Whether physical reality contains an actual infinity is usually discussed philosophically. We ask a narrower, quantitative question: what would it cost a finite experiment to certify it? We combine sharp results from quantum information with physical resource bounds. (1) Certification budget. Resolving a game value to ε requires N ≳ ln(1/4δ) w(1−w)/ε² rounds (Bretagnolle–Huber), so the cost of certifying a Hilbert-space dimension D is governed by the gap ε_G(D) between the optimal value and the best dimension-D value. Take the unspecified constants in the known bounds to be one. Then the best known two-player dimension witness needs at least 10⁸⁶ rounds to certify a dimension beyond the Bekenstein bound of a 1 kg, 1 m laboratory (2.6×10⁴³ qubits). Such a laboratory can perform only 2.4×10⁶⁸ operations in the age of the universe. Beyond the de Sitter entropy (4.8×10¹²² qubits), the witness needs at least 10²⁴⁵ rounds, against the 10¹²⁰ operations of the universe. Spatial parallelisation does not help, because the Margolus–Levitin bound limits the total rate summed over all subsystems that share the energy. For the I₃₃₂₂ inequality the requirement is doubly exponential. (2) Embezzlement asymptotics. The optimal fidelity of van Dam–Hayden embezzlement of one EPR pair is F_n = (H_{n/2} + σ_{n/2})/H_n. Hence 1 − F_n = (ln 2 − σ)/H_n + O(1/(n ln n)) with σ = 0.2992041. This holds for all n. It is confirmed to 10⁻⁶ for dimensions up to 2²⁶, and to 6×10⁻⁸ for odd n = 10⁷ + 1. The infinite-dimensional resource behind these witnesses is approached only logarithmically. (3) The Tsirelson loophole. MIP*=RE implies that commuting-operator correlations can exceed every finite-dimensional value by a finite gap, which finite statistics could in principle detect. However, the set of true bounds ω*(G) ≤ t is Π⁰₁ but not computably enumerable. No consistent theory therefore proves all of them, and this route is limited by provability rather than by statistics. In practice, no separating game with small question and answer sets and a computed gap is known. (4) Thermodynamic infinities. Kaufman's exact formula for the two-dimensional Ising model on a finite torus shows that a 1 cm sample rounds the critical singularity at |K − K_c|/K_c ~ 4×10⁻⁹. The logarithmic amplitude of the specific-heat peak equals that of the bulk singularity. The orbital helium lambda-point experiment, which reached |t| ≈ 10⁻⁹, stays about four orders of magnitude above the finite-size scale of its sample. These results do not show that nature is finite. They show the following: apart from one route whose cost is logical rather than statistical, finiteness bounded by holography cannot be refuted from inside a universe that respects that bound. Code, data and figures: https://github.com/Ruqing1963/certify-infinity
Whether physical reality contains an actual infinity is usually discussed philosophically. We ask a narrower, quantitative question: what would it cost a finite experiment to certify it? We combine sharp results from quantum information with physical resource bounds. (1) Certification budget. Resolving a game value to ε requires N ≳ ln(1/4δ) w(1−w)/ε² rounds (Bretagnolle–Huber), so the cost of certifying a Hilbert-space dimension D is governed by the gap ε_G(D) between the optimal value and the best dimension-D value. Take the unspecified constants in the known bounds to be one. Then the best known two-player dimension witness needs at least 10⁸⁶ rounds to certify a dimension beyond the Bekenstein bound of a 1 kg, 1 m laboratory (2.6×10⁴³ qubits). Such a laboratory can perform only 2.4×10⁶⁸ operations in the age of the universe. Beyond the de Sitter entropy (4.8×10¹²² qubits), the witness needs at least 10²⁴⁵ rounds, against the 10¹²⁰ operations of the universe. Spatial parallelisation does not help, because the Margolus–Levitin bound limits the total rate summed over all subsystems that share the energy. For the I₃₃₂₂ inequality the requirement is doubly exponential. (2) Embezzlement asymptotics. The optimal fidelity of van Dam–Hayden embezzlement of one EPR pair is F_n = (H_{n/2} + σ_{n/2})/H_n. Hence 1 − F_n = (ln 2 − σ)/H_n + O(1/(n ln n)) with σ = 0.2992041. This holds for all n. It is confirmed to 10⁻⁶ for dimensions up to 2²⁶, and to 6×10⁻⁸ for odd n = 10⁷ + 1. The infinite-dimensional resource behind these witnesses is approached only logarithmically. (3) The Tsirelson loophole. MIP*=RE implies that commuting-operator correlations can exceed every finite-dimensional value by a finite gap, which finite statistics could in principle detect. However, the set of true bounds ω*(G) ≤ t is Π⁰₁ but not computably enumerable. No consistent theory therefore proves all of them, and this route is limited by provability rather than by statistics. In practice, no separating game with small question and answer sets and a computed gap is known. (4) Thermodynamic infinities. Kaufman's exact formula for the two-dimensional Ising model on a finite torus shows that a 1 cm sample rounds the critical singularity at |K − K_c|/K_c ~ 4×10⁻⁹. The logarithmic amplitude of the specific-heat peak equals that of the bulk singularity. The orbital helium lambda-point experiment, which reached |t| ≈ 10⁻⁹, stays about four orders of magnitude above the finite-size scale of its sample. These results do not show that nature is finite. They show the following: apart from one route whose cost is logical rather than statistical, finiteness bounded by holography cannot be refuted from inside a universe that respects that bound. Code, data and figures: https://github.com/Ruqing1963/certify-infinity
