The Quantum Formalization Drive

We consider discrete-modulation protocols for continuous-variable quantum key distribution (CV-QKD) that employ a modulation constellation consisting of a finite number of coherent states and that use a homodyne or a heterodyne-detection receiver. We establish a security proof for collective attacks in the asymptotic regime, and we provide a formula for an…

A quantum channel transmits classical information with an efficiency limited by the noise it adds. One measure of that noise is the minimal output entropy: the smallest entropy the channel's output can have when the input is a pure state. The additivity conjecture asked whether using two channels in parallel, with inputs that may be entangled across them,…

We investigate the maximum rates for transmitting quantum information, distilling entanglement, and distributing secret keys between a sender and a receiver in a multipoint communication scenario, with the assistance of unlimited two-way classical communication involving all parties. First we consider the case where a sender communicates with an arbitrary…

We consider quantum and private communications assisted by repeaters, from the basic scenario of a single repeater chain to the general case of an arbitrarily-complex quantum network, where systems may be routed through single or multiple paths. In this context, we investigate the ultimate rates at which two end-parties may transmit quantum information,…

We identify a special information-theoretic property of quantum field theories with holographic duals: the mutual informations among arbitrary disjoint spatial regions A,B,C obey the inequality I(A:BC) >= I(A:B)+I(A:C), provided entanglement entropies are given by the Ryu-Takayanagi formula. Inequalities of this type are known as monogamy relations and are…

This paper addresses some general questions of quantum information theory arising from the transmission of quantum entanglement through (possibly noisy) quantum channels. A pure entangled state is prepared of a pair of systems $R$ and $Q$, after which $Q$ is subjected to a dynamical evolution given by the superoperator $\superop^{Q}$. Two interesting quantities can…

Mutual information is the reciprocal information that is common to or shared by two or more parties. Quantum mutual information for bipartite quantum systems is non-negative, and bears the interpretation of total correlation between the two subsystems. This may, however, no longer be true for three or more party quantum systems. In this paper, we propose an…

In this paper we study the problem of calculating the convex hull of certain affine algebraic varieties. As we explain, the motivation for considering this problem is that certain pure-state measures of quantum entanglement, which we call polynomial entanglement measures, can be represented as affine algebraic varieties. We consider the evaluation of…

The existing unconditional security definitions of quantum key distribution (QKD) do not apply to joint attacks over QKD and the subsequent use of the resulting key. In this paper, we close this potential security gap by using a universal composability theorem for the quantum setting. We first derive a composable security definition for QKD. We then prove…

We report a deterministic and exact protocol to reverse any unknown qubit-unitary operation, which simulates the time inversion of a closed qubit system. To avoid known no-go results on universal deterministic exact unitary inversion, we consider the most general class of protocols transforming unknown unitary operations within the quantum circuit model,…

Two new relative entropy quantities, called the min- and max-relative entropies, are introduced and their properties are investigated. The well-known min- and max- entropies, introduced by Renner, are obtained from these. We define a new entanglement monotone, which we refer to as the max-relative entropy of entanglement, and which is an upper bound to the…

We describe a modification of the Alicki-Fannes-Winter method which allows to prove uniform continuity on the set of quantum states with bounded energy of any locally almost affine function having limited growth with increasing energy. Some applications in quantum information theory are considered. The asymptotic continuity of the relative entropy of…

Quantum hypothesis testing is one of the most basic tasks in quantum information theory and has fundamental links with quantum communication and estimation theory. In this paper, we establish a formula that characterizes the decay rate of the minimal Type-II error probability in a quantum hypothesis test of two Gaussian states given a fixed constraint on…

The Hahn–Banach theorem extends every bounded linear functional on a linear manifold of a normed space to the whole space, with the same norm. For bounded linear operators into an arbitrary Banach space, and for bounded bilinear maps, nothing of the kind holds in general: there is no Hahn–Banach theorem for bilinear functionals. Extension problems for…

The state of a finite-dimensional quantum system is a density: a positive semidefinite matrix of trace one. Under coherent (Hamiltonian) control the evolution is unitary, $\rho\mapsto U\rho U^*$, and preserves the eigenvalues of $\rho$. Even with full unitary control, a state can therefore only be moved within its unitary orbit. A mixed state cannot be…

The trade-off capacity region of a quantum channel characterizes the optimal net rates at which a sender can communicate classical, quantum, and entangled bits to a receiver by exploiting many independent uses of the channel, along with the help of the same resources. Similarly, one can consider a trade-off capacity region when the noiseless resources are…

We present an efficient family of quantum circuits for a fundamental primitive in quantum information theory, the Schur transform. The Schur transform on n d-dimensional quantum systems is a transform between a standard computational basis to a labelling related to the representation theory of the symmetric and unitary groups. If we desire to implement the…

Machine-learned models of molecular energies and properties are trained on the output of quantum-chemistry calculations, and a common way to make them data-efficient is to build the physical symmetries of the problem into the architecture. When the input is not a point cloud but a set of matrices expressed in an atomic-orbital basis (Fock, density, overlap…

Lie 2-algebras allow a Lie bracket to satisfy its laws up to a specified higher operation. They arise when an ordinary Lie algebra extension has a cocycle in degree three rather than degree two. Baez, Crans, Stevenson, and Schreiber use this language to relate a finite-dimensional model attached to a compact simple Lie group to an infinite-dimensional…

Quantum key distribution (QKD) is the first quantum information task to reach the level of mature technology, already fit for commercialization. It aims at the creation of a secret key between authorized partners connected by a quantum channel and a classical authenticated channel. The security of the key can in principle be guaranteed without putting any…

Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $A^\circ$ in $R^A$.

A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call "weak" and "coherent" 2-groups. A weak 2-group is a…

We propose a general security definition for cryptographic quantum protocols that implement classical non-reactive two-party tasks. The definition is expressed in terms of simple quantum-information-theoretic conditions which must be satisfied by the protocol to be secure. The conditions are uniquely determined by the ideal functionality F defining the…

We provide a model of the String group as a central extension of finite-dimensional 2-groups in the bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This bicategory is a geometric incarnation of the bicategory of smooth stacks and generalizes the more naïve 2-category of Lie groupoids, smooth functors, and smooth natural…

The cyclotomic Birman-Murakami-Wenzl (BMW) algebras B_n^k, introduced by R. Häring-Oldenburg, are a generalisation of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (aka Ariki-Koike algebras) and type B knot theory. In this paper, we prove the algebra is free and of rank k^n (2n-1)!! over ground rings with parameters…