The Quantum Formalization Drive

In quantum information theory a state of a system with Hilbert space $\mathcal H$ is a density operator, and an ensemble of states is a probability measure on the set of states: a discrete ensemble $\{\pi_i,\rho_i\}$ assigns weight $\pi_i$ to the state $\rho_i$, and a continuous ensemble is an arbitrary Borel probability measure. The average state of an ensemble is…

A decoherence-free subspace (DFS) is a subspace of a quantum register on which a given noise process acts trivially, so that information stored there is protected without active error correction. Weak collective decoherence is the simplest noise with such subspaces: every qubit couples to the environment through the same operator, which after a fixed…

Noise on a quantum register is described by the operators through which the register couples to its environment. These operators, together with the identity, generate an algebra $A$ of operators on the register, which D. M. Bacon's thesis calls the OSR algebra (operator-sum representation; for Markovian noise the analogous SME algebra). Information stored…

Continuous-variable quantum key distribution (CV-QKD) lets two parties, Alice and Bob, establish a secret key by sending Gaussian-modulated coherent or squeezed states of light and detecting them with homodyne or heterodyne receivers (Grosshans–Grangier 2002; Grosshans et al. 2003). The protocols use standard telecom components, so the security question has…

Pure-state entanglement is a resource, and the basic question about a resource is which conversions between its forms are possible. For finite-dimensional bipartite pure states the answer under local operations and classical communication (LOCC) is Nielsen's majorization criterion (Nielsen 1999): $|\psi\rangle$ can be converted into $|\phi\rangle$ iff the…

A quantum channel can be used many times, and the inputs to different uses may be entangled. Whether entangled inputs ever help to send classical information was, around 2000, one of the central open questions of quantum Shannon theory. The single-use answer is the Holevo capacity $\chi_N$: by the Holevo–Schumacher–Westmoreland theorem the classical capacity…

Quantum communication protocols such as quantum key distribution and teleportation consume maximally entangled states shared between distant parties. Losses and decoherence in transmission mean that the parties can in practice only share partially entangled states, and entanglement purification (or concentration) converts many partially entangled copies…

A quantum memory stores a qubit in a noisy medium for a long time. The toric code of Kitaev stores it in the ground space of many physical qubits on an $L\times L$ torus, so that only errors along a path winding around the torus can change the stored information. Errors are detected by measuring local check operators, the syndrome, and corrected by a…

Entanglement shared between distant parties is the resource behind quantum key distribution and quantum teleportation. Sent directly through optical fibre, a photon survives a distance $L$ with probability $e^{-L/l_{\mathrm{att}}}$, so the expected number of photons needed to deliver one entangled pair grows exponentially in $L$. A quantum repeater splits the…

Schumacher's quantum data compression (Schumacher, 1995) encodes $n$ copies of a quantum source into about $nH$ qubits, where $H$ is the von Neumann entropy of the source. The encoder is a fixed unitary transformation on $n$ qubits. As Bennett described (Physics Today, 1995), one realization of it is a sorting permutation: order the $2^n$ basis states by their number…

The completely bounded trace and spectral norms in finite dimensions are shown to be expressible by semidefinite programs. This provides an efficient method by which these norms may be both calculated and verified, and gives alternate proofs of some known facts about them.

The use of entangled light to illuminate objects is shown to provide significant enhancements over unentangled light for detecting and imaging those objects in the presence of high levels of noise and loss. Each signal sent out is entangled with an ancilla, which is retained. Detection takes place via an entangling measurement on the returning signal…

Given an arbitrary single-qubit operation, an important task is to efficiently decompose this operation into an (exact or approximate) sequence of fault-tolerant quantum operations. We derive a depth-optimal canonical form for single-qubit quantum circuits, and the corresponding rules for exactly reducing an arbitrary single-qubit circuit to this canonical…

Shor and Preskill have provided a simple proof of security of the standard quantum key distribution scheme by Bennett and Brassard (BB84) by demonstrating a connection between key distribution and entanglement purification protocols with one-way communications. Here we provide proofs of security of standard quantum key distribution schemes, BB84 and the…

The quantum Fourier transform (QFT) is the principal algorithmic tool underlying most efficient quantum algorithms. We present a generic framework for the construction of efficient quantum circuits for the QFT by ``quantizing'' the separation of variables technique that has been so successful in the study of classical Fourier transform computations.…

Fawzi and Renner [Commun. Math. Phys. 340(2):575, 2015] recently established a lower bound on the conditional quantum mutual information (CQMI) of tripartite quantum states $ABC$ in terms of the fidelity of recovery (FoR), i.e. the maximal fidelity of the state $ABC$ with a state reconstructed from its marginal $BC$ by acting only on the $C$ system. The FoR…

Transmitting data reliably over noisy communication channels is one of the most important applications of information theory, and well understood when the channel is accurately modelled by classical physics. However, when quantum effects are involved, we do not know how to compute channel capacities. The capacity to transmit quantum information is essential…

We provide a necessary and sufficient condition for separability of Gaussian states of bipartite systems of arbitrarily many modes. The condition provides an operational criterion since it can be checked by simple computation. Moreover, it allows us to find a pure product-state decomposition of any given separable Gaussian state. Our criterion is…

We study the amount of communication needed for two parties to transform some given joint pure state into another one, either exactly or with some fidelity. Specifically, we present a method to lower bound this communication cost even when the amount of entanglement does not increase. Moreover, the bound applies even if the initial state is supplemented…

We show that it is possible for the so-called weak locking capacity of a quantum channel [Guha et al., PRX 4:011016, 2014] to be much larger than its private capacity. Both reflect different ways of capturing the notion of reliable communication via a quantum system while leaking almost no information to an eavesdropper; the difference is that the latter…

In [Berta et al., J. Math. Phys. 56, 022205 (2015)], we recently proposed Renyi generalizations of the conditional quantum mutual information of a tripartite state on $ABC$ (with $C$ being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to…