Universal Bell Correlations Do Not Exist Cole A. Graham 1 William M. - - PowerPoint PPT Presentation
Universal Bell Correlations Do Not Exist Cole A. Graham 1 William M. Hoza 2 December 4, 2016 CS395T Quantum Complexity Theory 1 Department of Mathematics, Stanford University 2 Department of Computer Science, UT Austin Quantum nonlocality
Cole A. Graham1 William M. Hoza2 December 4, 2016 CS395T – Quantum Complexity Theory
1Department of Mathematics, Stanford University 2Department of Computer Science, UT Austin
◮ Recall Bell’s theorem: Entanglement allows interactions that
can’t be simulated using shared randomness / hidden variables
◮ Recall Bell’s theorem: Entanglement allows interactions that
can’t be simulated using shared randomness / hidden variables
◮ Recall the no-communication theorem: Entanglement can’t be
used to send signals
◮ Recall Bell’s theorem: Entanglement allows interactions that
can’t be simulated using shared randomness / hidden variables
◮ Recall the no-communication theorem: Entanglement can’t be
used to send signals
◮ Contradictory?
x ∈ {0, 1} y ∈ {0, 1}
x ∈ {0, 1} y ∈ {0, 1} a ∈ {0, 1} b ∈ {0, 1}
x ∈ {0, 1} y ∈ {0, 1} a ∈ {0, 1} b ∈ {0, 1} (a, b) =
with probability 1/2 (1, 1 − xy) with probability 1/2
x ∈ {0, 1} y ∈ {0, 1} a ∈ {0, 1} b ∈ {0, 1} (a, b) =
with probability 1/2 (1, 1 − xy) with probability 1/2
◮ Cannot be used to communicate
x ∈ {0, 1} y ∈ {0, 1} a ∈ {0, 1} b ∈ {0, 1} (a, b) =
with probability 1/2 (1, 1 − xy) with probability 1/2
◮ Cannot be used to communicate ◮ But can be used to win CHSH game: a + b = xy (mod 2)
x ∈ X y ∈ Y a ∈ A b ∈ B
x ∈ X y ∈ Y a ∈ A b ∈ B
◮ A correlation box is a map
Cor : X × Y → {µ : µ is a probability distribution over A × B}
x ∈ X y ∈ Y a ∈ A b ∈ B
◮ A correlation box is a map
Cor : X × Y → {µ : µ is a probability distribution over A × B}
◮ Assume X, Y , A, B are countable
x ∈ X y ∈ Y a ∈ A b ∈ B
◮ A correlation box is a map
Cor : X × Y → {µ : µ is a probability distribution over A × B}
◮ Assume X, Y , A, B are countable ◮ Abuse notation and write Cor : X × Y → A × B
x ∈ X y ∈ Y a ∈ A b ∈ B
◮ Can think of a correlation box as a distributed sampling
problem – the problem of simulating the box
◮ SR: class of correlation boxes that can be simulated using just
shared randomness
◮ SR: class of correlation boxes that can be simulated using just
shared randomness
◮ Q: class of correlation boxes that can be simulated using
shared randomness + arbitrary bipartite quantum state
◮ SR: class of correlation boxes that can be simulated using just
shared randomness
◮ Q: class of correlation boxes that can be simulated using
shared randomness + arbitrary bipartite quantum state
◮ Obviously SR ⊆ Q
◮ SR: class of correlation boxes that can be simulated using just
shared randomness
◮ Q: class of correlation boxes that can be simulated using
shared randomness + arbitrary bipartite quantum state
◮ Obviously SR ⊆ Q ◮ Bell’s theorem: SR = Q
◮ NS: class of non-signalling correlation boxes
◮ NS: class of non-signalling correlation boxes ◮ No-communication theorem: Q ⊆ NS
◮ NS: class of non-signalling correlation boxes ◮ No-communication theorem: Q ⊆ NS ◮ Tsierelson bound: PR ∈ Q, so Q = NS
◮ Goal: Understand Q
◮ Goal: Understand Q ◮ Baby step: Understand BELL: class of correlation boxes that
can be simulated using shared randomness +
1 √ 2(|00 + |11)
+ projective measurements
◮ Goal: Understand Q ◮ Baby step: Understand BELL: class of correlation boxes that
can be simulated using shared randomness +
1 √ 2(|00 + |11)
+ projective measurements
◮ SR BELL Q
◮ Theorem (Toner, Bacon ’03): BELL can be simulated using
