Quantum teleportation does not move matter and does not transmit information faster than light. It transfers an unknown quantum state using a previously shared entangled pair plus two ordinary classical bits.
Superdense coding reverses the resource pattern: one shared entangled pair allows two classical bits to be communicated by sending one qubit. Entanglement swapping goes one step sideways: a Bell measurement can entangle two systems that have never interacted directly.
All three protocols are exercises in basis transformation and resource accounting. The mathematics is compact enough to derive by hand, yet rich enough to show why entanglement is a usable resource rather than merely a strange correlation.
Shared entanglement does not replace communication. It changes what communication can accomplish.
1. The Bell basis
The four Bell states are
|Φ+⟩=(|00⟩+|11⟩)/√2|Φ-⟩=(|00⟩−|11⟩)/√2|Ψ+⟩=(|01⟩+|10⟩)/√2|Ψ-⟩=(|01⟩−|10⟩)/√2.
They form an orthonormal basis for two qubits. A Bell-basis measurement therefore has four mutually exclusive outcomes and can encode two classical bits of result information.
2. Generate Bell states with H and CNOT
Starting from |00⟩, apply a Hadamard to the first qubit and then a CNOT from the first to the second:
|00⟩ → (|00⟩+|10⟩)/√2 → (|00⟩+|11⟩)/√2=|Φ+⟩.
The inverse circuit—a CNOT followed by H on the first qubit—maps the Bell basis to the computational basis. This is the standard Bell-measurement circuit.
3. The unknown qubit
Let Alice possess an unknown state
|ψ⟩=α|0⟩+β|1⟩
with |α|²+|β|²=1.
“Unknown” means Alice need not know the amplitudes well enough to describe or reproduce them classically. The teleportation protocol must work for every valid α and β by linearity.
4. Shared entanglement is prepared before the message
Alice and Bob share a Bell pair |Φ+⟩. Alice holds one half and Bob holds the other. Label the unknown qubit 1, Alice’s Bell-half 2 and Bob’s qubit 3.
The combined state is
|ψ⟩_1⊗|Φ+⟩_23.
The entangled pair can be distributed before Alice even receives |ψ⟩. This pre-shared resource does not contain the unknown state in advance.
5. Rewrite the three-qubit state in the Bell basis
The crucial identity is
|ψ⟩_1|Φ+⟩_23 = (1/2)[ |Φ+⟩_12 I|ψ⟩_3 + |Φ-⟩_12 Z|ψ⟩_3 + |Ψ+⟩_12 X|ψ⟩_3 + |Ψ-⟩_12 XZ|ψ⟩_3 ]
up to an irrelevant global phase convention on the final term.
The identity says that a Bell-basis measurement on qubits 1 and 2 projects Bob’s qubit into one of four Pauli-related versions of the original unknown state.
6. Derive the identity directly
Expand the initial state:
(1/√2)(α|000⟩+α|011⟩+β|100⟩+β|111⟩).
Use the inverse Bell-basis relations
|00⟩=(|Φ+⟩+|Φ-⟩)/√2|11⟩=(|Φ+⟩−|Φ-⟩)/√2|01⟩=(|Ψ+⟩+|Ψ-⟩)/√2|10⟩=(|Ψ+⟩−|Ψ-⟩)/√2.
Collect terms by Bell state. The four Bob-side states become I|ψ⟩, Z|ψ⟩, X|ψ⟩ and XZ|ψ⟩.
7. Alice measures two classical bits
Alice performs a Bell measurement on qubits 1 and 2. Each Bell outcome occurs with probability 1/4, independent of α and β.
She encodes the four outcomes as two classical bits and sends those bits to Bob through an ordinary classical channel.
The classical message does not describe α and β. It only tells Bob which Pauli frame his qubit occupies.
8. Bob’s correction table
| Alice Bell outcome | Bob has | Correction |
|---|---|---|
| Φ+ | |ψ⟩ | I |
| Φ− | Z|ψ⟩ | Z |
| Ψ+ | X|ψ⟩ | X |
| Ψ− | XZ|ψ⟩ | ZX or equivalent Pauli-frame update |
Because Pauli operators square to identity up to phase, Bob recovers |ψ⟩ exactly in the ideal protocol.
9. The original state is destroyed
Alice’s Bell measurement irreversibly removes the original standalone state from qubit 1. Teleportation is therefore consistent with the no-cloning theorem.
