Q06 — Superdense Coding
| Field | Value |
|---|---|
| Domain | Quantum Computing |
| System | Shared Bell pair + single-qubit encoding + Bell measurement |
| Group | SU(2) (local Pauli operations on one qubit) |
| H^k tier | H¹ |
| ISA | Forge (β ≈ β*) |
| Status | Validated |
| Opcodes | ORBIT · TWIST · BIND · LABEL · FLIP · FLOP |
| Papers | Paper 421, Paper 469 |
Physical system
Superdense coding (Bennett & Wiesner 1992) sends 2 classical bits of information by transmitting a single qubit over a quantum channel, using a pre-shared Bell pair as the resource.
| The protocol (sender Alice, receiver Bob, pre-shared | Φ⁺⟩): |
| 2 bits to send | Alice applies | Shared state becomes | |||
|---|---|---|---|---|---|
| 00 | I (nothing) | Φ⁺⟩ = ( | 00⟩+ | 11⟩)/√2 | |
| 01 | X (bit-flip) | Ψ⁺⟩ = ( | 01⟩+ | 10⟩)/√2 | |
| 10 | Z (phase-flip) | Φ⁻⟩ = ( | 00⟩− | 11⟩)/√2 | |
| 11 | iY = XZ | Ψ⁻⟩ = ( | 01⟩− | 10⟩)/√2 |
Alice sends her single qubit to Bob; Bob measures in the Bell basis and recovers 2 bits with certainty.
The key insight: Alice encodes 2 bits of classical information into which of the 4 orthogonal Bell states the joint system is in — but the four Bell states are only distinguishable jointly, not locally. Alice’s single-qubit operation selects among 4 globally distinguishable states using only H¹ local operations (Pauli gates) applied to her half of the shared BIND (Bell pair).
Target category
DenseCat — the dual of TeleportCat (Q05). Objects: two-qubit Bell states {|Φ±⟩, |Ψ±⟩}; morphisms: single-qubit Pauli operations on Alice’s qubit (the encoding map) and Bell measurements on both qubits (the decoding map). The superdense coding protocol is the canonical morphism Dense: 𝔽₂² → DenseCat that sends each 2-bit string to the corresponding Bell state, then recovers it.
Teleportation/superdense duality: Q05 and Q06 use the same Bell pair (BIND resource) in opposite directions:
- Q05 (teleportation): 1 qubit sent → 2 bits consumed; moves quantum information
- Q06 (superdense coding): 2 bits encoded → 1 qubit sent; moves classical information
This duality is not coincidental — it is the adjunction in TeleportCat between encoding (Q06) and decoding (Q05 Bell measurement). The same BIND (|Φ⁺⟩) mediates both; the direction of information flow depends on which party applies which operation.
Interpretation functor
F: C → DenseCat defined by:
| Opcode | F(opcode) | |||
|---|---|---|---|---|
| BIND | Bell pair preparation: | Φ⁺⟩ = ( | 00⟩+ | 11⟩)/√2; Alice holds qubit A, Bob qubit B; the shared H² BIND resource |
| FLIP | X encoding: Alice applies X to qubit A; | Φ⁺⟩ → | Ψ⁺⟩; encodes bit 1 in the X-type Bell state | |
| FLOP | Z encoding: Alice applies Z to qubit A; | Φ⁺⟩ → | Φ⁻⟩; encodes bit 1 in the Z-type Bell state | |
| ORBIT | Alice sends qubit A to Bob; this is a single-qubit quantum channel; the physical quantum communication step (the “dense” part: 1 qubit carries 2 bits) | |||
| TWIST | Bell measurement rotation: Bob applies CNOT(A,B) then H on A; rotates from Bell basis to computational basis; same TWIST as the inverse of Bell pair preparation | |||
| LABEL | Computational-basis measurement: Bob measures both qubits in | 0⟩/ | 1⟩ basis after TWIST; recovers 2 bits (b₁,b₂) with certainty |
ISA programme
PREPARE: BIND[|Phi+> = (|00>+|11>)/sqrt(2) | Bell pair shared]
ENCODE: FLIP[if bit0=1: X on Alice's qubit A | X-type encoding]
FLOP[if bit1=1: Z on Alice's qubit A | Z-type encoding]
SEND: ORBIT[Alice sends qubit A to Bob | 1 qubit over quantum channel]
DECODE1: ORBIT[CNOT(A,B) at Bob | entanglement-breaking CNOT]
DECODE2: TWIST[H on A | Hadamard disentangles]
MEASURE: LABEL[measure both qubits in comp. basis | 2 bits output]
OUTPUT: LABEL[(b1,b2) = original 2 bits | perfect recovery]
Computable output
-
Channel capacity: 2 classical bits per qubit channel use (when 1 ebit pre-shared). The Holevo bound says a single qubit can carry at most 1 classical bit without entanglement. Superdense coding achieves 2 bits — double the classical limit — by using the pre-shared BIND. This is the LABEL eigenvalue of DenseCat: capacity = 2 × (Holevo bound) when entanglement is available.
