Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Certified randomness
Marco Vitturi
School of Mathematics
PG Colloquium - October 2015
1 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Of the five sequences that follow, one is entirely predictable, the
others are randomly generated through electromagnetic noise. Can
you tell them apart?
04322064090121485549696874665351513128980999083511
38203756867149242223416591855288485103170971224226
76804671056687013738809537782125837388658730778041
77354137869930169765352621305637958016190771832531
20205690315959428539973816151144999076498629234049
88817922715553418382057863130901864558736093352581
46199157795260719418491995998673283213776396837207
90016145394178294936006671919157552224249424396156
71884010744949348486100419165963971824286076476295
81247992298299083700278456293496030721708535207323
79232002707207098616907996178175604036329039025233
41171924255913471871297541916176010354784586812258
26775875190962385252582863999188623000721625812577
82331436156308865680523426987445954289614391359127
12729622538718604663376624011381088353735595429830
45047771257611326573884655629590416854972221080566
69424704120669452577198919464333178987500119497687
84514657078138915141142666576615135275574650781094
81671978130114647619353593739517907796085328799270
35308165033838729110253111480329696956148424712501
2 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Of the five sequences that follow, one is entirely predictable, the
others are randomly generated through electromagnetic noise. Can
you tell them apart?
04322064090121485549696874665351513128980999083511
38203756867149242223416591855288485103170971224226
76804671056687013738809537782125837388658730778041
77354137869930169765352621305637958016190771832531
20205690315959428539973816151144999076498629234049
88817922715553418382057863130901864558736093352581
46199157795260719418491995998673283213776396837207
90016145394178294936006671919157552224249424396156
71884010744949348486100419165963971824286076476295
81247992298299083700278456293496030721708535207323
79232002707207098616907996178175604036329039025233
41171924255913471871297541916176010354784586812258
26775875190962385252582863999188623000721625812577
82331436156308865680523426987445954289614391359127
12729622538718604663376624011381088353735595429830
45047771257611326573884655629590416854972221080566
69424704120669452577198919464333178987500119497687
84514657078138915141142666576615135275574650781094
81671978130114647619353593739517907796085328799270
35308165033838729110253111480329696956148424712501
3 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Of the five sequences that follow, one is entirely predictable, the
others are randomly generated through electromagnetic noise. Can
you tell them apart?
04322064090121485549696874665351513128980999083511
38203756867149242223416591855288485103170971224226
76804671056687013738809537782125837388658730778041
77354137869930169765352621305637958016190771832531
ζ(3) = 1.20205690315959428539973816151144999076498629234049
88817922715553418382057863130901864558736093352581
46199157795260719418491995998673283213776396837207
90016145394178294936006671919157552224249424396156
71884010744949348486100419165963971824286076476295
81247992298299083700278456293496030721708535207323
79232002707207098616907996178175604036329039025233
41171924255913471871297541916176010354784586812258
26775875190962385252582863999188623000721625812577
82331436156308865680523426987445954289614391359127
12729622538718604663376624011381088353735595429830
45047771257611326573884655629590416854972221080566
69424704120669452577198919464333178987500119497687
84514657078138915141142666576615135275574650781094
81671978130114647619353593739517907796085328799270
35308165033838729110253111480329696956148424712501
4 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
“Quantum mechanics is just probability theory with respect to the
L2 norm instead of L1 ”
5 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Axioms
Non-relativistic quantum mechanics can be axiomatized:
Axiom 1: the state of an isolated system is described by an
element |ψi of a Hilbert space H (with | hψ|ψi |2 = 1)
Axiom 2: observables are self-adjoint operators on H (so you
have the Spectral Thm.)
P
Axiom 3: measuring observable A = n an |φn i hφn |
returns the eigenvalue an with probability | hφn |ψi |2 , and
leaves the system in state |φn i afterwards
Axiom 4: the time-evolution of a state is unitary:
|ψ(t)i = U(t) |ψ(0)i, with U(t) ∈ U(H) (⇒ Schrödinger’s
eqn.)
Axiom 5: the system composed of sub-systems H1 , H2
is described by H1 ⊗ H2
6 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Axioms
Non-relativistic quantum mechanics can be axiomatized:
Axiom 1: the state of an isolated system is described by an
element |ψi of a Hilbert space H (with | hψ|ψi |2 = 1)
Axiom 2: observables are self-adjoint operators on H (so you
have the Spectral Thm.)