After successful correction, Bob possesses the state that Alice’s input qubit had, while Alice no longer has an independent copy.
“Teleportation” means state transfer, not duplication.
10. Why teleportation cannot signal faster than light
Before Bob receives Alice’s two classical bits, his local state averaged over the four Bell outcomes is maximally mixed:
ρ_B=(1/4)Σ_{P∈{I,X,Z,XZ}} P|ψ⟩⟨ψ|P†=I/2.
It contains no locally accessible information about α and β.
Only after the two classical bits arrive can Bob select the correct Pauli frame and recover |ψ⟩. Relativistic signalling limits therefore remain intact.
11. Teleporting one qubit consumes one ebit and two classical bits
The ideal resource accounting is:
- one pre-shared maximally entangled qubit pair, often called one ebit;
- two classical bits from Alice to Bob;
- one unknown input qubit state;
- local quantum operations and measurement.
The ebit is consumed. The classical bits are indispensable.
12. Teleportation fidelity with imperfect entanglement
If the shared pair is noisy, the teleportation channel is noisy. The exact average fidelity depends on the shared state’s entanglement and the protocol.
For standard teleportation using a two-qubit resource state, the singlet fraction determines the optimal fidelity within the corresponding protocol family.
A classical measure-and-prepare strategy for uniformly distributed unknown qubit states has an average fidelity limit of 2/3. Exceeding that benchmark is evidence that entanglement is providing a genuine quantum resource, subject to the assumptions of the test.
13. Superdense coding reverses the resource direction
Alice and Bob begin with |Φ+⟩. Alice wants to send one of four classical messages: 00,01,10,11.
She applies one of four local Pauli operations to her half:
- I maps Φ+ to Φ+;
- Z maps Φ+ to Φ−;
- X maps Φ+ to Ψ+;
- XZ maps Φ+ to Ψ− up to global phase.
The four possible two-qubit states are orthogonal Bell states.
14. Alice sends one qubit
After encoding, Alice physically sends her qubit to Bob. Bob now holds both halves and performs a Bell-basis measurement.
Because the four Bell states are orthogonal, Bob identifies the message with certainty in the ideal model.
Thus one transmitted qubit plus one pre-shared ebit communicates two classical bits.
15. Dense coding does not violate the Holevo bound
The transmitted system is not the entire resource. Alice and Bob already share entanglement established earlier.
The two classical bits are encoded in the joint two-qubit state accessible to Bob after Alice sends her half.
Entanglement-assisted channel capacity differs from unassisted capacity. Guide 32 develops this distinction formally.
16. Teleportation and superdense coding are resource duals
Teleportation:
1 qubit state + 1 ebit + 2 classical bits → 1 remote qubit state.
Superdense coding:
1 ebit + 1 transmitted qubit → 2 classical bits.
These resource inequalities capture a deep symmetry between quantum communication and shared entanglement.
17. Entanglement swapping
Suppose qubits 1 and 2 form a Bell pair, and qubits 3 and 4 form another Bell pair. Qubits 1 and 4 have never interacted.
Perform a Bell measurement jointly on qubits 2 and 3.
Conditional on the Bell outcome, qubits 1 and 4 collapse into a corresponding Bell state. Entanglement has been swapped from pairs (1,2) and (3,4) to the remote pair (1,4).
18. Bell-basis identity for swapping
For two initial Φ+ pairs, one can rewrite
|Φ+⟩_12|Φ+⟩_34 = (1/2)[|Φ+⟩_23|Φ+⟩_14 + |Φ-⟩_23|Φ-⟩_14 + |Ψ+⟩_23|Ψ+⟩_14 + |Ψ-⟩_23|Ψ-⟩_14].
The Bell outcome on 2–3 labels which Bell state appears on 1–4.
If a standard target Bell state is desired, a known local Pauli correction can be applied after the classical outcome is communicated.
19. Why swapping matters for quantum networks
Directly distributing high-fidelity entanglement over long distances is difficult because loss grows with distance.
A quantum repeater architecture can create entanglement over shorter segments, store the pairs in quantum memories, purify or error-correct them, and perform entanglement swapping to extend the connection.
The swap operation is therefore the network analogue of joining two entangled links into one longer link.
20. Swapping does not create entanglement from nothing
The two original Bell pairs already contain entanglement. The Bell measurement redistributes that resource across a different partition of the four systems.
