BELL NONLOCALITY
NICOLAS BRUNNER
Nice Nov 2013
1. What is Bell nonlocality?
2. What can you do with quantum nonlocality?
CORRELATIONS
ALICE (Geneva)
BOB (Bristol)
CORRELATIONS
ALICE (Geneva)
BOB (Bristol)
CORRELATED BEHAVIOUR
CORRELATIONS
ALICE (Geneva)
BOB (Bristol)
CORRELATED BEHAVIOUR
HOW DOES IT WORK?
CLASSICAL CORRELATIONS
ALICE (Geneva)
BOB (Bristol)
SIGNAL
CLASSICAL CORRELATIONS
ALICE (Geneva)
BOB (Bristol)
SIGNAL
SPACE-LIKE SEPARATION
NO SIGNAL
CLASSICAL CORRELATIONS
ALICE (Geneva)
BOB (Bristol)
DEVICES HAVE A COMMON STRATEGY
PRE-ESTABLISHED CORRELATIONS
CLASSICAL CORRELATIONS
ALICE (Geneva)
BOB (Bristol)
DEVICES HAVE A COMMON STRATEGY
PRE-ESTABLISHED CORRELATIONS
CAN THIS BE TESTED?
GAME – BELL INEQUALITY
ALICE
BOB
TWO QUESTIONS X0 or X1 (Alice) Y0 or Y1 (Bob)
TWO ANSWERS +1 or -1
Bell 64,Clauser-Horne-Shimony-Holt 69
GAME – BELL INEQUALITY
BOB
ALICE
TWO QUESTIONS X0 or X1 (Alice) Y0 or Y1 (Bob)
TWO ANSWERS +1 or -1
X0
X1
SAME
Y0
OPPOSITE
Y1
Bell 64,Clauser-Horne-Shimony-Holt 69
GAME – BELL INEQUALITY
BOB
ALICE
TWO QUESTIONS X0 or X1 (Alice) Y0 or Y1 (Bob)
TWO ANSWERS +1 or -1
+1 X0
X1
SAME
Y0
OPPOSITE
Y1
Bell 64,Clauser-Horne-Shimony-Holt 69
GAME – BELL INEQUALITY
BOB
ALICE
TWO QUESTIONS X0 or X1 (Alice) Y0 or Y1 (Bob)
TWO ANSWERS +1 or -1
+1 X0
X1
SAME
Y0 +1
OPPOSITE
Y1
Bell 64,Clauser-Horne-Shimony-Holt 69
GAME – BELL INEQUALITY
BOB
ALICE
TWO QUESTIONS X0 or X1 (Alice) Y0 or Y1 (Bob)
TWO ANSWERS +1 or -1
+1 X0
+1 X1
SAME
Y0 +1
OPPOSITE
Y1
Bell 64,Clauser-Horne-Shimony-Holt 69
GAME – BELL INEQUALITY
BOB
ALICE
TWO QUESTIONS X0 or X1 (Alice) Y0 or Y1 (Bob)
TWO ANSWERS +1 or -1
+1 X0
+1 X1
SAME
Y0 +1
OPPOSITE
Y1 +1
Bell 64,Clauser-Horne-Shimony-Holt 69
GAME – BELL INEQUALITY
BOB
ALICE
TWO QUESTIONS X0 or X1 (Alice) Y0 or Y1 (Bob)
TWO ANSWERS +1 or -1
+1 X0
+1 X1
SAME
Y0 +1
OPPOSITE
Y1 +1
Bell 64,Clauser-Horne-Shimony-Holt 69
GAME – BELL INEQUALITY
BOB
ALICE
