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Quantum Teleportation

Quantum teleportation transfers an arbitrary qubit state from Alice to Bob using a shared Bell pair and two classical bits of communication. No quantum information travels faster than light — the classical channel is essential. Teleportation is the prototype for quantum networking and demonstrates that entanglement is a consumable resource.

Key Concepts

Teleportation Protocol
Alice wants to send an unknown qubit ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle = \alpha|0\rangle+\beta|1\rangle to Bob. They share a Bell pair ∣Φ+⟩|\Phi^+\rangle. Alice performs a Bell measurement on her two qubits and sends the result (2 classical bits) to Bob. Bob applies one of four correction unitaries (I,X,Z,XZI, X, Z, XZ) to recover ∣ψ⟩|\psi\rangle.
Bell Measurement
A measurement in the Bell basis {∣Φ+⟩,∣Φ−⟩,∣Ψ+⟩,∣Ψ−⟩}\{|\Phi^+\rangle, |\Phi^-\rangle, |\Psi^+\rangle, |\Psi^-\rangle\} performed by Alice on her two qubits. The four equally-probable outcomes (00, 01, 10, 11) tell Bob which correction to apply. Implemented as CNOT then H then two computational-basis measurements.
No Faster-Than-Light Signaling
Teleportation cannot transmit information faster than light. Bob's qubit is in a mixed state until he receives the 2 classical bits from Alice and applies the correction. The quantum and classical channels together are required — neither alone suffices.
Resource Consumption
Each teleportation consumes exactly one Bell pair (entanglement resource) and sends exactly 2 classical bits. The unknown quantum state is destroyed at Alice's end (consistent with the no-cloning theorem). The protocol can be reversed: quantum information goes Alice→Bob, entanglement is consumed.
Superdense Coding
The dual protocol: using one shared Bell pair and sending ONE qubit, Alice can communicate 2 classical bits to Bob. This doubles the classical capacity of a quantum channel. Teleportation sends 1 qubit using 2 classical bits + 1 Bell pair; superdense coding sends 2 classical bits using 1 qubit + 1 Bell pair.

Key Equations

Initial Teleportation State
∣ψ⟩∣Φ+⟩=(α∣0⟩+β∣1⟩)∣00⟩+∣11⟩2|\psi\rangle|\Phi^+\rangle = (\alpha|0\rangle+\beta|1\rangle)\frac{|00\rangle+|11\rangle}{\sqrt{2}}
Three-qubit state before Alice's Bell measurement.
Post-Measurement Decomposition
=12[∣Φ+⟩(α∣0⟩+β∣1⟩)+∣Φ−⟩(α∣0⟩−β∣1⟩)+∣Ψ+⟩(α∣1⟩+β∣0⟩)+∣Ψ−⟩(α∣1⟩−β∣0⟩)]= \tfrac{1}{2}[|\Phi^+\rangle(\alpha|0\rangle+\beta|1\rangle) + |\Phi^-\rangle(\alpha|0\rangle-\beta|1\rangle) + |\Psi^+\rangle(\alpha|1\rangle+\beta|0\rangle) + |\Psi^-\rangle(\alpha|1\rangle-\beta|0\rangle)]
Each Bell measurement outcome leaves Bob's qubit in a known transformation of |ψ⟩.
Bob's Corrections
Outcome 00: I,01:X,10:Z,11:XZ\text{Outcome 00: } I, \quad 01: X, \quad 10: Z, \quad 11: XZ
Bob applies the appropriate Pauli gate based on Alice's 2-bit message to recover |ψ⟩.
Superdense Coding Capacity
1 qubit+1 Bell pair⟶2 classical bits1 \text{ qubit} + 1 \text{ Bell pair} \longrightarrow 2 \text{ classical bits}
One qubit channel, augmented by a shared Bell pair, carries 2 bits of classical information.
Worked Example

Tracing Through Teleportation

Problem

Alice has ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle = \alpha|0\rangle+\beta|1\rangle and shares ∣Φ+⟩|\Phi^+\rangle with Bob. She measures outcome ∣Ψ+⟩|\Psi^+\rangle (result 10). What gate must Bob apply?

Solution

From the decomposition, outcome ∣Ψ+⟩|\Psi^+\rangle (bits: 10) leaves Bob's qubit in state α∣1⟩+β∣0⟩\alpha|1\rangle + \beta|0\rangle.

This is X∣ψ⟩X|\psi\rangle — the bit-flipped version of ∣ψ⟩|\psi\rangle.

Bob applies XX (the inverse of XX is XX itself, since X2=IX^2=I):

X(α∣1⟩+β∣0⟩)=α∣0⟩+β∣1⟩=∣ψ⟩X(\alpha|1\rangle + \beta|0\rangle) = \alpha|0\rangle + \beta|1\rangle = |\psi\rangle
Answer Bob applies X to recover |ψ⟩.
Practice

Exercises

5 problems
1 of 5

How many classical bits must Alice send Bob per teleported qubit?

bits
2 of 5

How many Bell pairs are consumed per qubit teleported?

Bell pairs
3 of 5

How many possible outcomes can Alice's Bell measurement produce?

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4 of 5

In superdense coding, how many classical bits does Alice transmit to Bob by sending 1 qubit (with a shared Bell pair)?

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5 of 5

How many qubits are physically transmitted from Alice to Bob during standard teleportation?

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Key Takeaways

  • Teleportation transmits one qubit using 1 Bell pair + 2 classical bits; no quantum channel needed after setup.
  • The no-cloning theorem is respected: Alice's original qubit is destroyed in the Bell measurement.
  • Teleportation cannot signal FTL — Bob's correction requires the classical 2-bit message.
  • Superdense coding is the dual: 2 classical bits transmitted by sending 1 qubit (using a shared Bell pair).
  • These protocols reveal that qubits, classical bits, and entanglement (ebits) are three distinct resources.
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