-
Bell state distinguishability: the four Bell states are mutually orthogonal (⟨Φ⁺|Ψ⁺⟩ = 0, etc.) — they are the 4 elements of a complete orthonormal basis for ℂ²⊗ℂ². Bob’s Bell measurement perfectly distinguishes them (unlike Alice’s partial measurement in Q05, Bob has both qubits and can do the full Bell measurement). Output fidelity = 1 for each of the 4 possible inputs.
-
Encoding map (FLIP × FLOP): the four Pauli operations {I, X, Z, iY} map |Φ⁺⟩ to the four Bell states bijectively. Since Alice operates on only one qubit (her half of the entangled pair), this requires H¹ local operations — no BIND is created by Alice; the existing BIND is used to “spread” Alice’s local operation into a globally distinguishable state.
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Experimental realisation (Mattle et al. 1996, Phys. Rev. Lett. 76, 4656): superdense coding demonstrated with photons; 1.58 bits per photon transmitted (limited by partial Bell measurement — only 3 of 4 Bell states distinguished with linear optics); full 2 bits achievable with complete Bell measurement.
Why 1 qubit sends 2 bits: the BIND amplification
The apparent paradox — how can 1 qubit carry 2 bits, when Holevo says only 1? — is resolved by the BIND:
- Without the pre-shared BIND: Alice’s qubit has a 2-dimensional Hilbert space; Holevo: at most log₂(2) = 1 bit per qubit
- With the pre-shared BIND: Alice’s qubit + Bob’s qubit form a 4-dimensional joint Hilbert space; the 4 Bell states span this space; log₂(4) = 2 bits are recoverable at Bob’s end when he holds both qubits
The BIND “stores” one bit: it pre-correlates Alice’s and Bob’s qubits so that Alice’s single-qubit X or Z operation has a 2-bit effect on the joint state. This is the ISA interpretation: the BIND acts as a “pre-computed” second bit, and Alice’s FLIP/FLOP operations select which of the 4 joint states is realised.
The BIND resource is consumed (the Bell pair is destroyed by Bob’s measurement) — just as in Q05. One ebit enables one superdense coding transmission.
Teleportation ↔ Superdense duality
The resource trade-off:
| Protocol | Sends | Uses | Uses |
|---|---|---|---|
| Q05 Teleportation | 1 qubit | 1 ebit + 2 classical bits | |
| Q06 Superdense | 2 classical bits | 1 ebit + 1 qubit channel |
These are related by “quantum resource” exchange: classical bits ↔ quantum channel uses, mediated by entanglement. The duality is an adjunction in the resource theory of quantum communication — the same BIND (1 ebit) either “upgrades” 2 classical bits into 1 qubit channel (Q05) or “upgrades” 1 qubit channel into 2 classical bits (Q06).
In ISA terms: Q05 = BIND × FLIP/FLOP → ORBIT (quantum state transfer); Q06 = FLIP/FLOP × ORBIT → LABEL (classical bit recovery).
Connections to other entries
- Q05 (Teleportation): exact dual; same BIND resource, opposite direction; together they implement quantum/classical communication exchange
- Q07 (CHSH game): superdense coding wins the “2-bit-from-1-qubit” game against any classical strategy; CHSH (Q07) measures the non-classicality of the BIND that makes superdense coding possible
- Q01 (Shor’s algorithm): superdense coding shows that quantum channels can carry twice the classical information when assisted by entanglement; this “quantum advantage” is the resource theory basis for quantum speedups
- K01 (RSA encryption): superdense coding is the communication-theoretic dual to cryptography: RSA uses classical bits to protect classical bits (H¹); superdense coding uses 1 ebit (H²) to double classical throughput
Validation
- Bennett & Wiesner (1992), PRL 69, 2881: original protocol; 2 bits per qubit proved using 1 ebit; Holevo-beating capacity proved.
- Mattle et al. (1996), PRL 76, 4656: first experimental superdense coding with photons; 1.58 bits/photon (3-of-4 Bell states); limited by linear-optics partial Bell measurement.
- Williams et al. (2017), Phys. Rev. Lett. 118, 050501: complete Bell state measurement in atomic ensemble; full 2 bits/photon achieved.
- Quantum capacity theorem (Lloyd 1997; Shor 2002): entanglement-assisted classical capacity of a channel = 2 × quantum capacity; superdense coding is the canonical protocol achieving this bound.