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ICREA	
  Colloquium,	
  Barcelona,	
  27	
  September	
  2016	
  
Antonio	
  Acín	
  
ICREA	
  Professor	
  at	
  ICFO-­‐I...
Our	
  goal:	
  to	
  “prove”	
  the	
  existence	
  
of	
  randomness	
  in	
  nature.	
  
DefiniDon	
  of	
  randomness	
  
bi
Observer	
  
DefiniDon	
  of	
  randomness	
  
bi
Observer	
   Eve	
  
A	
  process	
  is	
  (perfectly)	
  random	
  if	
  it	
  is	
  ...
DefiniDon	
  of	
  randomness	
  
bi
Observer	
   Eve	
  
A	
  process	
  is	
  (perfectly)	
  random	
  if	
  it	
  is	
  ...
No	
  randomness	
  from	
  scratch	
  
The	
  generaDon	
  of	
  randomness	
  from	
  scratch	
  is	
  impossible!	
  
No	
  randomness	
  from	
  scratch	
  
The	
  generaDon	
  of	
  randomness	
  from	
  scratch	
  is	
  impossible!	
  
T...
CerDfiable	
  physical	
  randomness	
  
Our	
  working	
  assumpDon	
  is	
  that	
  processes	
  are	
  physical	
  
and	...
¿Does	
  randomness	
  exist	
  in	
  
classical	
  physics?	
  
Randomness	
  in	
  classical	
  physics	
  
In	
  the	
  macroscopic	
  world,	
  there	
  is	
  no	
  such	
  thing	
  a...
Randomness	
  in	
  classical	
  physics	
  
In	
  the	
  macroscopic	
  world,	
  there	
  is	
  no	
  such	
  thing	
  a...
Randomness	
  is,	
  thus,	
  a	
  simple	
  consequence	
  of	
  our	
  limitaDons,	
  for	
  instance	
  
in	
  our	
  o...
Randomness	
  is,	
  thus,	
  a	
  simple	
  consequence	
  of	
  our	
  limitaDons,	
  for	
  instance	
  
in	
  our	
  o...
Randomness	
  is,	
  thus,	
  a	
  simple	
  consequence	
  of	
  our	
  limitaDons,	
  for	
  instance	
  
in	
  our	
  o...
¿What	
  happens	
  when	
  we	
  
move	
  to	
  the	
  quantum	
  world?	
  
Quantum	
  randomness	
  
Textbook:	
  the	
  outputs	
  of	
  a	
  quantum	
  measurement	
  are	
  random.	
  
Quantum	
  randomness	
  
Textbook:	
  the	
  outputs	
  of	
  a	
  quantum	
  measurement	
  are	
  random.	
  
50%
50%
T...
Quantum	
  randomness	
  
Textbook:	
  the	
  outputs	
  of	
  a	
  quantum	
  measurement	
  are	
  random.	
  
50%
50%
T...
Quantum	
  randomness	
  
Textbook:	
  the	
  outputs	
  of	
  a	
  quantum	
  measurement	
  are	
  random.	
  
50%
50%
T...
Can	
  the	
  presence	
  of	
  randomness	
  be	
  guaranteed	
  
by	
  any	
  physical	
  mechanism?	
  
Known	
  soluDons	
  
•  Classical	
  Random	
  Number	
  Generators	
  (CRNG).	
  All	
  of	
  them	
  are	
  of	
  
dete...
Problem	
  1:	
  cerDficaDon	
  
•  Good	
  randomness	
  is	
  usually	
  verified	
  by	
  a	
  series	
  of	
  staDsDcal	...
RANDU	
  
RANDU	
  is	
  an	
  infamous	
  linear	
  congruenDal	
  pseudorandom	
  number	
  generator	
  
of	
  the	
  P...
Problem	
  2:	
  privacy	
  
50%
50%
T
R
1r
2r
nr
.
.
.
Classical	
  
Memory	
  
The	
  provider	
  has	
  access	
  to	
 ...
50%
50%
T
R
1r
2r
nr
.
.
.
Classical	
  
