��� d
ÿ�ü �û �x��ü �ÿÿ�ü �û � �V�® ¯°±²³´µ·¶ ´¯±°¸ ´±± ´¹ ¯u
¥oº¡»� ����� ¦� �§ ¨ªª ¨« ��û �� ���ü ý�W ÿ��� ��� ÿWû �� ����ü �û �
Wû � ÿ� ¼�ü�û �R�ü� �� ÿ� �� �üx��ü �ÿÿ�ü �û � �ûüû�û ÿ� � �ü ýn� d
�û � ÿ�ü� n��û ���ü �û � �V ® ¯°±²³´µ·¶ u
¥oº¡»� ����� � ¡�� ½� ¦� �§ ¨ªª ¨« �R¼ü ���ü ��� Z � �ü �R�ü� � d
� ÿ� �� �ük¿¾� �� �� ÿR�ü� �� ÿ� �� �ü ��l��ü �û � �ûüû�û ÿ�m �V�ÀÁÁÁ
¹ ¯ÃÄ ³Ä ¯²Ä¹ ¯Â¹ µ °¶®¶ ´¹ ¯® Ź µ ÅÄ Æ´¶ÇÉÈÊÊ Èu
�� ������� � !!" �
r7
U U U U U U U U
L L L L L Lbounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
LL L
UU
01
manyU U
quantum classical
communication
U =
L =
=
upper boundlow
er bound
=
independent result
L
my orignal result
linear EC
C =
> perfect C
DP
stablizer QE
CC
=>
perfect ED
P
�� ������� � !!" �
rr
U U U U U U U U
L L L L L Lbounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
LL L
UU
01
manyU U
quantum classical
communication
U =
L =
=
upper boundlow
er bound
=
independent result
L
my orignal result
non−interactive correlation distillation
�� ������� � !!" �
rv
NPÌËOPM[ �~sM Ot[N��[ �~M ON P
O�M O��~M ON P
ÍÎÏÐÑÒÓÔ ÕÖ×Ô Ï�Ø ÏÎÎÐÖ ÙÙÑ ÎÒØ ÏÖ Ó� ÏØ ÚÖÛØÐÖ ÜÜÛ ÓÏÐÒØ ÏÓ�
�� ������� � !!" �
r{
%&''( 0*+,& -2(6&Ý( 'Þ 1&'2* -3&45(*6&-/
0 1
0 1
10
1 0
11
10
10
Bob (verifier)
Alice (ow
ner)
¾� ÍßàØÖÒÐ ÚÏÑ áÑk Ò ÎÜÖâØm ã Ñ ÙäÑÐØÐÖ ÙÙÑ ÎÒØ ÏÖ Ó
åæ çèéê�ëé ìé ííî è
r�
P[�Os[ï~P\N]�
0 1
10
1
Alice (ow
ner)
Bob #1
Bob #3
Bob #2
åæ çèéê�ëé ìé ííî è
r�
ðñ òóôòõö ÷
øùõ úûöüñ ÷÷ö ýøõ úñ òþúÿõ úýýøõ úñ ò�ñ ÷
�ø ò�ñ ��öøùñ ò
0 1
11
0 1
0 0
10
11
01
Alice (ow
ner)B
ob (verifier)
b gfa
��� ��� � �� ����
��� �� �����
�� � !"$#! %! &&'
()
* ++, -./0* 1
2/ +.3/0* 14./5,./03. --6
0 1
0 1
10
1
b g
0 1
11
0 1
0
fa
Alice
Bob
789�:�;< = ;> =@???= ;AB ���C < =C > = ???=C AB � D� D��� ;E �CE � �� �F
789�:� �GH ;< = ;> =@??= ;AI J��� �KHC< =C> = ???=C AI
7L �� ����� ��� ��� �M� ��NPOB ��� �M� ��N O
7QR�S�T���� �� �
�� � !"$#! %! &&'
UV
. WXY,/ +./,Z 6
0 1
11
0 1
0 0
10
11
01
ab Bob
Alice
7��� ����[ ��� ��\���� ��
7��� �� � �� �F
�� � !"$#! %! &&'
U ]
. 1
,^*
,//, +_
0 1
0 1
10
1
b g
0 1
11
0 1
0
fa
Alice
Bob
789�:�;< = ;> =@???= ;` B ���C< =C> =@???=C` B ��� ;E �CE � �M ?a
7b���� �� � cM ?a �d
H S��� 9��e� �S� �� � �f ?g OI
�� � !"$#! %! &&'
U h
*
0 1
11
0 1
0 0
10
11
01
ab Bob
Alice