TWO QUESTIONS X0 or X1 (Alice) Y0 or Y1 (Bob)
TWO ANSWERS +1 or -1
X0
X1
SAME
Y0
OPPOSITE
Y1
Score ≤ ¾ FOR ANY CLASSICAL STRATEGY
Bell 64,Clauser-Horne-Shimony-Holt 69
CHSH BELL INEQUALITY
BOB
ALICE
X0
X1
SAME
OPPOSITE
Y0
Y1
Correlation function: E(X0,Y1) = P(X0=Y1) - P(X0≠Y1)
Clauser-Horne-Shimony-Holt 69
CHSH BELL INEQUALITY
BOB
ALICE
X0
X1
SAME
OPPOSITE
Y0
Y1
Correlation function: E(X0,Y1) = P(X0=Y1) - P(X0≠Y1)
CHSH = E(X0,Y0) + E(X0,Y1) + E(X1,Y0) - E(X1,Y1) ≤ 2
Clauser-Horne-Shimony-Holt 69
LOCAL HIDDEN VARIABLES
x
y
λ
ALICE
a
λ
BOB
b
LOCALITY: P(a,b|x,y) = ∫ dλ P(a|x,λ) P(b|y,λ)
BELL 64
LOCAL HIDDEN VARIABLES
x
y
λ
ALICE
a
λ
BOB
b
LOCALITY: P(a,b|x,y) = ∫ dλ P(a|x,λ) P(b|y,λ)
LOCAL CORRELATIONS SATISFY ALL BELL INEQUALITIES
BELL 64
LOCAL HIDDEN VARIABLES
x
y
λ
ALICE
a
λ
BOB
b
LOCALITY: P(a,b|x,y) = ∫ dλ P(a|x,λ) P(b|y,λ)
LOCAL CORRELATIONS SATISFY ALL BELL INEQUALITIES
VIOLATION OF BELL INEQUALITY
NONLOCALITY
BELL 64
USING QUANTUM RESOURCES
ALICE
|Ψ>
QUANTUM STATEGY
BOB
USING QUANTUM RESOURCES
ALICE
|Ψ>
BOB
QUANTUM STATEGY
1. ENTANGLED STATE |Ψ> = |0,1> - |1,0>
2. LOCAL MEAS X0 = z X1 = x and Y0 = -x-z Y1= x-z
USING QUANTUM RESOURCES
ALICE
|Ψ>
BOB
QUANTUM STATEGY
1. ENTANGLED STATE |Ψ> = |0,1> - |1,0>
2. LOCAL MEAS X0 = z X1 = x and Y0 = -x-z Y1= x-z
X0
E(a,b) = <Ψ| a b |Ψ> = - a b
X1
Y0
Y1
USING QUANTUM RESOURCES
ALICE
|Ψ>
BOB
QUANTUM STATEGY
1. ENTANGLED STATE |Ψ> = |0,1> - |1,0>
2. LOCAL MEAS X0 = z X1 = x and Y0 = -x-z Y1= x-z
X0
E(a,b) = <Ψ| a b |Ψ> = - a b
X1
Y0
Y1
CHSH = E(X0,Y0) + E(X0,Y1) + E(X1,Y0) - E(X1,Y1)
USING QUANTUM RESOURCES
|Ψ>
ALICE
BOB
QUANTUM STATEGY
1. ENTANGLED STATE |Ψ> = |0,1> - |1,0>
2. LOCAL MEAS X0 = z X1 = x and Y0 = -x-z Y1= x-z
X0
E(a,b) = <Ψ| a b |Ψ> = - a b
X1
Y0
Y1
CHSH = E(X0,Y0) + E(X0,Y1) + E(X1,Y0) - E(X1,Y1)
= 1/√2
USING QUANTUM RESOURCES
ALICE
|Ψ>
BOB
QUANTUM STATEGY
1. ENTANGLED STATE |Ψ> = |0,1> - |1,0>
2. LOCAL MEAS X0 = z X1 = x and Y0 = -x-z Y1= x-z
