x,y ∈ {0,1} uniformly at random
o
/|\
x y
ඌ o
/|\ /|\
o
/|\
x y
ඌ o
/|\ /|\
o
/|\
ඌ o
/|\ /|\
o
/|\
ඌ o
/|\ a b /|\
a,b ∈ {0,1}
o
/|\
a b
ඌ o
/|\ /|\
a ⊕ b = x ∧ y?
o
/|\
a b
ඌ o
/|\ /|\
a ⊕ b = x ∧ y?
o
/|\
a b
ඌ o
/|\ /|\
If yes: Alice and Bob win; If no: they lose.
how to make charlie? use two coins x and y.
H = 0. T = 1.
a ⊕ b = x ∧ y?
(a and b are different) and (x and y are both T)
(a and b are same) and (x and y are not both T)
each player takes the coin and flips it
the coins are still there in the quantum version of the game.
Optimal classical deterministic strategy. success prob. 75%
deterministic strategy := a pair of functions fA, fB : {0,1} ↦ {0,1}, where fA is Alice's response
input
x = 0, y = 0
winning outputs
a=b=0 or a=b=1
a and b always return 0
win win lose
x = 0, y = 1 | x = 1, y = 0 | x = 1, y = 1
a and b always return 1
win win lose
x = 0, y = 1 | x = 1, y = 0 | x = 1, y = 1
a and b always return x and y
lose lose lose
x = 0, y = 1 | x = 1, y = 0 | x = 1, y = 1
a and b always return ¬x and ¬y
lose lose lose
x = 0, y = 1 | x = 1, y = 0 | x = 1, y = 1
input
x = 0, y = 1
winning outputs
a=b=0 or a=b=1
a and b always return 0
win win lose
x = 0, y = 0 | x = 1, y = 0 | x = 1, y = 1
a and b always return 1
win win lose
x = 0, y = 0 | x = 1, y = 0 | x = 1, y = 1
a always return x; b always return ¬y
lose win win
x = 0, y = 0 | x = 1, y = 0 | x = 1, y = 1
a always return ¬x; b always return y
lose win win
x = 0, y = 0 | x = 1, y = 0 | x = 1, y = 1
input
x = 1, y = 0
winning outputs
a=b=0 or a=b=1
a and b always return 0
win win lose
x = 0, y = 0 | x = 0, y = 1 | x = 1, y = 1
a and b always return 1
win win lose
x = 0, y = 0 | x = 0, y = 1 | x = 1, y = 1
a always return x; b always return ¬y
lose win win
x = 0, y = 0 | x = 0, y = 1 | x = 1, y = 1
a always return ¬x; b always return y
lose win win
x = 0, y = 0 | x = 0, y = 1 | x = 1, y = 1
input
x = 1, y = 1
winning outputs
a=0,b=1 or a=1,b=0
a always returns 0; b always return 1
lose lose lose
x = 0, y = 0 | x = 0, y = 1 | x = 1, y = 0
a always returns 1; b always return 0
lose lose lose
x = 0, y = 0 | x = 0, y = 1 | x = 1, y = 0
a always return ¬x; b always returns y
lose win win
x = 0, y = 0 | x = 0, y = 1 | x = 1, y = 0
a always return x; b always return ¬y
lose win win
x = 0, y = 0 | x = 0, y = 1 | x = 1, y = 0
for any strategy, at least one of the inputs always makes a loss.
can measure Bell pair w/ online quantum computer?
if communication is allowed
o
/|\
ඌ o
/|\ /|\
quantum strategy is like when communication is possible even when separated by a large distance
Optimal quantum strategy
o
/|\
ඌ o
/|\ /|\
o
/|\
ඌ o
/|\ ⇅ ................................ ⇅ /|\
Alice and Bob share Φ = |00⟩ + |11⟩ = |++⟩ + |−−⟩
o
/|\
x y
ඌ o
/|\ ⇅ ................................ ⇅ /|\
Alice: if x = 0: measure in basis |0⟩, |1⟩. respond 0 if |0⟩, and 1 if |1⟩.
o
/|\
x y
ඌ o
/|\ ⇅ ................................ ⇅ /|\
if x = 1: measure in basis |+⟩, |−⟩. respond 0 if |+⟩, and 1 if |−⟩.
Bob: if y = 0: measure in basis {|a₀⟩ = (cos π/8) |0⟩ + (sin π/8) |1⟩, |a₁⟩ = (− sin π/8) |0⟩ + (cos π/8) |1⟩} respond 0 if |a₀⟩, and 1 if |a₁⟩.
o
/|\
x y
ඌ o
/|\ ⇅ ................................ ⇅ /|\
if y = 1: measure in basis {|b₀⟩ = (cos π/8) |0⟩ − (sin π/8) |1⟩, |b₁⟩ = (sin π/8) |0⟩ + (cos π/8) |1⟩}. respond 0 if |b₀⟩, and 1 if |b₁⟩.
case x=y=0:
o
/|\
0 0
ඌ o
/|\ ⇅ ................................ ⇅ /|\
winning pairs are a=b=0 and a=b=1.
Prob[a=b=0] = |(⟨0| ⊗ ⟨a₀|) |Φ⟩|² = 1/2 cos²(π/8).
Prob[a=b=1] = |(⟨1| ⊗ ⟨a₁|) |Φ⟩|² = 1/2 cos²(π/8).
Prob[a=b=0 or a=b=1] = cos²(π/8).
Similarly for the other 3 cases.
average win probability for randomly chosen input = cos²(π/8) ≈ 85%.
real experiment with colored coins .. information exists simultaneously
real experiment with photons .. information does not exist simultaneously
theory predicts it
Quantum theory says: cannot know shape and color simultaneously.
quantum theory says to avoid contradictions we must admit that:
A property not measured need not exist. And Measurement alters"
"every particle has a local classical “cheat sheet” w/ outcome of every possible measurement? implies classical strategy
If quantum strategies can beat classical strategies then nature does not work that way!
we require no communication. we enforce the requirement by separating the particles farther that light can travel between them. but they still act like they communicate.
o "wtf?" -- Einstein
/|\
"EPR: The choice of measurement in one location appears to be affecting the state of the system in the other location."
"The “common sense” inference, which Einstein insisted should be part of any acceptable notion of physical reality, is inconsistent with experimental facts.
if qon had shape and color simultaneously, no spooky action at a distance?
experiment confirms it: CHSH quantum win prob is real.
demo
take two coins. coin for me. coin for S.
intro win criterion
(a and b are different) and (x and y are both T)
(a and b are same) and (x and y are not both T)
agree on a strategy
think about performance in the other cases
flip coins many times. count wins.
flip coin. don’t show it to me. choose response according to fixed strategy.
check if win or lose.
flip coin 10 times. count wins
think about other possible strategies. probabilistic strategies.
intro quantum strategy
consider coins that communicate
quantum strategy is like coins that communicate even when separated by a large distance