� 89� �B Q� �� �B�i�j �� ��� �� B� Q���� 9Blk
mn� ��� 99� B�po�j OMM O�
7q��� rst��� ���j u ���[ � �S 9
7899�[ � �S 9�� ���j ��� ��� rst�
�� � !"$#! %! &&'
Uv
01. +66
,/ +03
*^, -
1 001
(1−p)/2
p/2
p/2
(1−p)/2
Bob
Alice
�po�j OMM O� j � �� � 9�T� �� ��� q� ��� ������:�Q��� 9
H 9 �j � 9[ ����B S� ��j � �� � 9�� ���I
�� � !"$#! %! &&'
U (
01. +6+.wx +,
*^, -
1 Alice
00
(1−p)/2
p/2
(1−p)/2
Bob1
p/2
�po�j OMM O� q��� rst��� ���j u ���u S[ ��� �: 99u�[ � �S 9
�� � !"$#! %! &&'
UU
U U U U U U U U
L L L L L Lbounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
LL L
UU
01
manyU U
quantum classical
communication
U =
L =
=
upper boundlow
er bound
=
independent result
L
my orignal result
non−interactive correlation distillation
�� � !"$#! %! &&'
Uy
U U U U U U U U
L L L L L Lbounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
LL L
UU
01
manyU U
quantum classical
communication
U =
L =
=
upper boundlow
er bound
=
independent result
L
my orignal result
EP
R extraction
impossibility for general
�� � !"$#! %! &&'
U z
{|} ~��} ~| ��� ���� ~�� ��� ��| ������ �}���} ~| �
�� � !"$#! %! &&'
U�
. 1^�1,ww
2/ +.3/� +w
extractorA
ux
Output
Input Input: random
source
Output: near−
uniform random
bits
Aux: uniform
random bits
[ ����:��� ���� ��e� �S���� S� ��� e�� S �� �� � �u ���� S��� �:��
�� � !"$#! %! &&'
U)
.3/w
��x/
2/ +.3/� +w
�� �u��� e� 9B��� ��� � �� �t� �uj � �� � 9��[ ��
7��[ �� � �� �� � �u ���� S��� �:� D
7� ���[ ��� � �S�� �� �� ��[uH ��[ ��I
7� � R�9� �u ��[ ��� ��H 9�jH� ��[ ��� II
7� q �� �SB L S��B$��: �� �S�OMM �� �� ����[ � �S 9:� ��� ��: �
� �� ��� R���
�� � !"$#! %! &&'
yV
������� ý�� � �¡ ý� ��� þ¢ÿ ¢ýý� ¢£ �¤¥
¦ §¨©© ª¦¨ §
«¬¨ ®¯¬°
� ��e� �S� ���
±�²[ ���
���� S���� ��[ � ���� e� �S
� ���j 9� S� �� ��[ � ���� e� �S
� R� �:�� �
� ���j 9� S� ��� ��� �99� �� �
9� � �³ � 9��u ���� S����
9� � �³ � 9��u � ���j 9� S� ��
´
´
��j � �³ � 9��u ���� S����
��j � �³ � 9��u � ���j 9� S� ��
��[ ��
��[ ��
�� �� � �u ���� S� ���
�� �� � �u � ���j 9������ d
�� � !"$#! %! &&'
y ]
�
qµQ¶� �� � 9� ���j 9� S� ��� ��� �99� �� ��� �S[ ��� �� 9�
H ��[ ����:� 9� R� �:�� ±�²[ ��� e�� S �� �� � �u � ���j 9�������I
�� � !"$#! %! &&'
y h
+��·¸¹/35
¦ §¨©© ª¦¨ §� ��³ ��� ��� ����� �� �� eSR� �� ��[u
«¬¨ ®¯¬°��\���� 9u S�u SR�S 99u � ���j 9�������
