The Quantum Formalization Drive

Gaussian loss channels are of particular importance since they model realistic optical communication channels. Except for special cases, quantum capacity of Gaussian loss channels is not yet known completely. In this paper, we provide improved upper bounds of Gaussian loss channel capacity, both in the energy-constrained and unconstrained scenarios. We…

Quantum Machine Learning (QML) models are aimed at learning from data encoded in quantum states. Recently, it has been shown that models with little to no inductive biases (i.e., with no assumptions about the problem embedded in the model) are likely to have trainability and generalization issues, especially for large problem sizes. As such, it is…

We consider two remote parties connected to a relay by two quantum channels. To generate a secret key, they transmit coherent states to the relay, where the states are subject to a continuous-variable (CV) Bell detection. We study the ideal case where Alice's channel is lossless, i.e., the relay is locally situated in her lab and the Bell detection is…

With growing success in experimental implementations it is critical to identify a "gold standard" for quantum information processing, a single measure of distance that can be used to compare and contrast different experiments. We enumerate a set of criteria such a distance measure must satisfy to be both experimentally and theoretically meaningful. We then…

The Renyi entropy is a generalization of the usual concept of entropy which depends on a parameter q. In fact, Renyi entropy is closely related to free energy. Suppose we start with a system in thermal equilibrium and then suddenly divide the temperature by q. Then the maximum amount of work the system can do as it moves to equilibrium at the new…

In this paper we introduce principal 2-bundles and show how they are classified by non-abelian Cech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2-groups possess a natural smooth structure. In the…

We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the role of a band, or structure 2-group.…

Lie's third theorem says that every finite-dimensional Lie algebra is the Lie algebra of a Lie group. $L_\infty$-algebras (strongly homotopy Lie algebras) generalize Lie algebras by allowing brackets of every arity, related by the Jacobi identity only up to coherent homotopy. They occur as symmetries in deformation theory, in higher gauge theory and in string…

Bosonic channels are important in practice as they form a simple model for free-space or fiber-optic communication. Here we consider a single-sender two-receiver pure-loss bosonic broadcast channel and determine the unconstrained capacity region for the distillation of bipartite entanglement and secret key between the sender and each receiver, whenever they…

Fault-tolerant quantum computers implement Clifford gates cheaply and the non-Clifford T gate expensively, because T gates are obtained by magic-state distillation (Fowler, Stephens and Groszkowski). Single-qubit rotations by small, arbitrary angles appear throughout quantum simulation and must be compiled into Clifford+T circuits, so the T-count of a small…

Quantum information carried by continuous variables, such as the quadratures $x$ and $p$ of an optical mode, is cheap to manipulate with beam splitters, squeezers and homodyne detectors, but these Gaussian tools cannot protect Gaussian states against Gaussian noise: Niset, Fiurášek and Cerf proved in 2008 that no Gaussian error-correcting code can do better…

A Gelfand–Tsetlin basis of a representation of the last group in a tower $G_1 \subset G_2 \subset \cdots \subset G_n$ is a basis adapted to every restriction at once: each basis vector spans a line that is cut out by the isotypic decompositions at all levels $k = 1, \dots, n$. When every restriction $G_k \downarrow G_{k-1}$ is multiplicity-free, these lines are…

We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann…

In previous papers, we expressed the Entanglement of Formation in terms of Conditional Mutual Information (CMI). In this brief paper, we express the Entanglement of Distillation in terms of CMI.

We investigate entanglement measures in the infinite-dimensional regime. First, we discuss the peculiarities that may occur if the Hilbert space of a bi-partite system is infinite-dimensional, most notably the fact that the set of states with infinite entropy of entanglement is trace-norm dense in state space, implying that in any neighborhood of every…

An ordinary Lie algebra has a vector space and an antisymmetric bracket satisfying the Jacobi identity. In a Lie 2-algebra, the vectors become the objects of a linear category, and the Jacobi identity is witnessed by a morphism called the Jacobiator. This lets the Jacobi identity hold coherently rather than as a strict equation. Baez and Crans developed…

Quantum mechanics is usually written in linear-algebraic terms: states are vectors of a complex Hilbert space, observables are Hermitian operators, and expectation values pair the two. Classical mechanics is written geometrically: states form a symplectic manifold, observables are functions on it, and dynamics is a Hamiltonian flow. A line of work starting…

The extension problem asks for all ways of building a larger object from a given quotient and a given kernel. For groups, Schreier's theory (1926) describes every extension $1\to K\to E\to G\to 1$ by a factor system: a map $G\to\mathrm{Aut}(K)$ that is a homomorphism only up to inner automorphisms, together with a correction term $\sigma : G\times G\to K$. When $K$…

We present in this paper a set of routines constructed to compute the rank of a matrix Lie algebra and also to determine a Cartan subalgebra from a given list of elements

In this paper, we introduce the notion of a normal form of one qubit quantum circuits over the basis $\{H, P, T\}$, where $H$, $P$ and $T$ denote the Hadamard, Phase and $π/8$ gates, respectively. This basis is known as the {\it standard set} and its universality has been shown by Boykin et al. [FOCS '99]. Our normal form has several nice properties: (i) Every…

Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined presentation of Matsumoto and Amano's results, simplifying some proofs along the way. We also point out some corollaries…

A direct proof of the relation between the one-shot classical capacity and the minimal output entropy for covariant quantum channels is suggested. The structure of covariant channels is described in some detail. A simple proof of a general inequality for entanglement-assisted classical capacity is given.

This paper investigates properties of noisy quantum information channels. We define a new quantity called {\em coherent information} which measures the amount of quantum information conveyed in the noisy channel. This quantity can never be increased by quantum information processing, and it yields a simple necessary and sufficient condition for the…

Given a completely positive (CP) map $T$, there is a theorem of the Radon-Nikodym type [W.B. Arveson, Acta Math. {\bf 123}, 141 (1969); V.P. Belavkin and P. Staszewski, Rep. Math. Phys. {\bf 24}, 49 (1986)] that completely characterizes all CP maps $S$ such that $T-S$ is also a CP map. This theorem is reviewed, and several alternative formulations are given along…

We study some structural aspects of the subspaces of the non-commutative (Haagerup) L_p-spaces associated with a general (non necessarily semi-finite) von Neumann algebra A. If a subspace X of L_p(A) contains uniformly the spaces \ell_p^n, n>= 1, it contains an almost isometric, almost 1-complemented copy of \ell_p. If X contains uniformly the finite…