Coherence in Property Testing: Quantum-Classical Collapses and Separations

Abstract: Understanding the power and limitations of classical and quantum information and how they differ is a fundamental endeavor. In property testing of distributions, a tester is given samples over a typically large domain ${0,1}^n$. An important property is the support size both of distributions [Valiant and Valiant, STOC'11], as well, as of quantum states. Classically, even given $2^{n/16}$ samples, no tester can distinguish distributions of support size $2^{n/8}$ from $2^{n/4}$ with probability better than $2^{-Θ(n)}$, even promised they are flat. Quantum states can be in a coherent superposition of states of ${0,1}^n$, so one may ask if coherence can enhance property testing. Flat distributions naturally correspond to subset states, $|φS \rangle=1/\sqrt{|S|}\sum{i\in S}|i\rangle$. We show that coherence alone is not enough, Coherence limitations: Given $2^{n/16}$ copies, no tester can distinguish subset states of size $2^{n/8}$ from $2^{n/4}$ with probability better than $2^{-Θ(n)}$. The hardness persists even with multiple public-coin AM provers, Classical hardness with provers: Given $2^{O(n)}$ samples from a distribution and $2^{O(n)}$ communication with AM provers, no tester can estimate the support size up to factors $2^{Ω(n)}$ with probability better than $2^{-Θ(n)}$. Our result is tight. In contrast, coherent subset state proofs suffice to improve testability exponentially, Quantum advantage with certificates: With poly-many copies and subset state proofs, a tester can approximate the support size of a subset state of arbitrary size. Some structural assumption on the quantum proofs is required since we show, Collapse of QMA: A general proof cannot improve testability of any quantum property whatsoever. We also show connections to disentangler and quantum-to-quantum transformation lower bounds.

• The paper demonstrates that quantum coherence alone does not improve property testing, as classical lower bounds persist without structured proofs.
• It employs spectral analysis of the Johnson scheme to establish tight indistinguishability between random subset states and Haar random states.
• Structured quantum certificates are shown to yield an exponential advantage, enabling efficient detection of support size and construction of pseudorandom quantum states.

Coherence in Property Testing: Quantum-Classical Collapses and Separations

Introduction and Problem Context

This work systematically investigates the information-theoretic limits of property testing with an emphasis on the quantum-classical divide. The focus is on distinguishing properties of high-dimensional distributions and quantum states, with support size as the main example. Classically, it is well-established that some natural testing tasks (such as estimating the support size of flat distributions over exponentially large domains) require exponentially many samples. The extension to the quantum field naturally raises the question of whether the phenomenon of quantum coherence—namely, access to quantum superpositions—allows for more efficient property testing of analogous quantum properties (such as support size in flat superposition states).

Main Results

Classical Lower Bounds

The paper rigorously demonstrates that, in the classical regime, even with $2^{n/16}$ samples from flat distributions over $\{0,1\}^n$, no tester can distinguish distributions with support sizes $2^{n/8}$ and $2^{n/4}$ with advantage exceeding $2^{-\Theta(n)}$. This holds even if the tester is supplemented with exponentially long certificates and can interact with multiple independent AM-type provers [(2411.15148), Theorem (2)]. The lower bound technique leverages connections to high-dimensional expander mixing and quantifies the entropy left after exposing certificates, showing that any advantage rapidly vanishes unless the sample or certificate complexity is essentially as large as the support size.

Quantum Lower Bound: Limitations of Coherence

A core question addressed is whether quantum coherence—access to coherent superpositions (subset states)—can circumvent the classical lower bounds. The main technical result is that coherence alone does not afford any advantage: even with $2^{n/16}$ copies, one cannot distinguish uniform superpositions over sets of size $2^{n/8}$ and $2^{n/4}$ with non-negligible probability. The argument is based on a sharp indistinguishability theorem between ensembles of random subset states and Haar-random states, under trace norm, using a spectral analysis of the Johnson association scheme [(2411.15148), Theorem (1), (3)]. This collapses the separation between classical and quantum settings in the absence of structured proofs.