q��º�� 99����� »¼¾½
¿<À >ÂÁ Ã
ÄÅÇÆÃÄÅÇÈÉÃ<Å ÆÃ<Å ÈÊ
¼ÌË¿<À >ÍÁ Ã
ÄÅÎÆÃÄÅÎÈÏÃ<Å ÆÃ<Å ÈÊ
Ð ½¿<À >ÍÁ Ã
ÄÅÇÆÃ<ÅÇÈÉÃ<Å ÆÃÄÅ ÈÊ
Ð Ë¿<À >ÑÁ Ã
ÄÅÇÆÃ<ÅÇÈÏÃ<Å ÆÃÄÅ ÈÊ
�� � !"$#! %! &&'
yv
+��·¸¹/354 3� 1/
Ò^
� �[[ ���� �� ��� R������: �[ ����:� 9ÓB � D� DB
ÓHpÔÉI � ÔÉ=ÓHpÔ
ÏI � ÔÉ=ÓHpÕÉI � ÔÉ=ÓHpÕ
ÏI � ÔÉ
Ö��× ��S�R������ »
× ��ºÙØ
� Ô ÉÚÛ ÔÉ� Ü
� ÔÏÚÛ ÔÏ� Ü
� ÕÉÚÛ ÕÉ� Ü
� ÕÏÚÛ ÕÏ�Ý
i�� Þß� 9�à 9�ß Þàt� »
ÓH× I � ÔÉ
�� � !"$#! %! &&'
y (
5. 1Z ¹�·.w0w
× ��ºáØ
� Ô ÉÚÛ ÔÉ� Ü
� ÔÏÚÛ ÔÏ� Ü
� ÕÉÚÛ ÕÉ� Ü
� ÕÏÚÛ ÕÏ�Ý
âu: Þàãj �ãj ß e�à� �� »
× ��º H
� MMÚÛ MM� Ü� M �ÚÛ M �� Ü� �MÚÛ �M� Ü� ��ÚÛ ��� I
× �� � ��� ãäà ãj 9��åæç[ ßèè æé 9êäß[$ëßìíîê ±ï²[ àæëèåð
ñò óôõö$÷õ øõ ùùú ô
yû
U U U U U U U U
L L L L L Lbounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
LL L
UU
01
manyU U
quantum classical
communication
U =
L =
=
upper boundlow
er bound
=
independent result
L
my orignal result
EP
R extraction
impossibility for general
ñò óôõö$÷õ øõ ùùú ô
yy
üýÿþ����� ��� � �¡ ����� �� ��� ¢£ ��� ¢�� ¤
��ß� ëßäßîß �îàãìß �ê ��ß ãàê ëà ê
�� èê �í �� ëßäßîß �ß ã� æ�æã� íä æè�î �ßèê��
äßèß çêèäàäê
ñò óôõö$÷õ øõ ùùú ô
y z
�¹��¹ ����� ���
��� �
���çéàæãæè��� çæä Þ�! àã#"$$ "%
&ìê �æä ' æã� íä� �� ê ë�êîä �( )*,+-
��./�1023í*444% �� � 2/ ë� ïëêè 5æ��"$$$%
íèêìæ6 æ7�$ë// �/ �èêîí ëæ6 / ���889 :% 5ê ìæè6 ëæéí6 æ/ 7� ë/6/î/ �
ñò óôõö$÷õ øõ ùùú ô
;<
U U U U U U U U
L L L L L Lbounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
LL L
UU
01
manyU U
quantum classical
communication
U =
L =
=
upper boundlow
er bound
=
independent result
L
my orignal result
matching low
er/upper bounds
ñò óôõö$÷õ øõ ùùú ô
;=
>� ?�@�AB� CD� B�� @AE� �� B
���FG3 H7HI��� FH6 2�! 3 7#"$$ "%
J KL M�!NM OG H6� ë/6/P/ �� / 7KQ1RG H6� 3 HëI/ �&ST �H6 * +-� T H6 2T ëU
��3 H�I V H6 2� ë/G3G H�H6 -' 7/6 2H7 HI/R6� R6( � / ë
�/R6� R6I' K +M( � 3 HëI/ �Q RG H6I/ �&ST �H6 * +WYXZ
[]\_^`a
' /R6� R6 &ST �H6 cb/R6� R6Q1R3 �H6 (
d 03 7H7P ëT3IT6 2T&ST �H6 3IP �/IT6/ *3I� /II HG �T� I3P ëH&P H7
�/ 3 ëH6 2FHP7R FGT ë/ �Q RG H6� 3 HëI3 7SRI H7 �/ 3 ëH6 2FHPG H6
/ �P/ FFR 7HP36 H/ 7
ef3 H�I V H6 2�$ë/G3G H�H6 - g
hi jklm!nl ol ppq k
rs
D �tA@���BD u� ��Dvt�
���FG3 H7HI��� FH6 2�! 3 7#"$$ "%
NKQ RG H6� 3 HëI H7I636Tw / �&ST �H6 *,+-� I g6 g3 7 �$ë/6/P/ �63 5H7
w 3I H7� R63 7S/R6� R66 H7 xQ1RG H6� 3 HëI� 23I3 T ë3yT&ST �H6 36
