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📝 Abstract
Measurement-induced entanglement (MIE) captures how local measurements generate long-range quantum correlations and drive dynamical phase transitions in many-body systems. Yet estimating MIE experimentally remains challenging: direct evaluation requires extensive post-selection over measurement outcomes, raising the question of whether MIE is accessible with only polynomial resources. We address this challenge by reframing MIE detection as a data-driven learning problem that assumes no prior knowledge of state preparation. Using measurement records alone, we train a neural network in a self-supervised manner to predict the uncertainty metric for MIE-the gap between upper and lower bounds of the average post-measurement bipartite entanglement. Applied to random circuits with one-dimensional all-to-all connectivity and two-dimensional nearestneighbor coupling, our method reveals a learnability transition with increasing circuit depth: below a threshold, the uncertainty is small and decreases with polynomial measurement data and model parameters, while above it the uncertainty remains large despite increasing resources. We further verify this transition experimentally on current noisy quantum devices, demonstrating its robustness to realistic noise. These results highlight the power of data-driven approaches for learning MIE and delineate the practical limits of its classical learnability.
💡 Analysis
Measurement-induced entanglement (MIE) captures how local measurements generate long-range quantum correlations and drive dynamical phase transitions in many-body systems. Yet estimating MIE experimentally remains challenging: direct evaluation requires extensive post-selection over measurement outcomes, raising the question of whether MIE is accessible with only polynomial resources. We address this challenge by reframing MIE detection as a data-driven learning problem that assumes no prior knowledge of state preparation. Using measurement records alone, we train a neural network in a self-supervised manner to predict the uncertainty metric for MIE-the gap between upper and lower bounds of the average post-measurement bipartite entanglement. Applied to random circuits with one-dimensional all-to-all connectivity and two-dimensional nearestneighbor coupling, our method reveals a learnability transition with increasing circuit depth: below a threshold, the uncertainty is small and decreases with polynomial measurement data and model parameters, while above it the uncertainty remains large despite increasing resources. We further verify this transition experimentally on current noisy quantum devices, demonstrating its robustness to realistic noise. These results highlight the power of data-driven approaches for learning MIE and delineate the practical limits of its classical learnability.
📄 Content
Quantum entanglement is a central resource of quantum information science, enabling advantages in computation, communication, and sensing [1][2][3][4][5]. Importantly, entanglement need not arise solely from coherent unitary evolution: suitably chosen and adaptively processed measurements can also generate nonlocal correlations in many-body systems [6][7][8][9]. This measurement-induced entanglement (MIE) underlies measurement-based quantum computation [10], enables rapid preparation of longrange entangled states [11][12][13][14][15], and gives rise to novel non-equilibrium phases of matter [16][17][18][19][20]. In particular, in hybrid unitary-measurement dynamics, competition between scrambling and projective measurements produces a measurement-induced phase transition (MIPT), across which the scaling of MIE changes from volume law to area law [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].
Despite its conceptual and practical significance, directly characterizing MIE in experiments remains notoriously challenging [38]. The central obstacle is postselection: probing properties of a state conditioned on a specific measurement outcome requires repeating the experiment until that outcome reoccurs, an effort that grows exponentially with the number of measurements by Born’s rule. Several scalable diagnostics have been proposed to circumvent this bottleneck, including purification of an entangled reference qubit [39][40][41][42][43], learnability of conserved quantities [44] or of the pre-measurement state [45], cross-entropy benchmarks [46][47][48] and other machine-learning proxies [49][50][51]. Although these proxies successfully reflect the distinct behavior of MIE in different regimes and help reveal its critical behavior, they remain indirect witnesses rather than quantitative estimators of MIE itself. Recent works have made progress showing promise and limits [52,53]. On the one hand, MIE can in principle be estimated without post-selection by leveraging quantum-classical correlations, but this requires priori knowledge of the underlying quantum dynamics and the accuracy of estimation hinges on the fidelity of classical simulations. On the other hand, Ref. [53] proves that, without any knowledge of the pre-measurement state, no learning protocol can extract properties beyond ensemble averages using only a polynomial number of measurement shots. The practical limits of experimentally detectable MIE thus remain an open question.
Motivated by the recent success of data-driven quantum learning methods based solely on measurement records [54][55][56], we ask in this Letter: under what conditions can MIE be learned with polynomial resources from data alone-i.e., without post-selection and without any prior knowledge of state preparation? To address this question, we study the entanglement generated between two distant qubits A and B after all other qubits in a many-body state are measured. Using measurement outcomes only, we train a transformer-based neural network in a self-supervised manner to estimate the post-measurement state on AB ≡ A ∪ B conditioned on a given measurement outcome. Instead of reconstructing MIE directly, we estimate the total entanglement entropy of AB, which we use as an uncertainty metric quantifying the learnability of MIE. We focus on two settings-random one-dimensional (1D) allto-all circuits and two-dimensional (2D) nearest-neighbor circuits-and in both we observe a clear learnability transition with increasing circuit depth. In the learnable phase, the uncertainty decreases with additional measurement data and model parameters, reflecting increasing time and space complexity. In the unlearnable phase, the uncertainty saturates or even grows. We further probe this learnability transition in the presence of noise through simulations and experiments on IBM QPU ibm marrakesh, finding that the transition remains observable under realistic noise. Overall, our results demonstrate the power of data-driven methods to extract informative features of the post-measurement state that reveal MIE, while simultaneously clarifying the fundamental limitations of learning MIE from measurement data alone. We note that a recent study [57] investigated MIE learnability for GHZ and cluster states using a similar data-driven approach, where signatures of a learnability transition were observed for the latter upon tuning the measurement direction. In contrast, our work addresses generic states generated by random quantum circuits and provides an explicit characterization of the emergence of distinct learnable and unlearnable regimes.
Quantifying learnability-We quantify the learnability of MIE as follows. Consider an L-qubit pure state |ψ⟩. All qubits except two distant ones, A and B, are measured in the computational basis, yielding outcome m. The corresponding post-measurement state on AB is σ AB,m , with reduced state σ A,m = Tr B (σ AB,m ). The MIE between A and B is defined
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