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. The bound is attained in every dimension by the completely dephasing channel, and at $d=2$ by an explicit family with $c\neq0$. 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.
For a channel Φ on Md written in the generalised Bloch parameterisation r↦Ar+c, we prove that entanglement breaking implies ∥A∥∗2+2d(d−1)∣c∣2≤(d−1)2, where ∥⋅∥∗ denotes the nuclear norm. The bound is attained in every dimension by the completely dephasing channel, and at d=2 by an explicit family with c=0. The qubit case reads ∥A∥∗2+∣c∣2≤1 and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound ∥A∥∗≤1 by retaining the POVM completeness relation ∑kfk=0, which her argument discards; this recentres A 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.
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≥1, \[ \Phi\circ\tau+(r-1)\beta_2(\Phi) \left(\Delta-\frac1r\id\right) \] is r-positive, where τ 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 Kr. Regrouping by vertices costs its degree r−1, rather than its (2r) edges. Transpose symmetry protects one direction and is exactly what produces the term $-\id/r$. For Φ=ΛU⊗ΛV, with $\Lambda_E(X)=\Tr(X)I-X^T$, a sharp double-skew singular-value estimate gives β2≤1, with equality when both local dimensions are at least two. A map identity then yields, for every operator C of rank at most r on U⊗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 r 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−1 whenever the graph contains an r-clique. The Choi lifting chain through the partial-trace and graph results, together with its sharpness witnesses, is formalized in Lean~4.
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 ρ 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)) valid for any state σ, where Cρ is a constant depending on the rank of ρ.
The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state ρ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality EF(ρ)−EF(σ)≤Cρ(1−Trρσ) valid for any state σ, where Cρ is a constant depending on the Schmidt rank of ρ.
In both cases the optimal constant Cρ is equal to the optimal constant Kd in the log-Sobolev inequality for the complete graph with d vertices: in the first case d=rankρ, in the second one d=rankρA=rankρB.
The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.
In a recent paper by S. Pandey, V. Paulsen, J. Prakash, and M. Rahaman, the authors studied the entanglement breaking quantum channels $\Phi_t:\mbb{C}^{d\times d} \to \mbb{C}^{d \times d}$ for t∈[−d2−11,d+11] defined by $\Phi_t(X) = tX+ (1-t)\tr(X) \frac{1}{d}I$. They proved that Zauner's conjecture is equivalent to the statement that entanglement breaking rank of Φd+11 is d2. The authors made the extended conjecture that $\ebr(\Phi_t)=d^2$ for every t∈[0,d+11] and proved it in dimensions 2 and 3. In this paper we prove that for any t∈[−d2−11,d+11]∖{0} the equality $\ebr(\Phi_t)=d^2$ is equivalent to the existence of a pair of informationally-complete unit-norm tight frames $\{|x_i\ra\}_{i=1}^{d^2}, \{|y_i\ra\}_{i=1}^{d^2}$ in $\mbb{C}^d $ which are mutually unbiased in the following sense: for any i=j it holds that $|\la x_i|y_j\ra|^2 = \frac{1-t}{d}$ and $|\la x_i|y_i\ra|^2 = \frac{t(d^2-1)+1}{d}$. Moreover, it follows that $|\la x_i|x_j\ra\la y_i|y_j\ra|=|t|$ for i=j. However, our numerical searches for solutions were not successful in dimensions 4 and 5 for values of t other than 0 or d+11.
Let Φ1,Φ2:Md(C)→Mn(C) be two quantum channels with respective Stinespring isometries V1,V2:Cd→Cn⊗Cm on any common dilation space Cm. We prove that there exists a unitary U on Cm such that ∥V1−(1⊗U)V2∥∞≤2∥Φ1−Φ2∥⋄, thus resolving vom Ende's modification of the Kretschmann-Schlingemann-Werner conjecture in the affirmative.
Let ρ and σ be density operators on a separable Hilbert space. For 0<α<1, we study the directed trace-norm overlap Φα(ρ,σ)=∥ρασ1−α∥1 and its symmetrized form Fα(ρ,σ)=21(Φα(ρ,σ)+Φ1−α(ρ,σ)). At α=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) is log-convex. This yields a sharp lower bound for Fα in terms of the root fidelity, together with a stronger intermediate geometric-mean bound. We also show that Φα is the trace functional Qα,1/2 associated with the α-z Renyi divergence, which connects the directed overlap with the known α-z Renyi theory. We then determine the exact data-processing behavior of the symmetrized family. Universal monotonicity under quantum channels holds only at α=21; for every other value of α, 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 ρ and σ commute, without requiring either faithfulness or any additional support assumption.