The transpose map is the canonical nonphysical map in quantum information that is central to entanglement tests and to converse bounds for quantum capacity. It is a longstanding conjecture that transposition satisfies a dimension-independent strict submultiplicativity for physically relevant channel differences. We answer this affirmatively by proving that for every quantum channel $T$ on $\mathsf{L}(\mathbb{C}^d)$, $\|Θ\circ(\mathrm{id}-T)\|_\diamond \le \frac{1}{\sqrt{2}}\|Θ\|_\diamond \|\mathrm{id}-T\|_\diamond$. The same bound holds for all Hermiticity-preserving trace-annihilating maps. Because a trace-preserving channel produces a traceless deviation from the identity, transposition cannot act as strongly on such physically generated errors as it can on arbitrary perturbations. Thus, although transposition is extremal in the unrestricted diamond-norm geometry, physical constraints mandate a universal margin in the submultiplicative bound. As one immediate consequence, the finite-error Holevo--Werner converse extends from decoding errors $\varepsilon<1/2$ to $\varepsilon<1/\sqrt{2}$. The same tracelessness mechanism also yields rank-sensitive continuity refinements for negativity and logarithmic negativity, demonstrating its potential for broader application in quantum information.
The transpose map is the canonical nonphysical map in quantum information that is central to entanglement tests and to converse bounds for quantum capacity. It is a longstanding conjecture that transposition satisfies a dimension-independent strict submultiplicativity for physically relevant channel differences. We answer this affirmatively by proving that for every quantum channel T on L(Cd), ∥Θ∘(id−T)∥⋄≤21∥Θ∥⋄∥id−T∥⋄. The same bound holds for all Hermiticity-preserving trace-annihilating maps. Because a trace-preserving channel produces a traceless deviation from the identity, transposition cannot act as strongly on such physically generated errors as it can on arbitrary perturbations. Thus, although transposition is extremal in the unrestricted diamond-norm geometry, physical constraints mandate a universal margin in the submultiplicative bound. As one immediate consequence, the finite-error Holevo--Werner converse extends from decoding errors ε<1/2 to ε<1/2. The same tracelessness mechanism also yields rank-sensitive continuity refinements for negativity and logarithmic negativity, demonstrating its potential for broader application in quantum information.
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.
Transformations of quantum channels, such as the transpose, complex conjugate, and adjoint, are fundamental to quantum information theory. Given access to an unknown channel, a central problem is whether these transformations can be implemented physically with quantum supermaps. While such supermaps are known for unitary operations, the situation for general quantum channels is fundamentally different. In this work, we establish a strict hierarchy of physical realizability for the transposition, complex conjugation, and adjoint transformation of an unknown quantum channel. We present a probabilistic protocol that exactly implements the transpose with a single query. In contrast, we prove no-go theorems showing that neither the complex conjugate nor the adjoint can be implemented by any completely positive supermap, even probabilistically. We then overcome this impossibility by designing a virtual protocol for the complex conjugate based on quasi-probability decomposition, and show its optimality in terms of the diamond norm. As a key application, we propose a protocol to estimate the expectation values resulting from the Petz recovery map of an unknown channel, achieving an improved query complexity compared to existing methods.
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.