X0
E(a,b) = <Ψ| a b |Ψ> = - a b
X1
Y0
Y1
CHSH = E(X0,Y0) + E(X0,Y1) + E(X1,Y0) - E(X1,Y1)
= 1/√2
USING QUANTUM RESOURCES
ALICE
|Ψ>
BOB
QUANTUM STATEGY
1. ENTANGLED STATE |Ψ> = |0,1> - |1,0>
2. LOCAL MEAS X0 = z X1 = x and Y0 = -x-z Y1= x-z
X0
E(a,b) = <Ψ| a b |Ψ> = - a b
X1
Y0
Y1
CHSH = E(X0,Y0) + E(X0,Y1) + E(X1,Y0) - E(X1,Y1)
= 1/√2
USING QUANTUM RESOURCES
ALICE
|Ψ>
BOB
QUANTUM STATEGY
1. ENTANGLED STATE |Ψ> = |0,1> - |1,0>
2. LOCAL MEAS X0 = z X1 = x and Y0 = -x-z Y1= x-z
X0
E(a,b) = <Ψ| a b |Ψ> = - a b
X1
Y0
Y1
CHSH = E(X0,Y0) + E(X0,Y1) + E(X1,Y0) - E(X1,Y1)
= -1/√2
USING QUANTUM RESOURCES
ALICE
|Ψ>
BOB
QUANTUM STATEGY
1. ENTANGLED STATE |Ψ> = |0,1> - |1,0>
2. LOCAL MEAS X0 = z X1 = x and Y0 = -x-z Y1= x-z
X0
E(a,b) = <Ψ| a b |Ψ> = - a b
X1
Y0
Y1
CHSH = E(X0,Y0) + E(X0,Y1) + E(X1,Y0) - E(X1,Y1) = 2√2 > 2
USING QUANTUM RESOURCES
ALICE
|Ψ>
BOB
QUANTUM STATEGY
1. ENTANGLED STATE |Ψ> = |0,1> - |1,0>
2. LOCAL MEAS X0 = z X1 = x and Y0 = -x-z Y1= x-z
X0
E(a,b) = <Ψ| a b |Ψ> = - a b
X1
Y0
Y1
CHSH = E(X0,Y0) + E(X0,Y1) + E(X1,Y0) - E(X1,Y1) = 2√2 > 2
QUANTUM NONLOCALITY
BELL’S THEOREM
ALICE
|Ψ>
BOB
QUANTUM CORRELATIONS ARE NONLOCAL
STRONGER THAN ANY LOCAL CORRELATIONS
BELL’S THEOREM: PREDICTIONS OF QM ARE
INCOMPATIBLE WITH ANY THEORY
SATISFYING LOCALITY
BEST QUANTUM STRATEGY
ALICE
|Ψ>
BOB
CHSH ≤ 2 + || 2 [X0,X1] [Y0,Y1] || 1/2 ≤ 2√2
TSIRELSON’S BOUND
BEST QUANTUM STRATEGY
|Ψ>
ALICE
BOB
CHSH ≤ 2 + || 2 [X0,X1] [Y0,Y1] || 1/2 ≤ 2√2
TSIRELSON’S BOUND
BELL INEQ VIOLATION
1.  ENTANGLEMENT
2.  INCOMPATIBLE
OBSERVABLES
BEYOND QM
BOB
ALICE
X0
X1
SAME
OPPOSITE
BEST POSSIBLE SCORE ?
Y0
Y1
BEYOND QM
BOB
ALICE
X=0
X=1
SAME
OPPOSITE
Y=0
Y=1
BEST POSSIBLE SCORE ?
CHSH = E(X=Y=0) + E(X=0,Y=1) + E(X=1,Y=0) - E(X=Y=1) = 4
HOW TO REACH THIS ?
PR BOX
y ∈{0,1}
x ∈ {0,1}
a ⊕ b = xy
a ∈ {0,1}
b ∈ {0,1}
Ø  NON-SIGNALING
Ø  MAXIMALLY NONLOCAL CHSH = 4
WHY DOES THE PR BOX NOT EXIST IN NATURE ?
