import marimo __generated_with = "0.11.5" app = marimo.App(layout_file="layouts/rtn-nechita-fermi-days-2025.slides.json") @app.cell def _(mo): mo.md( r""" # Entanglement in random tensor networks **Ion Nechita** (CNRS, LPT Toulouse) Joint work with **Khurshed Fitter** (former intern at LPT, now EPFL), **Faedi Loulidi** (former PhD student at LPT, now OIST, Japan), and **Miao Hu** (PhD student, LPT) Talk mostly based on https://arxiv.org/abs/2407.02559 **Outline.** We explore the *entanglement properties* of *random tensor networks*. We introduce a model of random tensor networks and compute the entanglement entropy of a bipartition of the system. We will also visualize the *entanglement spectrum* as a function of the underlying graph structure of the network. """ ) return @app.cell def _(mo): mo.md( r""" ## Entanglement of quantum states Given a bipartite quanutm state $\ket \psi_{AB} \in \mathcal H_A \otimes \mathcal H_B$, we define its [**entropy of entanglement**](https://en.wikipedia.org/wiki/Entropy_of_entanglement) as $$ S_{\text{ent}}(\ket \psi) = -\sum_i \lambda_i \log \lambda_i,$$ where $\lambda_i$ are the *Schmidt coefficients* of the state: $$ \ket \psi = \sum_i \sqrt{\lambda_i} \ket{a_i} \otimes \ket{b_i}.$$ For example, a *separable state* has zero entropy of entanglement $$S_{\text{ent}}(\ket{00}) = 0$$ while a maximally entangled state of two qubits has unit entropy of entanglement $$S_{\text{ent}}\left(\frac{\ket{00} + \ket{11}}{\sqrt 2}\right) = \log 2 =1.$$ In general, for a state on two *qudits* $\ket \psi \in \mathbb C^d \otimes \mathbb C^d$, we have $$0 \leq S_{\text{ent}}(\ket \psi) \leq \log d.$$ """ ) return @app.cell def _(mo): mo.md( r""" ### Random states. Page formula We consider now random states $\ket \psi_{AB} \in \mathbb C^{d_A} \otimes \mathbb C^{d_B}$, obtained by randomly rotating a fixed state by a uniform *global* unitary rotation $$\ket \psi = U \cdot \ket{\psi_0},$$ where $U \in \mathcal U(d_Ad_B)$ is a [Haar-distributed](https://en.wikipedia.org/wiki/Haar_measure) **random unitary matrix**. Equivalently, $\ket \psi$ is chosen **uniformly at random from the unit sphere** of $\mathbb C^{d_A \cdot d_B}$. Page conjectured in 1993 (and it was later proved) that the *average* entropy of entanglement of a random state is given by (we assume wlog $d_A\leq d_B$) $$\mathbb E S_{\text{ent}}(\ket \psi) = \sum_{i=d_B+1}^{d_A d_B} \frac{1}{i} - \frac{d_A-1}{2d_B}.$$ In the limit $d_A = d_B = d\to \infty$, we have $$\boxed{\mathbb E S_{\text{ent}}(\psi) = \log d - \frac 1 2 + o(1).}$$ """ ) return @app.cell def _(mo): mo.md( r""" ### Entanglement spectrum One can also study the **entanglement spectrum** of a bipartite state $\ket \psi_{AB}$. The entanglement spectrum is defined as the set of Schmidt coefficients $\lambda_i$ or, equivalently, as the set of eigenvalues of the reduced density matrix $\rho_A = \text{Tr}_B \ket \psi_{AB} \bra \psi$. Note that the scalars $\lambda_i$ are non-negative $$ \lambda_i \geq 0 \quad \forall i$$ and normalized $$\sum_i \lambda_i = 1.$$ In other words, they form a *probability distribution*. The entropy of entanglement of the state is then just the [Shannon entropy](https://en.wikipedia.org/wiki/Entropy_(information_theory)) of the vector $\lambda = (\lambda_i)_i$. """ ) return @app.cell def _(mo): d_slider_biparite = mo.ui.slider(start=2, stop=1000, step=1, value=10, label="Local Hilbert space dimension d:") d_slider_FC = mo.ui.slider(start=2, stop=1000, step=1, value=10, label="Local Hilbert space dimension d:") return d_slider_FC, d_slider_biparite @app.cell def _(create_plot_MP, d_slider_biparite, mo): # Create the plot using the current slider value # This plot variable implicitly depends on 'amplitude_slider' spectrum = create_plot_MP(d_slider_biparite.value) # Arrange the slider and the plot vertically and display them # This is the last expression in the cell mo.vstack([d_slider_biparite, spectrum.gca()]) return (spectrum,) @app.cell def _(np, plt): # Define the Marchenko-Pastur