The Quantum Formalization Drive

Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are capacity-achieving for the task of sending classical data over a channel with classical inputs and quantum outputs. Although they demonstrated the existence of such codes, their proof does not provide an explicit construction of codes for this task. The aim of…

Let ${\cal H}_1,$ ${\cal H}_2$ be finite dimensional complex Hilbert spaces describing the states of two finite level quantum systems. Suppose $ρ_i$ is a state in ${\cal H}_i, i=1,2.$ Let ${\cal C} (ρ_1, ρ_2)$ be the convex set of all states $ρ$ in ${\cal H} = {\cal H}_1 \otimes {\cal H}_2$ whose marginal states in ${\cal H}_1$ and ${\cal H}_2$ are $ρ_1$ and $ρ_2$ respectively.…

We consider the problem of distributed compression for correlated quantum sources. The classical version of this problem was solved by Slepian and Wolf, who showed that distributed compression could take full advantage of redundancy in the local sources created by the presence of correlations. We show that, in general, this is not the case for quantum…

We explore the link between two concepts: the level of violation of a Bell inequality by a quantum state and discrimination between two states by means of restricted classes of operations, such as local operations and classical communication (LOCC) and separable ones. For any bipartite Bell inequality, we show that its value on a given quantum state cannot…

We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of identity. We show that if G and A are locally compact and second countable, then the second cohomology group based on…

Given a certain amount of entanglement available as a resource, what is the most efficient way to accomplish a quantum task? We address this question in the relevant case of continuous variable quantum teleportation protocols implemented using two-mode Gaussian states with a limited degree of entanglement and energy. We first characterize the class of…

We study asymptotic state transformations in continuous variable quantum resource theories. In particular, we prove that monotones displaying lower semicontinuity and strong superadditivity can be used to bound asymptotic transformation rates in these settings. This removes the need for asymptotic continuity, which cannot be defined in the traditional sense…

A new proof of the direct part of the quantum channel coding theorem is shown based on a standpoint of quantum hypothesis testing. A packing procedure of mutually noncommutative operators is carried out to derive an upper bound on the error probability, which is similar to Feinstein's lemma in classical channel coding. The upper bound is used to show the…

We prove the converse part of the theorem for quantum Hoeffding bound on the asymptotics of quantum hypothesis testing, essentially based on an argument developed by Nussbaum and Szkola in proving the converse part of the quantum Chernoff bound. Our result complements Hayashi's proof of the direct (achievability) part of the theorem, so that the quantum…

In this letter, first, we investigate the security of a continuous-variable quantum cryptographic scheme with a postselection process against individual beam splitting attack. It is shown that the scheme can be secure in the presence of the transmission loss owing to the postselection. Second, we provide a loss limit for continuous-variable quantum…

Despite enormous progress both in theoretical and experimental quantum cryptography, the security of most current implementations of quantum key distribution is still not established rigorously. One of the main problems is that the security of the final key is highly dependent on the number, M, of signals exchanged between the legitimate parties. While, in…

Shannon's noisy channel coding theorem identifies the largest rate at which classical messages can be sent reliably over a classical channel. When the channel is a quantum system, for example photons sent through an optical fibre, the sender may prepare arbitrary quantum states, entangled across many uses of the channel, and the receiver may perform…

Many protocols in quantum Shannon theory must make a message invisible to an eavesdropper rather than readable by a receiver. A sender who wants Eve's system to carry no information about a secret bit randomizes over a small set of codewords so that, whichever bit was sent, Eve receives (approximately) the same average state. The covering lemma quantifies…

The Heisenberg–Robertson uncertainty relation bounds a product of standard deviations by a quantity that depends on the state, and it can vanish even when both measurement outcomes are uncertain. Entropic uncertainty relations replace standard deviations by Shannon entropies of the outcome distributions and give a state-independent bound. Deutsch proved the…

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.