The Quantum Formalization Drive

We present a general necessary and sufficient criterion for the possibility of a state transformation from one mixed Gaussian state to another of a bi-partite continuous-variable system with two modes. The class of operations that will be considered is the set of local Gaussian completely positive trace-preserving maps.

In this paper we give several equivalent formulations of the additivity conjecture for constrained channels, which formally is substantially stronger than the unconstrained additivity. To this end a characteristic property of the optimal ensemble for such a channel is derived, generalizing the maximal distance property. It is shown that the additivity…

The hypothesis testing problem of two quantum states is treated. We show a new inequality between the error of the first kind and the second kind, which complements the result of Hiai and Petz to establish the quantum version of Stein's lemma. The inequality is also used to show a bound on the first kind error when the power exponent for the second kind…

We consider quantum channels with one sender and two receivers, used in several different ways for the simultaneous transmission of independent messages. We begin by extending the technique of superposition coding to quantum channels with a classical input to give a general achievable region. We also give outer bounds to the capacity regions for various…

Quantum channels can be described via a unitary coupling of system and environment, followed by a trace over the environment state space. Taking the trace instead over the system state space produces a different mapping which we call the conjugate channel. We explore the properties of conjugate channels and describe several different methods of…

It has been observed by numerous authors that a quantum system being entangled with another one limits its possible entanglement with a third system: this has been dubbed the "monogamous nature of entanglement". In this paper we present a simple identity which captures the trade-off between entanglement and classical correlation, which can be used to derive…

We show the equivalence of two different notions of quantum channel capacity: that which uses the entanglement fidelity as its criterion of success in transmission, and that which uses the minimum fidelity of pure states in a subspace of the input Hilbert space as its criterion. As a corollary, any source with entropy rate less than the capacity may be…

A longstanding open problem in quantum information theory is to find the classical capacity of an optical communication link, modeled as a Gaussian bosonic channel. It has been conjectured that this capacity is achieved by a random coding of coherent states using an isotropic Gaussian distribution in phase space. We show that proving a Gaussian minimum…

Characterizing genuine multipartite quantum correlations in quantum physical systems has historically been a challenging problem in quantum information theory. More recently however, the total correlation or multipartite information measure has been helpful in accomplishing this goal, especially with the multipartite symmetric quantum (MSQ) discord [Piani…

Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. While it can amplify the correctness of a quantum device with classical output, the analogous procedure for quantum output is not known. We introduce quantum majority vote as the following task: given a product state …

In quantum mechanics an observable is a self-adjoint operator, not just a symmetric one. A differential expression such as $-d^2/dx^2$ on an interval or a half-line first defines a symmetric operator on smooth functions with compact support, and this operator usually has many self-adjoint extensions or none. Each extension is a different physical theory, with…

Unitary representations turn symmetries into operators on Hilbert spaces. For ordinary locally compact groups, the associated group $C^*$-algebra lets one study these operators through a single algebraic object. A Lie supergroup adds odd infinitesimal symmetries to an ordinary Lie group. Its unitary representations must respect both the group action and those…

The relative entropy of entanglement measures how distinguishable a multipartite quantum state is from all fully separable states. In finite dimensions it is a standard entanglement measure, but infinite-dimensional systems raise a basic analytical problem: the measure may take the value $+\infty$, and finite-rank approximations do not automatically control…

A quantum repeater is judged by how much it beats direct transmission over a point-to-point link. The natural yardsticks are the two-way assisted capacities of the link's channel $\mathcal E$: the quantum capacity $Q_2(\mathcal E)$ and the secret-key capacity $K(\mathcal E)$, the best rates of entanglement and key that two parties can obtain over many uses of …

Measurements of long quantum spin chains can have outcomes far from their typical values. A large-deviation principle quantifies how quickly the probability of such outcomes decreases with the number of sites. In a classical lattice system this is a standard way to describe rare macroscopic fluctuations; in a quantum chain, observables on overlapping sets…

Optical quantum communication commonly encodes information in continuous variables such as the position and momentum quadratures of a light mode. Gaussian states and Gaussian operations are attractive in this setting because phase shifters, squeezers, beam splitters, and homodyne measurements are standard optical tools. Transmission through a lossy or noisy…

Quantum circuits whose gates respect a global symmetry are a standard model for quantum simulation, for symmetry-protected quantum computing, and for the study of conserved quantities in quantum dynamics. For SU(d) symmetry on $n$ qudits, the unitaries that commute with every $U^{\otimes n}$ are, by Schur–Weyl duality, exactly the unitaries in the linear span…

The walled Brauer algebra $B_{r,s}(\delta)$ is the centralizer algebra of mixed Schur–Weyl duality: for $\delta=n$ it acts on the mixed tensor space $V^{\otimes r}\otimes (V^*)^{\otimes s}$ of $V=\mathbb C^n$ and commutes with $\mathfrak{gl}_n$, and for $\delta=m-n$ it plays the same role for the general linear supergroup $GL(m|n)$. Its representation theory has been…

Statistical decision theory compares two experiments by asking which one is more useful: one experiment is deficient with respect to another if, in every decision problem, it achieves nearly the same payoff. The randomization criterion turns this operational comparison into a structural one: an experiment is $\epsilon$-deficient exactly when a Markov kernel…

Noncommutative $L_p$-spaces built from a von Neumann algebra $M$ come in two kinds. When $M$ carries a normal faithful trace, $L_p(M)$ is a space of (unbounded) operators affiliated with $M$, and much of classical analysis — martingale inequalities, square functions, maximal inequalities, interpolation — has been carried over to this tracial setting. Algebras of type…

Quantum information protocols often require a finite collection of unitary gates whose averages reproduce those of the full unitary group. A unitary design gives a precise finite-moment version of that requirement. When gates are sampled from an exact design, a practitioner needs to know how many independent draws make the empirical collection a reliable…

Positive maps give tests for entanglement in bipartite quantum states. A density matrix can pass one positive-map test and fail another, so the choice of map changes which states can be detected. Maps that behave predictably under unitary changes of basis are particularly structured, but that symmetry can appear restrictive. Bardet, Collins, and Sapra ask…