Private communication via zero-private-capacity quantum channels

• Unlike independent classical memoryless wiretap channels, quantum channels can exhibit this superactivation; earlier quantum-capacity superactivation had a factor already supporting private communication.Source: §1 · p. 1
• Two finite-dimensional memoryless quantum channels with zero private capacities separately have positive private capacity jointly.Source: §1 · p. 2
• A weak signal over a mixed background gives the receiver linear information gain and at most quadratic environmental leakage, with fixed joint measurement and classical coding.Source: §1 · p. 2
• Decoders with effects positive under partial transpose cannot achieve a positive private rate, even with collective processing within each output group.Source: §1 · p. 2
• Lean 4 formalizes the capacity inequalities and fixed-measurement information bound using Mathlib and Lean-QIT.Source: §1 · p. 2

• The four-level channel has an identity branch of weight 1/7 mixed with seven errors, which the encoding detects.The weights also admit the transpose simulator described in Appendix A.Source: §2 · p. 2
• The helper is a qubit erasure channel whose output reaches Bob with probability 1 − p or Eve with probability p, with orthogonal flags marking the alternatives.Source: §2 · p. 3
• No preshared key, entanglement, or auxiliary communication is supplied; encoder randomness is local and private, and each product use consumes one use of each channel.Source: §2 · p. 3
• The four-level channel and qubit erasure channel each have zero private capacity, while their joint use has positive private capacity.The zero-capacity statements allow arbitrary block encodings and collective decoding.Source: §2 · p. 3
• Every rate below the bound is achievable using a fixed binary measurement on each product output and classical wiretap coding across uses.Source: §2 · p. 3
• At half erasure, the joint channel achieves approximately 1.903 × 10^-4 private bits per product use.The bound is 3 ln 2/10927, and every rate below the bound is achievable.Source: §2 · p. 3

3 Proof of Theorem 1

• The proof compares Bob’s information from a fixed binary measurement with Eve’s leakage for the same input ensemble.The proof first establishes both zero-capacity statements.Source: §3 · p. 3

3.1 Why each factor has zero private capacity

• The four-level channel has zero private capacity by transpose simulation, and the erasure channel also has zero private capacity for p ≥ 1/2.The transpose simulator yields the blocklength result without restricting inputs to product states.Source: §3.1 · p. 4

3.2 A signal event absent from the background

• A click identifies U = 1 with certainty and contributes apt/2 to Bob’s mutual information; the no-click contribution is nonnegative.The click probabilities are zero for U = 0 and apt for U = 1.Source: §3.2 · p. 5

3.3 Bounding Eve’s leakage by a classical coin

• Eve’s Holevo information is bounded quadratically in t, including both erasure branches and the full quantum environment.For states with ω1 ⪯cω0, the coin bound is 3(c − 1)t^2/(32 ln 2) for 0 ≤ t ≤ 1/2 and c ≥ 3.Source: §3.3 · pp. 5–6

3.4 From an information advantage to a private rate

• For every fixed 1/2 ≤ p < 1, a sufficiently weak signal gives a positive information difference, and classical coding achieves every rate below I(U : Y) − χ(U : E′E).Fixing p and t before the coding limit establishes the operational rate.Source: §3.4 · p. 6

4 Why the measurement must be joint

• For a code with decoding error ε and secrecy distance δ, the bound gives 1 − ε − 1/M ≤ δ.The proof compares Eve’s joint guessing test success with its random-message success probability 1/M.Source: §4 · p. 7
• PPT decoding has zero private rate, including separable measurements and LOCC across the two factors.The protocol’s effect |w⟩⟨w| has partial-transpose eigenvalue −1/2.Source: §4 · p. 7
• Bob’s output remains PPT across the factors, but the protocol uses a non-PPT measurement within each product use; PPT entangled states can also have positive distillable key [18].Classical coding is used across product uses.Source: §4 · p. 7

5 Discussion

• The construction uses support asymmetry to certify privacy for mixed letters with a fixed receiver measurement.The background state fills the environment’s relevant support while leaving the receiver’s signal direction empty; signal dilution then keeps leakage below receiver gain.Source: §5 · p. 7
• The complement’s regularized less-noisy dominance does not ensure stability under tensor products in this construction, unlike the complete order described for the qutrit example [23].The complement is not degradable, and the qutrit example’s complete less-noisy order tensorizes, excluding activation with an antidegradable helper.Source: §5 · p. 7
• It remains open which zero-private-capacity channels admit the support asymmetry needed to turn failure of joint simulation into an explicit private protocol.Source: §5 · p. 7
• Whether quantum channels’ classical capacity can exhibit superadditivity is another open question raised by the authors.Source: §5 · p. 7