P
Axiom 3: measuring observable A = n an |φn i hφn |
returns the eigenvalue an with probability | hφn |ψi |2 , and
leaves the system in state |φn i afterwards
Axiom 4: the time-evolution of a state is unitary:
|ψ(t)i = U(t) |ψ(0)i, with U(t) ∈ U(H) (⇒ Schrödinger’s
eqn.)
Axiom 5: the system composed of sub-systems H1 , H2
is described by H1 ⊗ H2
6 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Axioms
Non-relativistic quantum mechanics can be axiomatized:
Axiom 1: the state of an isolated system is described by an
element |ψi of a Hilbert space H (with | hψ|ψi |2 = 1)
Axiom 2: observables are self-adjoint operators on H (so you
have the Spectral Thm.)
P
Axiom 3: measuring observable A = n an |φn i hφn |
returns the eigenvalue an with probability | hφn |ψi |2 , and
leaves the system in state |φn i afterwards
Axiom 4: the time-evolution of a state is unitary:
|ψ(t)i = U(t) |ψ(0)i, with U(t) ∈ U(H) (⇒ Schrödinger’s
eqn.)
Axiom 5: the system composed of sub-systems H1 , H2
is described by H1 ⊗ H2
6 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Axioms
Non-relativistic quantum mechanics can be axiomatized:
Axiom 1: the state of an isolated system is described by an
element |ψi of a Hilbert space H (with | hψ|ψi |2 = 1)
Axiom 2: observables are self-adjoint operators on H (so you
have the Spectral Thm.)
P
Axiom 3: measuring observable A = n an |φn i hφn |
returns the eigenvalue an with probability | hφn |ψi |2 , and
leaves the system in state |φn i afterwards
Axiom 4: the time-evolution of a state is unitary:
|ψ(t)i = U(t) |ψ(0)i, with U(t) ∈ U(H) (⇒ Schrödinger’s
eqn.)
Axiom 5: the system composed of sub-systems H1 , H2
is described by H1 ⊗ H2
6 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Axioms
Non-relativistic quantum mechanics can be axiomatized:
Axiom 1: the state of an isolated system is described by an
element |ψi of a Hilbert space H (with | hψ|ψi |2 = 1)
Axiom 2: observables are self-adjoint operators on H (so you
have the Spectral Thm.)
P
Axiom 3: measuring observable A = n an |φn i hφn |
returns the eigenvalue an with probability | hφn |ψi |2 , and
leaves the system in state |φn i afterwards
Axiom 4: the time-evolution of a state is unitary:
|ψ(t)i = U(t) |ψ(0)i, with U(t) ∈ U(H) (⇒ Schrödinger’s
eqn.)
Axiom 5: the system composed of sub-systems H1 , H2
is described by H1 ⊗ H2
6 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Axioms
Non-relativistic quantum mechanics can be axiomatized:
Axiom 1: the state of an isolated system is described by an
element |ψi of a Hilbert space H (with | hψ|ψi |2 = 1)
Axiom 2: observables are self-adjoint operators on H (so you
have the Spectral Thm.)
P
Axiom 3: measuring observable A = n an |φn i hφn |
returns the eigenvalue an with probability | hφn |ψi |2 , and
leaves the system in state |φn i afterwards
Axiom 4: the time-evolution of a state is unitary:
|ψ(t)i = U(t) |ψ(0)i, with U(t) ∈ U(H) (⇒ Schrödinger’s
eqn.)
Axiom 5: the system composed of sub-systems H1 , H2
is described by H1 ⊗ H2
6 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Entanglement
What is entanglement?
EPR paradox
“A spooky action at a distance”
A general element of H1 ⊗ H2 is NOT of the form |ψ1 i ⊗ |ψ2 i!
E.g. for two qubits
1
1
|Φ+ i = √ |0i ⊗ |0i + √ |1i ⊗ |1i
2
2
Since |Φ+ i = √12 (|++i + |−−i), deciding in which base to
measure changes the state of the other qbit.
⇒ a general state cannot be thought of as two separate states in
two separate systems. Instead, the two systems have correlation they are inseparable.
