Connes’ embedding problem, Tsirelson's problem, and MIP* = RE THOMAS VIDICK CALIFORNIA INSTITUTE OF TECHNOLOGY Joint work with Zhengfeng Ji (UTS), Anand Natarajan (MIT), John Wright (UT Austin) and Henry Yuen (Columbia) MIP* = RE • In complexity theory a problem is represented as a set 푆 ⊆ 0,1 ∗ • The problem associated to 푆 is, “given 푥, is 푥 ∈ 푆 ?” • Ex: 푆 = { binary representations of composite numbers} 푆 = { binary representations of connected graphs } 푆 = { binary representations of provable sentences in ZFC } • A complexity class is a collection of problems, i.e. a collection of sets that share some measure of “complexity” MIP* = RE BPP: Problems for which there is a (randomized) polynomial-time algorithm that always returns the correct answer. MA: Problems that can be verified in (randomized) polynomial time, given a polynomial-size proof MIP* = RE RE: Problems for which there is an algorithm that eventually terminates and returns ‘YES’ on positive instances (and doesn’t terminate/returns ‘NO’ on negative instances) MIP*: Problems that can be verified in polynomial time by interacting with quantum provers sharing entanglement Consequences of MIP* = RE • Negative answer to Tsirelson’s problem: the tensor and commuting models for quantum correlations are strictly distinct • Negative answer to Connes’ embedding problem: there exist type 퐼퐼1 von Neumann algebras that are not ‘hyperfinite’ • Verification of quantum systems: asymptotically efficient tests for arbitrarily high-dimensional entanglement Plan for the talk 1) Quantum correlations and Tsirelson’s problem 2) An approach to Tsirelson’s problem via algorithms & complexity 3) Quantum multiprover interactive proofs 4) Open questions Quantum correlations and Tsirelson’s problem Quantum nonlocality Millions of light years apart Local measurements on far-away particles can exhibit unexpected correlations... … almost as if one particle could instantaneously influence the other! Schrödinger called this phenomenon quantum entanglement. Nonlocal correlations 푥 푦 • Experimental data modeled as family of 2 2 distributions 푝 푎, 푏 푥, 푦 ∈ ℝ푚 푘 푥, 푦: 푚 possible measurement choices 푎 푎, 푏: 푘 possible measurement outcomes 푏 • Bell 1964: some {푝 푎, 푏 푥, 푦 } have a model in QM but no classical explanation • Classical correlations: 푝 푎, 푏 푥, 푦 = න푞 푎 푥, 휆 푞 푏 푦, 휆 푑휆 푝 푎, 푏 푥, 푦 = 퐴푞퐴 푎 푥 푞퐵퐵(푏|푦) 휆 Nonlocal correlations 푥 푦 • Experimental data modeled as family of 2 2 distributions 푝 푎, 푏 푥, 푦 ∈ ℝ푚 푘 푥, 푦: 푚 possible measurement choices 푎, 푏: 푘 possible measurement outcomes 푎 푏 • Bell 1964: some {푝 푎, 푏 푥, 푦 } have a model in QM but no classical explanation • Classical correlations: 푝 푎, 푏 푥, 푦 = න푞 푎 푥, 휆 푞 푏 푦, 휆 푑휆 푝 푎, 푏 푥, 푦 = 퐴푞퐴 푎 푥 푞퐵퐵(푏|푦) 휆 • Tsirelson ’80s: principled approach to study geometry of quantum correlations Tsirelson’s setup 푥 푦 • Correlation: family of distributions 푝 푎, 푏 푥, 푦 | 푥, 푦 ∈ 1, … , 푛 푎, 푏 ∈ 1, … 푘 푎 푏 • Quantum correlations (1): Pure state 휓 ∈ ℋ ⊗ ℋ, 휓 = 1 푥 푥 푦 For every 푥, psd operators 푃푎 푎 such that σ푎 푃푎 = 퐼. Similarly, {푄푏 } • 푥 푦 푝 푎, 푏 푥, 푦 = 휓 푃푎 ⊗ 푄푏 휓 • Quantum correlations (2): Pure state 휓 ∈ ℋ, 휓 = 1 푥 푥 푦 For every 푥, psd operators 푃푎 푎 such that σ푎 푃푎 = 퐼. Similarly, {푄푏 } 푥 푦 ∀푥푦푎푏, 푃푥, 푄푦 = 0 푝 푎, 푏 푥, 푦 = 휓 푃푎 푄푏 휓 푎 푏 Tsirelson’s problem Quantum Bell-type inequalities are defined in terms of two (or more) subsystems of a quantum system. The subsystems may be treated either via (local) Hilbert spaces, - tensor factors of the given (global) Hilbert space, or via commuting (local) operator algebras. The latter approach is less restrictive, it just requires that the given operators commute whenever they belong to different subsystems. Are these two approaches equivalent? Tsirelson’s problem 퐶 푚, 푘 = { 휓 푃푥 ⊗ 푄푦 휓 : |휓〉 ∈ ℋ ⊗ ℋ } ⊗ 푎 푏 푎푏푥푦 퐶 푚, 푘 = { 휓 푃푥푄푦 휓 : 푐표푚 푎 푏 푎푏푥푦 퐶푐표푚 푥 푦 |휓〉 ∈ ℋ, [푃푎 , 푄푏 ] = 0} • Both sets are convex 퐶⊗ • 퐶⊗ 푚, 푘 ⊆ 퐶푐표푚 푚, 푘 for all 푚, 푘. 푚2푘2 • 퐶푐표푚(푚, 푘) is closed, but [Slofstra’18] 퐶⊗(푛, 푘) is not! [0,1] Is 퐶⊗ 푚, 푘 = 퐶푐표푚 푚, 푘 for all 푚, 푘 ≥ 2 ? The connection with operator algebras ●A vonIn 1932, Neumann von Neumann algebra is put an Quantumalgebra 푀 Mechanics⊂ ℬ(ℋ) s.t.: • on퐼 ∈ a 푀firm mathematical basis • 푀○ isQuantum closed under state =adjoints vector in a complex Hilbert space ○ Measurement = linear operator on that space. • 푀 is closed in the weak operator topology ● Over the next decade, von Neumann (with F. Murray) wrote a series of papers that launched the field of operator algebras ● Important goal of operator algebras: classification of von Neumann factors ○ Main species: Type I, Type II, Type III ○ Within each type, there are subspecies. Type I factors were completely solved by Murray and von Neumann. They also made progress on Type II factors. ○ 1970s: Alain Connes won the Fields Medal for his contributions to the theory of operator algebra, including the classification of Type III factors. The connection with operator algebras ● In a 1976 paper, Connes makes a casual remark about Type II1 factors: “We now construct an approximate embedding of N into R. Apparently such an embedding ought to exist for all II1 factors.” ● This throwaway line became known as Connes’ Embedding Problem: roughly speaking, can every finite subset of a II1 factor be approximately embedded in the finite-dimensional matrices? ● Easier to state, weaker conjecture: are all countable groups hyperlinear? A countable group Γ is hyperlinear if ∀푛 ≥ 1 there is 휎푛: Γ → 푈푛 s.t. • ∀, ℎ ∈ Γ, 휎푛 ℎ − 휎푛 휎푛 ℎ 퐹 →푛→∞ 0 ( ⋅ 퐹 is the dimension- • ∀ ≠ 1 , 휎 − 퐼 → 1 normalized Frobenius Γ 푛 푛 퐹 푛→∞ norm) Equivalent formulations of CEP • CEP: “Every type II1 von Neumann algebra embeds in an ultrapower of the hyperfinite II1 factor ℛ” • Kirchberg in 1993 introduces QWEP conjecture and shows its equivalence to ∗ ∗ ? ∗ ∗ 퐶 퐹2 ⊗푚푖푛 퐶 퐹2 = 퐶 퐹2 ⊗푚푎푥 퐶 퐹2 • Kirchberg proved the equivalence CEP ↔ QWEP • Multiple other reformulations: free entropy, group theory, etc. • [Fritz,Junge et al.’11] QWEP/CEP imply a positive answer to Tsirelson’s problem • [Ozawa’12] Equivalence: 퐶푞푠 푚, 푘 = 퐶푞푐 푚, 푘 for all 푛, 푘 ≥ 2 iff CEP holds An approach to Tsirelson’s problem via algorithms & complexity Optimizing linear functionals over a convex set 퐶 푚, 푘 = { 휓 푃푥 ⊗ 푄푦 휓 : ⊗ 푎 푏 푎푏푥푦 휆 |휓〉 ∈ ℋ ⊗ ℋ } 휔∗ 휆 = sup 휆 ⋅ 푝 푝∈퐶⊗ 푚,푘 퐶 푚, 푘 = { 휓 푃푥푄푦 휓 : 푐표푚 푎 푏 푎푏푥푦 푥 푦 ` |휓〉 ∈ ℋ, [푃푎 , 푄푏 ] = 0} 휔푐표푚 휆 = sup 휆 ⋅ 푝 푝∈퐶푐표푚 푚,푘 2 2 푂 [0,1]푚 푘 Optimizing linear functionals over a convex set • Suppose exhaustive search & NPA 휆 converge to the same value, for all 휆 → Tsirelson’s problem has a positive answer • Suppose exhaustive search & NPA do not converge to the same value,for some 휆 → Tsirelson’s problem has a negative answer • [Fritz-NT’14] Suppose that 휔∗(휆) is uncomputable 2 2 푂 [0,1]푚 푘 → Tsirelson’s problem has a negative answer Quantum multiprover interactive proofs Interactive proofs 푧 ∈ 0,1 푛 푦푒푠/푛표 • BPP: efficient decision 퐴 Time: randomized/quantum poly(푛) Ex: Primality testing; Connectivity; Linear programming; Factoring 푃 • MA: efficient verification 휋 푧 is “yes” ⇒ ∃ 휋, accepted by 푉 whp 푧 ∈ 0,1 푛 푦푒푠/푛표 푉 푧 is “no” ⇒ ∀ 휋, rejected by 푉 whp Time: randomized/quantum poly(푛) Ex: Graph colorability; Local Hamiltonian Interactive proofs 푃 • IP: efficient interactive verification 푥 푎 푧 ∈ 0,1 푛 푦푒푠/푛표 Ex: Graph non-colorability, GO, … 푉 [Shamir’90] IP = PSPACE Time: randomized/quantum poly(푛) • MIP: efficient interactive verification 푃1 푃2 푎 with two provers 푏 푥 푦 푛 푦푒푠/푛표 Ex: Exponential-size graph coloring 푧 ∈ 0,1 푉 [Babai-FL’91] MIP = NEXP Time: randomized/quantum poly(푛) Interactive proof systems as optimization problems 푃1 푃2 푎 푏 푥 푦 푧 ∈ 0,1 푛 푉 푦푒푠/푛표 MIP*MIP: : efficient efficient interactive interactive verification verification with twowith provers two provers sharing entanglement • [Cleve-HTW’04] The class MIP* characterizes the complexity of optimizing over the sets 퐶⊗ 푚, 푘 Interactive proof systems as optimization problems 푃1 푃2 푎 푏 푥 푦 푧 ∈ 0,1 푛 푉 푦푒푠/푛표 Max acc(푉, 푧) = sup σ푥푦 휋 푥, 푦 σ Prob 푎, 푏 푥, 푦 strategy 푎푏:correct for 푥푦 푥 푦 = sup Σ푥푦푎푏 휋(푥, 푦)1푎푏: correct for 푥푦⟨휓|푃푎 ⊗ 푄푏 |휓⟩ strategy ∗ = 휔 (휆) for 휆푎푏푥푦 = 휋 푥, 푦 1푎푏: correct for 푥푦 Interactive proof systems as optimization problems 푃1 푃2 푎 푏 푥 푦 푧 ∈ 0,1 푛 푉 푦푒푠/푛표 MIP* : efficient interactive verification with two provers sharing entanglement • [Cleve-HTW’04] The class MIP* characterizes the complexity of optimizing over the sets 퐶⊗ 푚, 푘 • What can be said about problems in MIP*? • 휔∗(휆) uncomputable ↔ MIP* contains undecidable languages MIP* ⊇ RE • [Ito-V’12] MIP* contains all problems in MIP = NEXP • Proof shows that error correction-based probabilistically checkable proofs used in the proof of NEXP = MIP are sound in the presence of entanglement • [Natarajan-W’19] MIP* contains all problems in NEEXP • Proof leverages entanglement between the provers as a tool to aid verification • [Natarajan-JVYW’20] MIP* ⊇ RE by recursively applying technique from [NW’19] • [Turing’1936] RE contains the halting problem, which is undecidable • MIP* contains undecidable languages The compression theorem COMPRESS: 푉 ↦ 푉′ s.t.