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arXiv:2610.16018

Adaptive Entanglement Witnessing for Two-Qubit Systems

Lev Gruber, Ria Haapala, Elio Thadhani, Paco Navarro, Eritas Yang, Benjamin Hartley, Rebecca Verghese, Oscar Scholin, Isabel Godoy, Stuart Kerr, Brayden Mendoza, Alec Roberson, Theresa W. Lynn

quant-phSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations
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Abstract

Entanglement witnesses are a key instrument in determining the entanglement of unknown bipartite qubit systems without requiring a full tomography. Riccardi et al. (2020) proposed a set of two-qubit witnesses, denoted $\{W^3\}$, which require a set of local measurements in three bases. In this work we propose an additional set of witnesses, denoted $\{W^5\}$, that build on $\{W^3\}$ and require measurements in two additional bases. We present an adaptive two-step method that uses the measurement results obtained for $\{W^3\}$ to select which $W^5$ to evaluate. We computationally demonstrate that our adaptive two-step protocol can identify 72% of entangled states generated according to the methods of Roik et al. (2021) while requiring measurements in only five of the nine bases needed for full quantum tomography. Additionally, we experimentally verify the behavior of both sets of witnesses using polarization-entangled photon pairs.

Comments
11 pages, 6 figures

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Entanglement witnesses are a key instrument in determining the entanglement of unknown bipartite qubit systems without requiring a full tomography. Riccardi et al. (2020) proposed a set of two-qubit witnesses, denoted {W3}\{W^3\}{W3}, which require a set of local measurements in three bases. In this work we propose an additional set of witnesses, denoted {W5}\{W^5\}{W5}, that build on {W3}\{W^3\}{W3} and require measurements in two additional bases. We present an adaptive two-step method that uses the measurement results obtained for {W3}\{W^3\}{W3} to select which W5W^5W5 to evaluate. We computationally demonstrate that our adaptive two-step protocol can identify 72% of entangled states generated according to the methods of Roik et al. (2021) while requiring measurements in only five of the nine bases needed for full quantum tomography. Additionally, we experimentally verify the behavior of both sets of witnesses using polarization-entangled photon pairs.
We experimentally demonstrate a method for detection of entanglement via construction of entanglement witnesses from a limited fixed set of local measurements ( M ). Such a method does not require a priori knowledge about the form of the entanglement witnesses. It is suitable for a scenario where a full state tomography is not available, but the only resource is a limited set of M . We demonstrate the method on pure two-qubit entangled states and mixed two-qubit entangled states, which emerge from photonic implementation of controllable quantum noisy channels. The states we select are motivated by realistic experimental conditions, and we confirm it works well for both cases. Furthermore, possible generalizations to higher-dimensional bipartite systems have been considered, which can potentially detect both decomposable and indecomposable entanglement witnesses. Our experimental results show perfect validity of the method, which indicates that even a limited set of local measurements can be used for quick entanglement detection and further provide a practical test bed for experiments with entanglement witnesses.
Entanglement has been recognized as being crucial when implementing various quantum information tasks. Nevertheless, quantifying entanglement for an unknown quantum state requires nonphysical operations or post-processing measurement data. For example, evaluation methods via quantum state tomography require vast amounts of measurement data and likely estimation. Although a direct entanglement determination has been reported for the unknown pure state, it is still tricky for the mixed state. In this work, assisted by weak measurement and deep learning technology, we directly detect the entanglement (namely, the concurrence) of a class of two-photon polarization-entangled mixed states both theoretically and experimentally according to the local photon spatial distributions after weak measurement. In this way, the number of projective bases is much smaller than that required in quantum state tomography.
Entanglement witnesses are essential for certifying entanglement, yet constructing ones that are both noise-robust and economical in measurement settings remains challenging - particularly beyond qubits and for non-stabilizer ("magic") states. We present a machine-learning method that, given a target state and a user-specified number of measurement settings, generates an entanglement witness optimized for noise tolerance in the neighborhood of that state, requiring only local measurements. The approach is fully general, applying to multipartite qubit and qudit systems alike, including non-stabilizer states. For N qudits of dimension d, we train on the fully-separable eigenstates of each qudit's SU(d) generators to find a prototype witness, then tune the witness's bias term via gradient descent to maximize noise tolerance. Adversarial training further strengthens the witnesses, delivering greater noise tolerance with even fewer settings; critically, under this scheme the required training-set size becomes independent of system size. We package the entire pipeline as an automated script that, in every case we tested, produces witnesses surpassing all existing methods in noise tolerance and/or number of measurement settings. We demonstrate the method on Bell, GHZ, W, and hypergraph states, along with a range of qudit states, spanning 2-6 qubits, bipartite qudits up to d=10, and tripartite qutrits. Our witnesses achieve perfect accuracy across both physical experimental test states and large numerical sets of separable mixed states-including 30 million test states for a 3-qubit W-state witness and 10 million for a 4-qubit hypergraph-state witness-and we experimentally confirm the noise tolerance of Bell- and hypergraph state witnesses on both photonic and superconducting platforms, respectively.
We address the problem of optimizing entanglement witnesses when a limited fixed set of local measurements can be performed on a bipartite system, thus providing a procedure, feasible also for experiments, to detect entangled states using only the statistics of these local measurements. We completely characterize the class of entanglement witnesses of the form $W={P}^{\mathrm{\ensuremath{\Gamma}}}$, where $\mathrm{\ensuremath{\Gamma}}$ denotes partial transposition, that can be constructed from the measurements of the bipartite operators ${\ensuremath{\sigma}}_{x}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{x}, {\ensuremath{\sigma}}_{y}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{y}$, and ${\ensuremath{\sigma}}_{z}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{z}$ in the case of two-qubit systems. In particular, we consider all possible extremal decomposable witnesses within the considered class that can be defined from this set of measurements. Finally, we discuss possible extensions to higher-dimension bipartite systems when the set of available measurements is characterized by the generalized Gell-Mann matrices. We provide several examples of entanglement witnesses, both decomposable and indecomposable, that can be constructed with these limited resources.
The certification of entanglement is a vital task in quantum technology. To theoretically quantify and experimentally detect entanglement of many qubits, we suggest to identify quantum entanglement by measuring the statistical response of a quantum system to an arbitrary nonlocal parametric evolution. Our suggestion does not basing on the tomographic reconstruction of the quantum state, or the realization of witness operators. The protocol requires two collective settings for any number of parties and is robust against noise and decoherence occurring after the implementation of the parametric transformation. The bounds for witnessing quantum entanglement with strength of nonlocal interaction are shown with the help of Ising nonlocal Hamiltonian. Furthermore to illustrate its user friendliness we demonstrate multipartite entanglement in different experiments with ions and photons by analyzing published data on fidelity visibilities and variances of collective observables.
Entanglement is the key resource in quantum information processing, and an entanglement witness (EW) is designed to detect whether a quantum system has any entanglement. However, prior knowledge of the target states should be known first to design a suitable EW, which weakens this method. Nevertheless, a recent theory shows that it is possible to design a universal entanglement witness (UEW) to detect negative-partial-transpose (NPT) entanglement in unknown bipartite states with measurement-device-independent (MDI) characteristic. The outcome of a UEW can also be upgraded to be an entanglement measure. In this study, we experimentally design and realize an MDI UEW for two-qubit entangled states. All of the tested states are well-detected without any prior knowledge. We also show that it is able to quantify entanglement by comparing it with concurrence estimated through state tomography. The relation between them is also revealed. The entire experimental framework ensures that the UEW is MDI.