Quantum teleportation and entanglement swapping – moving states, not particles
Justin S
Introduction Quantum teleportation moves quantum information (i.e. states) not matter through entanglement to recreate an unknown quantum state while destroying the original. Meanwhile, entanglement swapping goes one step further: it entangles two systems that have never interacted, allowing for the theoretical long-distance networking via quantum repeaters.
This essay will explore what the protocols do; how the first experiments worked; and how repeaters scale alongside their real-world implementations. As there are many, many concepts, most will not be explained for brevity’s sake.
Dirac notation But before we begin, we have to explain Dirac notation.
Dirac Notation, also known as Bra-Ket notation, is a mathematical system introduced by Paul Dirac to concisely represent quantum states and operations: - | ψ ⟩ is the standard representation for a vector, with ψ being the label - ⟨ ψ |, on the other hand, is the Hermitian Adjoint (i.e. complex-conjugate and transpose) of | ψ ⟩ .
For example,
if
then
(1)
Furthermore, the tensor product builds the state space and operators for combined systems from the pieces. [1] When it acts on states for example:
( α |0 ⟩ + β |1 ⟩ ) ⊗ ( γ |0 ⟩ + δ |1 ⟩ ) = αγ |00 ⟩ + αδ |01 ⟩ + βγ |10 ⟩ + βδ |11 ⟩
(2)
When it acts on operators (matrices) for example:
(3)
Finally, a linear operator between spaces V and W is defined to be any function A : V →− B , which is linear in input [1]:
(4)
An important linear operator in any vector space V is the linear operator I :
I | v ⟩ ≡ | v ⟩
∀ V
(5)
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