Popescu & Rohrlich, Found. Phys. 1994
Barrett et al. PRA 2005
Popescu-Rohrlich 1994
GHZ PARADOX (1989)
X or Y
3-PARTY ENT: |ΨGHZ> = |0,0,0> + |1,1,1>
|ΨGHZ>
X or Y
X or Y
±1
±1
GHZ PARADOX (1989)
X or Y
3-PARTY ENT: |ΨGHZ> = |0,0,0> + |1,1,1>
X Y Y = -1
Y X Y = -1
Y Y X = -1
|ΨGHZ>
X or Y
X or Y
±1
±1
GHZ PARADOX (1989)
X or Y
3-PARTY ENT: |ΨGHZ> = |0,0,0> + |1,1,1>
X Y Y = -1
Y X Y = -1
Y Y X = -1
|ΨGHZ>
X or Y
X or Y
±1
±1
X X X = -1 CLASSICAL
GHZ PARADOX (1989)
X or Y
3-PARTY ENT: |ΨGHZ> = |0,0,0> + |1,1,1>
X Y Y = -1
Y X Y = -1
Y Y X = -1
|ΨGHZ>
X or Y
X or Y
X X X = -1 CLASSICAL
X X X = +1 QM
±1
±1
GHZ PARADOX (1989)
X or Y
3-PARTY ENT: |ΨGHZ> = |0,0,0> + |1,1,1>
X Y Y = -1
Y X Y = -1
Y Y X = -1
|ΨGHZ>
X or Y
X or Y
X X X = -1 CLASSICAL
X X X = +1 QM
±1
±1
OPPOSITE PREDICTIONS
`BELL’S THM WITHOUT INEQUALITY’
ENTANGLEMENT VS NONLOCALITY
CONCEPTUAL DIFFERENCE
ENTANGLEMENT
NONLOCALITY
CONCEPT OF
QUANTUM MECHANICS
BASED ON STATISTICS
MODEL INDEPENDENT
HOW TO COMPARE THEM?
ENTANGLEMENT = Q NONLOCALITY ?
ENTANGLEMENT
QUANTUM
NONLOCALITY
ENTANGLEMENT = Q NONLOCALITY ?
ENTANGLEMENT
???
QUANTUM
NONLOCALITY
ENTANGLEMENT = Q NONLOCALITY ?
ENTANGLEMENT
???
Q STATE
QUANTUM
NONLOCALITY
Q STATE + MEAS.
ENTANGLEMENT = Q NONLOCALITY ?
ENTANGLEMENT
???
Q STATE
QUANTUM
NONLOCALITY
Q STATE + MEAS.
DO ALL ENTANGLED STATE VIOLATE A BELL INEQUALITY?
PURE STATES
QUANTUM
NONLOCALITY
ENTANGLEMENT
GISIN 1991
2 PARTIES
POPESCU-ROHRLICH 1992 N PARTIES
MIXED STATES
ENTANGLEMENT
QUANTUM
NONLOCALITY
MIXED STATES
ENTANGLEMENT
QUANTUM
NONLOCALITY
THERE EXIST MIXED ENTANGLED STATE
WHICH ARE LOCAL WERNER 1989
MIXED STATES
QUANTUM
NONLOCALITY
ENTANGLEMENT
THERE EXIST MIXED ENTANGLED STATE
WHICH ARE LOCAL WERNER 1989
separable
Werner states
ρ = p |Ψ><Ψ| + (1-p) I/4
entangled
local
0
1/3
nonlocal
~2/3 1/√2
1 p
MIXED STATES
QUANTUM
NONLOCALITY
ENTANGLEMENT
THERE EXIST MIXED ENTANGLED STATE
WHICH ARE LOCAL WERNER 1989
separable
Werner states
ρ = p |Ψ><Ψ| + (1-p) I/4
entangled
local
0
1/3
?
nonlocal
~2/3 1/√2
1 p
MORE GENERAL SCENARIO
ONE COPY
ρ
MULTIPLE COPIES CAN BE PROCESSED JOINTLY
ρM
...
