What does "be to" mean?

Question

Harry swallowed and looked around him. He realized he must be in the hospital wing. He was lying in a bed with white linen sheets, and next to him was a table piled high with what looked like half the candy shop.
“Tokens from your friends and admirers,” said Dumbledore, beaming. “What happened down in the dungeons between you and Professor Quirrell is a complete secret, so, naturally, the whole school knows. I believe your friends Misters Fred and George Weasley were responsible for trying to send you a toilet seat. No doubt they thought it would amuse you. Madam Pomfrey, however, felt it might not be very hygienic, and confiscated it.”
“How long have I been in here?”
“Three days. Mr. Ronald Weasley and Miss Granger will be most relieved you have come round, they have been extremely worried.”
“But sir, the Stone —”
“I see you are not to be distracted. Very well, the Stone. Professor Quirrell did not manage to take it from you. I arrived in time to prevent that, although you were doing very well on your own, I must say.”
(Harry Potter and the Sorcerer's Stone)

If the sentence were ‘I see you are not distracted anymore’, I would think Dumbledore thought Harry could think properly, after having come round. Then what does the original mean?

Answer 1

The construction BE to VERB means, depending on context, BE expected/supposed/required to VERB. VERB may be any infinitive: simple, passive, perfect, or progressive (or any combination of these).

You are to leave immediately. = You are required to leave.
The prisoners were to be executed at dawn. = The prisoners were supposed to be executed at dawn.
She was to have been in New York on Thursday. = She was expected to have been there.
I am to be travelling all day tomorrow. = I am expected to be travelling—i.e. I expect to be travelling.

Negatives are a little tricky to paraphrase, because idiomatic use is unsymmetrical:

The prisoners were not to be executed until dawn. = They were not supposed to be executed. BUT
You are not to leave. = You are required not to leave, i.e. You are forbidden to leave.

And the negative passive has an additional possible meaning: BE not to be VERBen can mean BE impossible to VERB. So You are not to be distracted may be understood as either

1. You are required not be distracted, i.e., It is forbidden that you be distracted, OR
2. You are impossible to distract.

In this case it is clearly the latter which is meant: "I see I cannot distract you."

Answer 2

"To be distracted" is a passive infinitive, like "to distract" is an active one."

"You are not to be distracted" means you should not be the object of the verb to distract, i.e. Nobody must distract you.