For faithful finite-dimensional state pairs, globally optimized measured relative entropy is additive at any fixed copy number n >= 2 exactly when the states commute. This Letter proves the criterion through an explicit phase-optimized two-copy ascent direction, separates coherent collective access from the known fresh-copy classical-feed-forward bound, gives a faithful qubit construction, and states a thermodynamic corollary only for the Gibbs-prepared isolated unitary-drive setting.
For faithful finite-dimensional state pairs, globally optimized measured relative entropy is additive at any fixed copy number n >= 2 exactly when the states commute. This Letter proves the criterion through an explicit phase-optimized two-copy ascent direction, separates coherent collective access from the known fresh-copy classical-feed-forward bound, gives a faithful qubit construction, and states a thermodynamic corollary only for the Gibbs-prepared isolated unitary-drive setting.
Complete dynamics and entropy balances of all proper subsystems need not determine global irreversibility. A three-qubit construction keeps one initial product state and every one- and two-qubit reduced semigroup fixed while changing only the organisation of energy-preserving shared phase events. The global entropy production changes, and the thermodynamic ranking of the full system against its proper subsystems reverses. Both trajectories remain separable, every compared score is positive, and interval bounds certify the opposite rankings.The comparison uses the finite-window Thermodynamic Criterion Index, the entropy reduction per unit outward entropy transfer. An exact lift identity determines when relational-information change exceeds the component deficit and the signed nonadditivity of entropy transfer. With common lower-order marginals and equal global export, a larger increment of irreducible higher-order information means an exactly equal reduction of global entropy production. At a common temperature and for an additive Hamiltonian, that information has a precise nonequilibrium free-energy value.Within the parity-contact model, fixed lower-order dynamics leave an explicit physical fibre of higher-order generators. Selective control inside this fibre changes a chosen collective decay rate. Two controllers with equal amounts of parity correlation can produce opposite thermodynamic rankings. Finite adaptive protocols confined to any fixed proper subsystem with private memory cannot identify the change; outcomes jointly retained from all three sites can reveal it. A certified pair of compatible generators gives nonzero lower bounds on the minimax absolute errors for global entropy production and the full-versus-proper yield gap, even with exact lower-order data. Complete lower-order descriptions therefore leave a physically controllable coordinate of global irreversibility undetermined.
We study collective state preparation and channel encoding in fermionic systems under parity superselection and composition governed by the canonical anticommutation relations. For a two-mode Gaussian state τ, we obtain the exact entanglement of formation, EFphys(τ⊗m)=⌈m/2⌉log2 for every m≥1. Two copies have the same formation value as one, violating additivity and strong superadditivity, while regularization halves the single-copy value. For an associated Gaussian channel, we determine the minimum output entropy and Holevo information at every blocklength and prove nonadditivity of both. Together, these results give exact answers to the fermionic counterparts of the four additivity questions associated with Shor and Hastings. One physical use carries no classical information, whereas two uses transmit one bit perfectly. The unassisted classical and zero-error classical capacities both equal one half bit per mode. A sharp finite-block bound gives a strong converse: the decoding success probability vanishes at every rate above capacity. These optimizations cover all physical pure-state decompositions and channel inputs, with Gaussian ensembles and classical codes attaining the optima. The formation and minimum output entropy formulas hold for every Rényi order α∈(0,∞]. We also determine the unassisted quantum and zero-error quantum capacities, both equal to one half logical qubit per mode, and prove a strong converse: entanglement fidelity vanishes at every rate above capacity. For a noisy Gaussian family, we determine the one- and two-copy formation values and prove strict nonadditivity at full rank. For τ, Bell violation under local parity-preserving measurements first appears at three copies.
Copying an unknown quantum state into many approximate copies is known to become equivalent to estimating it and re-preparing: the best measure-and-prepare copier approaches the best quantum copier as the number of copies grows. We determine how fast. For equatorial qubit states, one input copy and m output copies, judged by the global fidelity, we prove F_EB/F_coh = 1 − 1/√(2m) + O(1/m). For phase states on d levels we prove, with explicit constants, the law 1 − F_EB/F_coh = (d − 1)/√(2m) + O(1/m), confirmed by exact computations up to m = 400; for qubits with T input copies the gap is √2·σ_T/√m, proved with explicit constants for T = 2, 3 (the upper bound conditionally on the optimality of the flat seed, R0(T); the lower bound unconditionally). The measure-and-prepare copier's deficit is the ground-state energy of a harmonic oscillator, one quantum per unknown phase. The exact quantum fidelity has plateaus at m = kd and kd + 1 (m = kd + 1 is the economical point). Numerical results suggest a period-d modulation of the ratio at order 1/m; a proof of its next-order coefficient remains open. This complements the known universality of the cloning fidelity itself, which depends only on the number of free parameters: within the phase-state family, the leading gap coefficient counts the independent phase directions. The classical values rest on the flat-seed theorem of the companion Paper 5, and the several-copy quantum values on the reduced coherent dual of the companion Paper 5. Core lemmas of the qubit proof and its explicit upper-bound constants are machine-checked in Lean 4; the remaining steps are proved on paper and independently refereed.