MORE GENERAL SCENARIO
ONE COPY
ρ
MULTIPLE COPIES CAN BE PROCESSED JOINTLY
ρM
...
IS NONLOCALITY ADDITIVE ?
SUPER-ACTIVATION OF NONLOCALITY
LOCAL
ρ
0
NONLOCAL
ρM
...
0+…+0>0
PALAZUELOS PRL 2012
SUPER-ACTIVATION OF NONLOCALITY
LOCAL
ρ
0
NONLOCAL
ρM
...
0+…+0>0
NONLOCALITY IS SUPER-ADDITIVE
ENTANGLED MEASUREMENTS
PALAZUELOS PRL 2012
NONLOCALITY AND TELEPORTATION
USEFUL FOR
TELEPORTATION
NONLOCAL
LARGE CLASS OF ENTANGLED STATES
Super-activation
K-copy nonlocal
separable
Werner states
ρ = p |Ψ><Ψ| + (1-p) I/4
local
0
1/3
nonlocal
~2/3 1/√2
1 p
CAVALCANTI, ACIN, NB, VERTESI PRA 2013
IS ENTANGLEMENT = NONLOCALITY ?
21
MANY POSSIBLE SCENARIOS
a.
c.
x
y
a
b
x
y
b.
y
x
a2
a1
b1
b2
d.
y
x
b
z
k
a
a
b
c
FIG. 5 The nonlocality of a quantum state ρ can be revealed in different scenarios. a) The simplest scenario: Alice and Bob
directly perform local measurements on a single copy of ρ. b) The hidden nonlocality scenario (Popescu, 1995): Alice and Bob
first apply a filtering to the state; upon successful operation of the filter, the perform the local measurements for the Bell test.
c) Many copies scenario: Alice and Bob measure collectively many copies of the state ρ. d) Network scenario: several copies of
IS ENTANGLEMENT = NONLOCALITY ?
PERES CONJECTURE (1999):
BOUND ENTANGLED STATES ARE LOCAL
BIPARTITE CASE ?
DISPROVED IN MULTIPARTITE CASE
NONLOCALITY
WITHOUT DISTILLABILITY
VERTESI & NB PRL 2012
What can you do
with quantum nonlocality?
EKERT’S PROTOCOL 1991
ALICE
|Ψ>
BOB
EKERT’S PROTOCOL 1991
|Ψ>
ALICE
X0
BOB
X2
X1
Y0
Y1
EKERT’S PROTOCOL 1991
|Ψ>
ALICE
X0
BOB
X2
X1
Y0
Y1
1.  CHSH TEST (X0,X1,Y0,Y1)
NONLOCALITY à SECURITY
2. SECRET KEY (X2,Y0) à CORRELATED BITS
USUAL QKD / BB84
|Ψ>
ALICE
BOB
σz
σz
σx
σx
USUAL QKD / BB84
|Ψ>
ALICE
BOB
σz
σz
σx
σx
1.  CORRELATIONS IN X AND Z à ENTANGLEMENT
ENTANGLEMENT MONOGAMY à SECURITY
2. SECRET KEY SAME BASIS à CORRELATED BITS
USUAL QKD / BB84
ALICE
|Ψ>
BOB
1. PARTICLES ARE QUBITS
ASSUMPTIONS
2. MEASUREMENTS ARE PAULI X,Z
USUAL QKD / BB84
ALICE
|Ψ>
BOB
1. PARTICLES ARE QUBITS
ASSUMPTIONS
2. MEASUREMENTS ARE PAULI X,Z
OTHERWISE SECURITY CANNOT BE GUARANTEED
QUANTUM HACKING
”0”
a)
mode
C
”0”
b)
Click!
C
”1”
”1”
ng
Single photon
I0
Ith
VAPD
I1
Ith
as to APD (Vbias )
I0
Ith
t
t
I1
Ith
t
t
T1
Rbias
FIG. 2. How Eve’s trigger pulses are detected by Bob.
