Given $n$ copies of an unknown quantum state $ρ\in\mathbb{C}^{d\times d}$, quantum state certification is the task of determining whether $ρ=ρ_0$ or $\|ρ-ρ_0\|_1>\varepsilon$, where $ρ_0$ is a known reference state. We study quantum state certification using unentangled quantum measurements, namely measurements which operate only on one copy of $ρ$ at a time. When there is a common source of shared randomness available and the unentangled measurements are chosen based on this randomness, prior work has shown that $Θ(d^{3/2}/\varepsilon^2)$ copies are necessary and sufficient. This holds even when the measurements are allowed to be chosen adaptively. We consider deterministic measurement schemes (as opposed to randomized) and demonstrate that $Θ(d^2/\varepsilon^2)$ copies are necessary and sufficient for state certification. This shows a separation between algorithms with and without shared randomness. We develop a unified lower bound framework for both fixed and randomized measurements, under the same theoretical framework that relates the hardness of testing to the well-established Lüders rule. More precisely, we obtain lower bounds for randomized and fixed schemes as a function of the eigenvalues of the Lüders channel which characterizes one possible post-measurement state transformation.
Given n copies of an unknown quantum state ρ∈Cd×d, quantum state certification is the task of determining whether ρ=ρ0 or ∥ρ−ρ0∥1>ε, where ρ0 is a known reference state. We study quantum state certification using unentangled quantum measurements, namely measurements which operate only on one copy of ρ at a time. When there is a common source of shared randomness available and the unentangled measurements are chosen based on this randomness, prior work has shown that Θ(d3/2/ε2) copies are necessary and sufficient. This holds even when the measurements are allowed to be chosen adaptively. We consider deterministic measurement schemes (as opposed to randomized) and demonstrate that Θ(d2/ε2) copies are necessary and sufficient for state certification. This shows a separation between algorithms with and without shared randomness. We develop a unified lower bound framework for both fixed and randomized measurements, under the same theoretical framework that relates the hardness of testing to the well-established Lüders rule. More precisely, we obtain lower bounds for randomized and fixed schemes as a function of the eigenvalues of the Lüders channel which characterizes one possible post-measurement state transformation.
We study quantum state testing where the goal is to test whether ρ=ρ0∈Cd×d or ∥ρ−ρ0∥1>ε, given n copies of ρ and a known state description ρ0. 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≥d and Pauli observables, a special class of 2-outcome measurements. Our work bridges this gap in the literature.
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 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.
We consider a fundamental problem of \emph{mixedness testing}: Given n copies of an N-qubit state ρ, determine whether ρ=Id/d or ∥ρ−Id/d∥1≥ε with high probability, where d=2N. In particular, we focus on performing this task in the practical setting of single-qubit measurements, where measurements are prepared independently on each qubit. We provide a nearly complete picture of single-qubit mixedness tesing by showing n=Θ~(10N/ε2). To establish our lower bound, we introduce a measurement-dependent lower bound framework for adaptive single-copy state certification. For the upper bound, we present a randomized Pauli basis measurement protocol, which relies on a new primitive for computationally efficient uniformity testing of correlation-concentrated distributions on the Boolean hypercube. In conjunction, we provide lower and upper bounds for mixedness testing with fixed Pauli measurement protocols.
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 introduce a framework for distributed quantum inference under communication constraints. In our model, m distributed nodes each receive one copy of an unknown d-dimensional quantum state ρ, before communicating via a constrained one-way communication channel with a central node, which aims to infer some property of ρ. This framework generalizes the classical distributed inference framework introduced by Acharya, Canonne, and Tyagi [COLT2019], by allowing quantum resources such as quantum communication and shared entanglement. Within this setting, we focus on the fundamental problem of quantum state certification: Given a complete description of some state σ, decide whether ρ=σ or ∥ρ−σ∥1≥ε. Additionally, we focus on the case of limited quantum communication between distributed nodes and the central node. We show that when each communication channel is limited to only nq≤logd qubits, then the sample complexity of distributed state certification is O(2nqε2d2) when public randomness is available to all nodes. Moreover, under the assumption that the channels used by the distributed nodes are mixedness-preserving, we prove a matching lower bound. We further demonstrate that shared randomness is necessary to achieve the above complexity, by proving an Ω(4nqε2d3) lower bound in the private-coin setting under the same assumption as above. Our lower bounds leverage a recently introduced quantum analogue of the celebrated Ingster-Suslina method and generalize arguments from the classical setting. Together, our work provides the first characterization of distributed quantum state certification in the regime of limited quantum communication and establishes a general framework for distributed quantum inference with communication constraints.