Memory	
   1r 2r nr…	
  
The	
  provider	
  has	
  access	
  to	
  a	
  proper	
 ...
Problem	
  3:	
  device	
  dependence	
  
All	
  the	
  soluDons	
  rely	
  on	
  the	
  details	
  of	
  the	
  devices	
...
No	
  randomness	
  for	
  single	
  systems	
  
It	
  is	
  impossible	
  to	
  cerDfy	
  that	
  the	
  outcomes	
  prod...
No	
  randomness	
  for	
  single	
  systems	
  
It	
  is	
  impossible	
  to	
  cerDfy	
  that	
  the	
  outcomes	
  prod...
No	
  randomness	
  for	
  single	
  systems	
  
It	
  is	
  impossible	
  to	
  cerDfy	
  that	
  the	
  outcomes	
  prod...
Let’s	
  then	
  move	
  to	
  more	
  
than	
  one	
  system…	
  
CerDfied	
  randomness	
  
y
a b
x
P(a,b x, y)
e=a?
z
Eve	
  
Observer	
  
The	
  observer	
  can	
  now	
  observe	
  the	...
A	
  crash	
  course	
  on	
  Bell	
  inequaliDes	
  
Example:	
  CHSH	
  Bell	
  inequality	
  
CHSH = A1B1 + A1B2 + A2B1 − A2B2
+1 -1 +1 -1
1 2 1 2
Source	
  
Example:	
  CHSH	
  Bell	
  inequality	
  
CHSH = A1B1 + A1B2 + A2B1 − A2B2
+1 -1 +1 -1
1 2 1 2
In	
  classical	
  physics...
Quantum	
  Bell	
  inequality	
  violaDon	
  
A2
A1
B1
B2
Φ =
1
2
00 + 11( )
Classical	
  values	
  are	
  now	
  replaced...
Quantum	
  Bell	
  inequality	
  violaDon	
  
CHSH = A1B1 Φ
+ A1B2 Φ
+ A2B1 Φ
− A2B2 Φ
= 2 2 > 2
A2
A1
B1
B2
Φ =
1
2
00 + ...
Quantum	
  non-­‐locality	
  
•  Bell	
  inequaliDes	
  are	
  condiDons	
  saDsfied	
  by	
  classical	
  models	
  in	
  ...
CerDfied	
  randomness	
  
y
a b
x
P(a,b x, y)
e=a?
z
Eve	
  
Observer	
  
Ask	
  the	
  provider	
  not	
  one	
  but	
  t...
CerDfied	
  randomness	
  
y
a b
x
P(a,b x, y)
e=a?
z
Eve	
  
Observer	
  
Ask	
  the	
  provider	
  not	
  one	
  but	
  t...
CerDfied	
  randomness	
  
The	
  randomness	
  in	
  the	
  outputs	
  can	
  be	
  esDmated	
  from	
  the	
  amount	
  o...
CerDfied	
  randomness	
  
The	
  randomness	
  in	
  the	
  outputs	
  can	
  be	
  esDmated	
  from	
  the	
  amount	
  o...
What	
  did	
  we	
  use?	
  
•  We	
  assume	
  the	
  validity	
  of	
  the	
  whole	
  quantum	
  formalism.	
  
•  We	...
What	
  did	
  we	
  use?	
  
•  We	
  assume	
  the	
  validity	
  of	
  the	
  whole	
  quantum	
  formalism.	
  
•  We	...
What	
  are	
  two	
  systems?	
  