Schemes show the last 50/50 coupler (C) and Bob’s detectors in a phase-encoded QKD system. I0 /I1 is the current
running through APD 0/1. a) Eve and Bob have selected
matching bases, and Eve detected the bit value 0. Therefore
the trigger pulse from Eve interferes constructively and its full
intensity hits detector 0. The current caused by Eve’s pulse
crosses the threshold current Ith and causes a click. b) Eve
and Bob have selected opposite bases. The trigger
pulse from
MAKAROV
et al. NAT PHOT 2010
Eve does not interfere constructively and half of its intensity
BLINDING ATTACK
EVE REMOTE CONTROLS BOB’S DETECORS
VHV ≈ 40 V
(c)
. a) In Geiger mode
e breakdown voltage
HACKED MOST USUAL QKD SYSTEMS
QUANTUM HACKING
DEVICE-INDEPENDENT QIP
GOAL: ACHIEVE INFORMATION-THEORETIC TASKS
WITHOUT PLACING ASSUMPTIONS ON
THE FUNCTIONING OF THE DEVICES USED
IN THE PROTOCOL
DEVICE-INDEPENDENT QIP
GOAL: ACHIEVE INFORMATION-THEORETIC TASKS
WITHOUT PLACING ASSUMPTIONS ON
THE FUNCTIONING OF THE DEVICES USED
IN THE PROTOCOL
NO ASSUMPTION ABOUT HILBERT SPACE DIMENSION
OR ALIGNMENT OF MEASUREMENT DEVICES
CERTIFIED RANDOMNESS
x
y
ρ
ALICE
a
BELL INEQ VIOLATION
b
TRULY RANDOM OUTCOMES
PIRONIO et al. NATURE 2010, COLBECK PhD 2007
CERTIFIED RANDOMNESS
x
y
ρ
ALICE
a
BELL INEQ VIOLATION
b
TRULY RANDOM OUTCOMES
OUTCOMES CANNOT BE CORRELATED
TO ANY OTHER PHYSICAL SYSTEM
PIRONIO et al. NATURE 2010, COLBECK PhD 2007
CERTIFIED RANDOMNESS
x
y
ρ
ALICE
a
BELL INEQ VIOLATION
b
TRULY RANDOM OUTCOMES
OUTCOMES CANNOT BE CORRELATED
TO ANY OTHER PHYSICAL SYSTEM
NO-SIGNALING + NONLOCALITY
RANDOMNESS
PIRONIO et al. NATURE 2010, COLBECK PhD 2007
of Alice’s device is now given by the conditional probability distributions p(a|xez). If Eve learns x, then for
any given z her best guess for a corresponds to the
CERTIFIED RANDOMNESS
most probable outcome maximizing p(a|xez). Maximizy a with probabiling over z thus x
means that Eve can guess
ity pguess (RA |EX = x). In the case where a can take on
ρ
two values and Alice and Bob’s
devices are characterized
ALICE
by a CHSH expectation value S it is shown in (Pironio
et al., 2010a) (see also (Masanes et al., 2011) for an alternative derivation),
that independently
a
b of the devices’
behaviours and Eve’s strategy
#
"
1!
pguess (RA |EX = x) ≤
1 + 2 − S 2 /4 .
(59)
2
√
In particular, when S = 2 2, we get as expected
Pguess (A|EX = x) ≤ 1/2 corresponding to 1 bit of minentropy Hmin (RA |EX = x), implying that Alice’s output is fully random. While when the CHSH expectation
achieves the local bound S = 2, we get the trivial bound
pguess (A|EX = x) ≤ 1. Using the SDP hierarchy (Navascues et al., 2007, 2008) (see Section II.C.1.d) it is possible
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CERTIFIED
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ing over
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pguess (RA |EX =
x) ≤
1 + 2 − S 2 /4 .
