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arXiv:2609.27906

A sharp norm inequality for entanglement-breaking channels

Eran Kopel

quant-phmath-phmath.OASubmitted Aug 21, 2026• Announced Aug 24, 2026• 0 citations• Revised Oct 17, 2026 (v2)
v1 · Aug 21, 2026v2 · Oct 17, 2026
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Abstract

For a channel $Φ$ on $M_d$ written in the generalised Bloch parameterisation $r \mapsto Ar+c$, we prove that entanglement breaking implies $\|A\|_*^2 + \frac{d(d-1)}{2}\,|c|^2 \;\le\; (d-1)^2 ,$ where $\|\cdot\|_*$ denotes the nuclear norm. Read on the Choi state of the channel, the inequality is an instance of the covariance-matrix criterion of Gittsovich et al.; what is added here is an elementary proof in Holevo form and a complete description of the equality cases. Apart from the constant channel onto a pure state, which shows that the coefficient $d(d-1)/2$ cannot be increased, equality holds exactly for the measure-and-prepare channels with rank-one effects $α_k|e_k\rangle\langle e_k|$ and pure outputs $|ψ_k\rangle\langleψ_k|$ whose overlaps are an affine image of those of the measurement directions, $|\langleψ_j|ψ_k\rangle|^2 = P + (1-P)\,(d|\langle e_j|e_k\rangle|^2 - 1)/(d-1)$ for a common $P = \mathrm{Tr}\,Φ(\mathbf{1}/d)^2$. These include the completely dephasing channel, the re-preparation channel of every rank-one POVM, a one-parameter family interpolating between dephasing and the constant channel, and non-unital non-projective examples already at $d = 2$. The qubit case reads $\|A\|_*^2+|c|^2\le1$ and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound $\|A\|_*\le1$ by retaining the POVM completeness relation $\sum_k f_k=0$, which her argument discards; this recentres $A$ from a second moment into a cross-covariance, and is the entire content of the improvement.

Comments
9 pages; v2: the equality cases of the inequality are characterised (Theorem 5); the constant channel onto a pure state settles the coefficient and c != 0 questions of v1 (pointed out by H. Ma and W. Zhang); equality conditions of Section 4 corrected; abstract states the relation to the covariance-matrix criterion; ancillary files updated

