Is Communication Complexity Physical? Samuel Marcovitch Benni Reznik Tel-Aviv University arxiv

2 Non-locality (NL) Bell: no local-hidden-variable (LHV) theory can simulate quantum-mechanical behavior –All possible outcomes are determined in advance –Hidden variables do not propagate superluminally Bell’s inequality –Measure of non-locality Causality Respecting NL LHVQuantum Maximal Non-locality

3 Main Result Nature is as non-local as QM as otherwise CC would be trivial Trivial Communication Complexity LHVQM Maximal Non-locality Conjecture Bipartite - Brassard et. al. PRL 96, (2006) Multipartite Trivial Communication Complexity LHVQM Maximal Non-locality NL

4 Space of all Physical Theories Causality Conjecture Causality  Non-Trivial CC  QM Non-Trivial CC LHV QM ?

5 More Non-local than the Quantum Non-local boxes (NLB) Popescu & Rohrlich Found. Phys. 24, 379 (1994) –Hypothetical devices –Respect causality –Computational power van-Dam quant-ph/ Brassard’s et. al. –Given Nature is sufficiently more non- local, CC is trivial LHV QM Non-local Boxes Trivial CC Causality LHV QM NLB NL Non-Trivial CC

6 More Non-local than the Quantum Physics is multipartite  Brassard’s et. al. Conjecture should be tested in the multipartite case Generalized Conjecture Quantum Theory is as non-local as it is in the multipartite case, since otherwise communication complexity would be trivial

7 Communication Complexity (CC) What is the minimal number of bit Alice should send to Bob? –Example: Boolean Inner-Product –Worse case O(k): Alice sends all her bits to Bob. –Proven: Shared randomness or shared entanglement do not help Trivial CC: only 1 bit of information for any function and input size

8 Non-local Boxes (NLB) A B Computing Inner-Product with trivial CC

9 Brassards’ et. al. Result Generalize CC to the probabilistic case –compute with probability  >½, independent of input size Trivial CC can be achieved with non-perfect NLB Trivial CC NLB probability LHV 75% Quantum Theory Non-perfect NLB 100%

10 NLB Probability = Non-locality Bell’s inequality:

11 Physics is Multipartite Multipartite CC –Example: Tripartite Inner-Product Trivial CC: N-1 bits of communication for N parties Brassard’s et al Conjecture generalized to multipartite Objection –Any multipartite function can be computed with trivial CC using bipartite NLB’s.

12 Physics is Multipartite Probabilistic CC –required operation probability of the non-local box Bipartite ~90.8% Tripartite ~96.7% Is Brassard’s el. al. Conjecture refuted?

13 Multipartite Non-local Boxes Examples of tripartite NLB Found a specific class of multipartite NLB that reduces CC to triviality effectively –Requires constant probability for all N>2. Trivial CC Multipartite NLB probability LHVQuantum Theory Non-perfect NLB 100%

14 Generalized Bell’s Inequalities More than two parties (More than two observables {x,y} per site) Suggested box corresponds to the generalized Bell’s inequality with maximal QM violation. Multipartite NLB Generalized Bell’s inequality Ardehali’s Inequality

15 Ardehali’s inequality N even: Ardehali’s inequality coincides with the maximal violating inequality (Klyshko’s inequality) N odd: Maximal violation among all CHSH inequalities that have corresponding NLB k is the number of y’s in

16 Summary of the results NLB probability Multipartite NLB probability LHVQuantum Theory Non-perfect multipartite NLB 100% Non-perfect bipartite NLB Ardehali’s Inequality Maximal quantum non-locality Bipartite Multipartite Trivial CC LHV 75% Quantum Theory Non-perfect NLB 100% Trivial CC

17 Space of all Physical Theories Causality Conjecture Causality  Non-Trivial CC  QM Non-Trivial CC LHV QM ?

18 Obtaining the Bound Bipartite case (Brassard et. al) Non-local computation = trivial CC With shared randomness can be computed non-locally with. Non-local Majority: If Alice and Bob can compute non- local Majority with, CC is trivial.

19 Obtaining the Bound Bipartite case (Brassard et. al) Non-local equality: Non-local equality can be computed with only two NLB’s Non-local Majority can be computed with only two NLB’s Required probability

20 Obtaining the Bound Multipartite case Non-local equality: N(N-1) bipartite NLB are required  3 optimal multipartite NLB are required

21 Obtaining the Bound Multipartite case Box 1: Box 2: Box 3:

22 Obtaining the Bound Multipartite case Non-local Majority is obtained in the same way: Required probability

23 Correspondence to Ardehali’s Inequality f = 1 for q 1’s such that q(q-1)/2 is odd Correspondence proved by induction Classic bound decreases as N increases –Classically, one can simulate f = 1 for odd q (or f = 1 for q equals 0/N) QM bound is constant and satisfied by the generalized GHZ state

24 Summary of the results Trivial communication complexity NLB Operation probability Classic Theory 75% Quantum Theory Non-perfect NLB 100% Trivial communication complexity Multipartite NLB Operation probability Classic TheoryQuantum Theory Non-perfect NLB 100% Non-perfect bipartite NLB Ardehali’s Inequality Maximal quantum non-locality Bipartite Multipartite