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.
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.
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 revisit the basic problem of quantum state certification: given copies of unknown mixed state ρ∈Cd×d and the description of a mixed state σ, decide whether σ=ρ or ∥σ−ρ∥tr≥ε. When σ is maximally mixed, this is mixedness testing, and it is known that Ω(dΘ(1)/ε2) copies are necessary, where the exact exponent depends on the type of measurements the learner can make [OW15, BCL20], and in many of these settings there is a matching upper bound [OW15, BOW19, BCL20]. Can one avoid this dΘ(1) dependence for certain kinds of mixed states σ, e.g. ones which are approximately low rank? More ambitiously, does there exist a simple functional f:Cd×d→R≥0 for which one can show that Θ(f(σ)/ε2) copies are necessary and sufficient for state certification with respect to any σ? Such instance-optimal bounds are known in the context of classical distribution testing, e.g. [VV17]. Here we give the first bounds of this nature for the quantum setting, showing (up to log factors) that the copy complexity for state certification using nonadaptive incoherent measurements is essentially given by the copy complexity for mixedness testing times the fidelity between σ and the maximally mixed state. Surprisingly, our bound differs substantially from instance optimal bounds for the classical problem, demonstrating a qualitative difference between the two settings.
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 we are given the description of a mixed state σ∈Cd×d,n copies of a mixed state ρ∈Cd×d, and ε>0, and we are asked to determine whether ρ=σ or whether ∥ρ−σ∥1>ε. When σ is the maximally mixed state d1Id, this is known as mixedness testing. We focus on algorithms which use incoherent measurements, i.e. which only measure one copy of ρ at a time. Unlike those that use entangled, multi-copy measurements, these can be implemented without persistent quantum memory and thus represent a large class of protocols that can be run on current or near-term devices. For mixedness testing, there is a folklore algorithm which uses incoherent measurements and only needs O(d3/2/ε2) copies. The algorithm is non-adaptive, that is, its measurements are fixed ahead of time, and is known to be optimal for non-adaptive algorithms. However, when the algorithm can make arbitrary incoherent measurements, the best known lower bound is only Ω(d4/3/ε2) [5], and it has been an outstanding open problem to close this polynomial gap. In this work: •We settle the copy complexity of mixedness testing with incoherent measurements and show that Ω(d3/2/ε2) copies are necessary. This fully resolves open questions of [15] and [5].•We show that the instance-optimal bounds for state certification to general σ first derived in [7] for non-adaptive measurements also hold for arbitrary incoherent measurements. Qualitatively, our results say that adaptivity does not help at all for these problems. Our results are based on new techniques that allow us to reduce the problem to understanding the concentration of certain matrix martingales, which we believe may be of independent interest.
In recent work Phys. Rev. X 16, 031057 (2026), we proposed a self-consistent approach to entanglement certification based on generalized noncontextuality. It requires no prior characterization of the measurement devices and, given access to all local measurements, can certify every entangled state in a Bell circuit. Here, we study the complexity of this protocol in a finite experiment. At fixed local dimension d, we show that the optimal number of distinct local effects needed to reach a trace-distance \textit{entanglement resolution} γ is Θd(γ−(d−1)). This scaling is necessary even when the measurements are tailored to the target state. Independent Haar-random projective measurements instead require Θd(γ−(d−1)log(1/γ)) effects. Finally, we show that, once the exact operational identities are known, Θd(γ−2log(1/δ)) copies of the target state are necessary and sufficient for certification with error probability at most δ. Determining the optimal sample complexity when those identities must instead be inferred from the same finite target-state data remains open.
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.