AI statement

• Interactions with large language models helped identify the initial activating example and explore proof strategies.Source: AI statement · p. 8
• The authors checked, rewrote, and organized the proofs and prepared the manuscript.Source: AI statement · p. 8
• Artificial intelligence was also used to prepare the initial manuscript and assist with revisions, with the authors retaining responsibility for the final work.Source: AI statement · p. 8

A An explicit transpose simulator

• The isometry V_N maps inputs into the receiver and environment outputs, and its full complement N^c is obtained by tracing out the receiver.Source: A An explicit transpose simulator · p. 8
• A CPTP map D is constructed to satisfy the required composition identity for the simulator.The simulator output C is identified with B in the numbered bases, with E=C8.Source: A An explicit transpose simulator · p. 8
• Positivity of the Choi operator J_D follows from the stated congruence.Source: A An explicit transpose simulator · p. 9
• The finite rational trace-preservation and composition identities establish that D is CPTP and D N^c=T_N.The square-root weights cancel in the second identity.Source: A An explicit transpose simulator · p. 9

B Environmental support and order

• Orthogonal sectors in each output make tracing out R_B remove cross terms even when R is correlated with A, giving the full-environment direct sum.The product dilation is J_p ⊗ V_N with outputs regrouped as RBB:E′E.Source: B Environmental support and order · p. 9
• The full sixteen-dimensional identities include all components and cross terms; positivity on the stated range and vanishing on its orthogonal complement support the environmental order argument.Source: B Environmental support and order · pp. 9–10
• σ1 ⪯ (205/9)σ0, with rank σ0=8 and rank σ1=4, and the support of σ1 is contained in that of σ0.Source: B Environmental support and order · p. 10
• The reduced-environment coordinate ratios are 1, 4, 25/9, 25/9, 1, 1, 0, 0, yielding ε1 ⪯4ε0.This sharper bound is why the coin cost is averaged over the two erasure branches.Source: B Environmental support and order · p. 10

C.1 The fixed measurement preserves Eve’s marginal

• The fixed measurement defines a memoryless wiretap channel with classical input U, classical receiver output Y, and Eve’s quantum output.Its product-channel isometry acts on the stated input states ρ0 and ρt.Source: §C.1 · p. 10
• Summing over Bob’s outcomes leaves Eve’s original marginal unchanged, and Bob’s outcome is not publicly disclosed.The private coding theorem achieves every rate below I(U : Y) − χ(U : E′E).Source: §C.1 · p. 10
• Because Bob’s output is classical, any block decoder can be implemented as classical processing of Y^n, with p and t fixed before n tends to infinity.Vanishing Holevo leakage implies the stated secrecy criterion by quantum Pinsker.Source: §C.1 · p. 10

C.2 The trace-distance capacity formula

• The secrecy error is half the trace distance between the message-environment state and the product of the uniform message state with the averaged environment state.Entropy continuity is applied in dimension M d_E^n to bound χ(U : E^n).Source: §C.2 · p. 10
• (1 − ε − δ) log M ≤ a_n + δn log d_E + h2(ε) + h2(δ) yields the converse after dividing by n and taking ε, δ to zero.The bound combines receiver information from Fano’s inequality and data processing with an entropy-continuity bound on environmental leakage.Source: §C.2 · p. 10
• Achievability follows by repeating a fixed block ensemble, applying the private coding theorem, and using quantum Pinsker to obtain vanishing trace-distance secrecy error.Optimizing over block sizes and ensembles gives the stated capacity formula.Source: §C.2 · p. 10

D Lean correspondence

• The formalization covers both zero-capacity statements for 1/2 ≤ p ≤ 1 and the positive lower bound for 1/2 ≤ p < 1.The channels, complements, input states, and receiver POVM are constructed explicitly; superactivation_main has only the stated bounds on p as hypotheses.Source: D Lean correspondence · p. 11
• The checked proof uses Lean 4.30.0, Mathlib, and Lean-QIT, with no admitted steps or additional axioms beyond propext, Classical.choice, and Quot.sound.The paper–Lean table marks definitions with def. and lists the remaining entries as proved statements.Source: D Lean correspondence · p. 11, Table 1 · p. 12
• Positivity uses Gram or LDL decompositions, while the flagged leakage bound uses a preparation channel for the two environmental branches.The argument’s dependency diagram groups Lean declarations and ends with the regularized information formula.Source: D Lean correspondence · p. 11, Figure 1 · p. 11
• The operational coding theorem, its trace-distance formulation, Proposition 3, and the Discussion’s further consequences are outside the formalization.Operational achievability relies on the external private coding theorem.Source: D Lean correspondence · p. 11