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For a channel ΦΦΦ on MdM_dMd​ written in the generalised Bloch parameterisation r↦Ar+cr \mapsto Ar+cr↦Ar+c, we prove that entanglement breaking implies ∥A∥∗2+d(d−1)2 ∣c∣2  ≤  (d−1)2,\|A\|_*^2 + \frac{d(d-1)}{2}\,|c|^2 \;\le\; (d-1)^2 ,∥A∥∗2​+2d(d−1)​∣c∣2≤(d−1)2, where ∥⋅∥∗\|\cdot\|_*∥⋅∥∗​ denotes the nuclear norm. Read on the Choi state of the channel, the inequality is an instance of the covariance-matrix criterion of Gittsovich et al.; what is added here is an elementary proof in Holevo form and a complete description of the equality cases. Apart from the constant channel onto a pure state, which shows that the coefficient d(d−1)/2d(d-1)/2d(d−1)/2 cannot be increased, equality holds exactly for the measure-and-prepare channels with rank-one effects αk∣ek⟩⟨ek∣α_k|e_k\rangle\langle e_k|αk​∣ek​⟩⟨ek​∣ and pure outputs ∣ψk⟩⟨ψk∣|ψ_k\rangle\langleψ_k|∣ψk​⟩⟨ψk​∣ whose overlaps are an affine image of those of the measurement directions, ∣⟨ψj∣ψk⟩∣2=P+(1−P) (d∣⟨ej∣ek⟩∣2−1)/(d−1)|\langleψ_j|ψ_k\rangle|^2 = P + (1-P)\,(d|\langle e_j|e_k\rangle|^2 - 1)/(d-1)∣⟨ψj​∣ψk​⟩∣2=P+(1−P)(d∣⟨ej​∣ek​⟩∣2−1)/(d−1) for a common P=Tr Φ(1/d)2P = \mathrm{Tr}\,Φ(\mathbf{1}/d)^2P=TrΦ(1/d)2. These include the completely dephasing channel, the re-preparation channel of every rank-one POVM, a one-parameter family interpolating between dephasing and the constant channel, and non-unital non-projective examples already at d=2d = 2d=2. The qubit case reads ∥A∥∗2+∣c∣2≤1\|A\|_*^2+|c|^2\le1∥A∥∗2​+∣c∣2≤1 and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound ∥A∥∗≤1\|A\|_*\le1∥A∥∗​≤1 by retaining the POVM completeness relation ∑kfk=0\sum_k f_k=0∑k​fk​=0, which her argument discards; this recentres AAA from a second moment into a cross-covariance, and is the entire content of the improvement.
Discarding an output subsystem is a local post-processing operation, so an entanglement-breaking channel remains entanglement breaking after output reduction. The converse need not hold: a reduced channel can be entanglement breaking even when a richer output still carries reference entanglement. This preprint gives an exactly solvable two-parameter benchmark that quantifies this asymmetry. On two qubits, let (\Pi_\pm=(I\pm Z\otimes I)/2), let (\mathcal L(X)=\sum_s\Pi_sX\Pi_s) be sector dephasing, let (\mathcal M(X)=\sum_s\operatorname{Tr}(\Pi_sX)\Pi_s/2) be measure-and-prepare within the two sectors, and let (\mathcal D_4(X)=\operatorname{Tr}(X)I_4/4). The channel family is [\Phi_{t,\eta}=(1-\eta)\big[(1-t)\mathcal L+t\mathcal M\big]+\eta\mathcal D_4,\qquad 0\le t,\eta\le1.] For the reduced task, a variable second-qubit input is embedded with an arbitrary fixed first-qubit state and the first output qubit is discarded. The induced qubit channel is depolarizing with parameter (q=(1-\eta)(1-t)), and is entanglement breaking exactly when (q\le1/3). For the full four-dimensional output, the partially transposed normalized Choi state has one potentially negative eigenvalue, [\lambda_- = \frac{-4(1-\eta)+6(1-\eta)t+\eta}{16},] with multiplicity two. The block structure makes PPT sufficient for separability in this family. Hence, for (\eta<1), [t_{\rm red}(\eta)=\max!\left{0,1-\frac{1}{3(1-\eta)}\right},\qquadt_{\rm full}(\eta)=\max!\left{0,\frac{4-5\eta}{6(1-\eta)}\right}.] The exact region in which the reduced channel is already entanglement breaking while the full channel is not is (t_{\rm red}(\eta)\le t<t_{\rm full}(\eta)). Its width is [w(\eta)=\begin{cases}\eta/[6(1-\eta)],&0\le\eta\le2/3,\(4-5\eta)/[6(1-\eta)],&2/3\le\eta<4/5,\0,&4/5\le\eta\le1,\end{cases}] and reaches the exact maximum (w_{\max}=1/3) at (\eta=2/3). The Choi negativities are [\mathcal N_{\rm red}=\frac{[3(1-\eta)(1-t)-1]+}{4},\qquad\mathcal N{\rm full}=\frac{[4-5\eta-6(1-\eta)t]_+}{8}.] Therefore the maximal full-output Choi negativity compatible with an already entanglement-breaking reduced output is (1/12), attained at ((\eta,t)=(2/3,0)). The result is a controlled benchmark for output-access dependence: the same physical channel family can cross the entanglement-breaking boundary at different noise values solely because an output subsystem is retained or discarded.