•  All	
  this	
  is	
  related	
  to	
  the	
  noDon	
  of	
  causality	
  and	
  spac...
Randomness	
  expansion	
  
a b
x y
Source	
  
Perfect	
  random	
  bits	
  are	
  available	
  to	
  choose	
  the	
  inp...
Randomness	
  amplificaDon	
  
k
Santha-­‐Vazirani	
  source:	
  a	
  device	
  that	
  produces	
  bits	
  with	
  the	
  ...
Randomness	
  amplificaDon	
  
k
Santha-­‐Vazirani	
  source:	
  a	
  device	
  that	
  produces	
  bits	
  with	
  the	
  ...
Randomness	
  amplificaDon	
  
k
Santha-­‐Vazirani	
  source:	
  a	
  device	
  that	
  produces	
  bits	
  with	
  the	
  ...
Randomness	
  amplificaDon	
  
0	
   1/2	
  
εi
εf
Randomness	
  amplificaDon	
  is	
  possible	
  using	
  quantum	
  non-­...
Experimental	
  realizaDon	
  
• 	
  The	
  two-­‐box	
  scenario	
  is	
  performed	
  by	
  two	
  atomic	
  parDcles	
 ...
NIST	
  Randomness	
  Beacon	
  
NIST	
  is	
  implemenDng	
  a	
  source	
  of	
  public	
  randomness.	
  The	
  service...
NIST	
  Randomness	
  Beacon	
  
Bell	
  cer5fied	
  randomess!	
  
Conclusions	
  
•  Randomness	
  can	
  be	
  cerDfied	
  using	
  the	
  non-­‐local	
  correlaDons	
  observed	
  when	
 ...
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72nd ICREA Colloquium "What can and cannot be said about randomness using quantum physics" by Antonio Acín

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What can and cannot be said about randomness using quantum physics
It is usually said that quantum physics is, contrary to classical physics, intrinsically random. The intrinsic randomness of quantum physics follows from the fact that it is possible to observe correlations among quantum particles for which there exists no classical and deterministic model. The observation of these correlations, however, requires some assumptions about the setup. In particular, it requires some initial randomness, which makes the whole argument apparently circular.
We discuss how it is possible to relax this circularity and conclude that an intrinsic form of randomness with no classical analogue does exist in the quantum world.

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72nd ICREA Colloquium "What can and cannot be said about randomness using quantum physics" by Antonio Acín

