Source
Fuchs (1995), PhD thesis, LIT-018, §4.3, read in full in NOTE-021. The section is “based largely on a manuscript submitted to Physical Review Letters” with Barnum, Caves, Jozsa and Schumacher (p. 113, fn. 3).
What was actually shown
Definitions. A broadcasting process sends ρ_s to a joint state whose two marginals both equal ρ_s (Eq. 4.49). Cloning is the product special case ρ_s ⊗ ρ_s (Eq. 4.50). The allowed processes are the most general ancilla, unitary and partial-trace maps (Eq. 4.51).
Theorem. {ρ0, ρ1} can be broadcast iff [ρ0, ρ1] = 0 (Eq. 4.101), and cloned iff F(ρ0, ρ1) ∈ {0, 1}, i.e. the states are identical or orthogonal (Eqs. 4.52, 4.81). The “if” directions are by construction (Eqs. 4.53–4.55).
Proof. Fidelity cannot decrease under partial trace (Eq. 4.74) or under the broadcasting map (Eq. 4.79). Broadcasting therefore forces F_A = F(ρ̃0, ρ̃1) = F_B (Eq. 4.80). The equality conditions force the operators in the fidelity bound to coincide with the optimal-fidelity operator M (Eqs. 4.89–4.97), hence [ρ0, M] = 0 and so [ρ0, ρ1] = 0 (Eqs. 4.98–4.103). Of the candidate broadcast states, the perfectly entangled one cannot be produced by any physical map when S(ρ0) ≠ S(ρ1) (Eqs. 4.105–4.109).
What could have come out otherwise. Some non-commuting mixed pair could have admitted a broadcast with correlated, non-product output, as mixed states escape the pure-state no-cloning argument. It does not.
What this does not say
- That non-invertible mixed pairs are covered. The proof assumes invertible ρ0 and ρ1 “to simplify the exposition” (p. 118). Pure states are covered separately by no-cloning, but mixed pairs that are not of full rank are covered by neither, as written (NOTE-021, Limitations).
- That no-broadcasting marks nonclassicality. The thesis says broadcasting “makes a communication theoretic cut between commuting and noncommuting density operators, and thus between classical and quantum state descriptions” (p. 114). It never considers restricted classical theories. Spekkens’ epistricted classical statistical theories (LIT-007) list no-broadcasting among the phenomena they reproduce. NOTE-021 records that both hold, since they answer different questions; LIT-007’s entry is asserted rather than derived, so the two cannot be compared at the level of proof.
- That eavesdroppers acting on whole strings are covered. The thesis extends the theorem to block systems informally, from its being dimension-free (p. 121; NOTE-021, C15, moderate).
- Any trade-off between information gained and disturbance. The general inference–disturbance relation (Eq. 4.31) is left unsolved.
Connections
- The fidelity properties the proof rests on are THEORY-584.
- Interference, another phenomenon often read as nonclassical, has a local epistemically restricted classical model in the fragment LIT-019 treats (THEORY-403). The same school’s toy bit theory is THEORY-474.