The Quantum Formalization Drive

A quantum error-correcting code protects information by embedding it in a larger space. The quantum Hamming bound limits how small that space can be: the images of the code under the correctable errors must fit into the available space. Niu, Chuang and Shapiro (arXiv:1709.05302) design bosonic codes for hardware whose only nonlinearity is coherent $\chi^{(2)}$…

There exist numerous proofs of Bell's theorem, stating that quantum mechanics is incompatible with local realistic theories of nature. Here we define the strength of such nonlocality proofs in terms of the amount of evidence against local realism provided by the corresponding experiments. This measure tells us how many trials of the experiment we should…

We present an operational framework for Einstein-Podolsky-Rosen steering as a physical resource. To begin with, we characterize the set of steering non-increasing operations (SNIOs) --i.e., those that do not create steering-- on arbitrary-dimensional bipartite systems composed of a quantum subsystem and a black-box device. Next, we introduce the notion of…

A convenient bicategory of topological stacks is constructed which is both complete and Cartesian closed. This bicategory, called the bicategory of compactly generated stacks, is the analogue of classical topological stacks, but for a different Grothendieck topology. In fact, there is an equivalence of bicategories between compactly generated stacks and…

Two parties, Alice and Bob, who share many copies of a pure entangled state $|\psi\rangle_{AB}$ and may act only by local operations and classical communication (LOCC) can transform their resource into many copies of another pure state $|\varphi\rangle_{AB}$. The question of how efficiently this can be done is the question of whether pure-state entanglement can…

A noisy quantum channel has several capacities, one for each resource it is asked to carry and each auxiliary resource allowed. Most of them are hard to compute: the classical capacity and the quantum capacity are given by regularized formulas, limits over many channel uses that are not known to be computable in general. The entanglement-assisted classical…

The classical capacity of a quantum channel is the largest rate, in bits per channel use, at which classical messages can be sent through the channel with vanishing error, when the sender may prepare quantum states entangled across many channel uses and the receiver may measure all outputs collectively. The Holevo–Schumacher–Westmoreland (HSW) theorem…

Device-independent key distribution asks whether two parties can establish a secret key without trusting the devices that prepare and measure their systems. If the outcomes of local measurements violate a Bell inequality, they cannot have been fixed in advance, and this limits what any third party can know about them. The only physical assumption needed is n…

A wiretap channel $W:\mathcal X\to\mathcal Y\times\mathcal Z$ models secure transmission over a noisy medium: a transmitter sends a symbol $x$, a legitimate receiver observes $Y$ and an eavesdropper observes a correlated $Z$, with joint probability $W(y,z|x)$. The transmitter wants to deliver a message $M$ reliably to the receiver while revealing (almost) nothing about…

A quantum system that is not perfectly isolated evolves, from its own point of view, by an operator-sum representation (OSR): its density matrix $\rho$ is mapped to $\sum_i A_i\rho A_i^\dagger$ for a family of OSR operators $A_i$ obtained by tracing out the environment. Generic OSRs describe decoherence, the loss of quantum information to the environment. Unitary…

A mode-locked laser emits a train of pulses whose spectrum is a frequency comb: many optical modes, equally spaced in frequency. The common phase offset of these modes, the carrier-envelope phase, is the quantity that has to be stabilized when femtosecond combs are used to define optical frequency standards (Jones et al., Science 2000; Udem et al., PRL 1999)…

A striking feature of quantum error correcting codes is that they can sometimes be used to correct more errors than they can uniquely identify. Such degenerate codes have long been known, but have remained poorly understood. We provide a heuristic for designing degenerate quantum codes for high noise rates, which is applied to generate codes that can be…

We present a resource-performance tradeoff of an all-optical quantum repeater that uses photon sources, linear optics, photon detectors and classical feedforward at each repeater node, but no quantum memories. We show that the quantum-secure key rate has the form $R(η) = Dη^s$ bits per mode, where $η$ is the end-to-end channel's transmissivity, and the…

We prove the security of the 1984 protocol of Bennett and Brassard (BB84) for quantum key distribution. We first give a key distribution protocol based on entanglement purification, which can be proven secure using methods from Lo and Chau's proof of security for a similar protocol. We then show that the security of this protocol implies the security of…

We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie algebra cohomology. We show that the secondary characteristic classes of Beilinson lead to differentiable cohomology…

We consider the problem of secure identification: user U proves to server S that he knows an agreed (possibly low-entropy) password w, while giving away as little information on w as possible, namely the adversary can exclude at most one possible password for each execution of the scheme. We propose a solution in the bounded-quantum-storage model, where U…

In the formalism of measurement based quantum computation we start with a given fixed entangled state of many qubits and perform computation by applying a sequence of measurements to designated qubits in designated bases. The choice of basis for later measurements may depend on earlier measurement outcomes and the final result of the computation is…

Macroscopic fields like electromagnetic, MHD, acoustic or gravitational waves are usually described by classical wave equations with possible additional damping terms and coherent sources. The aim of this paper is to develop a complete macroscopic formalism including random/thermal sources, dissipation and random scattering of waves by environment. The…

We show how the treatment of cellularity in families of algebras arising from diagram calculi, such as Jones' Temperley--Lieb wreaths, variants on Brauer's centralizer algebras, and the contour algebras of Cox et al (of which many algebras are special cases), may be unified using the theory of tabular algebras. This improves an earlier result of the first…

The sandwiched Rényi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched Rényi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical Rényi divergences for Rényi parameter $α>1$. The known proof…

We review recent results on the simulation of quantum channels, the reduction of adaptive protocols (teleportation stretching), and the derivation of converse bounds for quantum and private communication, as established in PLOB [Pirandola, Laurenza, Ottaviani, Banchi, arXiv:1510.08863]. We start by introducing a general weak converse bound for private…

Quantum memory is important to quantum information processing in many ways: a synchronization device to match various processes within a quantum computer, an identity quantum gate that leaves any state unchanged, and a tool to convert heralded photons to photons-on-demand. In addition to quantum computing, quantum memory would be instrumental for the…

This work is intended as an introduction to cryptographic security and a motivation for the widely used Quantum Key Distribution (QKD) security definition. We review the notion of security necessary for a protocol to be usable in a larger cryptographic context, i.e., for it to remain secure when composed with other secure protocols. We then derive the…

We present a MATLAB/Octave toolbox to decompose finite dimensionial representations of compact groups. Surprisingly, little information about the group and the representation is needed to perform that task. We discuss applications to semidefinite programming.

The quantum internet holds promise for performing quantum communication, such as quantum teleportation and quantum key distribution (QKD), freely between any parties all over the globe. Such a future quantum network, depending on the communication distance of the requesting parties, necessitates to invoke several classes of optical quantum communication…