  1. 1. ICREA  Colloquium,  Barcelona,  27  September  2016   Antonio  Acín   ICREA  Professor  at  ICFO-­‐InsDtut  de  Ciencies  Fotoniques,  Barcelona   What  can  and  cannot  be  said  about   randomness  in  quantum  physics  
  2. 2. Our  goal:  to  “prove”  the  existence   of  randomness  in  nature.  
  3. 3. DefiniDon  of  randomness   bi Observer  
  4. 4. DefiniDon  of  randomness   bi Observer   Eve   A  process  is  (perfectly)  random  if  it  is  unpredictable,  not  only  to  the  observer,  but  to   any  observer,  called  Eve  in  what  follows  and  possibly  correlated  to  the  process.  
  5. 5. DefiniDon  of  randomness   bi Observer   Eve   A  process  is  (perfectly)  random  if  it  is  unpredictable,  not  only  to  the  observer,  but  to   any  observer,  called  Eve  in  what  follows  and  possibly  correlated  to  the  process.   This  definiDon  is  saDsfactory  both  from  a  fundamental  and  applied  perspecDve.   •  From  a  fundamental  perspecDve  it  is  difficult  to  argue  that  a  process  is  random   if  there  could  exist  an  observer  able  to  predict  its  outcomes.     •  PracDcally,  by  demanding  that  the  results  should  look  random  to  any  observer,   the  generated  randomness  is  guaranteed  to  be  private.  
  6. 6. No  randomness  from  scratch   The  generaDon  of  randomness  from  scratch  is  impossible!  
  7. 7. No  randomness  from  scratch   The  generaDon  of  randomness  from  scratch  is  impossible!   This  follows  from  the  non-­‐falsifiable  hypothesis  of  the  existence  of  a  super-­‐ determinisDc  model  in  which  everything,  including  all  the  history  of  our  universe,   was  pre-­‐determined  in  advance  and  known  by  the  external  observer.     Any  protocol  for  randomness  genera5on  must  be  based  on  assump5ons.    
  8. 8. CerDfiable  physical  randomness   Our  working  assumpDon  is  that  processes  are  physical   and  therefore  obey  the  laws  of  physics.     The  random  numbers  should  be  unpredictable  to  any   physical  observer,  that  is,  any  observer  whose  acDons   are  constrained  by  the  laws  of  physics.  
  9. 9. ¿Does  randomness  exist  in   classical  physics?  
  10. 10. Randomness  in  classical  physics   In  the  macroscopic  world,  there  is  no  such  thing  as  true  randomness.  Any  random   process  is  simply  a  consequence  of:    1)  ImperfecDons  in  the  preparaDon  of  the  system  and/or    2)  ParDal  knowledge  
  11. 11. Randomness  in  classical  physics   In  the  macroscopic  world,  there  is  no  such  thing  as  true  randomness.  Any  random   process  is  simply  a  consequence  of:    1)  ImperfecDons  in  the  preparaDon  of  the  system  and/or    2)  ParDal  knowledge   Example:   One  can  never  exclude  the   existence  of  an  observer  with   perfect  knowledge  of  the  iniDal   posiDon  and  speed  of  the  ball   and  the  size  and  shape  of  the   roule^e,  who  can  predict  the   result  with  certainty.  
  12. 12. Randomness  is,  thus,  a  simple  consequence  of  our  limitaDons,  for  instance   in  our  observaDon  and  computaDonal  capabiliDes,  informaDon  storage  and   the  preparaDon  of  the  systems.   Randomness  in  classical  physics  
  13. 13. Randomness  is,  thus,  a  simple  consequence  of  our  limitaDons,  for  instance   in  our  observaDon  and  computaDonal  capabiliDes,  informaDon  storage  and   the  preparaDon  of  the  systems.   However,  the  theory  does  not  incorporate  any  form  of  intrinsic  randomness.   Given  a  perfect  knowledge  of  the  iniDal  condiDons  in  a  system,  it  is  in   principle  possible  to  predict  its  future  (and  past)  behaviour.   Randomness  in  classical  physics  
