optimize entropy

codes/plot_entropy_and_deviate.py

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+import numpy as np
+import matplotlib.pyplot as plt
+from quantum_states import sample_and_calculate
+from tqdm import tqdm
+
+# Set dimensions, keep db\geq da\geq 3
+db = 64
+da_values = [4, 8, 16, 32]
+da_colors = ['b', 'g', 'r', 'c']
+n_samples = 100000
+
+plt.figure(figsize=(10, 6))
+
+# Define range of deviations to test (in bits)
+deviations = np.linspace(0, 1, 50)  # Test deviations from 0 to 1 bits
+
+for i, da in enumerate(tqdm(da_values, desc="Processing d_A values")):
+    # Calculate maximal entropy
+    max_entropy = np.log2(min(da, db))
+    
+    # Sample random states and calculate their entropies
+    entropies = sample_and_calculate(da, db, n_samples=n_samples)
+    
+    # Calculate probabilities for each deviation
+    probabilities = []
+    theoretical_probs = []
+    for dev in deviations:
+        # Count states that deviate by more than dev bits from max entropy
+        count = np.sum(max_entropy - entropies > dev)
+        # Omit the case where count is 0
+        if count != 0:
+            prob = count / len(entropies)
+            probabilities.append(prob)
+        else:
+            probabilities.append(np.nan)
+        
+        # Calculate theoretical probability using concentration inequality
+        # note max_entropy - dev = max_entropy - beta - alpha, so alpha = dev - beta
+        beta = da / (np.log(2)*db)
+        alpha = dev - beta
+        theoretical_prob = np.exp(-(da * db - 1) * alpha**2 / (np.log2(da))**2)
+        # # debug
+        # print(f"dev: {dev}, beta: {beta}, alpha: {alpha}, theoretical_prob: {theoretical_prob}")
+        theoretical_probs.append(theoretical_prob)
+    
+    plt.plot(deviations, probabilities, '-', label=f'$d_A={da}$ (simulated)', color=da_colors[i])
+    plt.plot(deviations, theoretical_probs, '--', label=f'$d_A={da}$ (theoretical)', color=da_colors[i])
+
+plt.xlabel('Deviation from maximal entropy (bits)')
+plt.ylabel('Probability')
+plt.title(f'Probability of deviation from maximal entropy simulation with sample size {n_samples} for $d_B={db}$ ignoring the constant $C$')
+plt.legend()
+plt.grid(True)
+plt.yscale('log')  # Use log scale for better visualization
+plt.show()