shared randomness + 1 bit of communication
◮ Theorem (Toner, Bacon ’03): BELL can be simulated using
shared randomness + 1 bit of communication
◮ This is an upper bound on the power of BELL
◮ Theorem (Toner, Bacon ’03): BELL can be simulated using
shared randomness + 1 bit of communication
◮ This is an upper bound on the power of BELL ◮ Loose upper bound, since BELL ⊆ NS
◮ Theorem (Cerf et al. ’05): BELL can be simulated using
shared randomness + 1 PR box
◮ Theorem (Cerf et al. ’05): BELL can be simulated using
shared randomness + 1 PR box
◮ In other words, PR is BELL-hard with respect to 1-query
reductions
SR BELL Q NS BELL-hard PR
◮ Theorem: There does not exist a finite-alphabet
BELL-complete correlation box
◮ Theorem: There does not exist a finite-alphabet
BELL-complete correlation box
◮
◮ Theorem: There does not exist a finite-alphabet
BELL-complete correlation box
◮ ◮ Theorem:
◮ Theorem: There does not exist a finite-alphabet
BELL-complete correlation box
◮ ◮ Theorem:
◮ Suppose Cor : X × Y → A × B is in Q; X, Y countable; A, B
finite
◮ Theorem: There does not exist a finite-alphabet
BELL-complete correlation box
◮ ◮ Theorem:
◮ Suppose Cor : X × Y → A × B is in Q; X, Y countable; A, B
finite
◮ Then there exists a binary correlation box in BELL that does
not reduce to Cor
◮ Theorem: There does not exist a finite-alphabet
BELL-complete correlation box
◮ ◮ Theorem:
◮ Suppose Cor : X × Y → A × B is in Q; X, Y countable; A, B
finite
◮ Then there exists a binary correlation box in BELL that does
not reduce to Cor
◮
SR BELL Q NS BELL-hard PR
x ∈ {0, 1} y ∈ {0, 1} a ∈ {0, 1} b ∈ {0, 1}
◮ Goal: a + b = xy (mod 2)
x ∈ {0, 1} y ∈ {0, 1} a ∈ {0, 1} b ∈ {0, 1}
◮ Goal: a + b = xy (mod 2) ◮ Inputs x, y are chosen independently at random
x ∈ {0, 1} y ∈ {0, 1} a ∈ {0, 1} b ∈ {0, 1}
◮ Goal: a + b = xy (mod 2) ◮ Inputs x, y are chosen independently at random ◮ y is uniform, x is biased: Pr[x = 1] = p ∈ [1/2, 1]
x ∈ {0, 1} y ∈ {0, 1} a ∈ {0, 1} b ∈ {0, 1}
◮ Goal: a + b = xy (mod 2) ◮ Inputs x, y are chosen independently at random ◮ y is uniform, x is biased: Pr[x = 1] = p ∈ [1/2, 1] ◮ Theorem (Lawson, Linden, Popescu ’10): Optimal quantum
strategy can be implemented in BELL, wins with probability f (p) def = 1 2 + 1 2
p win prob 75% 100% 1 1/2 ≈ 85%
◮ Let Sp ∈ BELL be optimal quantum strategy
◮ Let Sp ∈ BELL be optimal quantum strategy ◮ Assume there is a reduction from Sp to Cor
◮ Let Sp ∈ BELL be optimal quantum strategy ◮ Assume there is a reduction from Sp to Cor ◮ Probability that reduction wins biased CHSH game is of form
1 − p 2 P00 + p 2P10 + 1 − p 2 P01 + p 2P11
◮ Let Sp ∈ BELL be optimal quantum strategy ◮ Assume there is a reduction from Sp to Cor ◮ Probability that reduction wins biased CHSH game is of form
1 − p 2 P00 + p 2P10 + 1 − p 2 P01 + p 2P11
◮ Affine function of p, for fixed reduction
◮ Fix shared randomness without decreasing win probability
◮ Fix shared randomness without decreasing win probability ◮ Recall Cor ∈ Q
◮ Fix shared randomness without decreasing win probability ◮ Recall Cor ∈ Q ◮ Win probability still exactly f (p) (Q is closed under reductions)
◮ Fix shared randomness without decreasing win probability ◮ Recall Cor ∈ Q ◮ Win probability still exactly f (p) (Q is closed under reductions) ◮ Recall Cor : X × Y → A × B with X, Y countable, A, B finite
◮ Fix shared randomness without decreasing win probability ◮ Recall Cor ∈ Q ◮ Win probability still exactly f (p) (Q is closed under reductions) ◮ Recall Cor : X × Y → A × B with X, Y countable, A, B finite ◮ Only countably many deterministic reductions!
