Second-order asymptotics for quantum hypothesis testing in settings beyond i.i.d. - quantum lattice systems and more

Quantum Hypothesis Testing

• Asymmetric quantum hypothesis testing minimizes type II error subject to a type I error constraint, unlike symmetric testing, which minimizes their sum.The two hypotheses are quantum states, and testing uses a measurement to decide which state was received.Source: Quantum Hypothesis Testing · p. 2

Quantum Stein’s lemma and its refinement: asymptotic i.i.d. setting

• The i.i.d. Stein limit gives a discontinuous threshold at relative entropy, because it retains only the linear-in-n term of the type II error exponent.Second-order asymptotics retain the order-√n term, and the minimal type I error then varies smoothly with the second-order parameter.Source: Quantum Stein’s lemma and its… · p. 3
• Quantum second-order asymptotics had been established only for i.i.d. resources, leaving the speed of finite-system convergence beyond i.i.d. open.This question matters for hypothesis testing, which underlies tasks including classical-information transmission through a quantum channel.Source: Quantum Stein’s lemma and its… · p. 4

Our contribution

• The extension treats non-tensor-power state sequences whose extractable information grows linearly with specified weights, including high-temperature spin Gibbs states and quasi-free fermionic lattice gases.The i.i.d. case is included, recovering the results of [44] and.Source: Our contribution · pp. 4–5
• For one-dimensional fermion Gibbs states at different temperatures, the minimal type I error has the i.i.d. second-order dependence, with relative entropy and variance rates and lattice size replacing their i.i.d. counterparts.The states correspond to the same Hamiltonian and are restricted to finite lattice subsets.Source: Our contribution · p. 5

• States are density matrices on a finite-dimensional Hilbert space, with notation set for operators, supports, and projections.Source: §2 · p. 5
• Self-adjoint operators have spectral decompositions, and A+ denotes the positive part of A.Source: §2 · p. 5
• For s in [0,1), Ψ_s is related to quantum Rényi entropy, and it equals −∞ when the states have orthogonal supports.Source: §2 · pp. 5–6
• Using the Hilbert-Schmidt operator space, the authors define relative modular operators for faithful states, whose supports coincide.The operator space uses the Hilbert-Schmidt scalar product and identifies operators with their left-multiplication maps.Source: §2 · pp. 6–7
• The spectral measure of −log Δρ|σ links Ψ_s to a classical random variable’s cumulant-generating function, enabling classical probability tools in the proof.Source: §2 · pp. 7–8
• The modular-operator formulation is used here in finite dimensions, while the authors describe its construction and a key relation as extending to infinite dimensions.Source: §2 · pp. 6, 8

3 Second order asymptotics for hypothesis testing

• The framework uses faithful state sequences on finite-dimensional spaces and an increasing positive weight sequence wn tending to infinity.The theorem's conditions require analytic extensions of the logarithmic moment generating functions and limits of their weight-normalized values.Source: §3 · pp. 8–9
• Theorem 1 bounds the second-order behavior on either side of a threshold t*, with a sub-√x slack in the stated bounds.The threshold is defined using the quantum information variance rate and the standard normal cumulative distribution function.Source: §3 · pp. 9–10
• When the relative-entropy deviation ΔnD is o(√wn), Corollary 1 gives the simplified second-order behavior covered by Theorem 1.The relative entropy has leading scale wn d(ρ̂, σ̂), with an o(wn) remainder.Source: §3 · p. 10

3.1 Ingredients of the proof of Theorem 1

• The proof applies Bryc’s theorem to obtain convergence in distribution, using analyticity, normalized limits, and a uniform bound as its conditions.For i.i.d. sums, the paper notes that analyticity near the origin suffices for Bryc’s conditions with wn=n, a stronger condition than existence of a second moment.Source: §3.1 · p. 11
• A symmetric-testing lemma bounds minimum total error through expectations of a classical random variable, including when the quantum states do not commute.The proof uses its lower bound; the lemma also supplies an upper bound.Source: §3.1 · pp. 12–13