  14. 14. Randomness  is,  thus,  a  simple  consequence  of  our  limitaDons,  for  instance   in  our  observaDon  and  computaDonal  capabiliDes,  informaDon  storage  and   the  preparaDon  of  the  systems.   However,  the  theory  does  not  incorporate  any  form  of  intrinsic  randomness.   Given  a  perfect  knowledge  of  the  iniDal  condiDons  in  a  system,  it  is  in   principle  possible  to  predict  its  future  (and  past)  behaviour.   LAPLACE   We  may  regard  the  present  state  of  the  universe  as  the  effect  of  its   past  and  the  cause  of  its  future.  An  intellect  which  at  a  certain   moment  would  know  all  forces  that  set  nature  in  moDon,  and  all   posiDons  of  all  items  of  which  nature  is  composed,  if  this  intellect   were  also  vast  enough  to  submit  these  data  to  analysis,  it  would   embrace  in  a  single  formula  the  movements  of  the  greatest  bodies   of  the  universe  and  those  of  the  Dniest  atom;  for  such  an  intellect   nothing  would  be  uncertain  and  the  future  just  like  the  past  would   be  present  before  its  eyes.     Randomness  in  classical  physics  
  15. 15. ¿What  happens  when  we   move  to  the  quantum  world?  
  16. 16. Quantum  randomness   Textbook:  the  outputs  of  a  quantum  measurement  are  random.  
  17. 17. Quantum  randomness   Textbook:  the  outputs  of  a  quantum  measurement  are  random.   50% 50% T R The  outputs  of  this  experiment  are   random  because:   1.  Devices  are  quantum…  
  18. 18. Quantum  randomness   Textbook:  the  outputs  of  a  quantum  measurement  are  random.   50% 50% T R The  output  randomnesss  does  not  rely  only  on  the  “quantumness”  of  the  process.   The  outputs  of  this  experiment  are   random  because:   1.  Devices  are  quantum…  but  also   2.  It  is  a  pure  single-­‐photon  state;   3.  The  transmission  coefficient  of   the  mirror  is  exactly  ½;   4.  The  detectors  do  not  have   memory  effects;   5.  …  
  19. 19. Quantum  randomness   Textbook:  the  outputs  of  a  quantum  measurement  are  random.   50% 50% T R The  output  randomnesss  does  not  rely  only  on  the  “quantumness”  of  the  process.   The  outputs  of  this  experiment  are   random  because:   1.  Devices  are  quantum…  but  also   2.  It  is  a  pure  single-­‐photon  state;   3.  The  transmission  coefficient  of   the  mirror  is  exactly  ½;   4.  The  detectors  do  not  have   memory  effects;   5.  …   It  is  unsaDsfactory  that  the  random  character  of  the  process  relies  on  our   knowledge  of  it.  How  can  we  know  if  our  descripDon  is  correct?  If  not  correct,  is   the  observed  randomness  again  an  arDfact  of  ignorance?  
  20. 20. Can  the  presence  of  randomness  be  guaranteed   by  any  physical  mechanism?  
  21. 21. Known  soluDons   •  Classical  Random  Number  Generators  (CRNG).  All  of  them  are  of   determinisDc  Nature.   •  Quantum  Random  Number  Generators  (QRNG).  There  exist  different   soluDons,  but  the  main  idea  is  encapsulated  by  the  following  example:   •  In  any  case,  all  these  soluDons  have  three  problems,  which  are  important   both  from  a  fundamental  and  pracDcal  point  of  view.   50% 50% T R Single  photons  are  prepared  and   sent  into  a  mirror  with   transmilvity  equal  to  ½.  The   random  numbers  are  provided  by   the  clicks  in  the  detectors.  
  22. 22. Problem  1:  cerDficaDon   •  Good  randomness  is  usually  verified  by  a  series  of  staDsDcal  tests.   •  There  exist  chaoDc  systems,  of  determinisDc  nature,  that  pass  all   exisDng  randomness  tests.   •  Do  these  tests  really  cerDfy  the  presence  of  randomness?  It  is  well   known  that  no  finite  set  of  tests  can  do  it.   •  Do  these  tests  cerDfy  any  form  of  quantum  randomness?  Classical   systems  pass  them!  