◮ Fix shared randomness without decreasing win probability ◮ Recall Cor ∈ Q ◮ Win probability still exactly f (p) (Q is closed under reductions) ◮ Recall Cor : X × Y → A × B with X, Y countable, A, B finite ◮ Only countably many deterministic reductions! ◮ Countably many affine functions, so ∃p where all the affine
functions disagree with f (p)
◮ Theorem:
◮ Theorem:
◮ Suppose Cor2 : X × Y → A × B is in Q; X, Y , A, B finite
◮ Theorem:
◮ Suppose Cor2 : X × Y → A × B is in Q; X, Y , A, B finite ◮ Then ∃ binary correlation box Cor1 ∈ BELL such that
◮ Theorem:
◮ Suppose Cor2 : X × Y → A × B is in Q; X, Y , A, B finite ◮ Then ∃ binary correlation box Cor1 ∈ BELL such that ◮ If there is a k-query ε-error reduction from Cor1 to Cor2, then
k4 · (2|X|)2|A|k · (2|Y |)2|B|k ≥ Ω(1/ε)
◮ Theorem:
◮ Suppose Cor2 : X × Y → A × B is in Q; X, Y , A, B finite ◮ Then ∃ binary correlation box Cor1 ∈ BELL such that ◮ If there is a k-query ε-error reduction from Cor1 to Cor2, then
k4 · (2|X|)2|A|k · (2|Y |)2|B|k ≥ Ω(1/ε)
◮ Upper bound: ∀ε > 0, ∃Cor2 : [T] × [T] → {0, 1} × {0, 1}
such that
◮ Theorem:
◮ Suppose Cor2 : X × Y → A × B is in Q; X, Y , A, B finite ◮ Then ∃ binary correlation box Cor1 ∈ BELL such that ◮ If there is a k-query ε-error reduction from Cor1 to Cor2, then
k4 · (2|X|)2|A|k · (2|Y |)2|B|k ≥ Ω(1/ε)
◮ Upper bound: ∀ε > 0, ∃Cor2 : [T] × [T] → {0, 1} × {0, 1}
such that
◮ T ≤ O(1/ε4)
◮ Theorem:
◮ Suppose Cor2 : X × Y → A × B is in Q; X, Y , A, B finite ◮ Then ∃ binary correlation box Cor1 ∈ BELL such that ◮ If there is a k-query ε-error reduction from Cor1 to Cor2, then
k4 · (2|X|)2|A|k · (2|Y |)2|B|k ≥ Ω(1/ε)
◮ Upper bound: ∀ε > 0, ∃Cor2 : [T] × [T] → {0, 1} × {0, 1}
such that
◮ T ≤ O(1/ε4) ◮ Cor2 ∈ BELL
◮ Theorem:
◮ Suppose Cor2 : X × Y → A × B is in Q; X, Y , A, B finite ◮ Then ∃ binary correlation box Cor1 ∈ BELL such that ◮ If there is a k-query ε-error reduction from Cor1 to Cor2, then
k4 · (2|X|)2|A|k · (2|Y |)2|B|k ≥ Ω(1/ε)
◮ Upper bound: ∀ε > 0, ∃Cor2 : [T] × [T] → {0, 1} × {0, 1}
such that
◮ T ≤ O(1/ε4) ◮ Cor2 ∈ BELL ◮ For every Cor1 ∈ BELL, there is a 1-query ε-error reduction
from Cor1 to Cor2
◮ Is there a countable-alphabet BELL-complete correlation box?
◮ Is there a countable-alphabet BELL-complete correlation box? ◮ What is the right relationship between |X|, |Y |, |A|, |B|, k, ε?
◮ Is there a countable-alphabet BELL-complete correlation box? ◮ What is the right relationship between |X|, |Y |, |A|, |B|, k, ε? ◮ Thanks for listening!
◮ This material is based upon work supported by the National Science
Foundation Graduate Research Fellowship under Grant No. DGE-1610403.
◮ Cole Graham gratefully acknowledges the support of the Fannie and John
Hertz Foundation.