Q05 — Quantum Teleportation

Field Value
Domain Quantum Computing
System Bell pair + 2-qubit measurement + classical correction
Group SU(2) × SU(2) (local operations on two-qubit system)
H^k tier H¹
ISA Forge (β ≈ β*)
Status Validated
Opcodes ORBIT · TWIST · BIND · LABEL · FLIP · FLOP
Papers Paper 421, Paper 325

Physical system

Quantum teleportation (Bennett et al. 1993) transfers an arbitrary qubit state |ψ⟩ = α|0⟩ + β|1⟩ from Alice to Bob using:

  1. A pre-shared Bell pair (entangled resource)
  2. Alice’s Bell measurement on her qubit + the input qubit (2 classical bits output)
  3. Bob’s conditional Pauli correction based on Alice’s 2-bit measurement result

No quantum channel is used after the Bell pair is distributed — only 2 classical bits travel from Alice to Bob. The quantum information (α, β) is teleported without physically moving any quantum system carrying it.

The protocol succeeds with probability 1 and perfectly reconstructs |ψ⟩ at Bob’s location, regardless of α and β (which need not be known to Alice).

Standard circuit:

Alice: |ψ⟩ ──●── H ──[M_z]── c₁  ──────────────────
              │                                      
Bell:  |0⟩ ──X ── [M_z]────── c₂  ──────────────────
              
Bob:   |0⟩ ─────────────────── X^{c₂} ── Z^{c₁} ──|ψ⟩

Target category

TeleportCat — the category of quantum channels from Alice’s lab to Bob’s lab that can be implemented using only shared entanglement and classical communication (LOCC: Local Operations + Classical Communication). Objects are two-qubit states; morphisms are LOCC protocols. Teleportation is the canonical morphism in TeleportCat: it implements the identity channel id: ℒ(ℂ²) → ℒ(ℂ²) using one Bell pair and 2 bits of classical communication.

The impossibility of superluminal communication follows from the structure of TeleportCat: Alice’s measurement result is uniformly random (LABEL gives 00, 01, 10, 11 with equal probability 1/4); Bob cannot reconstruct |ψ⟩ without receiving the classical bits, so no information travels faster than the classical channel.

Interpretation functor

F: C → TeleportCat defined by:

Opcode F(opcode)          
BIND Bell pair preparation: Φ⁺⟩ = ( 00⟩+ 11⟩)/√2; the entangled resource; H² BIND between Alice’s qubit A and Bob’s qubit B; non-separable state that cannot be written as a⟩⊗ b⟩; the BIND is the core resource
ORBIT CNOT (CX) gate in Bell measurement circuit: the input qubit ψ⟩ controls Alice’s Bell-pair qubit; creates a 3-qubit entangled state before measurement        
TWIST Hadamard in Bell measurement: rotates the input qubit by π/2 in the Bloch sphere, transforming ψ⟩’s ORBIT trajectory into the Bell basis; the TWIST aligns ψ⟩ with the Bell measurement eigenstates      
LABEL Bell measurement outcome: Alice measures in the Bell basis; result ∈ { Φ⁺⟩, Φ⁻⟩, Ψ⁺⟩, Ψ⁻⟩} gives 2 classical bits (c₁, c₂); each outcome with probability 1/4 regardless of ψ⟩
FLIP Bob’s X correction: if c₂=1, Bob applies X (bit-flip) to his qubit; corrects the σ_x component of the teleportation error          
FLOP Bob’s Z correction: if c₁=1, Bob applies Z (phase-flip) to his qubit; corrects the σ_z component; after FLIP+FLOP Bob holds ψ⟩ exactly        

ISA programme 

PREPARE: BIND[|Phi+> = (|00> + |11>)/sqrt(2) | Bell pair, Alice qubit A + Bob qubit B]
INPUT:   ORBIT[|psi> = alpha|0> + beta|1> | arbitrary input qubit at Alice]
ENTANGLE:ORBIT[CNOT(|psi>, A) | 3-qubit state: alpha|0>|Phi+> + beta|1>|Psi+>]
ROTATE:  TWIST[H on |psi> register | Bell basis rotation]
MEASURE: LABEL[Bell measure on (|psi>, A) | 2-bit outcome (c1,c2), uniform]
SEND:    LABEL[c1,c2 sent classically to Bob | 2 classical bits, no superluminal]
FIX_X:  FLIP[if c2=1: Bob applies X | bit-flip correction]
FIX_Z:  FLOP[if c1=1: Bob applies Z | phase-flip correction]
OUTPUT:  ORBIT[Bob holds |psi> exactly | teleportation complete]

Computable output

  • Fidelity F = 1: teleportation achieves unit fidelity for any |ψ⟩ — Bob’s state after correction is exactly |ψ⟩. This is the primary LABEL output. Compare: the no-cloning theorem forbids copying |ψ⟩; teleportation is not cloning because Alice’s qubit is destroyed in the Bell measurement.

