We extend the framework of quantum pushforward designs to the approximate setting, where averaging is achieved only up to finite precision. Using Schatten $p$-norms and Lipschitz continuity arguments, we derive bounds on the approximation parameters of pushforward designs obtained from complex projective spaces, including simplices, mixed states, and quantum channels. In the mixed-state case, we refine the bounds by exploiting the symmetric subspace structure, leading to asymptotically tighter estimates. Numerical simulations support our theoretical results, showing near-optimality in low-dimensional scenarios.
We extend the framework of quantum pushforward designs to the approximate setting, where averaging is achieved only up to finite precision. Using Schatten p-norms and Lipschitz continuity arguments, we derive bounds on the approximation parameters of pushforward designs obtained from complex projective spaces, including simplices, mixed states, and quantum channels. In the mixed-state case, we refine the bounds by exploiting the symmetric subspace structure, leading to asymptotically tighter estimates. Numerical simulations support our theoretical results, showing near-optimality in low-dimensional scenarios.
The question of how well a finite ensemble of quantum states can approximate Haar randomness is relevant in many areas of quantum physics. Here we consider this problem from a geometrical perspective, via the notion of Welch bounds. The latter are a set of inequalities, originally derived in the context of signal processing, which precisely characterize exact k-designs, i.e. a finite ensemble of states that exactly reproduces the k-th moment of the Haar measure. We derive stronger Welch bounds which enable a quantitative and operationally meaningful characterisation of approximate k-designs. Our new bounds are often tight, allowing us to characterize optimal approximate designs in regimes where the cardinality of the ensemble is small. A notable example is complete sets of mutually unbiased bases (MUBs), which form an optimal approximate 3-design among 1-designs of their cardinality. This leads to a new variational approach to determine the existence of complete sets of MUBs, which we explore numerically in dimension six. We highlight a connection to shadow tomography, relating our bounds to the shadow norm. We comment on how the method extends to other types of designs, including toric and unitary designs. A key technical ingredient, is the computation of the complete spectrum of the partially transposed symmetric-subspace projector, including multiplicities and eigenvectors, which may find further applications.
Designs, structures connected to averaging with respect to a given measure using finite sets of points, have proven themselves to be invaluable tools across the field of quantum information, finding their uses in state and process tomography, key distribution, and others. In this work, we introduce a concept of pushforward designs that allows us to obtain new structures from already existing ones by mapping them between the spaces, with specific examples including simplex designs and mixed state designs from complex projective designs. Based on the general concept, we put forward a structure called channel [t,k]-design, allowing for averaging over space of quantum channels for systems in contact with an environment of dimension k. Based on this notion, we introduce the concept of effective environment dimensionality k∗, which we estimate for the IBM Kyoto quantum computer to be below 2.2 for times up to $350\phantom{\rule{4pt}{0ex}}\textmu{}\text{s}$.
Standard finite quantum de Finetti theorems approximate the k-system marginals of permutation-invariant states of n-systems by mixtures of independent and identically distributed (iid) states, usually in trace distance. We prove both standard and Renner's exponential de Finetti theorems in the stronger form of operator inequalities, implying bounds in every Schatten norm and for every quantum Rényi divergence satisfying data processing. In the standard case, for fixed local dimension, our k/n error bound in max-relative entropy improves on the previously best known k2/n scaling, even in the classical setting. The operator-inequality approach is particularly suited to study channel de Finetti representations because operator order between Choi states is equivalent to completely positive (CP) order between the underlying channels. For permutation-covariant channels N(n):A⊗n→B⊗n, where dA=dimA, we prove that the k-system reduced channel is CP-dominated by a mixture of tensor-power channels with error O(k/n) and polynomial dependence on the local dimensions, addressing a question raised by Berta et al. [Math. Program. 194, 781-829 (2022)]. Under the no-signalling condition, we also prove an exponential channel de Finetti theorem where the approximating mixture consists of Choi-almost-iid channels, whose normalized Choi states are almost-iid in the sense of Mazzola-Sutter-Renner. In the case of r defects, the representation error is at most poly(n)(2dA4k3/(nr2))(r+1)/2 and decays exponentially in n for a suitable choice of parameters r and k.
We develop a theory of approximate quantum error correction (QEC) based on the error-set model, complemented by general methods for code construction. Exact QEC has a powerful error-set structure: by the Knill-Laflamme conditions, a code correcting a given error set automatically protects against every channel whose Kraus operators lie in their linear span. This linearity gives rise to code distance, the equivalence between erasures and general errors, and a theory of asymptotically good codes. A longstanding view has been that these features do not extend to AQEC, leaving the theory essentially channel-by-channel. We show instead that, although full Knill--Laflamme linearity fails, a restricted form survives and suffices to extend all three structural features to the approximate setting. Specifically, a common error-set criterion governs families of channels whose Kraus operators are linear combinations of a given error set and whose coefficient matrices satisfy a spectral constraint. Using the Bény-Oreshkov worst-case and Petz average-case frameworks, we derive uniform fidelity guarantees for these families in terms of two new code parameters--the \emph{environment-leakage distance}, controlling worst-case performance, and the \emph{Knill-Laflamme Hellinger distance}, characterizing the average-case performance of Petz recovery.
To demonstrate the scope of this model, we develop partition-based constructions across diverse quantum systems and geometries, placing exact and approximate correction on equal footing. These constructions lead to a metric--error alignment hierarchy for Hilbert spaces, metrics, and error families, which in turn characterizes the resulting recovery guarantees. They yield the first known asymptotically good code families for fermionic systems, one-dimensional Rydberg-blockaded systems, and deletion errors, and extend to other physical platforms.
Can an external environment be absorbed universally into a quantum process? This article identifies trace normalization as the obstruction and tracks it through higher orders. Exact ranks are derived for sequential-comb normalization at every order. Two concrete families of higher-order maps are defined, and their transformation-space dimensions are computed. In the family stable under arbitrary auxiliary combs, adding an environment to an input has neither a left nor a right adjoint from order three onward. Together with the channel and superchannel cases, this yields an all-order family of no-go theorems. The claim is restricted to specified process types: broader normalized type systems may still admit internal homs. Without normalization, a two-sided adjunction returns for normal completely positive maps. Exactly the finite-dimensional Hilbert spaces are dualizable, but trace scaling and rank-one evaluation prevent restriction to deterministic channels. The continuous scalar laws underlying composition- and tensor-additive dynamics are proved to be logarithmic boundary terms. Nonadditive quadratic actions are analyzed through exact stationarity equations and several exact qubit phase classifications. Finally, continuous channel encoding below the full channel-space dimension has worst-case diamond-norm error at least 1/(dAdB ), while normalized weighted cups have sharp defect 1 − 1/δ2. The results identify three ways around the obstruction: enlarge the type universe, remove normalization, or accept controlled approximation
We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries. Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologically-enriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.