(59)
pguess (RA |E) := max
p(e|z)
(58)
2 max p(a|e, z) .
z
√a
e
In particular, when S = 2 2, we get as expected
NUMBER OF FINAL RANDOM BITS
Pguess (A|EX = x) ≤ 1/2 corresponding to 1 bit of minThis guessing
alsoimplying
be expressed
as theoutentropy probability
Hmin (RA |EXcan
= x),
that Alice’s
min-entropy
Hminrandom.
(RA |E) While
= − log
(König
A |E)expectation
2 pguess
put is fully
when
the (R
CHSH
et al., 2009).
takes
values
between
0 the
andtrivial
log |Rbound
A |,
achievesItthe
localon
bound
S=
2, we get
corresponding
to the= cases
where
Eve
perfectly,
pguess (A|EX
x) ≤ 1.
Using
thecan
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(NavascuesEve’s
et al., probability
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CERTIFIED RANDOMNESS
1.4
a) SDP
b) Analytical lower−bound
c) No−signalling
Min−entropy bound f(I) (bits)
1.2
1.2
1
1
0.8
a
b
0.6
0.4
c
0.33
0.2
0
2
2.1
2.2
2.3
2.4
2.5
CHSH expectation
2.6
2.7
2.8
Figure 2: Plot of the function f (I). The function f (I) can be interpreted as a bound on the
min-entropy per use of the system for a given CHSH expectation I, in the asymptotic limit of
large n where finite statistics e↵ects (the parameter ✏ in (3)) can be neglected. The function f (I)
EXPERIMENTS
2010 NIST PIRONIO et al. Nature 2010
42 RANDOM BITS (1 MONTH)
EXPERIMENTS
2010 NIST PIRONIO et al. Nature
42 RANDOM BITS (1 MONTH)
2013 ILLINOIS KWIAT
4000 RANDOM BITS (3 HOURS)
DEVICE-INDEPENDENT Q CRYPTOGRAPHY
Alice
Eve
Bob
BELL INEQUALITY VIOLATION
LOCAL OUTCOMES ARE RANDOM
AND UNCORRELATED FROM EVE
ACIN, NB, GISIN, MASSAR, PIRONIO, SCARANI PRL 2007
DEVICE-INDEPENDENT Q CRYPTOGRAPHY
Alice
Eve
Bob
BELL INEQUALITY VIOLATION
LOCAL OUTCOMES ARE RANDOM
AND UNCORRELATED FROM EVE
SECURE EVEN IF EVE PREPARED THE DEVICES
MORE ROBUST TO DEVICE IMPERFECTIONS
ACIN, NB, GISIN, MASSAR, PIRONIO, SCARANI PRL 2007
DEVICE-INDEPENDENT Q CRYPTOGRAPHY
Alice
Eve
Bob
Proposal for a loophole-free Bell test based on spin-photon interactions in c
considers security against collective attacks, the key rate is given by
p
1 + (CHSH/2)2 1
)
KEY RATE R = 1 h(q) h(
2
where h(x) = x log2 x (1 x) log2 (1 x) is the binary entropy. Her
the fraction of the raw key that Alice and Bob must sacrifice in order to c
ERROR
note
that here CORRECTION
the error is given by q = sin2 ↵/(sin2 ↵
¯ + sin2 ↵) ' 4
plot the key rate as a function of the distance. We see that for distances up
PRIVACY
AMPLIFICATION
decent key rates, considering reasonable
parameters
for the reflectivity of t