  23. 23. RANDU   RANDU  is  an  infamous  linear  congruenDal  pseudorandom  number  generator   of  the  Park–Miller  type,  which  has  been  used  since  the  1960s.   Three-­‐dimensional  plot  of  100,000  values  generated  with  RANDU.  Each   point  represents  3  subsequent  pseudorandom  values.  It  is  clearly  seen   that  the  points  fall  in  15  two-­‐dimensional  planes.  
  24. 24. Problem  2:  privacy   50% 50% T R 1r 2r nr . . . Classical   Memory   The  provider  has  access  to  a  proper  RNG.  The  provider  uses  it  to  generate  a   long  sequence  of  good  random  numbers,  stores  them  into  a  memory  sDck   and  sells  it  as  a  proper  RNG  to  the  user.    
  25. 25. 50% 50% T R 1r 2r nr . . . Classical   Memory   1r 2r nr…   The  provider  has  access  to  a  proper  RNG.  The  provider  uses  it  to  generate  a   long  sequence  of  good  random  numbers,  stores  them  into  a  memory  sDck   and  sells  it  as  a  proper  RNG  to  the  user.       The  numbers  generated  by  the  user  look  random.    However,  they  can  be   perfectly  predicted  by  the  adversary.  How  can  one  be  sure  that  the  observed   random  numbers  are  also  random  to  any  other  observer,  possibly  adversarial?   Problem  2:  privacy  
  26. 26. Problem  3:  device  dependence   All  the  soluDons  rely  on  the  details  of  the  devices  used  in  the  generaDon.     How  can  imperfecDons  in  the  devices  affect  the  quality  of  the  generated   numbers?  Can  these  imperfecDons  be  exploited  by  an  adversary?   50% 50% T R Single  photons  are  prepared  and   sent  into  a  mirror  with   transmilvity  equal  to  ½.  The   random  numbers  are  provided  by   the  clicks  in  the  detectors.  
  27. 27. No  randomness  for  single  systems   It  is  impossible  to  cerDfy  that  the  outcomes  produced  by  a  single  system  are   random  without  making  assumpDons  about  its  internal  working.  
  28. 28. No  randomness  for  single  systems   It  is  impossible  to  cerDfy  that  the  outcomes  produced  by  a  single  system  are   random  without  making  assumpDons  about  its  internal  working.   This  follows  form  the  simple  fact  that  any  observed  probability  distribuDon   can  be  wri^en  in  terms  of  determinisDc  assignments:   P b =1,b = 2,…,b = r( )= p b = i( ) i=1 r ∑ δb.i The  observed  randomness  is  just  a  consequence  of  the  ignorance  of:   p b = i( )
  29. 29. No  randomness  for  single  systems   It  is  impossible  to  cerDfy  that  the  outcomes  produced  by  a  single  system  are   random  without  making  assumpDons  about  its  internal  working.   This  follows  form  the  simple  fact  that  any  observed  probability  distribuDon   can  be  wri^en  in  terms  of  determinisDc  assignments:   P b =1,b = 2,…,b = r( )= p b = i( ) i=1 r ∑ δb.i The  observed  randomness  is  just  a  consequence  of  the  ignorance  of:   p b = i( ) Any  staDsDcs  obtained  by  measuring  a  quantum  systems  can  be  simulated  classically.  
  30. 30. Let’s  then  move  to  more   than  one  system…  
  31. 31. CerDfied  randomness   y a b x P(a,b x, y) e=a? z Eve   Observer   The  observer  can  now  observe  the  correlaDons  between  the  two  systems.     From  the  point  of  view  of  correlaDons,  classical  and  quantum  physics  differ!  
  32. 32. A  crash  course  on  Bell  inequaliDes  
  33. 33. Example:  CHSH  Bell  inequality   CHSH = A1B1 + A1B2 + A2B1 − A2B2 +1 -1 +1 -1 1 2 1 2 Source  
  34. 34. Example:  CHSH  Bell  inequality   CHSH = A1B1 + A1B2 + A2B1 − A2B2 +1 -1 +1 -1 1 2 1 2 In  classical  physics,  observables  have  well-­‐defined  values,  now  +1  or  -­‐1.       Under  this  assumpDon:     Example:     So,  the  expectaDon  value  of  this  quanDty  also  saDsfies   Source   CHSH ≤ 2 A1 = A2 = B1 = B2 = +1⇒ CHSH = +2 CHSH ≤ 2
  35. 35. Quantum  Bell  inequality  violaDon   A2 A1 B1 B2 Φ = 1 2 00 + 11( ) Classical  values  are  now  replaced  by  operators.   -1 +1 -1 1 2 1 2 Source   +1