7 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Entanglement
What is entanglement?
EPR paradox
“A spooky action at a distance”
A general element of H1 ⊗ H2 is NOT of the form |ψ1 i ⊗ |ψ2 i!
E.g. for two qubits
1
1
|Φ+ i = √ |0i ⊗ |0i + √ |1i ⊗ |1i
2
2
Since |Φ+ i = √12 (|++i + |−−i), deciding in which base to
measure changes the state of the other qbit.
⇒ a general state cannot be thought of as two separate states in
two separate systems. Instead, the two systems have correlation they are inseparable.
7 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Entanglement
What is entanglement?
EPR paradox
“A spooky action at a distance”
A general element of H1 ⊗ H2 is NOT of the form |ψ1 i ⊗ |ψ2 i!
E.g. for two qubits
1
1
|Φ+ i = √ |0i ⊗ |0i + √ |1i ⊗ |1i
2
2
Since |Φ+ i = √12 (|++i + |−−i), deciding in which base to
measure changes the state of the other qbit.
⇒ a general state cannot be thought of as two separate states in
two separate systems. Instead, the two systems have correlation they are inseparable.
7 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Entanglement
What is entanglement?
EPR paradox
“A spooky action at a distance”
A general element of H1 ⊗ H2 is NOT of the form |ψ1 i ⊗ |ψ2 i!
E.g. for two qubits
1
1
|Φ+ i = √ |0i ⊗ |0i + √ |1i ⊗ |1i
2
2
Since |Φ+ i = √12 (|++i + |−−i), deciding in which base to
measure changes the state of the other qbit.
⇒ a general state cannot be thought of as two separate states in
two separate systems. Instead, the two systems have correlation they are inseparable.
7 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Entanglement
What is entanglement?
EPR paradox
“A spooky action at a distance”
A general element of H1 ⊗ H2 is NOT of the form |ψ1 i ⊗ |ψ2 i!
E.g. for two qubits
1
1
|Φ+ i = √ |0i ⊗ |0i + √ |1i ⊗ |1i
2
2
Since |Φ+ i = √12 (|++i + |−−i), deciding in which base to
measure changes the state of the other qbit.
⇒ a general state cannot be thought of as two separate states in
two separate systems. Instead, the two systems have correlation they are inseparable.
7 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Entanglement
What is entanglement?
EPR paradox
“A spooky action at a distance”
A general element of H1 ⊗ H2 is NOT of the form |ψ1 i ⊗ |ψ2 i!
E.g. for two qubits
1
1
|Φ+ i = √ |0i ⊗ |0i + √ |1i ⊗ |1i
2
2
Since |Φ+ i = √12 (|++i + |−−i), deciding in which base to
measure changes the state of the other qbit.
⇒ a general state cannot be thought of as two separate states in
two separate systems. Instead, the two systems have correlation they are inseparable.
7 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
We describe an unusual experiment: Alice and Bob are in different
labs, and they have labeled sets of three coins covered by cups
8 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
They can uncover any cup, revealing if the coin is Head or Tail
(which happens 50% − 50%), but doing so makes the other cups
disintegrate at once
9 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
They discover something interesting: every time they uncover the
same cup in the sets with the same label, they get the same face,
no matter which cup
10 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
They then expect to be able to determine the values of two coins
at once by lifting different cups.
11 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Of the three coins, at least two must show the same face (HH or
TT).
They test the prediction and measure:
P { 1 and 2 are the same } ≈ 1/4 ...
P { 1 and 3 are the same } ≈ 1/4 ...
P { 2 and 3 are the same } ≈ 1/4.
But P { at least two are the same } = 1 (!) what is going on?
It must be that it doesn’t make sense to consider simultaneously
the outcome of two mutually esclusive experiments
12 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Of the three coins, at least two must show the same face (HH or
TT).
They test the prediction and measure:
P { 1 and 2 are the same } ≈ 1/4 ...
P { 1 and 3 are the same } ≈ 1/4 ...
P { 2 and 3 are the same } ≈ 1/4.
But P { at least two are the same } = 1 (!) what is going on?
It must be that it doesn’t make sense to consider simultaneously
the outcome of two mutually esclusive experiments
12 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Of the three coins, at least two must show the same face (HH or
TT).
