Does fermionic quantum Shannon theory violate additivity of entanglement and capacities?

Nonadditivity means copies stop helping

Additivity would say that the entanglement needed to prepare several copies of a state is just the single-copy cost multiplied by the number of copies, and that encoding across several channel uses cannot beat encoding each use separately. This paper shows that neither statement holds for a specific fermionic setting: a two-mode Gaussian state τ, with one mode held by Alice and one by Bob, composed under the canonical anticommutation relations and constrained by parity superselection.

The exact result is E_F^phys(τ^⊗m) = ⌈m/2⌉ log 2 for every m ≥ 1. The single-copy value is log 2, and the regularized value is (log 2)/2. Because two copies give ⌈2/2⌉ log 2 = log 2, the same as one copy, additivity fails; because the two-copy value is below twice the one-copy value, strong superadditivity fails as well. The paper proves the same value for the Gaussian-restricted roof E_F^fG, so restricting decompositions to Gaussian states does not raise the cost.

Parity superselection changes how copies combine

Parity superselection forbids a coherent superposition of vacuum and occupation in an isolated mode, but it permits correlations between odd operators on different modes, since their product is even. That distinction is operational: an isolated-mode description does not fix the action on globally even products of odd operators drawn from different modes.

Composition here is the fermionic product, not an ordinary qubit tensor power. Regrouping modes by party carries occupation-dependent signs, so a matrix partial trace in an arbitrary interleaved ordering need not represent the physical restriction. The paper's channel erases all physical single-mode signals yet retains correlations across uses, and those retained Majoranas are what support both the classical codes and the noiseless logical quantum registers.

A symmetry bound fixes the formation cost

The upper bound comes from an explicit ensemble. The construction fixes the correlated Majorana pairs and matches the remaining Majoranas locally as far as possible, then averages over the signs on the free pairs to reproduce the full density operator.

The lower bound does not assume Gaussian decomposition states. For an arbitrary physical pure state in the same support, the marginal commutes with m+1 anticommuting Hermitian unitaries. Those symmetries force each marginal eigenvalue to occur at least 2^⌈m/2⌉ times, which yields the entropy lower bound. A separate covariance-rank argument first establishes optimality among Gaussian decompositions, and the two together give the exact value for both the physical and Gaussian roofs.

One use carries nothing, two uses send a bit

A Gaussian isometry maps input vectors onto the support of τ; tracing out one output defines a channel N for which entanglement entropy becomes output entropy. The same spectral argument gives S_min^phys(N^⊗m) = ⌈m/2⌉ log 2 and χ^phys(N^⊗m) = ⌊m/2⌋ log 2, so minimum output entropy and Holevo information are both nonadditive.

The operational consequence is sharp. 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, and a sharp finite-block bound gives a strong converse: the decoding success probability vanishes at every rate above capacity. An explicit Gaussian code has orthogonal outputs, attains the Holevo quantity, and achieves one half bit per mode with zero error.

Quantum capacity also lands at half a qubit

The correlations retained by the channel carry quantum information as well. Counting logical qubits encoded in a fixed-total-parity subspace, with no shared entanglement supplied, the largest exactly transmitted logical register for m uses has dimension 2^q_m with q_m = max{0, ⌊(m−1)/2⌋}. Every physical encoder and decoder transmitting a D-dimensional register obeys F_e ≤ min{1, 2^q_m / D}.

Consequently the unassisted quantum and zero-error quantum capacities both equal one half logical qubit per mode, and rates above that value have entanglement fidelity tending to zero. The paper attributes this to the physical output algebra: full logical matrix algebras with maximally mixed multiplicity subsystems, plus a classical sector label at even blocklength. That structure explains both the exact quantum codes and their strong converse.

Noise keeps nonadditivity, Bell violation needs three copies

The formation result survives noise. For the family τ_t = (1 − i t γ2 γ3)/4 with 0 ≤ t ≤ 1, the paper determines both the one-copy and two-copy formation values and proves strict nonadditivity for every t > 0, including the full-rank range 0 < t < 1. At full rank the pure-state support argument is unavailable; a local Gaussian transformation instead reduces the problem to an active one-mode-per-party state plus uncorrelated local modes whose convex-roof contribution can be removed exactly.

Bell nonlocality appears at a different copy number than nonadditivity. Under local parity-preserving measurements, one or two copies of τ cannot violate any Bell inequality, whereas three copies attain the maximal quantum CHSH value 2√2. The paper states that this corrects a multiple-copy claim in earlier work whose argument used ordinary qubit tensor powers instead of the fermionic product.

Exact answers stop at the solvable example

The formation and minimum output entropy formulas hold for every Rényi order α ∈ (0, ∞], and the optimizations cover all physical pure-state decompositions and channel inputs, with Gaussian ensembles and classical codes attaining the optima. The paper also notes that the proof works directly with the fermionic channel rather than invoking a fermionic extension of the general equivalence theorem.

The boundary is the noisy family. For τ_t the one- and two-copy formation values are exact, but the many-copy values and their regularization remain open; determining them would show how optimal collective preparation changes away from τ. The Pfaffian constraints in the appendix give a finite formulation of the Gaussian roof, and the paper identifies as a next step whether that formulation can yield useful approximation methods for larger systems.

Compare copy number before comparing capacities

For this state and channel, the most decision-relevant comparison is not between classical and quantum capacity, which coincide at one half per mode, but between copy numbers. The formation value is flat from one copy to two, the Holevo information is zero at one use and one half bit per mode asymptotically, and Bell violation switches on only at three copies. A reader assessing collective effects in fermionic Gaussian settings should therefore track the blocklength at which each quantity changes, not assume that all collective advantages appear at the same number of copies.

The paper's own framing supports that priority: collective resource manipulation in the Gaussian setting is distinctive and in some ways peculiar, and the exact answers here come from a single solvable example whose noisy extension is only partially resolved. The practical threshold to watch is the copy number at which a claimed advantage first appears, because for τ the answers differ at one, two, and three copies.

Sources used in this answer