The Quantum Formalization Drive

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…

Integrable spin chains with a Lie superalgebra symmetry $\mathfrak{gl}(M|N)$ are lattice regularizations of two-dimensional field theories with target-space supersymmetry, such as the $\mathfrak{gl}(M|N)$ Gross–Neveu models and sigma models on superspaces, which appear in the study of disordered systems, polymers and the AdS/CFT correspondence. Their spectra…

The time evolution of an open quantum system with no memory effects is described by a quantum dynamical semigroup (QDS): a family $\mathcal T^t$, $t\ge0$, of completely positive, trace preserving maps on the density operators of a Hilbert space $\mathcal H$, with $\mathcal T^{s+t}=\mathcal T^s\mathcal T^t$. Radiative decay, diffusion and dissipative state…

NOON states, $\tfrac{1}{\sqrt2}(|N,0\rangle + |0,N\rangle)$, put $N$ photons into one of two optical modes in coherent superposition. They are the standard example of a non-Gaussian, maximally path-entangled state of light and the resource behind phase estimation at the Heisenberg limit (Dowling, Contemp. Phys. 49, 2008). They are also fragile: they are hard to…

We show a continuity theorem for Stinespring's dilation: two completely positive maps between arbitrary C*-algebras are close in cb-norm iff we can find corresponding dilations that are close in operator norm. The proof establishes the equivalence of the cb-norm distance and the Bures distance for completely positive maps. We briefly discuss applications to…

Squashed entanglement [Christandl and Winter, J. Math. Phys. 45(3):829-840 (2004)] is a monogamous entanglement measure, which implies that highly extendible states have small value of the squashed entanglement. Here, invoking a recent inequality for the quantum conditional mutual information [Fawzi and Renner, Commun. Math. Phys. 340(2):575-611 (2015)]…

We classify extended Poincaré Lie super algebras and Lie algebras of any signature (p,q), that is Lie super algebras and Z_2-graded Lie algebras g = g_0 + g_1, where g_0 = so(V) + V is the (generalized) Poincaré Lie algebra of the pseudo Euclidean vector space V = R^{p,q} of signature (p,q) and g_1 = S is the spinor so(V)-module extended to a g_0-module…

The paper contains a detailed computation about the algebra of canonical commutation relation, the representation of the Weyl unitaries, the quasi-free states and their von Neumann entropy. The Markov triplet is defined by constant entropy increase. The Markov property of a quasi-free state is described by the representing block matrix. The proof is based…

We present a functional calculus approach to the study of rates of decay in mean ergodic theorems for bounded strongly continuous operator semigroups. A central role is played by operators of the form $g(A)$, where $-A$ is the generator of the semigroup and $g$ is a Bernstein function. In addition, we obtain some new results on Bernstein functions that are of…

Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results…

A host algebra of a (possibly infinite dimensional) Lie group $G$ is a $C^*$-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations $π\colon G \to \U(\cH)$. In this paper we present a new approach to host algebras for infinite dimensional Lie groups which is based on smoothing operators, i.e., operators…

In this letter we define a family of entanglement distribution protocols assisted by feedback classical communication that gives an operational interpretation to reverse coherent information, i.e., the symmetric counterpart of the well known coherent information. This lead to the definition of a new entanglement distribution capacity that exceeds the…

Since 1984, various optical quantum key distribution (QKD) protocols have been proposed and examined. In all of them, the rate of secret key generation decays exponentially with distance. A natural and fundamental question is then whether there are yet-to-be discovered optical QKD protocols (without quantum repeaters) that could circumvent this…

We derive a new entropic quantum uncertainty relation involving min-entropy. The relation is tight and can be applied in various quantum-cryptographic settings. Protocols for quantum 1-out-of-2 Oblivious Transfer and quantum Bit Commitment are presented and the uncertainty relation is used to prove the security of these protocols in the bounded…

We define "coherent communication" in terms of a simple primitive, show it is equivalent to the ability to send a classical message with a unitary or isometric operation, and use it to relate other resources in quantum information theory. Using coherent communication, we are able to generalize super-dense coding to prepare arbitrary quantum states instead…

It is not a problem to complement a classical bit, i.e. to change the value of a bit, a 0 to a 1 and vice versa. This is accomplished by a NOT gate. Complementing a qubit in an unknown state, however, is another matter. We show that this operation cannot be done perfectly. We define the Universal-NOT (U-NOT) gate which out of N identically prepared pure…

We construct the unique optimal quantum device for turning a finite number of d-level quantum systems in the same unknown pure state σinto M systems of the same kind, in an approximation of the M-fold tensor product of the state σ.

An action of an affine algebraic group $G$ on an affine variety $X$ is observable when invariant regular functions see every nonzero $G$-stable ideal of the coordinate ring: if $I\subseteq k[X]$ is a nonzero $G$-stable ideal, then $I^G\neq(0)$. The notion goes back to Grosshans' theory of observable subgroups (F. D. Grosshans, Observable groups and Hilbert's fourteenth…

Quantum Max Cut asks for the highest energy of a two-qubit interaction Hamiltonian placed on the edges of a weighted graph. It is a quantum analogue of Max Cut: a classical cut assigns one of two labels to each vertex, while a quantum state can use correlations and entanglement across the qubits. Lee studies an approximation algorithm that first solves a…