Quantum Teleportation Flashcards
Quantum Teleportation
sending q-bit information elsewhere
if have |phi> = a|0> + b|1>
can’t send a and b because they arent known and if we measure -> collapse to |0> or |1>, only know final state, the supperposition is destroyed and can’t measure again.
Instead share an entangled state |gamma> = 1/sqrt(2) |00> + 1/sqrt(2) |11>
will measure phi and first part of gamma, then will be able to deduce the amplitudes a and b from the ouputs. 4 different possible outputs: perform different unitary transformations on them.
The output of the transformation will be |phi> = a|0> + b|1>
No Cloning Theorem
you cannot clone a q-bit
It is not possible to find a Unitary transform that would make a circuit starting in |0> and |phi> to become both |phi>
so (a|0> + b|1>) (x) |0> -> (a|0> + b|1>) (x) (a|0> + b|1>)
but doesnt work (try with a , b = 0,1) contradiction
Quantum Teleportation 1st Approach
Quantum computation
start with |phi> = a|0> + b|1> and |0>
CNOT gate applied on the |0> state (this is the communication)
Then, CNOT( a|00> + b|10>) = a|00> + b|11>
=> CNOT entangles their qbits
Now measure 1st qbit in |+> and |-> state
(in 0 1 would just have |0> then |00> if have |1> then |11>)
rewrite : a|00> + b|11> = a(1/sqrt(2)|+> + 1/sqrt(2)|->) (x) |0> + b (1/sqrt(2)|+> - 1/sqrt(2)|->) (x) |1> = 1/sqrt(2) |+> (a|0> + b|1>) + 1/sqrt(2) |-> (a|0> - b|1>) . So if we measure either + or - we will collapse to these, if measure + (ie 0) then Bob gets phi and if - then rotate with the Z transform to get phi !
But this communication is unrealistic …
Quantum Teleportation 2nd Approach
Share a bell state gamma
and have phi = a|0> + b|1>
CNOT applied on phi and on the first part of the bell state. Control is phi, target is the 1st part of the composite state
composite state is (a|0> + b|1>) (x) gamma
a/sqrt(2)|000> + a/sqrt(2) |011> + b/sqrt(2) |100> + b/sqrt(2)|111>
CNOT =>
a/sqrt(2)|000> + a/sqrt(2) |011> + b/sqrt(2) |110> + b/sqrt(2)|101>
if get 0 then a|00> + b|11>, if get 1 then a|01> + b|10>
if 0 leave as it is, if 1 apply the flip X to get a|00> + b|11>
