bibkey: memarzadeh2016minimum authors: Laleh Memarzadeh; Stefano Mancini year: 2016 title: “Minimum output entropy of a non-Gaussian quantum channel” doi: 10.1103/PhysRevA.94.022341 url: https://arxiv.org/abs/1605.04525v1 claim: “Conjecture 1: truncated loss-dephasing minimum output entropy is attained by a binomial or kappa state.” strata_touched:

• D5/S3/Quantum/QuantumChannels/TruncatedLossDephasingOptimizerRefutation license: citation-only triage: anchor

Truncated loss-dephasing minimum output entropy

Conjecture 1, section IV, arXiv v1 PDF p. 4:

In a truncated Hilbert space of dimension K+1, the minimal output entropy of the quantum channel (3) is achieved either by binomial states of Eq.(14) or by states |κ_α⟩ of Eq. (11), depending on the values of ε and t.

The defining channel equations (1)–(7), pp. 1–2, give

$$\frac{d\rho }{dt}=(1-ϵ){\mathcal{L}}_{AD}(\rho )+ϵ{\mathcal{L}}_{PD}(\rho ),{\Phi }_{ϵ,t}(\rho )=\sum _{j,k=0}^{\infty }{A}_j{P}_k\rho P_k^{†}{A}_j^{†},$$

$$A_j=\sum _{l=j}^{\infty }\sqrt{\left(\genfrac{}{}{0ex}{}{l}{j}\right)}(1-f{)}^{(l-j)/2}{f}^{j/2}|l-j⟩⟨l|,f=1-e^{-2(1-ϵ)t},$$

$$P_k=\sum _{l=0}^{\infty }\sqrt{\frac{(2l^2ϵt{)}^k}{k!}}{e}^{-l^2ϵt}|l⟩⟨l|.$$

Equation (9) fixes the input energy, $\mathrm{Tr}(\rho a^{†}a)=N$. Equation (11) defines $|{\kappa }_{\alpha }⟩=\sqrt{1-N/K}|0⟩+\sqrt{N/K}{e}^{i\alpha K}|K⟩$. Equation (14) defines the binomial amplitudes ${\beta }_n=\sqrt{\left(\genfrac{}{}{0ex}{}{M}{n}\right){\mu }^n(1-\mu {)}^{M-n}}$; the energy condition is $M\mu =N$, and the allowed truncation sizes satisfy $\lceil N\rceil \le M\le K$.

The Lean encoding uses matrices on Fin (K+1), the Kraus equations above, and positive semidefinite trace-one inputs with number expectation N. For K ≥ 1, N ∈ [0,K], ε ∈ [0,1], and t ≥ 0, the claim says that some binomial or κ candidate has output entropy no greater than every admissible input. Its negation is proved. The spectral entropy uses natural logarithms, whereas the paper uses base two; multiplication by the positive constant 1/log(2) preserves the ordering. The definition extends entropy to all raw matrices by assigning zero to non-Hermitian matrices; the matrices used in the refutation are Hermitian and positive.

The counterexample uses K=2, N=3/2, ε=6/7, t=(7/2)log(2), and ψ=(|1⟩+|2⟩)/√2. Its output spectrum has a value above 1/2 and a value below 1/8. The only admissible binomial candidate is M=2, μ=3/4; its output spectrum and every κ candidate’s output spectrum lie strictly between 1/8 and 1/2. Their common trace is one. Strict three-coordinate majorization gives smaller entropy for ψ than for every candidate; no assertion that ψ is the global minimizer is made.

The displayed κ output in section III uses $(1-f{)}^K$ in its off-diagonal coefficient. The defining Kraus equations give $(1-f{)}^{K/2}$ instead. The encoding follows the Kraus equations and derives the K=2 off-diagonal coefficient √3/32768 at the counterexample parameters.

Verified locator

• DOI: https://doi.org/10.1103/PhysRevA.94.022341
• URL: https://arxiv.org/abs/1605.04525v1
• PDF: https://arxiv.org/pdf/1605.04525v1 (channel pp. 1–2; Conjecture 1 p. 4).