PDF for c=1, sigma^2=1 # lambda_minus = (1 - sqrt(c))^2 = (1-1)^2 = 0 # lambda_plus = (1 + sqrt(c))^2 = (1+1)^2 = 4 # Support is [0, 4] def marchenko_pastur_pdf(x): # Avoid division by zero and sqrt of negative numbers # Use np.clip to keep x within a safe range slightly inside (0, 4) x_safe = np.clip(x, 1e-9, 4.0 - 1e-9) pdf = np.sqrt(x_safe * (4.0 - x_safe)) / (2 * np.pi * x_safe) # Set density to 0 outside the support [0, 4] pdf[x < 0] = 0 pdf[x > 4] = 0 return pdf def create_plot_MP(d): n_samples = 1 # Generate the complex Wishart matrix and compute eigenvalues eigenvalues = [] for _ in range(n_samples): # Generate a d x d matrix with complex standard normal entries # E[|X_ij|^2] = E[(Re(X_ij))^2] + E[(Im(X_ij))^2] = 1/2 + 1/2 = 1 X = (np.random.randn(d, d) + 1j * np.random.randn(d, d)) / np.sqrt(2) # Compute the Wishart matrix W = (1/d) * X^dagger * X W = (X.conj().T @ X) / d # Compute the eigenvalues (W is Hermitian, so eigenvalues are real) evals = np.linalg.eigvalsh(W) eigenvalues.extend(evals) eigenvalues = np.array(eigenvalues) # Generate x values for the theoretical curve x_theory = np.linspace(0, 4.1, 500) # Extend slightly beyond 4 for plotting y_theory = marchenko_pastur_pdf(x_theory) # Create the plot plt.style.use('default') plt.figure(figsize=(10, 6)) # Plot the histogram of the eigenvalues # Use density=True to normalize the histogram count, bins, ignored = plt.hist(eigenvalues, bins=50, density=True, label='Entanglement spectrum', alpha=0.7) # Plot the theoretical Marchenko-Pastur density plt.plot(x_theory, y_theory, '-r', linewidth=2, label='Marchenko-Pastur PDF') # Add labels and title plt.xlabel('Eigenvalue') plt.ylabel('Density') plt.title(f'Entanglement spectrum of a random state (d={d}) vs Marchenko-Pastur Law') plt.legend() plt.grid(True) plt.ylim(bottom=0, top=3) # Ensure y-axis starts at 0 plt.xlim(left=-0.1, right = max(4.1, np.max(eigenvalues)*1.05)) # Adjust x-axis limits return plt return create_plot_MP, marchenko_pastur_pdf @app.cell def _(mo): mo.md( r""" We observe that as the dimension $d$ grows to infinity, the probability distribution $\mu$ converges to a deterministic distribution, the [**Marchenko-Pastur distribution**](https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution). This distribution is supported on the interval $[0, 4]$ and has a density given by $$\mathrm{d}MP(x) = \frac{\sqrt{x(4-x)}}{2\pi x} \mathbb{1}_{(0,4)}(x) \, \mathrm{d}x,$$ For such a distribution of the entanglement spectrum, the entanglement entropy behaves asymptotically like $\log d$, a behavior commonly known as the **volume law**: the entropy scales like the size (volume) of the system. More precisely, we have $$\boxed{S_{\text{ent}}(\ket \psi) = \log d - c + o(1)}$$ where $c$ is a constant correction given by the entropy of the limiting distribution of the entanglement spectrum $$c = \int_0^4 x \log x \, \mathrm{d}MP(x) = \frac 1 2,$$ which fits the value predicted by the Page formula. """ ) return @app.cell def _(mo): mo.md( r""" ### Beyond uniform random states Uniform random quantum states are unphysical due to volume-law entanglement (entropy scales with system size), unlike physical states (ground states of local Hamiltonians) which obey the [**area law**](https://en.wikipedia.org/wiki/Entropy_of_entanglement#Area_law_of_bipartite_entanglement_entropy) (entropy scales with boundary area) and occupy a small, structured corner of the Hilbert space. ![area law](public/area-law.png) """ ) return @app.cell def _(mo): mo.md( r""" ## Tensor networks Tensor networks, by their construction, efficiently parameterize states that naturally obey the area law. Their inherent structure limits the amount of entanglement a state can possess, making them a much more physically realistic model for describing the low-energy states of many-body quantum systems and capturing the essential physics without the exponential complexity of generic Hilbert space vectors. ![tensor-networks](public/tensor-networks.png) """ ) return @app.cell def _(mo): mo.md( r""" A tensor network is a graphical representation of a quantum state, where the vertices (nodes) correspond to tensors and edges are of two types: - *bulk edges* (or bonds) that connect tensors to each other and corespond to tensor contractions - *boundary edges* (or legs) that connect tensors to the physical degrees of freedom of the system. ![fig1](public/fig1.png) It is common to partition the boundary edges in two parts $\textcolor{orange}{A} \sqcup \textcolor{green}{B}$ corresponding to a bipartition of the physical degrees of freedom. The entanglement of the tensor network state shall be understood with respect to this bipartion. """ ) return @app.cell def _(mo): mo.md( r""" ### Cuts in tensor networks ![fig1-cut](public/fig1-cut.png) A **cut** in a tensor network (with a given partition of the boundary edges $A \sqcup B$) is a set of edges that, if removed, disconnect $A$ from $B$. Any cut induces an *upper bound* on the rank of the induced linear map $A \to B$ $$ \operatorname{rank}L_{A \to B} \leq d^{\text{ size of the cut}}$$ In the example above, although $\dim \mathcal H_A = \dim \mathcal H_B = d^{5}$, the blue cut induced a bound of $d^4$ on the rank, and thus on the entanglement entropy $$ S_{\text{ent}}(\ket \psi_{AB}) \leq 4 \cdot \log d.$$ """ ) return @app.cell def _(mo): mo.md( r""" ### Random tensor networks Start with **independent Gaussian tensors** $g_x$ at each verte of the network. ![fig1-tensors](public/fig1-tensors.png) Then, contract these tensors along the bulk edges $e \in E_b$: $$ \ket{\psi_G}:=\Big\langle \bigotimes_{e\in E_b}\Omega_e \; \Big | \; \bigotimes_{x\in V} g_x \Big\rangle \in \big(\mathbb C^D\big)^{\otimes|E_{\partial}|}$$ """ ) return @app.cell def _(mo): mo.md( r""" ## Main result **Theorem.** In the limit $d \to \infty$, the average entropy of entanglement of a random tensor network state $\ket{\psi_G}$ behvaes as $$\mathbb E S_{\text{ent}}(\ket{\psi_G}) = \operatorname{mincut}(G_{A|B}) \cdot \log d -\int t\,\log t\,\mathrm{d}\mu_{G_{A|B}}+o(1),$$ where $\operatorname{mincut}(G_{A|B})$ is the **minimal cut** in the network $G_{A|B}$ and $\mu_{G_{A|B}}$ is a probability distribution that depends on the network. More precisely, this measure depends on the number of minimal cuts and on the inclusion structure of the minimal cuts.
Note that in any network, $\operatorname{mincut}(G_{A|B}) = \operatorname{maxflow}(G_{A|B})$, the **maximum flow** that can be sent in the network from nodes in $A$ to nodes in $B$. By duality, the result above can be formulated in terms of maximal flows in the network. """ ) return @app.cell def _(mo): mo.md( r""" ### Series-parallel networks The measure $\mu_{G_{A|B}}$ can be *explicitly computed* in the case of **[series-parallel](https://en.wikipedia.org/wiki/Series%E2%80%93parallel_graph)** networks, using tools from [free probability theory](https://en.wikipedia.org/wiki/Free_probability). Series-parallel networks can be obtained by recursively by the two operations below. ![series-parallel](public/series-parallel.png) The network discussed before is series-parallel, and we have $$\mu_{G_{A|B}} = \left\{\left[ \mathrm{MP}^{\boxtimes 3} \boxtimes (\mathrm{MP}^{\boxtimes 2} \times \mathrm{MP}) \right] \times \left[ (\mathrm{MP} \times \mathrm{MP}) \boxtimes \mathrm{MP} \right] \right\} \boxtimes \mathrm{MP}^{\boxtimes 2}.$$ """ ) return @app.cell def _(mo): mo.md( r""" ### An example The completely random state that follows the Page law corresponds to the structureless network ![N1](public/N1.png) Consider now the next simplest network ![N1](public/N2.png) This network also corresponds to a random state $\ket \psi_{AB} \in \mathbb C^d \otimes \mathbb C^d$, with an internal structure: one bulk edge connects the two independent random tensors (here: matrices) $g_1$ and $g_2$. Using our tools, we can show that the limiting distribution of the entanglement spectrum of $\ket \psi_{AB}$ is given by the **free multiplicative convolution** of two Marchenko-Pastur distributions, and that $$\boxed{S_{\text{ent}}(\ket \psi_{AB}) = 1 \cdot \log d - \frac 5 6 +o (1).