The Quantum Formalization Drive

Quantum error correction (QEC) is an essential element of physical quantum information processing systems. Most QEC efforts focus on extending classical error correction schemes to the quantum regime. The input to a noisy system is embedded in a coded subspace, and error recovery is performed via an operation designed to perfectly correct for a set of…

Many quantum information tasks use inputs of the form $\rho^{\otimes m}$, which naturally induce permutation and unitary symmetries. We classify all quantum channels that respect both symmetries - i.e. unitary-equivariant and permutation-invariant quantum channels from $(\mathbb{C}^{d})^{\otimes m}$ to $(\mathbb{C}^{d})^{\otimes n}$ - via their extremal points.…

Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks, we need to determine the optimal point-to-point rates that are achievable by two remote parties at the ends of a quantum channel, without restrictions on their local operations…

We ask when qubit controls can produce an effective generator that drives every initial state to the same pure state. Lie-algebra closure alone cannot answer this question because it loses the distinction between reversible control and irreversible dissipation. Lie wedges retain this distinction by enforcing nonnegative strengths for dissipative generators.…

Coherences between different energy levels are strongly constrained by thermodynamics. Here we ask a related question: if several coherence transfers can be optimized separately, can they also be optimized simultaneously by the same thermodynamic process? We show that this is in general a compatibility problem. Using an energy-resolved Stinespring…

The use of quantum resources in the work extraction from a quantum system is an emerging research topic. Recently, [arXiv:2602.22893] established the necessity of measurement coherence for obtaining advantage in work extraction from an unknown isolated quantum system. However, a quantitative relation between the magnitude of this advantage and established…

We investigate the relaxation dynamics and steady state of a qubit interacting with a coherent environment through an energy-conserving flip-flop interaction within the Markovian repeated-interaction framework. We derive an exact analytical solution for the evolving and stationary state, nonperturbative in the environmental coherence. We show that…

Randomized measurement protocols give experimental access to low-degree polynomial invariants of a quantum state rather than to the state itself, which raises the question of what the best entanglement criterion is that can be built from finitely many such invariants. In this work we answer this question through convex duality. For each degree we introduce…

Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini--Study metric on the projective Hilbert…

This is a continuation of the expository article \cite{krp} with some new remarks. Let $S_n$ denote the set of all Gaussian states in the complex Hilbert space $L^2 (\mathbb{R}^n),$ $K_n$ the convex set of all momentum and position covariance matrices of order $2n$ in Gaussian states and let $\mathcal{G}_n$ be the group of all unitary operators in $L^2 (\mathbb{R}^n)$ c…

Simple optical instruments are linear optical networks where the incident light modes are turned into equal numbers of outgoing modes by linear transformations. For example, such instruments are beam splitters, multiports, interferometers, fibre couplers, polarizers, gravitational lenses, parametric amplifiers, phase-conjugating mirrors and also black…

We give a mathematical formulation for the Choi-Jamiolkowski (CJ) correspondence in the infinite-dimensional case in the form close to one used in quantum information theory. We show that there is no need to use a limiting procedure to define "unnormalized maximally entangled state" and the corresponding analog of the Choi matrix since they can be defined…

Fast and fault-tolerant logical measurements: Auxiliary hypergraphs and transversal surgery.

A simple formula for the average gate fidelity of a quantum dynamical operation establishes the selected mathematical guarantees below. The mission turns those original paper results into precise Lean proof targets.

Approximating a partition function $Z(\beta)=\sum_{x\in\Omega}e^{-\beta H(x)}$ is a central task in statistical physics and in counting: it covers the Ising and Potts models, and, at $\beta=\infty$, counting the zero-energy configurations of a Hamiltonian, for instance the proper $k$-colourings of a graph. Computing $Z(\infty)$ exactly is #P-complete in many cases,…

Quantum error correction and quantum communication both ask the same question: after a known noise process $\mathcal A$ has acted on a quantum system, how well can its effect be undone? For a given noise model the best recovery is the solution of a semidefinite program, which gives a number but no structure and no formula. Barnum and Knill showed in 2000 that…