They test the prediction and measure:
P { 1 and 2 are the same } ≈ 1/4 ...
P { 1 and 3 are the same } ≈ 1/4 ...
P { 2 and 3 are the same } ≈ 1/4.
But P { at least two are the same } = 1 (!) what is going on?
It must be that it doesn’t make sense to consider simultaneously
the outcome of two mutually esclusive experiments
12 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Of the three coins, at least two must show the same face (HH or
TT).
They test the prediction and measure:
P { 1 and 2 are the same } ≈ 1/4 ...
P { 1 and 3 are the same } ≈ 1/4 ...
P { 2 and 3 are the same } ≈ 1/4.
But P { at least two are the same } = 1 (!) what is going on?
It must be that it doesn’t make sense to consider simultaneously
the outcome of two mutually esclusive experiments
12 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Of the three coins, at least two must show the same face (HH or
TT).
They test the prediction and measure:
P { 1 and 2 are the same } ≈ 1/4 ...
P { 1 and 3 are the same } ≈ 1/4 ...
P { 2 and 3 are the same } ≈ 1/4.
But P { at least two are the same } = 1 (!) what is going on?
It must be that it doesn’t make sense to consider simultaneously
the outcome of two mutually esclusive experiments
12 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Of the three coins, at least two must show the same face (HH or
TT).
They test the prediction and measure:
P { 1 and 2 are the same } ≈ 1/4 ...
P { 1 and 3 are the same } ≈ 1/4 ...
P { 2 and 3 are the same } ≈ 1/4.
But P { at least two are the same } = 1 (!) what is going on?
It must be that it doesn’t make sense to consider simultaneously
the outcome of two mutually esclusive experiments
12 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Of the three coins, at least two must show the same face (HH or
TT).
They test the prediction and measure:
P { 1 and 2 are the same } ≈ 1/4 ...
P { 1 and 3 are the same } ≈ 1/4 ...
P { 2 and 3 are the same } ≈ 1/4.
But P { at least two are the same } = 1 (!) what is going on?
It must be that it doesn’t make sense to consider simultaneously
the outcome of two mutually esclusive experiments
12 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Physically, the sets of three coins correspond to a pair of entangled
electrons and measuring their spin along the following directions:
13 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Bell’s inequality
Formally, the experiment has disproven the existence of a
hidden-variable model:
Suppose the faces are predetermined according to a distribution
P(x, y, z),then
P { 1 and 2 are the same } =
P(H, H, H) + P(H, H, T ) + P(T , T , H) + P(T , T , T )
and similarly for the others. So
P { at least 2 are the same } = 1 + 2P(H, H, H) + 2P(T , T , T ) > 1
This is Bell’s inequality.
14 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Bell’s inequality
Formally, the experiment has disproven the existence of a
hidden-variable model:
Suppose the faces are predetermined according to a distribution
P(x, y, z),then
P { 1 and 2 are the same } =
P(H, H, H) + P(H, H, T ) + P(T , T , H) + P(T , T , T )
and similarly for the others. So
P { at least 2 are the same } = 1 + 2P(H, H, H) + 2P(T , T , T ) > 1
This is Bell’s inequality.
14 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Bell’s inequality
Formally, the experiment has disproven the existence of a
hidden-variable model:
Suppose the faces are predetermined according to a distribution
P(x, y, z),then
P { 1 and 2 are the same } =
P(H, H, H) + P(H, H, T ) + P(T , T , H) + P(T , T , T )
and similarly for the others. So
P { at least 2 are the same } = 1 + 2P(H, H, H) + 2P(T , T , T ) > 1
This is Bell’s inequality.
14 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Bell’s inequality
Formally, the experiment has disproven the existence of a
hidden-variable model:
Suppose the faces are predetermined according to a distribution
P(x, y, z),then
P { 1 and 2 are the same } =
P(H, H, H) + P(H, H, T ) + P(T , T , H) + P(T , T , T )
and similarly for the others. So
P { at least 2 are the same } = 1 + 2P(H, H, H) + 2P(T , T , T ) > 1
This is Bell’s inequality.