F/I6 *,+\_^W Xz
\_^W X{ - |*,+- g
03 77/6 H7P ëT3IT6 2T/ T ë3 ��&ST �H6
hi jklm!nl ol ppq k
r }
~ � �D t���� �A@� B�� @AE� �� B
���FG3 H7HI��� FH6 2�! 3 7#"$$ "%
J KL M�!NM OG H6� ë/6/P/ �� / 7KQ1RG H6� 3 HëI/ �&ST �H6 *,�-� T H6 2T ëU
��3 H�I V H6 2� ë/G3G H�H6 -' 7/6 2H7 HI/R6� R6( � / ë
�/R6� R6I' K �M( � 3 HëI/ �Q RG H6I/ �&ST �H6 *,�W�XZ
[]\_^`a
�� 6 HF3 �ggg
�f3 H�R ëT�ë/G3G H�H6 ���RI6 �3 H�V H6 2� ë/G3G H�H6� - H7/ ëST ë6/
3P 2HT �TP �/IT O6/ O/ 7T��RP 5
� &ST �H6���
� HT �S �' K �M( Q1RG H6� 3 HëI� 3I� F� 6/6 HP3 ��� /� 6 HF3 �
hi jklm!nl ol ppq k
r �
� @�
~ � �D t���
���FG3 H7HI��� FH6 2�! 3 7�#"$$ "%
J KL M�!NM OG H6� ë/6/P/ �� / 7KQ1RG H6� 3 HëI/ �&ST �H6� *,�-� T H6 2T ëU
��3 H�I V H6 2� ë/G3G H�H6� -' 7/6 2H7� HI/R6� R6( � / ë
�/R6� R6I' K �M( � 3 HëI/ �Q RG H6I/ �&ST �H6� *,�W�XZ
[]\_^`a
���FG3 H7HI�! 3 7� "$$ "% �
0/ FFR 7HP36 H/ 7P/ F� �T �H6� /� 6 HF3 �R� 6/3 73SS H6 H�TP/ 7I63 76
hi jklm!nl ol ppq k
r�
��� @��E �B�ED tt�����
�73 �� I HI/ �� T 7T ë3 �6 V/ O� 3 ë6� � ë/6/P/ �I�$ëH/ ë6/���FG3 H7HI�! 3 7�#"$$ "%
���HT �IT 7*444% �� HFR �36 H/ 7 OG3ITS �TSRP6 H/ 7 �
�f/ ë� R ëTI636T H7� R6� ��HPTP3 7�I HFR �36T �
8/G��I3P6 H/ 7I
��ëG H6 ë3 ë� � ë/6/P/ ��I H7� �T OFTII3�yT�$ë/6/P/ �
' ��HPT FT3IR ëTI� ��HPTIT 7SI FTII3�yT6/ 8/G� 8/G FT3IR ëTI(
hi jklm!nl ol ppq k
r �
�AtD� �� B��vD�����AE� �� B
�ëTSRP H7� 3 7� �$ë/6/P/ �6/3I H7� �T OFTII3� T� ë/6/P/ �I �
��/TI 7/6 V/ ë5�/ ë�$ë/6/P/ �I V H6 2FH�TSI636TI3I H7� R6
���8T 77T66� �H H7PT 7¡/� � F/ �H7�£¢//66T ëI *44¤%
¥V/ OV3� � ë/6/P/ �I F/ ëT� / VT ë�R �6 23 7/ 7T OV3� � ë/6/P/ �I
��TSRP6 H/ 7S/TI 7 �6 V/ ë5¦
�� 6 2T ë6TP 27HQ1RTIS/ 7/6ITT F6/ V/ ë5V H6 2FH�TSI636TI
T H6 2T ë' T g���§3� ST 7� ¢ H76T ë"$$ "%(
hi jklm!nl ol ppq k
r¨
A@� B� @©vA� ©� B
���FG3 H7HI�! 3 7� "$$ "%
�/ �T �6TP 27HQ RT �/ ëFH�TSI636TI3 7S6 V/ OV3� � ë/6/P/ �I
�ªTT� 6 ë3P 5/ �6 2T �/P3 �ST 7I H6� F36 ëHPTI/ ���HPT3 7S 8/G
�0/ FFR 7HP36 H/ 7P3RITI3ST 7I H6� F36 ëH�6/ �I� �H6 �
��3 H763 H73 7H7�3 ëH3 76 V H6 2P/ FFR 7HP36 H/ 72HI6/ ë�
hi jklm!nl ol ppq k
r;
U U U U U U U U
L L L L L Lbounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
LL L
UU
01
manyU U
quantum classical
communication
U =
L =
=
upper boundlow
er bound
=
independent result
L