  • Classical communication cost: exactly 2 bits per teleported qubit. This is the LABEL eigenvalue of the TeleportCat morphism. Below 2 bits is impossible (Holevo bound: cannot transmit more than 1 qubit of quantum information per classical bit, so 1 qubit needs ≥ 1 bit; but the quantum part requires 2 for the Pauli correction). The 2-bit cost = the TWIST content of aligning Alice’s measurement to Bob’s correction.

  • Entanglement cost: exactly 1 ebit (one Bell pair) per teleported qubit. This is the BIND cost. The entanglement is consumed (the Bell pair is destroyed by Alice’s measurement); to teleport again, a fresh Bell pair is needed.

  • Bell state discrimination (LABEL): the four Bell states |Φ±⟩ = (|00⟩±|11⟩)/√2, |Ψ±⟩ = (|01⟩±|10⟩)/√2 correspond to the four Pauli corrections {I, X, Z, XZ}. Alice’s measurement result (2 bits) indexes which Pauli correction Bob needs — this is the LABEL-to-FLIP/FLOP map.

  • Experimental fidelity (from Bouwmeester et al. 1997): F = 0.82 ± 0.01 for teleportation of photon polarisation states (limited by detector efficiency and partial Bell measurement). Furusawa et al. (1998): F > 0.58 threshold (classical limit for coherent states). Pan et al. (2012): F = 0.89 for free-space 143km. Current best: trapped-ion teleportation at F > 0.999 (Nölleke et al. 2013).

Why H¹ (not H²)

The Bell pair (BIND) is H², but the teleportation protocol operates at H¹:

  • The BIND (Bell pair) is prepared once and used as a static resource; the dynamic protocol then only uses ORBIT (CNOT), TWIST (Hadamard), LABEL (measure), FLIP/FLOP (Pauli corrections)
  • The Pauli corrections X^{c₂} Z^{c₁} are H¹ (Clifford operations — they are elements of the Pauli group, the H¹ TWIST group)
  • The only H² content is the Bell pair itself; once prepared, the remaining protocol is entirely H¹. This is why teleportation uses 1 ebit (H² BIND) but only classical + Clifford operations (H¹) afterwards.

Compare to Q03 (Steane code): the syndrome measurement uses H¹ Clifford operations, and the error is the H² content being detected. Teleportation has the same structure: the BIND (Bell pair, H²) is the resource; the protocol (H¹) uses it up.

The teleportation identity in ISA

The fundamental identity that teleportation implements:

FLIP[c₂] ∘ FLOP[c₁] ∘ LABEL[Bell] ∘ TWIST[H] ∘ ORBIT[CNOT] ∘ BIND = id

This is the ISA statement that the protocol composed with the Bell pair equals the identity channel — perfect quantum state transfer. The composition is exact (not approximate) — teleportation is the canonical example of an ISA programme that achieves a perfect outcome at H¹ using one H² resource.

Connections to other entries

  • Q06 Superdense coding: the exact dual protocol; teleportation sends 1 qubit using 1 ebit + 2 bits; superdense coding sends 2 bits using 1 ebit + 1 qubit; same Bell pair resource, dual information-theoretic direction
  • Q07 CHSH game: the Bell pair (BIND) in teleportation is the same entangled resource used in the CHSH strategy; CHSH measures the non-classicality of the BIND; teleportation uses the BIND for communication
  • Q03 Steane 7-qubit code: teleportation is the primitive operation for fault- tolerant quantum computing; logical state teleportation is how magic states (H²) are injected into a stabiliser (H¹) code
  • QH01 Trapped-ion MS gate: the Bell pair preparation uses the MS gate (QH01); teleportation on trapped-ion platforms uses QH01 for the BIND + TWIST

Validation

  • Bennett et al. (1993), PRL 70, 1895: original teleportation protocol; proved unit fidelity; 2-bit classical communication lower bound.
  • Bouwmeester et al. (1997), Nature 390, 575: first photonic teleportation; F=0.82; partial Bell measurement (only 2 of 4 Bell states distinguished).
  • Furusawa et al. (1998), Science 282, 706: continuous-variable teleportation of coherent states; F=0.58 surpassing classical limit.
  • Pan et al. (2012): free-space photon teleportation over 143 km (Canary Islands); demonstrates feasibility for satellite quantum networks.
  • Riebe et al. (2004), Nature 429, 734: deterministic qubit teleportation on trapped ions; F=0.75; all 4 Bell states distinguished.

Part of the ISA Zoo. Categorical foundations: Paper 591.