14 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Bell’s inequality
Formally, the experiment has disproven the existence of a
hidden-variable model:
Suppose the faces are predetermined according to a distribution
P(x, y, z),then
P { 1 and 2 are the same } =
P(H, H, H) + P(H, H, T ) + P(T , T , H) + P(T , T , T )
and similarly for the others. So
P { at least 2 are the same } = 1 + 2P(H, H, H) + 2P(T , T , T ) > 1
This is Bell’s inequality.
14 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
This is a cooperative game.
Alice and Bob can agree on any strategy beforehand (share
any amount of correlation)
then they are taken to separate rooms with no possibility of
communication between them
they open an envelope containing either a RED card or a
BLUE card at random
to win each of them has to write down a 0 or a 1 in such a
way that:
if both cards are RED then they write different numbers
otherwise, they have to write the same numbers
A moment’s thought reveals the best strategy is for both to always
write 0 ⇒ win 75% of the times
15 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
This is a cooperative game.
Alice and Bob can agree on any strategy beforehand (share
any amount of correlation)
then they are taken to separate rooms with no possibility of
communication between them
they open an envelope containing either a RED card or a
BLUE card at random
to win each of them has to write down a 0 or a 1 in such a
way that:
if both cards are RED then they write different numbers
otherwise, they have to write the same numbers
A moment’s thought reveals the best strategy is for both to always
write 0 ⇒ win 75% of the times
15 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
This is a cooperative game.
Alice and Bob can agree on any strategy beforehand (share
any amount of correlation)
then they are taken to separate rooms with no possibility of
communication between them
they open an envelope containing either a RED card or a
BLUE card at random
to win each of them has to write down a 0 or a 1 in such a
way that:
if both cards are RED then they write different numbers
otherwise, they have to write the same numbers
A moment’s thought reveals the best strategy is for both to always
write 0 ⇒ win 75% of the times
15 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
This is a cooperative game.
Alice and Bob can agree on any strategy beforehand (share
any amount of correlation)
then they are taken to separate rooms with no possibility of
communication between them
they open an envelope containing either a RED card or a
BLUE card at random
to win each of them has to write down a 0 or a 1 in such a
way that:
if both cards are RED then they write different numbers
otherwise, they have to write the same numbers
A moment’s thought reveals the best strategy is for both to always
write 0 ⇒ win 75% of the times
15 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
This is a cooperative game.
Alice and Bob can agree on any strategy beforehand (share
any amount of correlation)
then they are taken to separate rooms with no possibility of
communication between them
they open an envelope containing either a RED card or a
BLUE card at random
to win each of them has to write down a 0 or a 1 in such a
way that:
if both cards are RED then they write different numbers
otherwise, they have to write the same numbers
A moment’s thought reveals the best strategy is for both to always
write 0 ⇒ win 75% of the times
15 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
This is a cooperative game.
Alice and Bob can agree on any strategy beforehand (share
any amount of correlation)
then they are taken to separate rooms with no possibility of
communication between them
they open an envelope containing either a RED card or a
BLUE card at random
to win each of them has to write down a 0 or a 1 in such a
way that:
if both cards are RED then they write different numbers
otherwise, they have to write the same numbers
A moment’s thought reveals the best strategy is for both to always
write 0 ⇒ win 75% of the times
15 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
This is a cooperative game.
Alice and Bob can agree on any strategy beforehand (share
any amount of correlation)
then they are taken to separate rooms with no possibility of
communication between them
they open an envelope containing either a RED card or a
BLUE card at random
to win each of them has to write down a 0 or a 1 in such a
way that:
if both cards are RED then they write different numbers
otherwise, they have to write the same numbers
A moment’s thought reveals the best strategy is for both to always
write 0 ⇒ win 75% of the times
15 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
This is a cooperative game.
Alice and Bob can agree on any strategy beforehand (share
any amount of correlation)
then they are taken to separate rooms with no possibility of
communication between them
they open an envelope containing either a RED card or a
BLUE card at random
to win each of them has to write down a 0 or a 1 in such a
way that:
if both cards are RED then they write different numbers
otherwise, they have to write the same numbers
A moment’s thought reveals the best strategy is for both to always
write 0 ⇒ win 75% of the times
15 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
However, if Alice and Bob are allowed to share entanglement then
they can do better!
