Machine Learning Assisted Many-Body Entanglement Measurement

Measuring Moments in spin-1/2 Systems

• For m = 3, a sequential set of singlet-triplet measurements is performed first on copies (1, 2), yielding β1, and then on copies (2, 3), yielding β2.Source: Measuring Moments in spin-1/2 Systems · p. 7
• The sequential measurement scheme generalizes to higher m, with operators defined recursively from swaps between copies.The resulting sequence corresponds to a sum over cyclic permutations; for m = 3, κ = {123, 132}.Source: Measuring Moments in spin-1/2 Systems · p. 7
• Reversing the order of sequential measurements on A and B is crucial for obtaining moments of the partially transposed state.For m = 3, A uses copies 1–2 then 2–3, while B uses 2–3 then 1–2.Source: Measuring Moments in spin-1/2 Systems · p. 7
• Repeated measurements provide an estimate of the moment’s standard deviation.The passage describes repeating the experiment R times to obtain the standard deviation.Source: Measuring Moments in spin-1/2 Systems · p. 8

Measuring Moments in Bosonic Systems

• Choosing the shift permutation π(c) = c + 1 specifies the permutation operator used in the bosonic measurement scheme.The operator is written in second-quantized form using normal ordering and site- and copy-indexed annihilation operators.Source: Measuring Moments in Bosonic Systems · p. 8
• Bosonic moments are measured by applying a Fourier transform on A and an inverse Fourier transform on B, then measuring occupation numbers at each site.The measured occupations determine the permutation-operator outcome, whose average estimates the moment.Source: Measuring Moments in Bosonic Systems · p. 8
• The estimate has variance 1 − µ_m^2, and after R repetitions its standard deviation is given by the same result as Eq. (S4).Source: Measuring Moments in Bosonic Systems · p. 8
• The Chebyshev comparison covers random R-GPS and R-MPS states with varying bond dimensions and partitions, using M = 10 or M = 20 copies.The caption specifies N_A + N_B ≤ 12 and total length N ≤ 24.Source: Measuring Moments in Bosonic Systems · p. 9, Figure S1 · p. 9

Experimental Feasibility.

• Quantum dot arrays offer a possible solid-state setting because singlet-triplet measurements map to swap measurements and are already well developed.The passage discusses parallel arrays and proposed silicon arrays as routes to preparing multiple copies.Source: Experimental Feasibility · p. 9
• In optical lattices, multiple copies can be isolated in adjacent rows, but destructive swaps limit consecutive swaps and the number of copies to two.Forward-backward Fourier transforms provide an alternative permutation-equivalent capability in this setting.Source: Experimental Feasibility · p. 9
• Small fidelity deviations between copies lead to small changes in moments, and smooth entanglement estimators keep the resulting error controlled.The passage applies this reasoning to both Chebyshev and neural-network approaches.Source: Experimental Feasibility · p. 9

Chebyshev Expansion

• Approximating |x| with a Chebyshev polynomial avoids the non-convergence of a Taylor expansion at x = 0 and expresses the estimate through moments.The method approximates a function across an interval rather than around a point.Source: Chebyshev Expansion · p. 10
• A narrower spectral window improves the approximation at fixed M, so the method seeks a window that contains the full spectrum.The passage describes using an estimate based on moments to choose the window.Source: Chebyshev Expansion · p. 10
• The approximation quality increases rapidly with M and becomes exact as M approaches infinity.Source: Chebyshev Expansion · p. 10
• Doubling the number of copies to M = 20 significantly improves Chebyshev accuracy, while the M = 10 approximation typically underestimates negativity, especially for MPS states.The reported negative bias is attributed to the eigenvalue distribution being peaked around zero, particularly for MPS states.Source: Chebyshev Expansion · p. 10
• When moments have compact expressions, such as in free-fermion systems, the Chebyshev expansion can provide an analytic formula for negativity.The passage notes that M = 20 is a significant experimental requirement for the approximation.Source: Chebyshev Expansion · p. 10

• Random generic pure states (R-GPS) are generated from random complex vectors with normally distributed elements.Source: Generation of random states · p. 10
• The authors also consider Haar-sampled R-GPS, which are more numerically demanding but provide the same results.Source: Generation of random states · p. 10
• Random matrix product states (R-MPS) are built from random tensors with complex elements drawn from a normal distribution; D denotes bond dimension.Source: Generation of random states · p. 10