We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems. The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state ρρρ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality  S(ρ)−S(σ)≤Cρ(1−F(ρ,σ)) \,S(ρ)-S(σ)\leq C_ρ(1-F(ρ,σ))\,S(ρ)−S(σ)≤Cρ​(1−F(ρ,σ)) valid for any state σσσ, where CρC_ρCρ​ is a constant depending on the rank of ρρρ. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state ρρρ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality  EF(ρ)−EF(σ)≤Cρ(1−Trρσ) \,E_F(ρ)-E_F(σ)\leq C_ρ(1-\mathrm{Tr}ρσ)\,EF​(ρ)−EF​(σ)≤Cρ​(1−Trρσ) valid for any state σσσ, where CρC_ρCρ​ is a constant depending on the Schmidt rank of ρρρ. In both cases the optimal constant CρC_ρCρ​ is equal to the optimal constant KdK_{d}Kd​ in the log-Sobolev inequality for the complete graph with ddd vertices: in the first case d=rankρd=\mathrm{rank}ρd=rankρ, in the second one d=rankρA=rankρBd=\mathrm{rank}ρ_A=\mathrm{rank}ρ_Bd=rankρA​=rankρB​. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.
Let $\Phi:M_d(\C)\to M_d(\C)$ be a completely positive map with a transpose-symmetric Kraus representation. Its Choi operator determines the rank-two constant \[ \beta_2(\Phi) =\frac12\sup_{\substack{\norm z=1\\ \SR(z)\le2}} \ip z{J(\Phi)z}. \] We prove that, for every r≥1r\ge1r≥1, \[ \Phi\circ\tau+(r-1)\beta_2(\Phi) \left(\Delta-\frac1r\id\right) \] is rrr-positive, where τ\tauτ is transposition and $\Delta(X)=\Tr(X)I$. The proof turns the negative spectrum of a partially transposed rank-one operator into the edges of the complete graph KrK_rKr​. Regrouping by vertices costs its degree r−1r-1r−1, rather than its (r2)\binom r2(2r​) edges. Transpose symmetry protects one direction and is exactly what produces the term $-\id/r$. For Φ=ΛU⊗ΛV\Phi=\Lambda_U\otimes\Lambda_VΦ=ΛU​⊗ΛV​, with $\Lambda_E(X)=\Tr(X)I-X^T$, a sharp double-skew singular-value estimate gives β2≤1\beta_2\le1β2​≤1, with equality when both local dimensions are at least two. A map identity then yields, for every operator CCC of rank at most rrr on U⊗VU\otimes VU⊗V, \[ \norm{\Tr_U C}_2^2+\norm{\Tr_V C}_2^2 \le r\norm C_2^2+\frac1r\abs{\Tr C}^2, \] with dimension-uniform coefficients that are sharp whenever one local dimension is at least rrr and the other is at least two. A second family, built from the edges of an arbitrary graph, has the same local constant and attains the universal coefficient r−1r-1r−1 whenever the graph contains an rrr-clique. The Choi lifting chain through the partial-trace and graph results, together with its sharpness witnesses, is formalized in Lean~4.
Let Φ1,Φ2:Md(C)→Mn(C)Φ_1, Φ_2 : \mathbb{M}_d(\mathbb{C})\to \mathbb{M}_n(\mathbb{C})Φ1​,Φ2​:Md​(C)→Mn​(C) be two quantum channels with respective Stinespring isometries V1,V2:Cd→Cn⊗CmV_1, V_2 : \mathbb{C}^{d}\to \mathbb{C}^{n} \otimes \mathbb{C}^{m}V1​,V2​:Cd→Cn⊗Cm on any common dilation space Cm\mathbb{C}^{m}Cm. We prove that there exists a unitary UUU on Cm\mathbb{C}^{m}Cm such that ∥V1−(1⊗U)V2∥∞≤2∥Φ1−Φ2∥⋄,\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond},∥V1​−(1⊗U)V2​∥∞​≤2∥Φ1​−Φ2​∥⋄​​, thus resolving vom Ende's modification of the Kretschmann-Schlingemann-Werner conjecture in the affirmative.
Let ρρρ and σσσ be density operators on a separable Hilbert space. For 0<α<10<α<10<α<1, we study the directed trace-norm overlap Φα(ρ,σ)=∥ρασ1−α∥1Φ_α(ρ,σ)=\|ρ^ασ^{1-α}\|_1Φα​(ρ,σ)=∥ρασ1−α∥1​ and its symmetrized form Fα(ρ,σ)=12(Φα(ρ,σ)+Φ1−α(ρ,σ))\mathcal F_α(ρ,σ)=\frac12\bigl(Φ_α(ρ,σ)+Φ_{1-α}(ρ,σ)\bigr)Fα​(ρ,σ)=21​(Φα​(ρ,σ)+Φ1−α​(ρ,σ)). At α=12α=\tfrac12α=21​, both quantities coincide with the root Uhlmann--Jozsa fidelity. Our aim is not to introduce a new notion of fidelity, but to understand how these overlaps vary with the parameter and when equality occurs in the resulting inequalities. We first prove that α↦Φα(ρ,σ)α\mapstoΦ_α(ρ,σ)α↦Φα​(ρ,σ) is log-convex. This yields a sharp lower bound for Fα\mathcal F_αFα​ in terms of the root fidelity, together with a stronger intermediate geometric-mean bound. We also show that ΦαΦ_αΦα​ is the trace functional Qα,1/2Q_{α,1/2}Qα,1/2​ associated with the ααα-zzz Renyi divergence, which connects the directed overlap with the known ααα-zzz Renyi theory. We then determine the exact data-processing behavior of the symmetrized family. Universal monotonicity under quantum channels holds only at α=12α=\tfrac12α=21​; for every other value of ααα, it already fails under diagonal pinching of faithful real qubit states. In finite dimensions, we give a complete spectral description of the equality cases and provide explicit noncommuting examples. We also discuss some natural questions about overlap-preserving maps. Finally, we prove that the difference between the Petz overlap and the directed trace-norm overlap vanishes exactly when ρρρ and σσσ commute, without requiring either faithfulness or any additional support assumption.