We prove that there exists an absolute constant $α<1$ such that for every finite dimension $d$ and every quantum channel $T$ on $\mathsf{L}(\mathbb{C}^d)$, $\left\|Θ\circ(\mathrm{id}-T)\right\|_\diamond \le α\,\left\|Θ\right\|_\diamond\,\left\|\mathrm{id}-T\right\|_\diamond$, where $Θ$ is the transposition map. In fact we show the explicit choice $α=1/\sqrt{2}$ works.
We prove that there exists an absolute constant α<1 such that for every finite dimension d and every quantum channel T on L(Cd), ∥Θ∘(id−T)∥⋄≤α∥Θ∥⋄∥id−T∥⋄, where Θ is the transposition map. In fact we show the explicit choice α=1/2 works.
In this paper, we present a class of maps for which the multiplicativity of the maximal output p-norm holds for p=2 and p≥4. The class includes all positive trace-preserving maps from B(C3) to B(C2). In this sense, the result is a generalization of the corresponding result in the work of King and Koldan [“New multiplicativity results for qubit maps,” J. Math. Phys. 47, 042106 (2006)], where the multiplicativity was proved for all positive trace-preserving maps from B(C2) to B(C2) with p=2 and p≥4. Interestingly, by contrast, the multiplicativity of p-norm was investigated in the context of quantum information theory and shown not to hold, in general, for high dimensional quantum channels [Hayden, P. and Winter, A., “Counterexamples to the maximal p-norm multiplicativity conjecture for all p>1,” Commun. Math. Phys. 284, 263 (2008)]. Moreover, the Werner–Holevo channel, which is a map from B(C3) to B(C3), is a counterexample for p>4.79 [Werner and Holevo, J. Math. Phys. 43, 4353 (2002).].
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.
Evaluating the quantum capacity Q(N) of a noisy quantum channel is generally intractable due to the superadditivity of coherent information, which necessitates an infinite-letter regularization over tensor powers N⊗n. Building on recent discoveries of non-degradable channels with strictly additive coherent information and the conditional mutual information (CMI) continuity framework of Winter (2016) and Shirokov (2017), we establish an explicit, dimension-independent approximate additivity and single-letter capacity bounding framework for quantum channels in the diamond-norm neighborhood of any strongly additive channel D. While the general Alicki-Fannes-Winter-Shirokov (AFWS) CMI method yields a symmetric single-step continuity modulus fWS(ϵ)=2ϵlog2dB+2g(ϵ) for arbitrary perturbations 21∥N−D∥⋄≤ϵ—already avoiding the O(ϵlog(1/ϵ)) Stinespring square-root penalty of Leung and Smith (2009)—its orthogonal state decomposition incurs a 2g(ϵ) entropic overhead. Here we prove that for physical convex-mixture noise perturbations Nϵ=(1−ϵ)D+ϵM, direct state-level mixture concavity eliminates the orthogonal decomposition, cutting the entropic overhead in half to h2(ϵ) (fmix(ϵ)=2ϵlog2dB+h2(ϵ)). Furthermore, when M is a replacer noise process (such as depolarizing, erasure, or thermal-reset noise), its CMI vanishes identically and the quantum data-processing inequality restricts the linear slope to dmin:=min(dA,dB), yielding a strongly asymmetric two-sided bound −2ϵlog2dmin−h2(ϵ)≤Q(1)(Nϵ⊗M)−Q(1)(D⊗M)≤h2(ϵ). Telescoping over Nϵ⊗k⊗D⊗(n−k) tightly sandwiches the regularized capacity Q(Nϵ) via Q(1)(Nϵ)≤Q(Nϵ)≤Q(1)(Nϵ)+2ϵlog2dmin+2h2(ϵ). Numerical benchmarks on depolarizing-perturbed erasure channels confirm that our sharpened bound comes within 1.45× of the exact numerical shift at ϵ=10−3, improving upon the general Winter-Shirokov CMI bound by nearly 2× and the Leung-Smith Stinespring bound by over 45×.
This work studies the placement of an entanglement source along a communication line formed by two noisy qubit channels. Recent work argued, on analytical and numerical grounds, that midpoint placement should be at least as favorable as endpoint placement. Here it is shown that, for arbitrary qubit channels, if a sequential composition can preserve entanglement, then the corresponding parallel action cannot annihilate all entanglement. The proof uses the transpose-factorization criterion introduced in that recent work. Quantum Sinkhorn scaling converts every strictly positive qubit channel into a unital representative, and the special normal form of unital qubit channels then yields the required factorization of the transposed map through the original channel. Depolarizing regularization and the closedness of the set of entanglement-breaking channels extend the result to arbitrary channels. Consequently, Λ1⊗Λ2∈EA implies Λ2∘Λ1∈EB, and, by exchanging the two channels, the same holds for the opposite composition order. This proves the recent conjecture that midpoint placement is optimal for all qubit channels, in the feasibility sense in which that optimality was originally defined.
Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels — that is, not completely positive — can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.
One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity: