Near the university, there's an old café called "Shannon." Drawn by the name, Yuki and Aoi visited for the first time.
A quiet middle-aged owner greeted them from behind the counter.
"Interested in information theory?" the owner asked while serving coffee.
"How did you know?" Yuki was surprised.
"Students who enter because of this café's name usually are," the owner smiled. "I'm Professor S. Just a café master now."
Aoi's eyes widened. "Could you be from the university..."
"I'm retired. Now I quietly brew coffee."
Yuki gathered courage to ask. "What does channel capacity mean?"
Professor S answered while wiping the counter. "Imagine a pipe. There's a limit to how much water can flow. Communication channels are the same. There's an upper limit to the amount of information that can be sent."
"That upper limit is the channel capacity?"
"Yes. One of the most important principles Shannon discovered. No matter how cleverly you encode, there's a wall you can't cross."
Aoi supplemented. "C = B log₂(1 + S/N). Determined by bandwidth B and signal-to-noise ratio S/N."
"Accurate," Professor S nodded. "Bandwidth is like the pipe's thickness. Signal-to-noise ratio is like water purity."
Yuki pondered. "So more noise means less capacity?"
"Exactly. Without any noise, theoretically infinite capacity. But real communication channels always have noise."
The owner refilled their coffee.
"For example, talking in this café. If it's quiet around us, whispers work. But if it's noisy, you need to speak loudly to be heard."
"You increase the signal strength," Aoi said.
"That's one method. But smarter is encoding. Choose phrasings resistant to noise."
Yuki suddenly thought of something. "Like repeating yourself?"
"Exactly. Adding redundancy. But that takes time. It's a tradeoff between communication speed and reliability."
Professor S drew a diagram with chalk on a blackboard.
"Shannon's channel coding theorem. At rates below channel capacity C, codes exist that can make error probability arbitrarily small. But exceeding C, any code will necessarily err."
"There's a limit," Yuki murmured.
"But knowing that limit is important. It saves wasted effort."
Aoi asked. "Are modern communication systems close to that limit?"
"5G, Wi-Fi, optical fiber. Remarkably close. Turbo codes and LDPC codes reach within a few percent of Shannon's limit."
"A few percent..." Yuki was impressed.
"The distance between theory and implementation has shrunk. That was the communication revolution of the late 20th century."
The café became a bit livelier. Other customers entered.
Professor S said quietly. "Information theory teaches the boundary between possibility and impossibility. What can be done and what cannot."
"Like philosophy," Yuki said.
"Same as physical laws. Just as you can't exceed light speed, you can't exceed channel capacity."
Aoi took a sip of coffee. "But we can do our best within those limits."
"Yes. That's the beauty of engineering," Professor S smiled.
Yuki took notes. "Knowing limits is the first step to freedom."
"Good words," Professor S acknowledged. "Shannon would surely have thought so too."
Outside the café, countless radio waves fly through the air. Each carries maximum information within the invisible wall of Shannon's limit.
"Thank you very much," Yuki and Aoi left the café.
"Please come again. Next time, let's talk about coding theory."
The door chime rang. Silence returned to Café Shannon once more.