Some applications of hypercontractive inequalities in quantum information theory

1 Introduction

• Hypercontractivity generalizes the contraction property of the boolean-cube noise operator by allowing its output Lq norm to be bounded by an input Lp norm for q > p.The noise operator suppresses higher-order Fourier coefficients, and applying noise smooths extreme values.Source: §1 · pp. 1–2

1.1 Hypercontractivity in quantum information

• The paper uses hypercontractivity in three settings to give proofs concerning local-measurement bias, k-local Hamiltonian spectra, and multiplayer XOR-game bias.The settings are the real n-sphere, Hermitian operators on n qubits, and the boolean cube.Source: §1.1 · pp. 2–3

1.1.1 Bias achievable by local measurements

• A real n-sphere hypercontractive proof reproduces Lancien and Winter’s bound for multipartite 4-design measurements, with a worse constant.The authors describe the alternative proof as based on hypercontractivity on the real n-sphere.Source: §1.1.1 · p. 3

1.1.2 Spectral concentration for k-local Hamiltonians

• For a k-local Hermitian operator M on n qubits with tr M^2 = 2^n, Theorem 4 gives a bound on its spectrum for t ≥ (2e)^(k/2).Source: §1.1.2 · pp. 3–4

1.1.3 General bounds on multiplayer XOR games

• Replacing a component of the proof behind the general XOR-game bias bound with Corollary 2 yields a simple, explicit bound.The section states that Theorem 5 implies a special case of Aaronson and Ambainis’s conjecture about influential variables in bounded low-degree boolean-cube polynomials [1].Source: §1.1.3 · p. 4

2 The bias of local 4-design measurements

• Theorem 6 gives a bias bound for a single 4-design measurement, proved using the fourth-moment method and Hölder’s inequality.The proof uses the design property to replace measurement moments with sphere integrals, then relates the resulting quantities through hypercontractivity.Source: §2 · p. 5

2.1 Hypercontractivity and spherical harmonics

• Writing a quantum state in real coordinates makes f(ξ) = tr ∆|ψ⟩⟨ψ| a homogeneous degree-2 polynomial, to which spherical hypercontractivity applies.The Poisson semigroup on the sphere relates norms of low-degree polynomials; the section states a corresponding q-norm bound for degree-d polynomials.Source: §2.1 · p. 6

2.2 Hypercontractivity on k copies of the sphere

• Tensorizing spherical hypercontractivity yields Theorem 9 for k-fold products of local 4-designs, reproducing Lancien and Winter’s result [43] with slightly worse constants.The argument uses multiplicativity of the Poisson operator norm to extend the inequality to k copies of the sphere.Source: §2.2 · pp. 7–9
• The multipartite bound is described as almost optimal: the cited example has bias that decreases exponentially with k.The paper attributes the exponential prefactor in Theorem 9 to multiplicativity of the Poisson semigroup’s operator norm.Source: §2.2 · p. 9
• The method gives bounds for all q-norms, whereas [43] considered the L4 norm; for k = 1, earlier methods can give better constants.The paper notes that the broader q-norm bounds may be useful for studying q-designs with q > 4.Source: §2.2 · p. 9

3 Spectral concentration for k-local Hamiltonians

• The depolarizing channel on n qubits satisfies a noncommutative hypercontractive inequality, yielding Lp-to-Lq norm bounds for k-local operators.In the Pauli expansion, k-local operators have no coefficients of weight greater than k.Source: §3 · pp. 9–10
• Theorem 15 gives a less precise concentration bound for a k-local Hermitian operator on n qubits with ∥M∥2 = 1, without interaction-geometry assumptions; its stated condition is t ≥ (2e)^(k/2).Source: §3 · p. 10
• Theorem 15 implies that a random state has exponentially small probability of having high energy under a k-local Hamiltonian.The paper further uses spectral concentration to bound how likely a random initial state is to change significantly under time evolution.Source: §3 · pp. 10–11
• Every nonzero k-local Hermitian operator on n qubits has rank at least 2^(n−2.89k).The stated exact exponent is n − (2 log2 e)k.Source: §3 · p. 11

Bohnenblust–Hille inequality

• The Bohnenblust–Hille inequality has C_k=O(klog2 e)≈O(k1.45) and yields a general lower bound on XOR-game bias.The section uses the inequality to turn bounds on a multilinear form’s coefficient norm into a bias bound.Source: §4 · p. 12

XOR-game bias

• β(G)=Ω(k^-3/2 n^-(k−1)/2) for a k-player XOR game with n inputs per player.The game corresponds to a multilinear form whose supremum norm is its bias.Source: §4 · p. 12

Hypercontractive proof

• The proof uses Corollary 2’s hypercontractive estimate as a black box, replacing an equivalent step in Pellegrino and Seoane-Sepúlveda’s proof with worse constants.The author says this makes the proof somewhat simpler and more transparent.Source: §4 · pp. 12–13
• When k is not a power of two, increasing it to the next power of two incurs at most a constant increase in C_k.Source: §4 · p. 14

Influence bound

• For multilinear forms with ∥f∥∞≤1 and coefficients ˆf_i1,...,ik=±α, every variable has influence Ω(Var(f)^2/k^3).This is a special case of the Aaronson–Ambainis conjecture discussed in the section.Source: §4 · p. 14

• The note presents several quantum-information results as provable using hypercontractive inequalities, with proofs that may be more concise and intuitive but less explicit and sometimes yield worse constants.Source: §5 · p. 15