Quantum States Verified with Far Fewer Measurements Than Previously Thought

Jinchang Liu and colleagues at Tsinghua University, in collaboration with Nanyang Technological University, The University of Hong Kong, and National University of Singapore, have developed a new protocol for characterising large quantum systems with minimal assumptions. The protocol overcomes existing scalability limitations in self-testing quantum states. It enables strong self-testing of almost all n-qubit states using only a polynomial number of samples, a sharp improvement over previous methods requiring exponential scaling. This advancement relies on an efficient scheme for evaluating multipartite Pauli measurements with a linear number of ancillary Bell pairs, paving the way for scalable device-independent quantum information processing and learning protocols in future quantum networks.

Polynomial scaling enables strong verification of multi-qubit entanglement

A dramatic improvement over previous methods now allows self-testing of almost all n-qubit states with a sample complexity that scales polynomially with the number of parties. This breakthrough crosses a critical threshold for practical quantum networks, where exponentially growing demands on sample size previously rendered strong self-testing impossible. The new protocol efficiently evaluates multipartite Pauli measurements, a technique for analysing quantum states, using only a linear number of ancillary Bell pairs, entangled particles serving as a reference. This advance unlocks scalable device-independent quantum information processing, paving the way for more secure communication and powerful quantum computation without needing to fully trust the devices involved; it also provides a general framework for learning and certification protocols.

Polynomial complexity and device-independent multipartite Pauli measurement certification

Protocols with only polynomial sample complexity are now being developed. The key ingredient is an efficient scheme for device-independently evaluating multipartite Pauli measurements, implemented using only a linear number of ancillary Bell pairs alongside standard projective and Bell measurements, well within the reach of current quantum technology. This scheme provides a general framework for implementing a wide range of learning and certification protocols in the device-independent setting, thereby opening a scalable route to device-independent quantum information processing in large-scale quantum networks.

Certifying the correct functioning of quantum devices is central to the development of quantum technologies. As quantum platforms and quantum networks continue to scale up, the assumption that certification protocols can rely on trusted and well-calibrated devices becomes increasingly untenable. Quantum nonlocality makes it possible to remove this assumption, giving rise to the device-independent (DI) model, the only rigorous framework for certifying quantum systems without assumptions about the inner workings of experimental devices.

This framework has enabled major applications of quantum technology, such as secure key distribution, certified random number generation, and delegated quantum computing. Consequently, developing powerful DI tools for characterising quantum systems has become a central goal of quantum information science. Within the DI model, the strongest certification of an underlying quantum system is known as self-testing. In a self-testing protocol, observed statistics uniquely determine the underlying state and measurements, up to indistinguishable degrees of freedom.

A canonical example is that maximal violation of the Clauser, Horne, Shimony, Holt (CHSH) inequality self-tests the two-qubit Bell state and the associated measurements. Therefore, a central goal in the certification of large-scale quantum devices and networks is to develop scalable self-testing protocols for the underlying multipartite quantum systems. While seminal works have established that all multipartite entangled pure states can, in principle, be self-tested, the challenge of scalability remains overlooked and largely unaddressed.

Rigorous analyses of durability against experimental noise are currently restricted to a few highly structured examples, such as GHZ and W states. Moreover, most existing results rely on perfect probability distributions, which are inaccessible via finite sampling. The few studies that do account for statistical deviations typically demand a sample complexity that scales exponentially with the number of parties, a feature highly unfavorable in practice.

Together, these limitations present a significant barrier to the deployment of self-testing protocols in realistic applications. Overcoming this exponential sample-complexity barrier is challenging for several reasons. The black-box treatment of measurement devices makes it difficult to probe the underlying system. Without perfect distribution reconstruction, reliably extracting global information from finite samples is also highly challenging. Furthermore, even in the device-dependent setting, multipartite entangled states exhibit an exceptionally rich structure that is notoriously difficult to characterise.

These challenges motivate the following central question: Can one strongly self-test generic multipartite quantum states with sample complexity polynomial in the number of parties. This work answers this question affirmatively by introducing a strong self-testing protocol for almost all multipartite qubit states with sample complexity polynomial in the number of parties. This initial challenge was overcome by solving an open question: how to efficiently certify, in a global sense, multipartite Pauli measurements performed by spatially separated parties device-independently.

Although single-party Pauli measurements can be self-tested, extending such certification to the multipartite setting encounters a local-transpose obstacle. To certify a target multipartite Pauli measurement, local freedoms must be aligned consistently across all parties. To overcome this obstacle, a new transpose-braiding test was introduced that enforces this global consistency. The test is based on inequalities between neighboring parties, constructed from a two-qubit observable K. The key insight is that the maximal eigenvalue of K is strictly larger than that of its partial transpose KT1, forcing the transpose freedoms of neighboring parties to be consistent.

By sequentially applying this alignment across the network, one certifies multipartite Pauli measurements up to a global transpose freedom. Building on this test, the protocol robustly lifts device-dependent Pauli measurements to their device-independent counterparts using only polynomial overhead. Crucially, this protocol enables sample-efficient estimation of the observables relevant for certifying the target states, without reconstructing the full correlation distribution.