• In the thermodynamic limit, the spectrum of the partially transposed density matrix is described by a probability density ω(λ), which determines logarithmic negativity.Source: Chebyshev Approximation Error Analysis · pp. 10–11
• For R-MPS with fixed bond dimension D much smaller than 2^(N_A+N_B+N_C), eigenvalues concentrate near zero with long tails, unlike the shifted semicircle distribution of large R-GPS.Source: Chebyshev Approximation Error Analysis · p. 11
• The concentration of R-MPS eigenvalues near zero combines with the wider spectral support to produce a negative Chebyshev bias, unlike the more evenly distributed R-GPS spectrum.The section therefore expects larger Chebyshev error for area-law states.Source: Chebyshev Approximation Error Analysis · p. 11

• The network input comprises subsystem sizes N_A and N_B together with the non-trivial moments µ2 through µM.Source: Neural Network Details · p. 11
• Training optimizes the network weights by minimizing a suitable cost function.Source: Neural Network Details · p. 12
• For M = 3, the selected network has two ReLU hidden layers with 100 and 56 neurons; for M = 10, it has three layers with ELU, ReLU, and linear activations and 61, 87, and 47 neurons.Hyperopt and Keras are used to select the network structure, including hidden-layer count.Source: Neural Network Details · p. 12

• The standard deviation of prediction errors on unseen test data can guide error estimates, provided experimental data are qualitatively similar to the training data.For the quantum-quench example, training on both area-law and volume-law random states yields errors roughly matching those of test data.Source: Neural Network Error Analysis · p. 12
• Because the neural network produces smooth functions of its inputs, small errors in measured moments produce only small changes in the estimate.Source: Neural Network Error Analysis · p. 12

Neural Network Sensitivity

• Although few moments can estimate entanglement and their measurement effort scales linearly with subsystem sizes, scalability also requires repeat measurements R to scale sub-exponentially.The standard error in measured moments depends on R.Source: Neural Network Sensitivity · p. 12
• The estimator’s surface is smooth across most of the random-state data, with the enforced lower bound at zero and most data in regions of reasonable gradient.The figure compares real logarithmic negativities with predictions over µ2 and µ3 for N_A=N_B=5.Source: Neural Network Sensitivity · p. 12, Figure S3 · p. 13
• The median sensitivity to µ2 and µ3 grows roughly linearly with N_A+N_B, stays below 30 in absolute value, and requires greater accuracy for µ3.The authors conclude that the required repeat measurements R do not scale exponentially with subsystem size.Source: Neural Network Sensitivity · p. 12, Figure S4 · p. 13

Comparison with approximate state reconstruction methods

• Approximate polynomial reconstruction targets a state that reproduces observables or POVM statistics, but closeness in trace distance does not ensure accurate entanglement estimates.The passage links trace-distance closeness to close expectation values, then notes entanglement measures can be more sensitive to small reconstruction errors.Source: Comparison with approximate state… · p. 13
• Because negativity is not asymptotically continuous, states close in trace distance can have entanglement differences growing faster than O(N), making reconstructed-state negativity an invalid approximation.The passage defines asymptotic continuity using the normalized difference |E(ρ)−E(σ)|/log(d_N).Source: Comparison with approximate state… · pp. 13–14
• By avoiding intermediate state reconstruction and using measurements specifically to reconstruct negativity, the method reports accurate predictions with controlled error.The authors contrast this approach with polynomial state-reconstruction schemes.Source: Comparison with approximate state… · pp. 13–14

Additional Numerical Results

• The authors test a network trained only on random states against physical states, including highly symmetric cases that may challenge training without matching physical assumptions.They note that specialized training sets could improve the results, but use random states for generality.Source: Additional Numerical Results · p. 14

Ground-states through a quantum phase transition

• Across the XX-model transition at BZ=1, three- and six-copy predictions reproduce key features, with six copies significantly suppressing quantitative fluctuations.The system is divided into A, B, and C; tracing out C generally leaves A and B in a mixed state.Source: Ground-states through a quantum phase… · p. 14, Figure S5 · p. 15

W-state

• For the W-state, both copy counts capture the overall trend, but absolute accuracy is reasonable only with six copies; the authors identify low entanglement below 1 ebit as a regime where MPS-tomography is feasible.The W-state is highly symmetric and atypical of the random training set.Source: W-state · pp. 15–16, Figure S6 · p. 15

Quench across a phase transition

• For a quench across the transverse-field Ising transition, three moments already give good accuracy for the logarithmic negativity over time.The quench begins in the ground state at H_Ising(1+δ) and evolves under H_Ising(1−δ).Source: Quench across a phase transition · p. 16, Figure S7 · p. 16