We consider the problem of quantum state certification, where one is given n copies of an unknown d-dimensional quantum mixed state ρ, and one wants to test whether ρ is equal to some known mixed state σ or else is є-far from σ. The goal is to use notably fewer copies than the Ω(d2) needed for full tomography on ρ (i.e., density estimation). We give two robust state certification algorithms: one with respect to fidelity using n = O(d/є) copies, and one with respect to trace distance using n = O(d/є2) copies. The latter algorithm also applies when σ is unknown as well. These copy complexities are optimal up to constant factors.
We consider the problem of quantum state certification, where one is given n copies of an unknown d-dimensional quantum mixed state ρ, and one wants to test whether ρ is equal to some known mixed state σ or else is є-far from σ. The goal is to use notably fewer copies than the Ω(d2) needed for full tomography on ρ (i.e., density estimation). We give two robust state certification algorithms: one with respect to fidelity using n = O(d/є) copies, and one with respect to trace distance using n = O(d/є2) copies. The latter algorithm also applies when σ is unknown as well. These copy complexities are optimal up to constant factors.
In the quantum state tomography problem, one wishes to estimate an unknown d-dimensional mixed quantum state ρ, given few copies. We show that O(d/ε) copies suffice to obtain an estimate ρ that satisfies ||ρ − ρ||F2 ≤ ε (with high probability). An immediate consequence is that O((ρ) · d/ε2) ≤ O(d2/ε2) copies suffice to obtain an ε-accurate estimate in the standard trace distance. This improves on the best known prior result of O(d3/ε2) copies for full tomography, and even on the best known prior result of O(d2log(d/ε)/ε2) copies for spectrum estimation. Our result is the first to show that nontrivial tomography can be obtained using a number of copies that is just linear in the dimension.
In this work, we consider the fundamental task of quantum state certification: given copies of an unknown quantum state ρ, test whether it matches some target state σ or is ε-far from it. For certifying d-dimensional states, Θ(d/ε2) copies of ρ are known to be necessary and sufficient. However, the algorithm achieving this complexity makes fully entangled measurements over all O(d/ε2) copies of ρ. Often, one is interested in certifying states to a high precision; this makes such joint measurements intractable even for low-dimensional states. Thus, we study whether one can obtain optimal rates for quantum state certification and related testing problems while only performing measurements on t copies at once, for some 1<t≪d/ε2. While it is well-understood how to use intermediate entanglement to achieve optimal quantum state learning, the only protocol known to achieve optimal testing is the one using fully entangled measurements.
Our main result is a smooth copy complexity upper bound for state certification as a function of t, which achieves a near-optimal rate at t=d2. In the high-precision regime, i.e., for ε<d1, this is a strict improvement over the entanglement used by the aforementioned optimal protocol. We also extend our techniques to develop new algorithms for the related tasks of mixedness testing and purity estimation, and show tradeoffs achieving the optimal rates for these problems at t=d2 as well. Our algorithms are based on novel reductions from testing to learning and leverage recent advances in quantum state tomography in a non-black-box fashion. We complement our testing upper bounds with a (nearly) matching lower bound that holds in the high-precision regime, implying that one must take t≥Ω(d2) to achieve the optimal rate for certification.
We consider the task of quantum state certification: given a description of a hypothesis state σ and multiple copies of an unknown state ρ, a tester aims to determine whether the two states are equal or є-far in trace distance. It is known that Θ(d/є2) copies of ρ are necessary and sufficient for this task, assuming the tester can make entangled measurements over all copies. However, these bounds are for a worst-case σ, and it is not known what the optimal copy complexity is for this problem on an instance-by-instance basis. While such instance-optimal bounds have previously been shown for quantum state certification when the tester is limited to measurements unentangled across copies, they remained open when testers are unrestricted in the kind of measurements they can perform.
A central task in quantum information science is state certification: testing whether an unknown state is є1-close to a fixed target state, or є2-far. Recent work has shown that surprisingly simple measurement protocols – comprising only single-qubit measurements – suffice to certify arbitrary n-qubit states. However, these certification protocols are not robust: rather than allowing constant є1, they can only positively certify states within є1=O(1/n) trace distance of the target. In many experimental settings, the appropriate error tolerance is constant as the system size grows, so this lack of robustness renders existing tests inapplicable at scale, no matter how many times the test is repeated.
A central task in quantum information science is state certification: testing whether an unknown state is ε1-close to a fixed target state, or ε2-far. Recent work has shown that surprisingly simple measurement protocols--comprising only single-qubit measurements--suffice to certify arbitrary n-qubit states [Huang, Preskill, Soleimanifar '25; Gupta, He, O'Donnell '25]. However, these certification protocols are not robust: rather than allowing constant ε1, they can only positively certify states within ε1=O(1/n) trace distance of the target. In many experimental settings, the appropriate error tolerance is constant as the system size grows, so this lack of robustness renders existing tests inapplicable at scale, no matter how many times the test is repeated.
Here we present robust certification protocols based on few-qubit measurements that apply to all but a O(2−n)-fraction of pure target states. Our first protocol achieves constant robustness, i.e. ε1=Θ(1), using a single O(logn)-qubit measurement along with single-qubit measurements in the Z or X basis on the other qubits. As a corollary of its robustness, this protocol also achieves constant (in n) copy complexity, which is optimal. Our second protocol uses exclusively single-qubit measurements and is nearly robust: ε1=Ω(1/logn). Our tests are based on a new uncertainty principle for conditional fidelities, which may be of independent interest.
We study the problems of quantum state certification, equivalence testing and independence testing. In certification, given samples of an unknown quantum state ρ and the description of a state σ, the goal is to test whether ρ=σ, or whether ρ and σ are far in a given distance measure. In equivalence testing, σ is also unknown and only accessible via samples. Independence testing decides whether ρAC=ρA⊗ρC, or is far from being a product. The sample complexities of these problems are now well-understood for a decision gap ε in trace distance: in the single-copy measurement setting with d-dimensional states, all three tasks can be solved using the same non-adaptive approach, which uses Θ(d3/2/ε2) samples and is optimal in general, even without adaptivity.
In this work, we consider decision gaps expressed in fidelity and study possible separations between these problems and how adaptivity can help. We prove that certification with respect to fidelity for a state σ of rank r does not benefit from adaptivity and requires Θ(r3/2/ε) samples. For equivalence testing and independence testing, we provide adaptive algorithms using O(min{d3/2/ε2,d9/4/ε}) and O(min{(dAdC)3/2/ε2,dA9/4dC3/4/ε}) samples, for dA≥dC, respectively. Our main technique is a framework that uses partial learning and a reduction to testing in ℓ2-distance, adapted from the distribution testing literature. We show that adaptivity matters for equivalence testing in fidelity by proving that Ω(1/ε2) samples are necessary in the non-adaptive case even for qubits, showing a separation from certification.