photo-detector efficiency. We believe that this opens promising perspectives
DI-QKD using the present scheme. A detailed study of the performance o
interface (and its implementation in different physical platforms) would be o
ACIN, NB, GISIN, MASSAR, PIRONIO, SCARANI PRL 2007
REVIEW ARTICLE
NB, Cavalcanti, Pironio, Scarani, Wehner
arXiv:1303.2849
(EPR) STEERING
BACK TO SCHRODINGER (1935)
BY PERFORMING A LOCAL MEASUREMENT
ALICE CAN STEER THE STATE OF BOB
|Ψ> = |0,1> - |1,0>
(EPR) STEERING
BACK TO SCHRODINGER (1935)
BY PERFORMING A LOCAL MEASUREMENT
ALICE CAN STEER THE STATE OF BOB
|Ψ> = |0,1> - |1,0>
σz
0
(EPR) STEERING
BACK TO SCHRODINGER (1935)
BY PERFORMING A LOCAL MEASUREMENT
ALICE CAN STEER THE STATE OF BOB
|Ψ> = |0,1> - |1,0>
σz
0
|1>
(EPR) STEERING
BACK TO SCHRODINGER (1935)
BY PERFORMING A LOCAL MEASUREMENT
ALICE CAN STEER THE STATE OF BOB
|Ψ> = |0,1> - |1,0>
σz
1
|0>
(EPR) STEERING
BACK TO SCHRODINGER (1935)
BY PERFORMING A LOCAL MEASUREMENT
ALICE CAN STEER THE STATE OF BOB
|Ψ> = |0,1> - |1,0>
σx
+x
|-x> = |0>-|1>
(EPR) STEERING
BACK TO SCHRODINGER (1935)
BY PERFORMING A LOCAL MEASUREMENT
ALICE CAN STEER THE STATE OF BOB
|Ψ> = |0,1> - |1,0>
σx
-x
~ REMOTE STATE PREPARATION
|+x> = |0>+|1>
STEERING AS INFORMATION TASK
DISTRIBUTION OF ENTANGLEMENT
FROM AN UNTRUSTED PARTY
A
B
WISEMAN, JONES, DOHERTY PRL 2007
STEERING AS INFORMATION TASK
DISTRIBUTION OF ENTANGLEMENT
FROM AN UNTRUSTED PARTY
B
A
1. A SENDS STATE TO B
2. B CHOOSES MEAS BASIS AND TELLS A
3. A GUESSES OUTCOME OF B
WISEMAN, JONES, DOHERTY PRL 2007
STEERING AS INFORMATION TASK
DISTRIBUTION OF ENTANGLEMENT
FROM AN UNTRUSTED PARTY
B
A
1. A SENDS STATE TO B
2. B CHOOSES MEAS BASIS AND TELLS A
3. A GUESSES OUTCOME OF B
B ESTIMATES CORRELATIONS
WISEMAN, JONES, DOHERTY PRL 2007
STEERING INEQUALITIES
B
A
LOCAL UNCERTAINTY RELATION
H(σx) + H(σz) ≥ 1
STEERING INEQUALITY H(σx|A) + H(σz|A) ≥ 1
HOLDS FOR ANY CHEATING STRATEGY
WITH ENTANGLED STATE
H(σx|A) + H(σz|A) = 0
3 FORMS OF INSEPARABILITY IN QM
ENTANGLEMENT
STEERING
NONLOCALITY
WISEMAN, JONES, DOHERTY PRL 2007
3 DIFFERENT CONCEPTS
DO WE TRUST MEAS. DEVICES OR NOT
ρ
ENTANGLEMENT
< W >ρ ≤ 0 FOR ANY SEPARABLE ρ
TRUST MEAS
3 DIFFERENT CONCEPTS
DO WE TRUST MEAS. DEVICES OR NOT
ρ
ENTANGLEMENT
< W >ρ ≤ 0 FOR ANY SEPARABLE ρ
TRUST MEAS
ρ
NONLOCALITY
BELL ≤ L
FOR ANY LOCAL ρ
DO NOT TRUST MEAS
3 DIFFERENT CONCEPTS
DO WE TRUST MEAS. DEVICES OR NOT
ρ
ENTANGLEMENT
< W >ρ ≤ 0 FOR ANY SEPARABLE ρ
TRUST MEAS
ρ
NONLOCALITY
BELL ≤ L
FOR ANY LOCAL ρ
DO NOT TRUST MEAS
ρ
STEERING
Σ < A σ > ≤ L IF A CHEATS
TRUST B BUT NOT A
END