Almost one bit violation for the additivity of the minimum output entropy

• Kk,t describes the large-dimension limit of possible output-state eigenvalues for a natural family of random quantum channels.It is the intersection of the probability simplex with the dual unit ball of the (t)-norm.Source: §1 · p. 1
• The maximum of ℓp norms on Kk,t occurs at (a, b, …, b), with a > b depending on k and t, and the maximizing vector is independent of p.Source: §1 · p. 1
• For conjugate random quantum channels and the Bell state, additivity violations can occur iff the output dimension is at least 183.Almost surely, the defect is less than log 2 and can be made arbitrarily close to log 2.Source: §1 · p. 2
• The minimization result yields the precise limit value of the minimum output entropy for a single random quantum channel and optimizes the bounds available from the random techniques considered.Source: §1 · p. 2

2.1. The (t)-norm.

• The (t)-norm is defined using compression by a projection of normalized trace t that is free from the vector being considered.The quantity is a norm invariant under coordinate permutations, though it can be difficult to compute explicitly.Source: §2.1 · pp. 2–3

2.2. Definition of the convex body Kk,t.

• Kk,t is the intersection of the probability simplex with the unit ball of the dual norm of the (t)-norm.Source: §2.2 · p. 4

2.3. Main result.

• For any p > 1, the maximum ℓp norm on Kk,t is attained at x*_t.The maximizing point is the same across p, and the maximum value can be computed.Source: §2.3 · p. 4

3.1. Strategy of the proof.

• K_{k,t} is the image of the simplex under the subdifferential of the (t)-norm; at differentiability points its gradient lies in K_{k,t}, and differentiability points are dense for t < 1 − 1/k.The subgradient map is increasing and preserves coordinate monotonicity.Source: §3.1 · p. 5

3.2. Some technical results about the (t)-norm.

• For nonconstant x ∈ R^k, whenever m_x/k + t ≤ 1 (in particular t ≤ 1/k), max supp(μ ⊞ 1/t) coincides with the largest point of non-analyticity of F_{μ⊞1/t}.Under the strict condition t + m_x/k < 1, w(x) > x_1.Source: §3.2 · pp. 6–7

3.3. Some properties of the Hessian matrix.

• The Hessian annihilates x and the all-ones vector; for two-valued x it has block structure, and H(x)1_k is positive unless x is constant or two-valued.For x = (1, 0, . . . , 0), the first row and column of H are null.Source: §3.3 · pp. 7, 9

3.4. Local maxima of ∥∇∥·∥(t)∥p

• Every differentiable point that is not two-valued admits an ascent direction, so local maxima of g on the differentiability set must be at most two-valued.Two-valued vectors are critical points of g; the ascent result rules out differentiable, non-two-valued points as local maxima.Source: §3.4 · pp. 9–10, §3.5 · p. 12

3.5. Maximum of g on two-valued vectors.

• The maximum over two-valued vectors is attained at x*_t = ∇∥(1, 0, . . . , 0)∥(t), with one coordinate ϕ(1/k, t) and the other k−1 coordinates equal.This holds for p > 1 and t ∈ (0, 1 − 1/k).Source: §3.5 · p. 12
• The ℓp norms of vectors with m entries a and the remaining entries b are maximized at m = 1.This is the target of the two-step proof.Source: §3.5 · p. 13
• For c = a/b > 1, the first step establishes the required derivative inequality by showing h′(c) is positive for p, c > 1.The proof defines h(c) as the left-hand side of an equivalent inequality and notes h(1+) = 0.Source: §3.5 · pp. 13–14
• For the second step, the inequality at p → 1 is reduced to showing an expression is nonnegative, using a change of variables that simplifies its derivative.The variables are t = cos^2(r) and x = sin^2(s), with r ∈ (0, π/2) and s ∈ (0, r); details are cited in. Also, 0 < b < 1 < a.Source: §3.5 · pp. 13–14

2. Our function from equation (35) becomes

• For t ≤ 1/k, g(x) = ∥∇∥x∥(t)∥p^p reaches its global maximum at e1.The result also gives the maximum value in terms of ϕ(1/k, t).Source: §2 · p. 15