Recall
1
1
√ (|00i + |11i) = √ (|++i + |−−i)
2
2
if Alice’s card is red she measures in base |+i , |−i
if Alice’s card is blue she measures in base |0i , |1i
if Bob’s card is red he measures in base
cos(π/8) |0i + sin(π/8) |1i , − sin(π/8) |0i + cos(π/8) |1i
if Bob’s card is blue he measures in base
cos(−π/8) |0i + sin(−π/8) |1i , − sin(−π/8) |0i + cos(−π/8) |1i
then both output the result of their measurement.
16 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
However, if Alice and Bob are allowed to share entanglement then
they can do better!
Recall
1
1
√ (|00i + |11i) = √ (|++i + |−−i)
2
2
if Alice’s card is red she measures in base |+i , |−i
if Alice’s card is blue she measures in base |0i , |1i
if Bob’s card is red he measures in base
cos(π/8) |0i + sin(π/8) |1i , − sin(π/8) |0i + cos(π/8) |1i
if Bob’s card is blue he measures in base
cos(−π/8) |0i + sin(−π/8) |1i , − sin(−π/8) |0i + cos(−π/8) |1i
then both output the result of their measurement.
16 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
However, if Alice and Bob are allowed to share entanglement then
they can do better!
Recall
1
1
√ (|00i + |11i) = √ (|++i + |−−i)
2
2
if Alice’s card is red she measures in base |+i , |−i
if Alice’s card is blue she measures in base |0i , |1i
if Bob’s card is red he measures in base
cos(π/8) |0i + sin(π/8) |1i , − sin(π/8) |0i + cos(π/8) |1i
if Bob’s card is blue he measures in base
cos(−π/8) |0i + sin(−π/8) |1i , − sin(−π/8) |0i + cos(−π/8) |1i
then both output the result of their measurement.
16 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
However, if Alice and Bob are allowed to share entanglement then
they can do better!
Recall
1
1
√ (|00i + |11i) = √ (|++i + |−−i)
2
2
if Alice’s card is red she measures in base |+i , |−i
if Alice’s card is blue she measures in base |0i , |1i
if Bob’s card is red he measures in base
cos(π/8) |0i + sin(π/8) |1i , − sin(π/8) |0i + cos(π/8) |1i
if Bob’s card is blue he measures in base
cos(−π/8) |0i + sin(−π/8) |1i , − sin(−π/8) |0i + cos(−π/8) |1i
then both output the result of their measurement.
16 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
However, if Alice and Bob are allowed to share entanglement then
they can do better!
Recall
1
1
√ (|00i + |11i) = √ (|++i + |−−i)
2
2
if Alice’s card is red she measures in base |+i , |−i
if Alice’s card is blue she measures in base |0i , |1i
if Bob’s card is red he measures in base
cos(π/8) |0i + sin(π/8) |1i , − sin(π/8) |0i + cos(π/8) |1i
if Bob’s card is blue he measures in base
cos(−π/8) |0i + sin(−π/8) |1i , − sin(−π/8) |0i + cos(−π/8) |1i
then both output the result of their measurement.
16 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
The bases have been chosen to minimize the probability of
measuring the same state in the case both cards are RED, and to
maximize the probability otherwise:
⇒ Alice and Bob win in every case with probability
| cos(π/8)|2 = 12 + √12 ≈ 85% (!)
17 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
CHSH game
The bases have been chosen to minimize the probability of
measuring the same state in the case both cards are RED, and to
maximize the probability otherwise:
⇒ Alice and Bob win in every case with probability
| cos(π/8)|2 = 12 + √12 ≈ 85% (!)
17 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
Conclusion
If we have two players that are able to win the game 85% of the
time, we must conclude:
that they have access to entanglement
that their outputs must be TRULY RANDOM, since no
hidden-variable model is possible! It is, by any mean,
impossible to predict their outputs.
We seem to have found a source of true randomness!
18 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
[Vazirani-Vidick] exponential expansion
Initial seed: Cn random bits
Have Alice and Bob play 2n n2 rounds, divided in blocks of n2
In each block they get the same cards over and over n2 times
In almost all blocks they get (RED, RED)
except in O(1) of them, selected at random (using Cn − O(1)
bits of the seed)
in these they get random cards (using the remaining bits of
the seed).
