The Quantum Formalization Drive

The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. We give new ways to compute this index. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this…

Lecture notes of a minicourse given at the Summer School on Large Coulomb Systems - QED in Nordfjordeid, 2003, devoted to representations of the CCR and CAR. Quasifree states, the Araki-Woods and Araki-Wyss representations, and the lattice of von Neumenn algebras in a bosonic/fermionic Fock space are discussed in detail.

The concept of reflection positivity has its origins in the work of Osterwalder--Schrader on constructive quantum field theory and duality between unitary representations of the euclidean motion group and the Poincare group. On the mathematical side this duality can be made precise as follows. If $\g$ is a Lie algebra with an involutive automorphism $τ$.…

Generalising in the sense of Hahn's spin echo, we completely characterise those unitary propagators of effective multi-qubit interactions that can be inverted solely by {\em local} unitary operations on $n$ qubits (spins-$\tfrac{1}{2}$). The subset of $U\in \mathbf{SU}(2^n)$ satisfying $U^{-1}=K_1 U K_2$ with pairs of local unitaries …

The scheme of quantum teleportation, where Bob has multiple (N) output ports and obtains the teleported state by simply selecting one of the N ports, is thoroughly studied. We consider both deterministic version and probabilistic version of the teleportation scheme aiming to teleport an unknown state of a qubit. Moreover, we consider two cases for each…

Nazarov \cite{Nazarov:brauer} introduced an infinite dimensional algebra, which he called the \textit{affine Wenzl algebra}, in his study of the Brauer algebras. In this paper we study certain ``cyclotomic quotients'' of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these…

We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of…

This paper addresses the problem of designing universal quantum circuits to transform $k$ uses of a $d$-dimensional unitary input-operation into a unitary output-operation in a probabilistic heralded manner. Three classes of protocols are considered, parallel circuits, where the input-operations can be simultaneously, adaptive circuits, where sequential uses of…

The cyclotomic Birman-Wenzl-Murakami algebras are quotients of the affine BMW algebras in which the affine generator satisfies a polynomial relation. We show that the cyclotomic BMW algebras are free modules over any (admissible, integral) ground ring, and that they are isomorphic to cyclotomic versions of the Kauffman tangle algebras

We consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states. This is motivated in part by results of Hayden et al., which show that in large d x d--dimensional systems there exist random subspaces of dimension almost d^2, all of whose states have entropy of entanglement…

We consider a quantum state shared between many distant locations, and define a quantum information processing primitive, state merging, that optimally merges the state into one location. As announced in [Horodecki, Oppenheim, Winter, Nature 436, 673 (2005)], the optimal entanglement cost of this task is the conditional entropy if classical communication is…

In this work, various versions of the so-called Omega-Lemma are provided, which ensure differentiability properties of pushforwrds between spaces of C^r-sections (or compactly supported C^r-sections) in vector bundles over finite-dimensional base manifolds whose fibres are (possibly infinite-dimensional) locally convex spaces. Applications are given,…

The Differential-Phase-Shift (DPS) and the Coherent-One-Way (COW) are among the most practical protocols for quantum cryptography, and are therefore the object of fast-paced experimental developments. The assessment of their security is also a challenge for theorists: the existing tools, that allow to prove security against the most general attacks, do not…

We review the quantum version of the linear Boltzmann equation, which describes in a non-perturbative fashion, by means of scattering theory, how the quantum motion of a single test particle is affected by collisions with an ideal background gas. A heuristic derivation of this Lindblad master equation is presented, based on the requirement of…

We present a generalization of quantum Stein's Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d.. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in…

We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of $(\mathbb{C}^d)^{\otimes n}$ a…

We investigate a general class of quantum key distribution (QKD) protocols using one-way classical communication. We show that full security can be proven by considering only collective attacks. We derive computable lower and upper bounds on the secret key rate of those QKD protocol involving only entropies of two--qubit density operators. As an…

We obtain an explicit characterization of linear maps, in particular, quantum channels, which are covariant with respect to an irreducible representation ($U$) of a finite group ($G$), whenever $U \otimes U^c$ is simply reducible (with $U^c$ being the contragradient representation). Using the theory of group representations, we obtain the spectral decomposition of…

Variational quantum machine learning is an extensively studied application of near-term quantum computers. The success of variational quantum learning models crucially depends on finding a suitable parametrization of the model that encodes an inductive bias relevant to the learning task. However, precious little is known about guiding principles for the…

Shannon's entropy power inequality (EPI) can be viewed as a statement of concavity of an entropic function of a continuous random variable under a scaled addition rule:

We show that the quantum $α$-relative entropies with parameter $α\in (0,1)$ can be represented as generalized cutoff rates in the sense of [I. Csiszar, IEEE Trans. Inf. Theory 41, 26-34, (1995)], which provides a direct operational interpretation to the quantum $α$-relative entropies. We also show that various generalizations of the Holevo capacity, defined in…