Quantum Certificates: Exponential Advantage with Structure

The situation changes dramatically if the tester is furnished with quantum certificates—structured subset states serving as proofs. Under this model (QMA with structured subset-state witnesses), the authors demonstrate that with just polynomially many copies and polynomial-size subset-state certificates, the support size of a flat state can be multiplicatively approximated or detected as malicious with high probability. The testing protocol recursively certifies a chain of nested subset-states, using swap tests and iterative "support halving", to achieve an approximation within $(1 \pm 1/\mathrm{poly}(n))$ [(2411.15148), Theorem (3)]. This exponential separation is not replicable in the classical setting, where all certificates can yield at most a linear trade-off with sample complexity.

Limitations of General Proofs: Collapse of QMA

If arbitrary (unstructured) quantum proofs are allowed, they do not provide any improvement in the information-theoretic property testing regime for any property. That is, any QMA tester can be simulated, up to polynomial overhead, by a BQP tester using more copies of the input state but no proof at all. This collapse is formalized via a de-Merlinization procedure extending quantum union bounds for gap amplification [(2411.15148), Section 6]. Thus, only highly structured quantum proofs (e.g., subset states) provide meaningful quantum-classical separation in property testing.

Connections and Implications

• Pseudorandomness: The indistinguishability results directly yield new constructions of information-theoretic pseudorandom quantum states and pseudoentangled states, notably resolving an open problem by Ji, Liu, and Song regarding subset-state-based pseudorandomness [(2411.15148), Section 2.2].
• Quantum Cryptography: The new pseudorandom state families can be efficiently generated from classically pseudorandom permutations, providing base primitives for cryptographic protocols, quantum coin flipping, and efficient indistinguishability obfuscation.
• Quantum-to-Quantum Lower Bounds: The arguments extend to quantum-to-quantum state transformation lower bounds (e.g., the impossibility of absolute-value or conjugation transformations from multiple copies), providing a new method for deriving quantum information processing lower bounds from property testing impossibilities.
• Complexity Theoretic Landscape: The work draws precise boundaries between BQP, QMA, AM, and IP variants of property testing, both in their information-theoretic and computational incarnations, presenting a taxonomy of quantum versus classical and interactive versus non-interactive proof systems.

Technical Approach

The indistinguishability results for subset states versus Haar random states are established through spectral properties of the Johnson scheme, providing explicit bounds on the trace norm between $k$-fold copy state ensembles of flat subset states and Haar states. This analysis is tight except in the regime of very small support size, where collisions can be efficiently detected. For the lower bounds with AM-type classical proofs, the approach leverages the entropy contraction under random walks in high-dimensional expanders, showing that unless the certificate reveals nearly the whole set, the resulting distribution is still indistinguishable from uniform over small samples.

For the QMA (subset-state proof) testers, the algorithm inductively verifies a nesting sequence of supports using swap tests and acceptance thresholds, tolerating adversarial certificates by employing symmetry tests and bounding the potential for malicious deviation. This results in robust certification even when the proof is not perfectly honest.

Implications and Future Directions

The analysis rigorously establishes the boundary where quantum coherence alone does not enhance property testing, and where quantum proofs (with specific structure) enable exponentially more efficient testing than any classical or unstructured quantum protocol. Practically, this distinction has implications for the design of quantum verification protocols and quantum cryptographic primitives that rely on testing global properties of high-dimensional quantum systems.

Theoretically, these results provide a new paradigm for extracting quantum information processing lower bounds from property testing, suggest new regimes in which interactive proof complexity diverges (especially regarding the role of public-coin versus private-coin protocols), and prompt further inquiry into the power of unentangled multiprover quantum proofs.

Open directions include the complexity theory of property testing with separable versus general quantum proofs, seeking analogous bounds for more general quantum properties (beyond support size), and leveraging property testing techniques for disentangling and pseudoentanglement conjectures.

Conclusion

The work gives a comprehensive, tight characterization of coherence in property testing, revealing both fundamental limitations and powerful separations between classical, quantum, and interactive proof models. Coherence, without structure, manifests no quantitative advantage for property testing of support size; when combined with structured quantum certificates, it entails exponential quantum-classical separations. The broader implications unify themes in quantum property testing, cryptographic primitives, and quantum complexity theory, charting a systematic map of when and how quantum information exhibits superiority over the classical paradigm in the regime of property testing.