  36. 36. Quantum  Bell  inequality  violaDon   CHSH = A1B1 Φ + A1B2 Φ + A2B1 Φ − A2B2 Φ = 2 2 > 2 A2 A1 B1 B2 Φ = 1 2 00 + 11( ) Classical  values  are  now  replaced  by  operators.   -1 +1 -1 1 2 1 2 Source   +1
  37. 37. Quantum  non-­‐locality   •  Bell  inequaliDes  are  condiDons  saDsfied  by  classical  models  in  which   measurement  outputs  are  pre-­‐determined.   •  CorrelaDons  observed  when  measuring  entangled  states  may  lead  to  a  violaDon   of  Bell  inequality  and,  therefore,  do  not  have  a  classical  counterpart.  These   correlaDons  are  usually  called  non-­‐local.   •  If  some  observed  correlaDons  violate  a  Bell  inequality,  the  outcomes  could  not   have  pre-­‐determined  in  advance  è  They  are  random.   •  If  some  observed  correlaDons  violate  a  Bell  inequality,  they  cannot  be   reproduced  classically    è  The  devices  are  quantum.  
  38. 38. CerDfied  randomness   y a b x P(a,b x, y) e=a? z Eve   Observer   Ask  the  provider  not  one  but  two  devices.  If  a  Bell  inequality  violaDon  is   observed,  the  outputs  contain  some  randomness.    
  39. 39. CerDfied  randomness   y a b x P(a,b x, y) e=a? z Eve   Observer   Ask  the  provider  not  one  but  two  devices.  If  a  Bell  inequality  violaDon  is   observed,  the  outputs  contain  some  randomness.     The  cerDficaDon  is  device-­‐independent,  in  the  sense  that  it  does  not  rely  on   any  assumpDon  on  the  internal  working  of  the  device.  
  40. 40. CerDfied  randomness   The  randomness  in  the  outputs  can  be  esDmated  from  the  amount  of  observed   Bell  violaDon.  At  no  violaDon,  there  is  no  randomness.  
  41. 41. CerDfied  randomness   The  randomness  in  the  outputs  can  be  esDmated  from  the  amount  of  observed   Bell  violaDon.  At  no  violaDon,  there  is  no  randomness.     This  randomness  is  not  a  consequence  of  ignorance!  This  region  is  impossible   within  quantum  physics.  
  42. 42. What  did  we  use?   •  We  assume  the  validity  of  the  whole  quantum  formalism.   •  We  needed  two  different  systems.  What  does  it  mean?  Do  two  devices  define   two  systems?  Not  if  they  could  be  jointly    prepared  in  advance.   a b
  43. 43. What  did  we  use?   •  We  assume  the  validity  of  the  whole  quantum  formalism.   •  We  needed  two  different  systems.  What  does  it  mean?  Do  two  devices  define   two  systems?  Not  if  they  could  be  jointly    prepared  in  advance.   •  We  need  the  inputs,  processes  that  happen  in  one  locaDon  and  is  not  known   at  the  other  locaDon.  Then,  we  can  idenDfy  two  separate  systems.   a b x y
  44. 44. What  are  two  systems?   •  All  this  is  related  to  the  noDon  of  causality  and  space-­‐Dme.  We  think  of   regions  in  space-­‐Dme  that  are  staDsDcally  meaningful  and  independent  of  the   rest.  We  need  these  noDons  for  making  scienDfic  predicDons!   •  It  may  however  argued  that  assuming  that  something  happens  in  a  region  in   space-­‐Dme  means  that  cannot  be  predicted  by  the  rest  of  observers  in  the   remaining  space-­‐Dme  and,  therefore,  it  is  random.   •  It  adds  a  form  of  circularity  in  the  argument:  randomness  is  needed  to  run  the   Bell  test,  which  is  in  turn  used  to  cerDfy  the  presence  of  randomness.   •  From  a  pracDcal  perspecDve,  or  even  from  a  reasonable  fundamental  point  of   view,  we  can  assume  that  there  are  independent  events  (again,  the  whole   noDon  of  causality  is  based  on  it,  otherwise  everything  would  be  connected).   •  Yet,  it  is  a  logical  possibility  and  a  limitaDon  in  the  proofs  of  randomness.  