}$$ Hence, the correction to the volume law is *larger* than in the uniform case. """ ) return @app.cell def _(np, plt): # Define the Marchenko-Pastur PDF for c=1, sigma^2=1 # lambda_minus = (1 - sqrt(c))^2 = (1-1)^2 = 0 # lambda_plus = (1 + sqrt(c))^2 = (1+1)^2 = 4 # Support is [0, 4] def FC_pdf(x): """Fuss-Catalan P_2(x) distribution density.""" density = np.zeros_like(x, dtype=float) mask = (x > 1e-6) & (x < 27./4.) xm = x[mask] sqrt_term = np.sqrt(np.maximum(0., 81. - 12. * xm)) term_brackets = np.maximum(0., 27. + 3. * sqrt_term) term_pow_1_3 = np.power(term_brackets, 1./3.) term_pow_2_3 = np.power(term_brackets, 2./3.) x_pow_1_3 = np.power(xm, 1./3.) x_pow_2_3 = np.power(xm, 2./3.) const = (np.sqrt(3.) * np.power(2., 1./3.)) / (12. * np.pi) num = (np.power(2., 1./3.) * term_pow_2_3 - 6. * x_pow_1_3) den = x_pow_2_3 * term_pow_1_3 valid_den = np.abs(den) > 1e-15 density_masked = np.zeros_like(xm) density_masked[valid_den] = const * num[valid_den] / den[valid_den] density[mask] = density_masked return np.maximum(0, np.nan_to_num(density)) def create_plot_FC(d): n_samples = 1 # Generate the complex Wishart matrix and compute eigenvalues eigenvalues = [] for _ in range(n_samples): # Generate a d x d matrix with complex standard normal entries # E[|X_ij|^2] = E[(Re(X_ij))^2] + E[(Im(X_ij))^2] = 1/2 + 1/2 = 1 X = (np.random.randn(d, d) + 1j * np.random.randn(d, d)) / np.sqrt(2) Y = (np.random.randn(d, d) + 1j * np.random.randn(d, d)) / np.sqrt(2) # Compute the Wishart matrix W = (1/d) * X^dagger * X W = (X @ Y @ Y.conj().T @ X.conj().T) / d**2 # Compute the eigenvalues (W is Hermitian, so eigenvalues are real) evals = np.linalg.eigvalsh(W) eigenvalues.extend(evals) eigenvalues = np.array(eigenvalues) # Generate x values for the theoretical curve x_theory = np.linspace(0, 27/4.0+0.1, 500) # Extend slightly beyond 4 for plotting y_theory = FC_pdf(x_theory) # Create the plot plt.style.use('default') plt.figure(figsize=(10, 6)) # Plot the histogram of the eigenvalues # Use density=True to normalize the histogram count, bins, ignored = plt.hist(eigenvalues, bins=50, density=True, label='Entanglement spectrum', alpha=0.7) # Plot the theoretical Marchenko-Pastur density plt.plot(x_theory, y_theory, '-r', linewidth=2, label='Fuss-Catalan(2) PDF') # Add labels and title plt.xlabel('Eigenvalue') plt.ylabel('Density') plt.title(f'Entanglement spectrum of a RTN [--1--2--] (d={d}) vs Fuss-Catalan(2) Law') plt.legend() plt.grid(True) plt.ylim(bottom=0, top=2) # Ensure y-axis starts at 0 plt.xlim(left=-0.1, right = max(27/4.0+1, np.max(eigenvalues)*1.05)) # Adjust x-axis limits return plt return FC_pdf, create_plot_FC @app.cell def _(create_plot_FC, d_slider_FC, mo): # Create the plot using the current slider value # This plot variable implicitly depends on 'amplitude_slider' spectrumFC = create_plot_FC(d_slider_FC.value) # Arrange the slider and the plot vertically and display them # This is the last expression in the cell mo.vstack([d_slider_FC, spectrumFC.gca()]) return (spectrumFC,) @app.cell def _(mo): mo.md( r""" ## Take home message Tthe average entropy of entanglement of a random tensor network state $\ket{\psi_G}$ behvaes as $$\mathbb E S_{\text{ent}}(\ket{\psi_G}) = \operatorname{mincut}(G_{A|B}) \cdot \log d -\int t\,\log t\,\mathrm{d}\mu_{G_{A|B}}+o(1),$$ where $\operatorname{mincut}(G_{A|B})$ is the **minimal cut** in the network $G_{A|B}$ and $\mu_{G_{A|B}}$ is a probability distribution that depends on the network. Our **main contribution** is the explicit computation, in the *series-parallel case*, of the limiting measure $\mu_{G_{A|B}}$ which gives the constant correction to the area law. ![fig1-cut](public/fig1-cut.png) """ ) return @app.cell(hide_code=True) def _(): import marimo as mo import numpy as np import matplotlib.pyplot as plt import math return math, mo, np, plt @app.cell def _(): return if __name__ == "__main__": app.run()