MIP*=RE implies that correlations of commuting observables on an infinite-dimensional Hilbert space (C_qc) can lie at a finite distance from every limit of finite-dimensional tensor-product correlations (C_qa). Such correlations would let a finite experiment detect an actual infinity with a modest number of rounds. In earlier work this was identified as the only route that escapes the resource bounds on certifying Hilbert-space dimension. Here we ask whether relativistic quantum field theory allows Bell correlations in this gap. Under standard structural assumptions, it does not. Suppose the theory has the split property, which follows from the Buchholz–Wichmann nuclearity condition and holds for free fields. Then the correlations of local measurements in strictly spacelike separated regions are tensor-product correlations and lie in C_qa. If the local algebras are hyperfinite, the same holds for regions that are merely spacelike, with no gap between them. Buchholz, D'Antoni and Fredenhagen showed this hyperfiniteness under a scaling-limit assumption. We give a self-contained proof based on semidiscreteness. Alice's measurements are compressed through matrix algebras by unital completely positive maps, and the resulting correlations converge to the original ones. The same applies to the type II₁ observer algebra of a de Sitter static patch, constructed by Chandrasekaran, Longo, Penington and Witten. This algebra is a corner of a crossed product of such an algebra by ℝ, so it is again injective. Within these frameworks, local measurements in spacelike separated regions cannot realise the gap found by MIP*=RE. This excludes the fourth of the logical possibilities listed by Cabello, Quintino and Kleinmann. We then bound, from the state side, how much dimension a Bell test can certify. Compressing each party to the support of its ε-smoothed reduced state reproduces all correlations to within 2(√ε_A + √ε_B) + ε_A + ε_B. Since N rounds resolve correlations only to N^(-1/2), an experiment with N rounds certifies at most the smooth max-entropy of the local state at smoothing ε ~ 1/N. We compute this effective dimension exactly for the vacuum of a lattice free scalar field. The single-particle entanglement energies form two towers, one per block endpoint, with a spacing set by the mass, so only a handful of modes are active. The effective dimension follows an area law: it is at most 9 bits for a 128-site block at ε = 10⁻⁶, against 553 bits from naive counting. As ε decreases from 10⁻¹ to 10⁻¹², it grows only from 2.3 to 13.1 bits. A product bound over thermal modes shows that the growth is polylogarithmic in 1/ε for any regularised block. In the continuum, the corresponding object is the canonical splitting type I factor of Doplicher and Longo, whose vacuum entropy coincides with the reflected entropy. Evaluating its smooth max-entropy as a function of the splitting distance is left for future work. Together with the certification budget of the companion paper, the result gives a coherent picture. Physics uses infinite-dimensional Hilbert spaces and type III algebras. Yet, within these frameworks, the correlations that local measurements can produce are reproduced to any finite precision by finite-dimensional models. The size of these models is controlled by entanglement entropy. Code, data and figures: https://github.com/Ruqing1963/qft-tsirelson-loophole
Quantum information has the potential to disrupt the present computational landscape. Much of this potential rests on the existence of efficient quantum algorithms for classically intractable problems and of quantum cryptographic protocols for tasks that are provably impossible to realize classically. At the heart of many quantum advantages is one of the most counterintuitive features of quantum mechanics, known as entanglement. The central motivating question of this thesis is the following: if quantum devices will perform tasks that are beyond the reach of classical devices, can we hope to certify that they are performing these tasks correctly? Bell's theorem, a landmark result in physics, provides a partial answer to this question: it asserts that measurements on spatially isolated, but entangled, particles can result in outcomes