Quantum state testing with restricted measurements
Yuhan Liu, Jayadev Acharya
TL;DR
This work establishes the copy complexity of quantum state testing under restricted unentangled measurements by introducing the measurement information channel (MIC) framework that links distinguishability to the MIC’s spectral norms. It provides tight bounds for randomized non-adaptive k-outcome measurements, ns=Θ(d^2/(ε^2√min{ab,d})), and for fixed non-adaptive k-outcome measurements, ns=Θ(d^3/(ε^2 min{ab,d})), revealing a pronounced separation in which randomness yields sublinear copy complexity while fixed schemes can be substantially harder. The paper also develops concrete algorithms for randomized ab-outcome tests and fixed Pauli/ab-outcome tests, including a Pauli-based Θ(d^3/ε^2) bound and a general ab=d design achieving Θ(d^2/ε^2), as well as an ab<d strategy via η-simulation with ns=Θ(d^3/(ab ε^2)). By extending Paninski-style lower bounds to the quantum setting, the authors illuminate the critical role of randomness and measurement finiteness in quantum state certification, with practical implications for distributed quantum devices and finite-outcome measurement architectures.
Abstract
We study quantum state testing where the goal is to test whether $ρ=ρ_0\in\mathbb{C}^{d\times d}$ or $\|ρ-ρ_0\|_1>\varepsilon$, given $n$ copies of $ρ$ and a known state description $ρ_0$. In practice, not all measurements can be easily applied, even using unentangled measurements where each copy is measured separately. We develop an information-theoretic framework that yields unified copy complexity lower bounds for restricted families of non-adaptive measurements through a novel measurement information channel. Using this framework, we obtain the optimal bounds for a natural family of $k$-outcome measurements with fixed and randomized schemes. We demonstrate a separation between these two schemes, showing the power of randomized measurement schemes over fixed ones. Previously, little was known for fixed schemes, and tight bounds were only known for randomized schemes with $k\ge d$ and Pauli observables, a special class of 2-outcome measurements. Our work bridges this gap in the literature.