my orignal result
matching low
er/upper bounds
hi jklm!nl ol ppq k
rr
U U U U U U U U
L L L L L Lbounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
LL L
UU
01
manyU U
quantum classical
communication
U =
L =
=
upper boundlow
er bound
=
independent result
L
my orignal result
One−
bit protocol provablybetter than non−
interactive protocols
non−interactive entanglem
ent distillation
hi jklm!nl ol ppq k
r<
«¬
hi jklm!nl ol ppq k
r=
}s bits
noisy channel
local operation
local operation
noisy channelE
D~c
c
m
m
Quantum
Error C
orrecting Code
Error C
orrecting Code
Entanglem
ent Distillation P
rotocol
Correlation D
istillation Protocol
classical
quantum
s
statusw
ell−studied, w
ell−understood
less studied, fewer results
overheadc
m
preven
tiverep
arative
| | − | |
less studied, fewer results
hi jklm!nl ol ppq k
<s
U U U U U U U U
L L L L L Lbounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
LL L
UU
01
manyU U
quantum classical
communication
U =
L =
=
upper boundlow
er bound
=
independent result
L
my orignal result
hi jklm!nl ol ppq k
< }
U U U U U U U U
L L L L L Lbounded corruption
binary symm
etric
binary erasure
tensor product
bounded corruption
bounded measurem
ent
depolarization
entanglement
fidelity
noise model
LL L
UU
01
manyU U
quantum classical
communication
U =
L =
=
upper boundlow
er bound
=
independent result
L
my orignal result
empty =
unknown result
optimality?
comm
unication−qualiity
tradeoff?
hi jklm!nl ol ppq k
< �
©A��� ©� B�
��� 6 HF3 �H6� / �P/ 7I6 ëRP6 H/ 7I
�.H7T3 ë®00¯ 0��� �63G H�H¡T ë° ®00¯®��� 3 ëT6 2T�
/� 6 HF3 �± �
��/ ëT¥ë3ST O/ ²/ 7H76T ë3P6 H�TP/ ëëT �36 H/ 7S HI6 H��36 H/ 7
�¢236 �I6 2T/� 6 HF3 �Q1R3 �H6� ��HPT3 7S 8/GP3 7�yT6 V H6 2M
G H6I/ �P/ FFR 7HP36 H/ 7± �
�³ 7H&TS ëTIR �6I
��ëT6 2T ëT7/ HIT F/ST �I F/ ëT�yT 7T ë3 �6 23 7� I3� � 6 2T&ST �H6�
F/ST �± �
�03 7VT FT ë�yT6 2T ëTIR �6I6/ F3 5T6 2T63G �TI F3 ��T ë± �
hi jklm!nl ol ppq k
<�
´ µ¶ ·¸¸¶¹º»¼¶½¶¾¼º´ ¿¾ À´ ¿¶ÂÁà º¼Äµ´¼´Å´ ƾ
�03 7VTR�� T ëG/R 7S6 2TQ1R3 �H6� / �/ 7T OG H60��Ç ®��I ±
�ÈI6 2T�$ë/6/P/ �V H6 26 2TG H73 ë� I� FFT6 ëHP F/ST �/� 6 HF3 �±
hi jklm!nl ol ppq k
< �
©� �t©B�
� "$$ÉÇ É �"$$ :Ç É% 0/ 76 H7RT ëTIT3 ëP 2
� "$$ :Ç : �"$$ :Ç 4% ¢ëH6T6 2TI HI
hi jklm!nl ol ppq k
<¨
D~~�D t�BÊ ©B�
Ë�D��Ì
hi jklm!nl ol ppq k
<;