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.

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. Our
goal:
to
“prove”
the
existence
of
randomness
in
nature.

3. DefiniDon
of
randomness
bi
Observer

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. 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. No
randomness
from
scratch
The
generaDon
of
randomness
from
scratch
is
impossible!

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. 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. ¿Does
randomness
exist
in
classical
physics?

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. 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. 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. 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. 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. ¿What
happens
when
we
move
to
the
quantum
world?

16. Quantum
randomness
Textbook:
the
outputs
of
a
quantum
measurement
are
random.

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. 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. 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. Can
the
presence
of
randomness
be
guaranteed
by
any
physical
mechanism?

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. 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. 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. 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. 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. 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. 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. 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. 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. Let’s
then
move
to
more
than
one
system…

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. A
crash
course
on
Bell
inequaliDes

33. Example:
CHSH
Bell
inequality
CHSH = A1B1 + A1B2 + A2B1 − A2B2
+1 -1 +1 -1
1 2 1 2
Source

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. NIST
Randomness
Beacon
Bell
cer5fied
randomess!

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.