3.6. The general t case.

• For general t, exposed points of Kk,t are images of points where the (t)-norm is differentiable.The proof uses this characterization to extend the maximum result beyond t ≤ 1/k.Source: §3.6 · p. 15
• Among differentiability points, ∥∇∥x∥(t)∥p^p reaches its maximum at e1, completing the proof for general t.The argument uses Theorem 3.7 and Proposition 3.9.Source: §3.6 · p. 15

4. Minimum output entropy for quantum channels

• Quantum channels are completely positive, trace-preserving maps, and their classical capacity requires regularization of the one-shot capacity.The channel’s Holevo capacity is described as its one-shot capacity.Source: §4 · pp. 15–16
• Additivity of the Holevo quantity is equivalent to related properties including additivity of minimum output entropy, while additivity of minimum p-output entropy was disproved.The cited work reports a series of disproofs culminating with Hastings’ counterexample [20].Source: §4 · pp. 16–17
• The section’s two results give a limit value for minimum p-output entropy of random quantum channels and address counterexamples to additivity of Hmin.These are the section’s stated main results.Source: §4 · p. 17

5.1. Random quantum channels and the subspace model.

• A random channel is induced by a Haar-distributed isometry into Ck ⊗ Cn; its image subspace determines the possible output eigenvalues used to compute minimum output entropy.For p ≥ 1, the entropy minimum is attained on pure input states.Source: §5.1 · pp. 17–18

5.2. The large n asymptotics.

• Almost surely, the finite-dimensional sets Kn,k,t eventually contain every compact subset of the interior of Kk,t and lie inside any open set containing Kk,t.This is the stated large-n asymptotic behavior of the singular-value sets.Source: §5.2 · p. 18

5.3. Convergence result for the minimum output entropy.

• Theorem 5.2 combines the set convergence and the maximization result to establish almost-sure convergence of minimum output p-entropies as n tends to infinity.It assumes fixed output dimension k and input dimensions dn ∼ tkn.Source: §5.3 · p. 18

6.1. The MOE additivity problem.

• Earlier additivity counterexamples mostly used random channels, and the Bell phenomenon pairs conjugate channels with a maximally entangled input to bound minimum output entropy.The method has been the most successful approach to the additivity problem; the section relates its origins to [32] and notes evidence that the Bell input may be near-optimal [7] [16].Source: §6.1 · p. 19, §6.2 · p. 19

6.2. The Bell phenomenon.

• Theorem 6.2 improves the previous lower bound [19] on the largest eigenvalue of the conjugate-channel Bell-state output by (1−t)k−2, yielding tighter bounds on violation dimensions.The improved eigenvalue result supports the subsequent analysis of the entropy difference for random channels.Source: §6.2 · pp. 19–20

6.3. Macroscopic violations for the minimum output entropy of random quantum channels.

• For output dimension k ≥ 183, almost all random channels violate von Neumann minimum-output-entropy additivity in the limit, with large-k violations approaching 1 bit.Numerical analysis finds the first negative entropy difference at k = 183 and t ≈ 0.11; for k < 183, the Bell-state output entropy is almost surely larger than twice the single-channel minimum output entropy.Source: §6.3 · p. 20
• At fixed t, the entropy difference approaches −H(t, 1−t) as k grows; for t = 1/2, violations occur for k ≥ 276, while varying t yields the threshold k = 183.The maximal violation approaches log 2 for t = 1/2 and large k; for von Neumann entropy, violations remain bounded rather than growing with k.Source: §6.3 · pp. 20–21
• Using the Hayden–Winter bound instead of the exact output eigenvalue result gives first violations at k = 184, one dimension above the improved threshold.That bound can still yield violations arbitrarily close to 1 bit.Source: §6.3 · p. 22
• The result does not rule out violations below k = 183 from other input states, and the techniques do not bound the environment dimension n.The below-threshold conclusion is limited to use of the Bell state with conjugate random channels.Source: §6.3 · p. 22