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A counterexample to Zhang–Wu–Fei superadditivity

Abstract

A maximally mixed two-qubit input and a full-rank correlated diagonal reference contradict relative-entropy superadditivity.

Definition 1.1 (The unitary superadditivity conjecture).

Formalization. D5/S3/Quantum/Information/ZhangWuFeiSuperadditivityRefutation.claim (✓ std3).

Citation. Lin Zhang, Junde Wu, and Shao-Ming Fei (2013). Comment on ‘Convergence of macrostates under reproducible processes’ [Phys. Lett. A 374: 3715–3717 (2010)]. DOI: 10.1016/j.physleta.2013.05.047. URL: https://arxiv.org/abs/1210.4720v4.

Commentary.

Zhang, Wu and Fei, arXiv:1210.4720v4, p. 6, §4: “Based on the above result, we propose the following conjecture: There exist unitary operators U_A ∈ U(H_A), U_B ∈ U(H_B) and U_AB ∈ U(H_A ⊗ H_B) such that S(U_AB ρ_AB U_AB† ‖ σ_AB) ⩾ S(U_A ρ_A U_A† ‖ σ_A) + S(U_B ρ_B U_B† ‖ σ_B), (4.2) where the reference state σ_AB is required to be full-ranked state. Indeed, our numerical calculations show that inequality (4.2) is true.”

The dimensions a and b range over natural numbers. DensityState means a positive semidefinite complex matrix of trace one. The first subsystem is Fin a and the second is Fin b. marginalRight retains A, while marginalLeft retains B. PosDef expresses full rank of the reference. quantumRelativeEntropy is Re Tr(ρ(log ρ − log σ)), with natural logarithms. The three matrices and their membership proofs express independent joint and local unitaries.

Theorem 1.2 (The conjecture fails for two qubits).

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/ZhangWuFeiSuperadditivityRefutation.result (✓ std3). ∎

Resolves. Problems/zhang-wu-fei-2012-relative-entropy-superadditivity-conjecture-4-2 (refuted) by D5/S3/Quantum/Information/ZhangWuFeiSuperadditivityRefutation.result.

Source. Repository-derived.

Acknowledgement. Lin Zhang, Junde Wu, and Shao-Ming Fei (2013). Comment on ‘Convergence of macrostates under reproducible processes’ [Phys. Lett. A 374: 3715–3717 (2010)]. DOI: 10.1016/j.physleta.2013.05.047. URL: https://arxiv.org/abs/1210.4720v4.

Commentary.

Take a = b = 2, ρ = I/4 and σ = diag(61,1,1,1)/64 in product order (0,0), (0,1), (1,0), (1,1). All four reference eigenvalues are positive. The two input marginals equal I/2, and the two reference marginals equal diag(31,1)/32. Every unitary fixes the input and its marginals.

The joint relative entropy is log(1/4) − log(61/64)/4 − 3 log(1/64)/4. Each marginal relative entropy is log(1/2) − log(31/32)/2 − log(1/32)/2. Four times the difference between the joint value and twice the marginal value is log(923521/3997696), which is negative. Thus no choice of the three unitaries satisfies the displayed inequality.

References