The Quantum Formalization Drive

We obtain a complete characterization of eventually entanglement breaking (EEB) unital channels in terms of some simple commutation properties of their eigenvectors. Specifically, the following properties are equivalent for a unital channel: (i) it is EEB; (ii) it is eventually positive partial transpose (PPT); (iii) it is eventually mixed twisted…

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…

Encrypted quantum cloning distributes an unknown $k$-qubit state among $m$ encrypted clones so that no individual clone reveals the input, yet the state can be recovered from any one clone together with a common quantum key. We ask the inverse question: for a fixed encoding architecture, which multipartite pure states can serve as exact resources for this task?…

For a channel $Φ$ on $M_d$ written in the generalised Bloch parameterisation $r \mapsto Ar+c$, we prove that entanglement breaking implies $\|A\|_*^2 + \frac{d(d-1)}{2}|c|^2 \le (d-1)^2$, where $\|\cdot\|_*$ denotes the nuclear norm. The bound is attained in every dimension by the completely dephasing channel, and at $d=2$ by an explicit family with $c\neq0$. The qubit…

Suppose Alice and Bob can transform one shared mixed state into another by local operations and classical communication, and can also recover the first state from the second. What can change in such a conversion? We give a complete classification of reversible conversions between two-qutrit states. We consider exact deterministic LOCC transformations of a…

We study entanglement distillation under operations that remain completely positive under partial-transpose conjugation, a.k.a. PPT channels, in the standard quantum Shannon theory regime of asymptotically vanishing but non-zero error. Our main results are: (a) a regularised formula for the PPT distillable entanglement in terms of a…

The notion of weak-degradability of quantum channels is introduced by generalizing the degradability definition given by Devetak and Shor. Exploiting the unitary equivalence with beam-splitter/amplifier channels we then prove that a large class of one-mode Bosonic Gaussian channels are either weakly degradable or anti-degradable. In the latter case this…

The class of subsets of locally convex spaces called $\mu$-compact sets is considered. This class contains all compact sets as well as several noncompact sets widely used in applications. It is shown that many results well known for compact sets can be generalized to $\mu$-compact sets. Several examples are considered. The main result of the paper is a…

We characterize the class of all physical operations that transform Gaussian states to Gaussian states. We show that this class coincides with that of all operations which can be performed on Gaussian states using linear optical elements and homodyne measurements. For bipartite systems we characterize the processes which can be implemented by local…

We describe an universal method for quantitative continuity analysis of entropic characteristics of energy-constrained quantum systems and channels. It gives asymptotically tight continuity bounds for basic characteristics of quantum systems of wide class (including multi-mode quantum oscillators) and channels between such systems under the energy…

The paper is devoted to systematic study of the $\chi$-capacity (underlying the classical capacity) of infinite dimensional quantum channels. An essential feature of this case is the natural appearance of the input constraints and infinite, in general, …

Several important measures of quantum correlations of a state of a finite-dimensional composite system are defined as linear combinations of marginal entropies of this state. This paper is devoted to the infinite-dimensional generalizations of such quantities and to the analysis of their properties. We introduce the notion of faithful extension of a linear…

Open Markovian quantum systems with fast and full Hamiltonian control can be reduced to an equivalent control system on the standard simplex modelling the dynamics of the eigenvalues of the density matrix describing the quantum state. We explore this reduced control system for answering questions on reachability and stabilizability with immediate…

Operator-sum or Kraus representations for single-mode Bosonic Gaussian channels are developed, and several of their consequences explored. Kraus operators are employed to bring out the manner in which the unphysical matrix transposition map when accompanied by injection of a threshold classical noise becomes a physical channel. The action of the…

The relative entropy of entanglement $E_R$ is defined as the distance of a multi-partite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a closest separable state, even in infinite dimensions; also, $E_R$ is everywhere lower semi-continuous. We…

We extend the definition of the conditional min-entropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended…

Quantum computers must correct errors faster than they accumulate, and quantum low-density parity-check (LDPC) codes are the leading candidates for doing so with little overhead: every check involves a bounded number of qubits, so syndromes can be measured by local circuits. In 2021–2022 the first asymptotically good quantum LDPC codes appeared, with…

Approximate counting asks for the number $K$ of marked items in a list of $N$ items, to within a multiplicative factor $1\pm\varepsilon$, when the marked set can only be accessed through a membership oracle. Brassard, Høyer, Mosca and Tapp (arXiv:quant-ph/0005055) gave a quantum algorithm that uses $O(\sqrt{N/K}\,\varepsilon^{-1})$ queries, which is optimal. Their…

Characterizing the state $\rho$ of an $n$-qubit device is a basic task in quantum computing and simulation. Full quantum state tomography reconstructs $\rho$ completely, but it needs a number of measurements exponential in $n$ (Haah et al., 2017; O'Donnell–Wright, 2016). In practice one usually wants only a list of properties, such as the expectation values …

NLTS Hamiltonians from good quantum codes establishes the selected mathematical guarantees below. The mission turns those original paper results into precise Lean proof targets.

Classical statistical mechanics describes a particle by a probability density on phase space; quantum mechanics replaces this by a wavefunction, and the uncertainty principle forbids any honest probability interpretation of simultaneous position and momentum. In 1932, Eugene Wigner introduced the transform that has become the standard way of placing quantum…

Qubit Entanglement Breaking Channels.

Quantum signal processing transforms the eigenphases of a unitary operator by a sequence of controlled operations and single qubit rotations. To know which transformations such a sequence can realize, one must characterize the matrix functions produced by the sequence. Haah's product decomposition theorem gives such a characterization without the…

Many quantum algorithms act on a matrix $A$ that is not itself unitary: a Hamiltonian to be simulated, a linear system to be inverted, a projector to be amplified. The standard way to hand such a matrix to a quantum computer is to embed it as a block of a unitary $U$, so that $A=\widetilde\Pi U\Pi$ for two orthogonal projectors $\Pi,\widetilde\Pi$. The question is…

Quantum error correction needs families of codes that protect a growing number of logical qubits while using checks that touch few physical qubits. In a quantum Tanner code, the local constraints come from two smaller binary codes. The distance analysis requires those local codes to satisfy a geometric condition called robustness: a low-weight word in a…