optimize entropy

codes/plot_entropy_and_da.py

@@ -0,0 +1,52 @@
+"""
+plot the probability of the entropy of the reduced density matrix of the pure state being greater than log2(d_A) - alpha - beta
+
+for different d_A values, with fixed alpha and d_B Note, d_B>d_A
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+from quantum_states import sample_and_calculate
+from tqdm import tqdm
+
+# Set dimensions
+db = 32
+alpha = 0
+da_range = np.arange(2, 10, 1)  # Range of d_A values to plot
+n_samples = 1000000
+
+plt.figure(figsize=(10, 6))
+
+predicted_probabilities = []
+actual_probabilities = []
+
+for da in tqdm(da_range, desc="Processing d_A values"):
+    # Calculate beta according to the formula
+    beta = da / (np.log(2) * db)
+    
+    # Calculate probability according to the formula
+    # Ignoring constant C as requested
+    prob = np.exp(-((da * db - 1) * alpha**2 / (np.log2(da)**2)))
+    predicted_probabilities.append(prob)
+    # Calculate actual probability
+    entropies = sample_and_calculate(da, db, n_samples=n_samples)
+    count = np.sum(entropies < np.log2(da) - alpha - beta)
+    # early stop if count is 0
+    if count != 0:
+        actual_probabilities.append(count / n_samples)
+    else:
+        actual_probabilities.extend([np.nan] * (len(da_range) - len(actual_probabilities)))
+        break
+    # debug
+    print(f'da={da}, theoretical_prob={prob}, threshold={np.log2(da) - alpha - beta}, actual_prob={actual_probabilities[-1]}, entropy_heads={entropies[:10]}')
+    
+# plt.plot(da_range, predicted_probabilities, label=f'$d_A={da}$', linestyle='--')
+plt.plot(da_range, actual_probabilities, label=f'$d_A={da}$', linestyle='-')
+
+plt.xlabel(r'$d_A$')
+plt.ylabel('Probability')
+plt.title(r'$\operatorname{Pr}[H(\psi_A) < \log_2(d_A)-\alpha-\beta]$ vs $d_A$ for fixed $\alpha=$'+str(alpha)+r' and $d_B=$' +str(db)+ r' with $n=$' +str(n_samples))
+# plt.legend()
+plt.grid(True)
+plt.yscale('log')  # Use log scale for better visualization
+plt.show()