In order to succeed ≈ cos(π/8) times in every block they are
forced to produce a random output everytime (if they get a red
card they don’t know if it’s one of the selected rounds or not!)
19 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
[Vazirani-Vidick] exponential expansion
Initial seed: Cn random bits
Have Alice and Bob play 2n n2 rounds, divided in blocks of n2
In each block they get the same cards over and over n2 times
In almost all blocks they get (RED, RED)
except in O(1) of them, selected at random (using Cn − O(1)
bits of the seed)
in these they get random cards (using the remaining bits of
the seed).
In order to succeed ≈ cos(π/8) times in every block they are
forced to produce a random output everytime (if they get a red
card they don’t know if it’s one of the selected rounds or not!)
19 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
[Vazirani-Vidick] exponential expansion
Initial seed: Cn random bits
Have Alice and Bob play 2n n2 rounds, divided in blocks of n2
In each block they get the same cards over and over n2 times
In almost all blocks they get (RED, RED)
except in O(1) of them, selected at random (using Cn − O(1)
bits of the seed)
in these they get random cards (using the remaining bits of
the seed).
In order to succeed ≈ cos(π/8) times in every block they are
forced to produce a random output everytime (if they get a red
card they don’t know if it’s one of the selected rounds or not!)
19 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
[Vazirani-Vidick] exponential expansion
Initial seed: Cn random bits
Have Alice and Bob play 2n n2 rounds, divided in blocks of n2
In each block they get the same cards over and over n2 times
In almost all blocks they get (RED, RED)
except in O(1) of them, selected at random (using Cn − O(1)
bits of the seed)
in these they get random cards (using the remaining bits of
the seed).
In order to succeed ≈ cos(π/8) times in every block they are
forced to produce a random output everytime (if they get a red
card they don’t know if it’s one of the selected rounds or not!)
19 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
[Vazirani-Vidick] exponential expansion
Initial seed: Cn random bits
Have Alice and Bob play 2n n2 rounds, divided in blocks of n2
In each block they get the same cards over and over n2 times
In almost all blocks they get (RED, RED)
except in O(1) of them, selected at random (using Cn − O(1)
bits of the seed)
in these they get random cards (using the remaining bits of
the seed).
In order to succeed ≈ cos(π/8) times in every block they are
forced to produce a random output everytime (if they get a red
card they don’t know if it’s one of the selected rounds or not!)
19 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
[Vazirani-Vidick] exponential expansion
Initial seed: Cn random bits
Have Alice and Bob play 2n n2 rounds, divided in blocks of n2
In each block they get the same cards over and over n2 times
In almost all blocks they get (RED, RED)
except in O(1) of them, selected at random (using Cn − O(1)
bits of the seed)
in these they get random cards (using the remaining bits of
the seed).
In order to succeed ≈ cos(π/8) times in every block they are
forced to produce a random output everytime (if they get a red
card they don’t know if it’s one of the selected rounds or not!)
19 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
[Vazirani-Vidick] exponential expansion
Initial seed: Cn random bits
Have Alice and Bob play 2n n2 rounds, divided in blocks of n2
In each block they get the same cards over and over n2 times
In almost all blocks they get (RED, RED)
except in O(1) of them, selected at random (using Cn − O(1)
bits of the seed)
in these they get random cards (using the remaining bits of
the seed).
In order to succeed ≈ cos(π/8) times in every block they are
forced to produce a random output everytime (if they get a red
card they don’t know if it’s one of the selected rounds or not!)
19 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
[Vazirani-Vidick] exponential expansion
Initial seed: Cn random bits
Have Alice and Bob play 2n n2 rounds, divided in blocks of n2
In each block they get the same cards over and over n2 times
In almost all blocks they get (RED, RED)
except in O(1) of them, selected at random (using Cn − O(1)
bits of the seed)
in these they get random cards (using the remaining bits of
the seed).
In order to succeed ≈ cos(π/8) times in every block they are
forced to produce a random output everytime (if they get a red
card they don’t know if it’s one of the selected rounds or not!)
19 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
[Coudron, Yuen] Infinite expansion
20 / 21
Quantum Mechanics primer
Bell’s theorem
CHSH game
Randomness expansion
21 / 21