  45. 45. Randomness  expansion   a b x y Source   Perfect  random  bits  are  available  to  choose  the  inputs.     One  can  prove  that  one  can  generate  using  the  outputs  more  random  bits  than   are  used  for  the  inputs.     There  even  exist  protocols  for  unbounded  randomness  expansion.   Colbeck,  Kent,  Pironio,  Massar  and  others…  
  46. 46. Randomness  amplificaDon   k Santha-­‐Vazirani  source:  a  device  that  produces  bits  with  the  promise   ε 1 2 −ε ≤ P k = 0 rest( )≤ 1 2 +ε
  47. 47. Randomness  amplificaDon   k Santha-­‐Vazirani  source:  a  device  that  produces  bits  with  the  promise   ε 1 2 −ε ≤ P k = 0 rest( )≤ 1 2 +ε 0   1/2   εi εf Randomness  amplificaDon:  improve  the  randomness  of  the  source.  
  48. 48. Randomness  amplificaDon   k Santha-­‐Vazirani  source:  a  device  that  produces  bits  with  the  promise   ε 1 2 −ε ≤ P k = 0 rest( )≤ 1 2 +ε 0   1/2   εi εf Randomness  amplificaDon:  improve  the  randomness  of  the  source.   Randomness  amplificaDon  is  impossible  classically.  
  49. 49. Randomness  amplificaDon   0   1/2   εi εf Randomness  amplificaDon  is  possible  using  quantum  non-­‐local   correlaDons.  Colbeck  and  Renner     Idea:  use  the  imperfect  source  to  choose  the  inputs  in  a  Bell  test   define  the  final  source  from  the  outputs  of  the  experiment.     Full  randomness  amplificaDon  is  possible:  arbitrarily  weak  random   bits  can  be  mapped  into  arbitrarily  good  random  bits.   Gallego  et  al.  
  50. 50. Experimental  realizaDon   •   The  two-­‐box  scenario  is  performed  by  two  atomic  parDcles  located  in  two  distant   traps.   •   Using  our  theoreDcal  techniques,  we  can  cerDfy  that  42  new  random  bits  are   generated  in  the  experiment.   • It  is  the  first  Dme  that  randomness  generaDon  is  cerDfied  without  making  any   detailed  assumpDon  about  the  internal  working  of  the  devices.   •   Similar  Bell  experiments  with  photons  have  recently  been  performed.  
  51. 51. NIST  Randomness  Beacon   NIST  is  implemenDng  a  source  of  public  randomness.  The  service  (at  h^ps:// beacon.nist.gov/home)  uses  two  independent  commercially  available  sources   of  randomness,  each  with  an  independent  hardware  entropy  source  and  SP   800-­‐90-­‐approved  components.   Commercially  available  physical  sources  of  randomness  are  adequate  as   entropy  sources  for  currently  envisioned  applicaDons  of  the  Beacon.  However,   demonstrably  unpredictable  values  are  not  possible  to  obtain  in  any  classical   physical  context.  Given  this  fact,  our  team  established  a  collaboraDon  with  NIST   physicists  from  the  Physical  Measurement  Laboratory  (PML).  The  aim  is  to  use   quantum  effects  to  generate  a  sequence  of  truly  random  values,  guaranteed  to   be  unpredictable,  even  if  an  a^acker  has  access  to  the  random  source.  In   August  2012,  this  project  was  awarded  a  mulD-­‐year  grant  from  NIST's   InnovaDons  in  Measurement  Science  (IMS)  Program.  
  52. 52. NIST  Randomness  Beacon   Bell  cer5fied  randomess!  
  53. 53. Conclusions   •  Randomness  can  be  cerDfied  using  the  non-­‐local  correlaDons  observed  when   measuring  quantum  states.   •  The  cerDficaDon  is  device-­‐independent:  it  does  not  rely  on  any  assumpDon  on  the   internal  working  of  the  devices.   •  The  argument  requires  two  different  devices.     •  Independent  (random?)  inputs  are  needed  to  define  the  two  different  devices.   This  requirement  may  introduce  some  circularity  in  the  argument.   •  Despite  this  circularity,  using  non-­‐local  quantum  correlaDons  randomness  can  be   arbitrarily  expanded  or  amplified.     •  The  device-­‐independent  approach  can  be  used  to  design  novel  devices  producing   cer5fied  quantum  randomness.  

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