that are correlated in a way that cannot be explained by any local hidden variable theory (such as Newtonian physics). A direct operational consequence of this theorem is that one can devise a statistical test to certify the presence of entanglement (and hence of genuine quantumness). Remarkably, nature allows us to take this certification one step further: in some cases, the correlation of measurement outcomes is sufficient to single out a unique quantum setup compatible with this correlation. This phenomenon is often referred to as self-testing, and is the central topic of this thesis. In the first part of this thesis, we review the basic terminology and results in the theory of self-testing. We then explore a concrete application to the problem of verifiably delegating a quantum computation. Our main technical contribution is a test that robustly certifies products of single-qubit Clifford measurements on many EPR pairs. We employ this test to obtain a protocol which allows a classical user to verifiably delegate her quantum computation to two spatially isolated quantum servers. The overall complexity of our protocol is near-optimal, requiring resources that scale almost linearly in the size of the circuit being delegated. In the second part of this thesis, the driving question is the following: what is the class of quantum states and measurements that can be certified through self-testing? Does self-testing only apply to a few special cases, like EPR pairs or copies of EPR pairs, or are these instances of a more general phenomenon? One of the main results of this thesis is that we settle this question for the case of bipartite states. We show the existence of a self-testing correlation for any pure bipartite entangled state of any finite local dimension. We then move on to explore the multipartite case, and we show that a significantly larger class of states can be self-tested than was previously known. This includes all multipartite partially entangled GHZ states, and more generally all multipartite qudit states which admit a Schmidt decomposition. In the final part of this thesis, we explore connections of the theory of self-testing to basic questions about entanglement and quantum correlation sets. In particular, we set out to understand the expressive power of infinite-dimensional quantum systems. We consider two questions: can spatially isolated quantum systems of infinite dimension produce correlations that are unattainable by finite-dimensional systems? Does there exist a correlation that cannot be attained exactly by spatially isolated quantum systems (not even infinite-dimensional ones), but can be approximated arbitrarily well by a sequence of finite or infinite-dimensional systems? The first question was posed by Tsirelson in 1993, and its answer has been elusive. One of the main results of this thesis is a resolution of this question. The second question is better known as the "non-closure of the set of quantum correlations", and was answered affirmatively in a breakthrough of Slofstra. We give a new elementary proof of this result which leverages one of our self-testing results and a phenomenon known as embezzlement.
In a nonlocal game of quantum value one that no finite-dimensional strategy wins, near-perfect play needs growing dimension, which separable states can have. We ask how much usable entanglement it needs: the demand at loss ε is the least one-way distillable entanglement of the original state of a strategy of loss at most ε. For synchronous games with tested diagonals and for constraint-variable consistency games, demand diverges exactly when no finite-dimensional strategy is perfect; then, for any number of Bell pairs and any fidelity below one, every sufficiently good mixed strategy yields that many at that fidelity by local isometries. For rational synchronous or binary constraint games, a loss guaranteeing this is computable under a promise, but boundedness of demand is undecidable among games of value one. Demand is at least logarithmic if near-winners have a sign relation and an even dilation mirrored by both players, as in a game from Abels' group. For agreement games the directions agree up to a factor three in loss, optimal among Bob-fixed deterministic local symmetrizations. The geometry of the group behind a game, not its presentation, sets the rates. Fixed games demand αlog(1/ε) for every α=21∑ilogmiqi with qi≥2 a power of two dividing mi, a dense set containing all positive rationals and some transcendentals, such as 21log62; a configuration-lamp game demands ε−1+o(1); Pauli lamps on the points of Houghton's three rays give demand between ε−0.0690 and ε−1/4. If group relations hold exactly and Alice matches Bob to error e, demand counts qubits addressable by short words: Θ(1/e), Θ(e−k) for integers k≥1, or 2Θ(1/e). Conditional on our attainment and tail bounds, the I3322 optimum needs bounded usable entanglement.
