Homework 8 - Exercise 2

Homework 8 - Exercise 2

par Yago Pérez Pérez,
Nombre de réponses : 9

What is meant by a perfect Bell basis? Does a) correspond to doing (|B_00⟩⟨B_00|) |Ψ⟩ for all the corresponding basis?

For b), if Alice sends Bob 01, is the teleported state then |φ⟩ = α|1⟩ + β|0⟩, or is it X * |Ψ⟩? Or is it something else since the entangled pair is corrupted


En réponse à Yago Pérez Pérez

Re: Homework 8 - Exercise 2

par Nicolas Macris,
So the thing is: You have to imagine that Alice and Bob do not know that the entangled state is corrupted. So they just follow the steps of the usual protocol. This means:

For a) yes as you sau, the perfect Bell projectors on the basis are |B_00>
For b) now you have to compute the teleported state by acting with the 01 projector on the shared entangled corrupted state x |phi>. Then you will find something different than in class.
En réponse à Nicolas Macris

Re: Homework 8 - Exercise 2

par Nicolas Macris,
I mean the ‘projectors’ on the perfect Bell basis are the ket-bra
| B_00>< B_00|
and the other ones.
The |B_00> is the perfect ‘state’
En réponse à Nicolas Macris

Re: Homework 8 - Exercise 2

par Yago Pérez Pérez,
I don't quite understand, if the perfect Bell basis refers only to B_00 (not B_01,B_10, B_11) then how can there be possible outcomes for a) for the same measurement of B_00
En réponse à Yago Pérez Pérez

Re: Homework 8 - Exercise 2

par Nicolas Macris,
Perfect Bell states refer to the 4 of them. So the measurement gives one of these 4 at random.
En réponse à Nicolas Macris

Re: Homework 8 - Exercise 2

par Yago Pérez Pérez,
"Alice does a measurement in the perfect Bell basis in her lab" and you said "The |B_00> is the perfect ‘state’". Does this mean that Alice measures using B_00?
To find the global state after the measurement I believe I'm supposed to do (|B_00⟩⟨B_00|) (|φ⟩⊗|Ψ⟩), where |φ⟩ = α|0⟩ + β|1⟩.

Do I then compute probability by doing | (⟨φ|⊗⟨Ψ|) | (|B_00⟩⟨B_00|) (|φ⟩⊗|Ψ⟩) |^2 ?
En réponse à Yago Pérez Pérez

Re: Homework 8 - Exercise 2

par Nicolas Macris,
Yes exactly.

And then same thing for the other three Bell sates which will give the other three probabilities.

For a sanity check, the four probabilities should sum to 1
En réponse à Nicolas Macris

Re: Homework 8 - Exercise 2

par Yago Pérez Pérez,

After a lot of tedious algebra I came to this result. Does this look somewhat correct? Because I spent like 40 mins just computing the global state and prob of B_00, and I don't wanna do this 3 more times and later find out that it was a waste of time


En réponse à Yago Pérez Pérez

Re: Homework 8 - Exercise 2

par Nicolas Macris,
Hi,
First sanity check: a probability must be a real number between 0 and 1. Your result appears to be complex so cannot be correct. Second sanity check: for delta =0 you should recoverthe perfect protocol result so this prob should become 1.
Maybe you did a midtake when computing the modulus square of the Dirac bracket ?
Or maybe somewhere else.
Best
N.M
En réponse à Nicolas Macris

Re: Homework 8 - Exercise 2

par Yago Pérez Pérez,
That was simply (⟨φ|⊗⟨Ψ|) | (|B_00⟩⟨B_00|) (|φ⟩⊗|Ψ⟩) without doing mod square, I did it and got something that's still wrong. I believe the mistake is in how I expanded the tensor products.

The top half is my computation of ⟨φ|⊗⟨Ψ|.
The bottom half if my computation of |B_00⟩⟨B_00|) (|φ⟩⊗|Ψ⟩, note there is a common factor of ((1+delta^2)^(-1/2))/sqrt(2) which I don't show in the computation, but I do later use it for the computation of |(⟨φ|⊗⟨Ψ|) | (|B_00⟩⟨B_00|) (|φ⟩⊗|Ψ⟩)|^2 correctly.