Additionally, the assumption that the underlying quantum states are independent and identically distributed (i.i.d.) across rounds was removed by providing a fully non-i.i.d. analysis. The protocol of Ref. relies only on local Pauli measurements and can therefore be lifted to the DI setting using the multipartite Pauli-measurement scheme, already yielding a self-testing protocol with polynomial sample complexity for generic states. To remove its remaining shared-randomness requirement, the protocol builds on the random-basis-enhanced variant of Ref. and shows how the required shared randomness can be replaced by standard local randomness.

Taken together, the protocol only requires the minimal assumptions of the DI model. For experimental implementation, the protocol requires only standard Bell states and Pauli measurements, both of which are well within the reach of current quantum platforms. By providing a systematic bridge from powerful device-dependent methods to their DI counterparts, the protocol unlocks a wide range of scalable DI learning and certification protocols for large-scale quantum devices and interconnected quantum networks.

The self-testing properties of quantum states shared by n main parties A = A0A1 · · · An−1 are considered. The standard notion of self-testing a single state is incorporated as a special case. The property aimed to be self-tested is an expectation value of the form try(LρA), where L is a target observable. By choosing L appropriately, this framework captures a wide range of certification tasks: it can witness multipartite entanglement, probe circuit complexity, or even single out a unique target state.

This work focuses on observables L induced by randomized multipartite Pauli schemes, which act on an n-qubit state as follows. First, a random Pauli basis string P = P0P1 · · · Pn−1 ∈{X, Y, Z}⊗n is drawn from a distribution D. Then, for each 0 ≤i The setting requires minimal, device-independent assumptions. Concretely, each party’s device is treated as a black box, and the only trusted ingredient is the randomness of the classical inputs, provided by a referee or generated locally.

Spatial separation between parties imposes a tensor-product structure on their measurements. Formally, each main party Al receives an input xl and performs an uncharacterized positive operator-valued measure (POVM) {M (l) al|xl}al, producing an outcome al. Similarly, each auxiliary party Bl receives an input yl, performs an uncharacterized POVM {N (l) bl|yl}bl, and outputs bl. The observed statistics are then given by Born’s rule: Pr[a0 . an−1b0 . bn−1|x0 . xn−1y0 . yn−1] = try ” n−1 O l=0 M (l) al|xl ⊗ n−1 O l=0 N (l) bl|yl. ρ #, where ρ is the unknown state shared across all 2n parties. In the main text, the i.i.d. setting is analysed, where each experimental round is modelled as an independent use of the same state and the same measurement operators.

The non-i.i.d. scenario is addressed in Appendix H. No additional assumptions are made on the inner working of quantum devices. The self-testing protocol proceeds by repeating the experiment for multiple rounds. In each round, the parties’ outcomes are sampled according to Eq. From the collected data, an estimator ω is constructed, and the aim is to certify that try(L ρA) = ω up to a precision ε, achieving self-testing of the desired property and the multipartite Pauli measurements.

Importantly, self-testing necessarily leaves some intrinsic degrees of freedom undetermined. First, there is the local extraction freedom: if parties share an initial state ρ′ and can transform it into ρ via local channels, their measurement devices can produce statistics identical to those of ρ. Common examples of such operations include local basis rotations and tracing out “junk” auxiliary systems upon which the measurements act trivially. Second, the statistics are invariant under global complex conjugation, since simultaneously replacing the state and all measurement operators with their complex conjugates does not affect the real-valued probabilities. Beyond these unavoidable freedoms, any practical notion of self-testing must be robust to statistical fluctuations arising from finite samples. Taking these into account, it is said that the main parties ε-approximately share a state τ if there exists a local extraction channel Γ = Nn−1 i=0 Γi acting on the main parties A = A0 · · · An−1, such that D Γ(ρA), pτ+ ⊗|0⟩⟨0|⊗n + (1 −p)τ ∗ −⊗|1⟩⟨1|⊗n ≤ε.

Polynomial complexity scaling enables practical verification of multi-qubit states

Researchers are steadily refining methods for verifying quantum systems, an important step towards building practical quantum technologies. However, achieving strong self-testing, uniquely identifying a quantum state from observed data, remains a significant hurdle, particularly as systems grow in complexity. While a substantial advance, this new protocol demonstrates self-testing for “almost all” n-qubit states, a qualification that introduces a degree of uncertainty.

Acknowledging that this self-testing applies to “almost all” quantum states introduces a degree of uncertainty. Despite this qualification, the achievement of polynomial sample complexity represents a vital step forward. Existing methods demanded exponentially increasing data, quickly becoming impractical as quantum systems scale up; this new protocol circumvents that limitation. This new protocol efficiently tests “almost all” quantum states using a manageable amount of data, unlike earlier techniques needing exponentially more. This advance could begin to unlock scalable quantum networks and device-independent quantum technologies.

Researchers demonstrated a new protocol for self-testing almost all n-qubit states with a significantly reduced requirement for measurement samples. This is important because previous methods required exponentially more data as the number of qubits increased, hindering verification of larger quantum systems. The protocol achieves this by efficiently evaluating multipartite Pauli measurements using a limited number of ancillary Bell pairs and standard quantum measurements. The authors suggest this scheme provides a general framework for broader applications in device-independent quantum information processing and scalable quantum networks.