Can uncomputability occur in the resource law of one fixed finite nonlocal game, rather than in a problem whose input is a game? We construct a finite two-player Boolean constraint-system game with rational question probabilities, quantum value one, and no finite-dimensional perfect strategy. If D(ε) is the least local dimension needed to win with probability at least 1−ε, then limn→∞log2D(2−n)/n=(3+θK)/4, where θK=∑e∈K4−(e+1) and K is the halting set. This exponent has the Turing degree of K. The construction combines finite operator relations for three Cantor clocks, a counter program with forced initialization, and a conditional clock that charges dimension for halting inputs. A Boolean compiler preserves weighted precision linearly and converts flat models to strategies with quadratic loss. The lower bound holds in every density matrix. Explicit finite models attain the same rate. The flag-density lower bound uses compactness without an effective modulus; convergence to the dimension exponent has no computable certified rate.
How expensive is an optimum that a finite quantum system can approach but never attain? For a fixed finite rational nonlocal game of quantum value one, write A_G(n)=log2 D_G(2^(−n)), where D_G is the least local dimension at loss 2^(−n). We prescribe log2 A_G(n) at every logarithmic-space, nondecreasing unit-increment scale F with F(n)/log n → ∞, up to an additive O(log n+log²(F(n)+2)) error, and match the one-way distillable entanglement of the original mixed strategy state. At every asymptotically flat such scale, the positive finite lower and upper normalized dimension endpoints are exactly the ordered pairs with lower endpoint in Σ⁰₂ and upper endpoint in Π⁰₂; existing positive limits are exactly Δ⁰₂. A single uniform spectral-tail construction supplies the density-free profile law behind these results. It also classifies stretched-scale pairs in the unit square and gives quasi-polynomial dimension windows of width three in the exponent. A computable unit-increment target outside every rational game's profile shows that computability alone is insufficient. We locate the boundaries through a quantitative locality cap, an honest-linear compression cap, and an explicitly conditional route to finite exponents above one. A separate exact-marginal correlation theorem identifies a Farey law and its Diophantine endpoint curve. The dimension arithmetic necessity applies to all rational games; the operational realization statements concern the constructed games.
How much entanglement does near-optimal play of a two-player nonlocal game require? We give an explicit binary linear system game, with 1687 equations in 1993 variables and at most three variables in each equation, whose quantum value is one, and in which every strategy that wins with probability 1−ε needs entanglement entropy ε−1+o(1), hence local dimension exp(ε−1+o(1)). The same bound holds for the entanglement of formation of mixed strategies, and a strategy on a maximally entangled state of local dimension exp(Cε−1log(1/ε)) suffices, so the exponent is sharp. For explicit two-player games of quantum value one, the best previous lower bound was polynomial in 1/ε. The game comes from a finitely generated, elementary amenable group: a lamplighter whose lamps are Pauli pairs indexed by finite sets of points on the three rays on which Houghton's group acts. Exponentially many of these pairs share one central involution, yet the relation that any two of them commute can be derived from a fixed finite set of relations in a number of steps only quasi-polynomial in the number of points. Two tools may be of independent use. The first counts qubits from approximate Pauli relations measured in an arbitrary state, with the errors of different pairs adding in squares. The second identifies, in both directions and at the natural precision ε, the near-optimal strategies of a linear system game with approximate representations of its solution group measured in the players' actual state, so that no power of ε is lost by passing to a flat state. The same tools give sharp rates for two earlier examples: the CHSH(n) games need Θ(min(n,1/ε)) ebits, and the linear system game of Slofstra and Vidick needs local dimension Θ(ε−1/2).