The Quantum Formalization Drive

We consider the problem of distinguishing, with minimum probability of error, two optical beam-splitter channels with unequal complex-valued reflectivities using general quantum probe states entangled over M signal and M' idler mode pairs of which the signal modes are bounced off the beam splitter while the idler modes are retained losslessly. We obtain a…

This paper presents self-contained proofs of the strong subadditivity inequality for quantum entropy and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein's inequality and one of Lieb's less well-known concave trace…

A short quantum Markov chain is a tripartite state $ρ_{ABC}$ such that system $A$ can be recovered perfectly by acting on system $C$ of the reduced state $ρ_{BC}$. Such states have conditional mutual information $I(A;B|C)$ equal to zero and are the only states with this property. A quantum channel $\mathcal{N}$ is sufficient for two states $ρ$ and $σ$ if there exists a…

What is the ultimate performance for discriminating two arbitrary quantum channels acting on a finite-dimensional Hilbert space? Here we address this basic question by deriving a general and fundamental lower bound. More precisely, we investigate the symmetric discrimination of two arbitrary qudit channels by means of the most general protocols based on…

Two parts of an entangled quantum state can have a correlation in their joint behavior under measurements that is unexplainable by shared classical information. Such correlations are called non-local and have proven to be an interesting resource for information processing. Since non-local correlations are more useful if they are stronger, it is natural to…

Numerous quantum many-body systems are characterized by either fundamental or emergent constraints---such as gauge symmetries or parity superselection for fermions---which effectively limit the accessible observables and realizable operations. Moreover, these constraints combine non-trivially with the potential requirement that operations be performed…

We prove that the quantum relative entropy decreases monotonically under the application of any positive trace-preserving linear map, for underlying separable Hilbert spaces. This answers in the affirmative a natural question that has been open for a long time, as monotonicity had previously only been shown to hold under additional assumptions, such as…

We show that all quantum states that do not have a positive partial transpose are distillable via channels, which preserve the positivity of the partial transpose. The question whether NPT bound entanglement exist is therefore closely related to the connection between the set of separable superoperators and PPT-preserving maps.

We study and solve the problem of classical channel simulation with quantum side information at the receiver. This is a generalization of both the classical reverse Shannon theorem, and the classical-quantum Slepian-Wolf problem. The optimal noiseless communication rate is found to be reduced from the mutual information between the channel input and output…

Entanglement entropy for a spatial partition of a quantum system is studied in theories which admit a dual description in terms of the anti-de Sitter (AdS) gravity one dimension higher. A general proof of the holographic formula which relates the entropy to the area of a codimension 2 minimal hypersurface embedded in the bulk AdS space is given. The…

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…