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feat: add key quantum protocols & concepts

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Kristofers Solo 2025-06-05 18:02:10 +03:00
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@ -348,3 +348,65 @@ $
U_i ket(z_1 ... z_n)=ket(z_1 ... z_n)
$
ja $z_1 ... z_n != x_1 ...x_n$, $z_1 ... z_n != y_1 .. y_n$.
= Key Quantum Protocols & Concepts
== No-Cloning Theorem
Impossible to create and identical copy of an arbitrary unknown quantum state.
== Quantum Teleprotation
Transmits $ket(psi)=a ket(0) + b ket(1)$ using an entangled pair
$ket(Phi^+)_(A B)=1/sqrt(2)(ket(00)+ket(11))$ and $2$ classical bits.
Initial state (Alice has $ket(psi)_C$ and qubit $A$, Bob has $B$):
$
ket(psi)_C tensor ket(Phi^+)_(A B)= \ =
(a ket(0)_C+b ket(1)_C)1/sqrt(2)(ket(0_A 0_B)+ket(1_A 1_B))= \ =
a/sqrt(2)ket(000)+a/sqrt(2)ket(011)+b/sqrt(2)ket(100)+b/sqrt(2)
$
(qubits $C$, $A$, $B$)
Alice applied $CNOT$ ($C$ is control, $A$ is target):
$
a/sqrt(2) ket(000)+a/sqrt(2)ket(011)+b/sqrt(2)ket(110)+b/sqrt(2)ket(101)
$
Alice applies $H$ to qubit $C$:
$
1/2[
a(ket(0)+ket(1))ket(11)+
a(ket(0)+ket(1))ket(11)+ \ +
b(ket(0)-ket(1))ket(10)+
b(ket(0)-ket(1))ket(01)
]
$
Regroup by Alice's qubits $C A$:
$
1/2[
ket(00)_(C A) (a ket(0) + b ket(1))+
ket(01)_(C A) (a ket(1) + b ket(0))+ \ +
ket(10)_(C A) (a ket(0) - b ket(1))+
ket(11)_(C A) (a ket(1) - b ket(0))
]
$
Alice measures $C A$, sends 2 classical bits to Bob. Bob applies correction to
his qubit $B$:
- Alice gets $00 ==>$ Bob has $a ket(0)+b ket(1)$ (Needs $I$).
- Alice gets $01 ==>$ Bob has $a ket(1)+b ket(0)$ (Needs $X$).
- Alice gets $10 ==>$ Bob has $a ket(0)-b ket(1)$ (Needs $Z$).
- Alice gets $11 ==>$ Bob has $a ket(1)-b ket(0)$ (Needs $Z X$).
== Dense Coding (Bļīvā kodēšana)
Sends 2 classical bits of information From Alice to Bob by sending only 1 qubit,
using pre-shared entangled pair.
=== Steps
+ Alice and Bob share $ket(Phi^+)_(A B)$
+ To send classical bits $x y$:
- $00$: Alice does nothing (applies $I$) to her qubit.
- $01$: Alice applies $X$ to her qubit.
- $10$: Alice applies $Z$ to her qubit.
- $11$: Alice applies $X$ then $Z$ (or $i Y$) to her qubit.
+ Alice sends her modified qubit to Bob.
+ Bob performs a Bell measurement on the two qubits he now possesses to recover
$x y$.