Quantum Teleportation Logic

Quantum states cannot be copied. This fundamental principle presents a significant challenge for network design. If you cannot copy data, how do you send it from a processor to memory, or from one quantum computer to another? The answer lies in a protocol known as quantum teleportation.

Despite the science fiction name, quantum teleportation is a rigorous networking protocol. It does not transport matter. Instead, it transfers the state of a qubit (the information) from one location to another using entanglement and classical communication. By the end of the process, the original qubit is destroyed (measured), and the destination qubit inherits the exact state of the original. This adheres strictly to the No-Cloning Theorem because the information never exists in two places simultaneously.

The Three-Qubit System

To understand the logic, imagine two parties, Alice and Bob. Alice has a qubit she wants to send to Bob. Let us call this the payload qubit ($\psi$). Its state is unknown to Alice, and it can be represented as:

$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$

Alice cannot simply measure this qubit and tell Bob the result. A measurement would return only $0$ or $1$, destroying the superposition coefficients $\alpha$ and $\beta$. The quantum information would be lost.

To transfer the full quantum state, the system requires three qubits in total:

1. The Payload ($\psi$): Held by Alice.
2. Alice's Resource Qubit ($A$): Part of an entangled pair.
3. Bob's Resource Qubit ($B$): The other part of the entangled pair.

Before the protocol begins, qubits $A$ and $B$ must be prepared in a Bell State (specifically $|\Phi^+\rangle$) and distributed so that Alice holds $A$ and Bob holds $B$. This entangled pair acts as the quantum bridge between the two parties.

The Protocol Workflow

The teleportation process involves a specific sequence of gates and measurements. It relies on the fact that operations performed on one half of an entangled pair affect the global state of the system.

Logic flow of the quantum teleportation protocol demonstrating the transfer of state information via entanglement and classical bits.

Step 1: Alice's Operations

Alice now holds two qubits: the payload ($\psi$) and her half of the entangled pair ($A$). She performs two operations to mix the information of the payload into the entangled pair:

1. CNOT Gate: She uses the payload as the control and her entangled qubit as the target.
2. Hadamard Gate: She applies a Hadamard gate to the payload qubit.

At this stage, no information has left Alice's location. However, mathematical entanglement now links the payload, Alice's resource qubit, and Bob's resource qubit into a single three-qubit state.

Step 2: Measurement and Collapse

Alice measures both of her qubits. This is the critical step where the original quantum state is destroyed.

• Alice measures the payload qubit.
• Alice measures her resource qubit.

These measurements result in two classical bits of information. There are four possible outcomes: $00$, $01$, $10$, or $11$. Because of the entanglement, the moment Alice measures her qubits, Bob's qubit ($B$) instantly collapses into a state that is mathematically related to the original payload, though usually slightly rotated.

Step 3: Classical Transmission

At this specific moment, Bob's qubit resembles the original payload, but it has been altered based on the outcome of Alice's measurement. To fix this, Bob needs to know how it was altered.

Alice must send her two classical bits to Bob. She can do this via an Ethernet cable, fiber optics, or radio waves. This step proves that quantum teleportation cannot happen faster than the speed of light. Bob cannot reconstruct the state until he receives the classical data.

The Correction Logic

Once Bob receives the two bits, he applies a specific quantum gate to his qubit to complete the transfer. The two bits act as instructions for correcting the state.

The logic follows a simple decision tree:

1. If 00: Bob does nothing (Identity gate). His qubit is already in the state $|\psi\rangle$.
2. If 01: Bob applies an X gate (Bit flip).
3. If 10: Bob applies a Z gate (Phase flip).
4. If 11: Bob applies a Z gate and then an X gate.

By applying these conditional operations, Bob reverses the specific rotation caused by Alice's measurement. His qubit $B$ transforms exactly into $\alpha|0\rangle + \beta|1\rangle$. The information has successfully moved from Alice to Bob.

Visualizing the Outcome Probabilities

When simulating this circuit, you will not see the intermediate "traveling" of the state. Instead, simulators usually validate teleportation by measuring Bob's qubit at the end. If the protocol works, Bob's measurement statistics should match what we expected from the original payload.

The correction table outlines how Bob interprets the two classical bits to recover the original quantum state.

Why This Matters for Engineering

Understanding this logic is important for building quantum networks. In a classical network, we use repeaters to boost signals over long distances. We cannot use classical repeaters for quantum data because they would copy (and thus destroy) the state.

Instead, engineers propose quantum repeaters that utilize teleportation. By creating a chain of entangled pairs, we can "teleport" a state from one node to the next, preserving the quantum information without ever copying it. This protocol is the foundation of the theoretical Quantum Internet.