Quantum Teleportation

Teleportation immediatly conjures up images from popular Sci-Fi scenarios as in Star Trek

The notion implies that it is possible to take each atom/molecule in an object, disassemble it, and
transport it to some other location in space/time where it is re-assembled.

There are obvious dangers associated with such types of teleportation. One of which was
dramatically illustrated in the 1958 Sci-Fi classic The Fly (starring Vincent Price)

Hyperlink["http://www.youtube.com/watch?v=NzTGSKNjrp4"]

Alice sets up a state |ψ⟩ in her Lab. With the quantum teleportation protocol she can send
this state to Bob. Though Bob might be hundreds of Km away.

Hyperlink["http://phys.org/news/2012-09-km-physicists-quantum-teleportation-distance.html"]

Bell States

In order to see how this is achieved we need to investigate the notion of a Bell state (named after 
John Bell, the quantum pioneer). Given two qubits we distinguished between those described by

|ψ⟩ = |0⟩|1⟩

|ψ⟩= |1 ⟩ ( |0⟩-i|1⟩ )/(Sqrt[2])

|ψ⟩= ( |0⟩-i|1⟩ )( |0⟩+ |1⟩ )/2

etc.

and those that look like (called entangled, or Bell, states)

|ψ⟩ = |0⟩|1⟩+|1⟩|0⟩    etc.

Notice that the first set involve states which is a product of Qubit 1 in a definite state
and Qubit 2 in a definite state. So if Alice performs a measurement in the first Qubit. How
does that effect what Bob measures for Qubit 2?

Example 1: Suppose Alice an Bob have access to the 2-Qubit  (e.g. two electrons, photons etc. )
that is described by the state

|ψ⟩ =  ((Sqrt[2/3]) |0⟩-i/(Sqrt[1/3])|1⟩ )( |0⟩ + |1⟩ )/(Sqrt[2])


Alice makes a measurement on Qubit 1

(i) What results could she measure ?
(ii) What is the probability that she obtains 0?
(iii) What is the quantum state immediately after she has made the measurement that
results in that 0 ?
(iv) Now Bob performs a measurement on his Qubit, what is the probability that Bob gets
the value 1?
(v) Suppose now that Alice obtains 1 for her measurement. Will this affect the answer that
Bob gets?
(vi) Suppose Alice does not do anything. Will this affect the answer that Bob gets?

Example 2: Suppose Alice an Bob have access to the 2-Qubit  (e.g. two electrons, photons etc. ) 
that is described by the state

|Ω⟩= (Sqrt[2/3]) |0⟩|1⟩ - i (Sqrt[1/3])|1⟩|0⟩


Alice makes a measurement on Qubit 1

(i) What results could she measure ?
(ii) What is the probability that she obtains 0?
(iii) What is the quantum state immediately after she has made the measurement that
results in that 0 ?
(iv) Now Bob performs a measurement on his Qubit, what is the probability that Bob gets
the value 1?
(v) Suppose now that Alice obtains 1 for her measurement. Will this affect the answer that
Bob gets?
(vi) Suppose Alice does not do anything. Will this affect the answer that Bob gets?

From examples 1,2 we note that there is a big qualitative difference between measurements
made with |ψ⟩, and |Ω⟩.  The former set of states can be factored and the latter can’t

e.g can we find two states for each Qubit so that their product is equal to Bell-like state?

|ψ⟩= a|0⟩ + b |1⟩    |a((|)^2)+ |b((|)^2) =1
|ϕ⟩= c|0⟩ + d |1⟩    |c((|)^2)+ |d((|)^2) =1

|ψ⟩|ϕ⟩ =( a|0⟩ + b |1⟩)(c|0⟩ + d |1⟩ ) = a c |0⟩|0⟩ + b c |1⟩|0⟩ + a d |0⟩|1⟩ + b d |1⟩|1⟩

we want that state to be the following Bell state

(|0⟩|1⟩+|1⟩|0⟩)/(Sqrt[2])

so we require 
ac = 0      
bd = 0   

possible solutions a =0 b=0 but then ad |0⟩|1⟩ vanishes
                             a =0 d=0 but then ad |0⟩|1⟩ vanishes
                             c =0  b=0 but then bc |1⟩|0⟩ vanishes
                             c =0  d=0 but then bc |1⟩|0⟩ vanishes

And so we see that it is it is impossible to express the 2-Qubit Bell state
(|0⟩|1⟩+|1⟩|0⟩)/(Sqrt[2])   as a product of definite state Qubits.

Some common Bell states

|(Subscript[β, 00])⟩ = ( |00⟩ + |11⟩ )/(Sqrt[2])
|(Subscript[β, 01])⟩ = ( |01⟩ + |10⟩ )/(Sqrt[2])
|(Subscript[β, 10])⟩ = ( |00⟩ - |11⟩ )/(Sqrt[2])
|(Subscript[β, 11])⟩ = ( |01⟩ - |10⟩ )/(Sqrt[2])

Question: Do such states appear in nature?

Answer: Heck yeah! (an emphatic yes) 

Every Helium atom in its ground state has two electrons. The electrons have a property called
spin, which we label as spin up or spin down. This state looks like

|↑⟩|↓⟩ - |↓⟩|↑⟩  =  |↑ ↓⟩ - |↓ ↑⟩

It is an entangled, or Bell like state.

Can Alice and Bob distinguish which electrons belongs to them ?

Many atoms can decay and emit two photons that separate in opposite directions 




These photons can travel for large distances (e.g 143 km) but still be entangled, i.e. be
in Bell like state

(|R L⟩ - |L R ⟩)/(Sqrt[2])

The ability for quantum objects to stay in such states for long times and distances is called
coherence.

Hyperlink["https://journals.aps.org/pr/pdf/10.1103/PhysRev.47.777"]