3.2 Proof of Theorem 1

• In the proof of the upper bound, convergence of the test’s type I error to a normal probability below ε ensures that the test eventually satisfies the error constraint.The resulting bound establishes the corresponding part of Theorem 1.Source: §3.2 · p. 15

4 Examples

• The examples satisfy the paper’s condition for applying Theorem 1, and follow examples in under stronger conditions than the first-order Stein lemma requires.This frames the i.i.d. and lattice-system cases that follow.Source: §4 · p. 15

4.1 Quantum i.i.d. states

• In the i.i.d. case, Theorem 1 and Corollary 1 follow from the analysis; earlier derivations by Tomamichel and Hayashi [44] and Li used the CLT.The Berry–Esseen theorem can also provide information about the third-order term in −log β_n(ε).Source: §4.1 · p. 16

4.2 Quantum Spin Systems

• At sufficiently high temperatures, finite-volume Gibbs states from translation-invariant, finite-range interactions satisfy the paper’s condition with weights equal to the volume |Λ_n|.The condition is stated for inverse temperatures β1 and β2 small enough.Source: §4.2 · p. 18
• The spin-system proof uses analyticity and convergence of functions derived from cluster expansions to establish the required properties.The argument connects the target function to a cluster expansion whose weights are analytic and bounded under suitable conditions.Source: §4.2 · pp. 18–19

4.3 Free fermions on a lattice

• For shift-invariant fermionic Gibbs states restricted to finite lattice regions, the sequence satisfies the paper’s condition, and Corollary 1 applies when the stated additional assumption holds.The local restrictions are faithful when the symbols have essential range in [δ, 1 − δ].Source: §4.3 · pp. 23–24
• For free fermions under the stated condition, the second-order term is n(t*_2(ε)+C) in d=2 and n^(d−1)C in d≥3.For d=1, Remark 5 states that the assumptions of Corollary 1 hold when Equation (4.22) is satisfied.Source: §4.3 · p. 24

A Vitali’s theorem and Bryc’s theorem

• The appendix presents these results as the one-variable versions of Vitali’s theorem and Bryc’s theorem.The section explicitly states that it proves both results in this setting.Source: A Vitali’s theorem and Bryc’s theorem · pp. 29–30
• Under its assumptions, the theorem establishes an analytic extension of the limiting function and uniform convergence of the sequence and its derivatives on compact subsets.The proof uses equicontinuity and Arzelà–Ascoli to obtain subsequential limits, then uniqueness to establish convergence.Source: A Vitali’s theorem and Bryc’s theorem · p. 30
• When the analytic functions are normalized cumulant-generating functions under the stated conditions, the scaled random variables converge in law to N(0, H''(0)).The proof establishes convergence of cumulants and moments, then uses that the normal distribution is determined by its moments.Source: A Vitali’s theorem and Bryc’s theorem · p. 30

B Proof of Lemma 2 and Corollary 2

• The proof establishes a bound using the trace inequality in Equation (B.1), then obtains the lower bound in Equation (3.11).The upper-bound argument also invokes Markov’s inequality.Source: B Proof of Lemma 2 and Corollary 2 · pp. 31–32
• For the lower bound, the minimizing test can be taken as an orthogonal projection, and the proof uses spectral decompositions of A and B.The projection test is denoted P in the proof.Source: B Proof of Lemma 2 and Corollary 2 · p. 31

C Proof of Proposition 1

• The proof constructs a test by projecting onto the support of a positive operator, then uses spectral subspaces of ρ and σ to analyze it.The construction considers projections of vectors from eigenspaces of ρ onto eigenspaces of σ.Source: C Proof of Proposition 1 · p. 32
• Positivity of Δ_ρ|σ establishes the first inequality in Equation (3.14).Source: C Proof of Proposition 1 · p. 33
• The proof groups eigenspaces of ρ into mutually orthogonal subspaces using Gram–Schmidt orthogonalization, then concludes the argument.The constructed subspaces satisfy the stated cumulative decomposition of the grouped eigenspaces.Source: C Proof of Proposition 1 · pp. 33–34