If both initial pairs are separable, local measurement and classical communication at the middle station cannot generate remote entanglement between independent endpoints.
This connects directly to the LOCC resource theory in Guide 31.
21. Teleportation through a Pauli frame
Bob need not always apply the physical Pauli correction immediately. In a Clifford-dominated fault-tolerant circuit, the required X/Z correction can often be tracked classically as a Pauli frame.
This reduces physical gate count and integrates teleportation naturally with error-corrected architectures.
Gate teleportation generalises the idea: resource states and measurements can teleport not just a state but the action of a logical gate.
22. Common misconception: teleportation scans and rebuilds a qubit
Alice never learns α and β. The two classical bits carry only a Bell-outcome label. The unknown amplitudes are transferred through the joint entangled quantum state and measurement structure.
23. Common misconception: superdense coding sends two bits in one bare qubit
It uses one transmitted qubit plus pre-shared entanglement. Without the entangled half already held by Bob, the four local Pauli encodings are not four perfectly distinguishable states of Alice’s one qubit.
24. Common misconception: entanglement swapping sends a signal to the endpoints
Before the Bell-measurement result is known, the endpoints do not know which Bell state they share. The classical outcome remains necessary for a deterministic standardised resource.
25. Worked synthesis problem
Alice teleports |ψ⟩=(√3/2)|0⟩+(1/2)|1⟩ and obtains Bell outcome Ψ+.
Step 1: Bob’s conditional state. From the teleportation identity, Bob has X|ψ⟩.
Step 2: Apply X.
X|ψ⟩=(√3/2)|1⟩+(1/2)|0⟩.
Applying X again returns
|ψ⟩=(√3/2)|0⟩+(1/2)|1⟩.
Step 3: Resource accounting. One Bell pair is consumed and Alice sends two classical bits identifying Ψ+.
Step 4: No-signalling check. Before those bits arrive, Bob cannot distinguish this input from any other unknown qubit using his local state alone.
26. Practice set
- List the four Bell states.
- What resources are consumed by ideal one-qubit teleportation?
- Why are Alice’s Bell outcomes each probability 1/4?
- What correction follows a Φ− outcome?
- Why is teleportation consistent with no-cloning?
- Why is classical communication required?
- How does superdense coding encode four messages?
- What does Bob measure in dense coding?
- What resources allow two classical bits to be sent through one transmitted qubit?
- What is entanglement swapping?
- Why can endpoints become entangled without directly interacting?
- What role does the Bell-measurement result play in swapping?
Answers
- Φ+, Φ−, Ψ+, Ψ− as defined above.
- One shared ebit and two classical bits, plus local operations and measurement.
- The Bell-basis expansion has equal coefficient magnitude 1/2 for all four orthogonal branches.
- Z.
- Alice’s Bell measurement destroys the original standalone input state.
- Bob must know which Pauli correction or frame update corresponds to Alice’s outcome.
- Alice applies I, Z, X or XZ to her half of a shared Bell pair.
- The Bell basis after receiving Alice’s qubit.
- One pre-shared ebit plus one transmitted qubit.
- A Bell measurement on middle qubits redistributes two existing entangled links into a remote entangled endpoint pair.
- The joint measurement projects the remote pair into a Bell state through the four-qubit entangled expansion.
- It identifies which Bell state the endpoints share and therefore which correction/frame convention is needed.
Sources and further study
[1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres and William K. Wootters, Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels, Physical Review Letters 70, 1895 (1993). The original quantum teleportation protocol.
[2] Charles H. Bennett and Stephen J. Wiesner, Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states, Physical Review Letters 69, 2881 (1992). The dense-coding protocol.
[3] Marek Żukowski, Anton Zeilinger, Michael A. Horne and Artur K. Ekert, Event-ready-detectors Bell experiment via entanglement swapping, Physical Review Letters 71, 4287 (1993). The canonical entanglement-swapping proposal.
Continue through Quantum Mathematics
Guide 29: Bell Inequalities, CHSH, Tsirelson Bounds and No-Signalling establishes the correlation limits. Guide 31: Entanglement Transformations, LOCC, Majorisation and Distillation formalises entanglement as a convertible resource. Guide 32: Quantum Shannon Theory, Holevo Bounds, Channel Capacities and Coding Theorems places these protocols inside information-rate limits.