Let $Φ_1, Φ_2 : \mathbb{M}_d(\mathbb{C})\to \mathbb{M}_n(\mathbb{C})$ be two quantum channels with respective Stinespring isometries $V_1, V_2 : \mathbb{C}^{d}\to \mathbb{C}^{n} \otimes \mathbb{C}^{m}$ on any common dilation space $\mathbb{C}^{m}$. We prove that there exists a unitary $U$ on $\mathbb{C}^{m}$ such that $\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond},$ thus resolving vom Ende's modification of the Kretschmann-Schlingemann-Werner conjecture in the affirmative.
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.
Given any pair of quantum channels Φ1,Φ2 such that at least one of them has Kraus rank one, as well as any respective Stinespring isometries V1,V2, we prove that there exists a unitary U on the environment such that $\|V_1-(\mathbbm1\otimes U)V_2\|_\infty\leq\sqrt{2\|\Phi_1-\Phi_2\|_\diamond}$. Moreover, we provide a simple example which shows that the factor 2 on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.
The geometric channel fidelity, the geometric Rényi divergences and the Belavkin--Staszewski relative entropy give converse bounds for capacities, discrimination and estimation. Channels are known only approximately, yet two pairs of channels with equal sums can be arbitrarily close in the diamond norm and have fidelities 0 and 1. The remedy is the Arveson--Radon--Nikodym derivative T of one map with respect to the sum Ω, whose transpose is the environment effect selecting that map in a Stinespring dilation of Ω. In T, these quantities and all maximal f-divergences of operator convex f are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most 2∥T1−T2∥∞1/2, which gives Hölder continuity in the diamond norm on such splittings. The fidelity is continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes: noisy channels of full Kraus rank are continuity points, unitary gates are not.
Given any pair of quantum channels Φ1,Φ2 such that at least one of them has Kraus rank one, as well as any respective Stinespring isometries V1,V2, we prove that there exists a unitary U on the environment such that ∥V1−(1⊗U)V2∥∞≤2∥Φ1−Φ2∥⋄. Moreover, we provide a simple example which shows that the factor 2 on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.
Earlier, we proved a duality between two optimizations problems [Phys. Rev. Lett. 104, 120501 (2010)]. The primary one is, given two quantum channels M and N, to find a quantum channel R such that $\mathcal{R}\ensuremath{\circ}\mathcal{N}$ is optimally close to M as measured by the worst-case entanglement fidelity. The dual problem involves the information obtained by the environment through the so-called complementary channels $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{M}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{N}}$, and consists in finding a quantum channel ${\mathcal{R}}^{\ensuremath{'}}$ such that ${\mathcal{R}}^{\ensuremath{'}}\ensuremath{\circ}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{M}}$ is optimally close to $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{N}}$. It turns out to be easier to find an approximate solution to the dual problem in certain important situations, notably when M is the identity channel---the problem of quantum error correction---yielding a good near-optimal worst-case entanglement fidelity as well as the corresponding near-optimal correcting channel. Here we provide more detailed proofs of these results. In addition, we generalize the main theorem to the case where there are certain constraints on the implementation of R, namely, on the number of Kraus operators. We also offer a simple algebraic form for the near-optimal correction channel in the case M=id. For approximate error correction, we show that any $\ensuremath{\varepsilon}$-correctable channel is, up to appending an ancilla, $\ensuremath{\varepsilon}$-close to an exactly correctable one. We also demonstrate an application of our theorem to the problem of minimax state discrimination.