The entanglement-assisted capacity of a quantum channel admits an additive single-letter characterization, implying that joint encodings across channel uses cannot increase the ultimate communication rate. Here, we show that this additive picture does not extend to communication reliability. Specifically, we prove that the Petz-Rényi channel information can be strictly superadditive for every α∈[1/2,1), yielding a genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent, even though the entanglement-assisted capacity remains additive. We establish this phenomenon analytically already for measurement channels, which are entanglement-breaking and have additive unassisted capacity. Remarkably, this strict superadditivity is witnessed by a separable, classically correlated two-copy channel-input marginal, demonstrating that no entanglement between the transmitted systems is required. Our results show that, although correlations across channel uses cannot increase the ultimate rate of entanglement-assisted communication, they can enhance its reliability.
We determine exactly how well a measure-and-prepare ("measure-first") forger can copy qudit phase states. For one input copy and every number m of output copies, in every dimension d, we prove that the flat seed is the optimal measure-first strategy for the continuous family of phase states, the optimum being the top eigenvalue of an explicit matrix on the symmetric subspace (for example (7d − 8 + √(25d² − 64d + 48))/(2d³) at 1→2). The proof reduces the question to a negative-correlation inequality for closed lattice walks in symmetric time-dependent windows, which we prove by summation by parts; this combinatorial core is verified in the Lean 4 proof assistant. Consequently, for cyclic phase-state codebooks ψ_x = d^(−1/2) Σ_j ω^(x·k_j) |j⟩, the states used in quantum fingerprinting and quantum hashing, in every dimension and every 1→m task the measure-first optimum equals the continuous one if and only if the keys form a B_{m+1} set, whereas at 1→2 the coherent optimum does if and only if they form a Sidon set. Around this law we compare, for every copying task T→T+k, the optimal coherent copier (any quantum channel) and the optimal measure-first copier on a finite codebook with the continuous family: both finite-codebook optima are at least the continuous ones; if the keys have no additive collisions among multisets of size 2T+k, both are equal for every dimension; and for codebooks that are collision-free at size T, every collision at size 2T+k strictly increases the value of the natural measure-first strategy. For cyclic qubit codebooks of q states and every 1→m task (m ≥ 2) we prove an exact leak-order theorem: the measure-first optimum equals the continuous one if and only if q ≥ m+2, while the coherent optimum already equals it when q ≥ m+1, so q = m+1 separates the two attackers for every m. Pre-registered experiments on IBM superconducting processors, in which the strategies are emulated by classically recombining two circuit stages, confirm the predicted qubit gaps at 1→2 and 1→3. We give an explicit polynomial-phase hash with a guaranteed (sufficient) copy budget and prove a separation for it. Whether measure-first attackers leak no later than coherent ones for every codebook remains open; the stronger statement that the measure-first gap always exceeds the coherent gap is false for general finite phase codebooks.
We study Haar-Bayesian prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after an arbitrary collective measurement on n observed copies. Performance is evaluated by quantum relative entropy. For a fixed measurement, the Bayes predictive state is the posterior mean and the optimized conditional loss is its entropy. We then optimize the measurement over all POVMs on the symmetric subspace. For every nonzero positive effect E, the corresponding posterior predictive state is μE=(I+nρE)/(n+d), where ρE is the normalized one-particle marginal of E. Since a pure spectrum majorizes every density-operator spectrum, this identity gives an outcome-wise entropy lower bound. Coherent rank-one effects attain the bound, and their Haar orbit yields the highest-weight covariant POVM. Hence this POVM is globally Bayes optimal over all collective measurements and, by covariance, globally minimax. Its exact risk is hd((n+1)/(n+d)), where hd(r)=−rlogr−(1−r)log((1−r)/(d−1)). The same arbitrary-effect reduction shows that the highest-weight POVM also maximizes the joint overlap between the latent pure state and its posterior predictive state, equivalently the mean posterior purity, with optimum ((n+1)2+d−1)/(n+d)2.