Distilling common randomness from bipartite quantum states

The problem of converting noisy quantum correlations between two parties into noiseless classical ones using a limited amount of one-way classical communication is addressed. A single-letter formula for the optimal trade-off between the extracted common randomness and classical communication rate is obtained for the special case of classical-quantum correlations. The resulting curve is intimately related to the quantum compression with classical side information trade-off curve $Q^*(R)$ of Hayden, Jozsa and Winter. For a general initial state we obtain a similar result, with a single-letter formula, when we impose a tensor product restriction on the measurements performed by the sender; without this restriction the trade-off is given by the regularization of this function. Of particular interest is a quantity we call ``distillable common randomness'' of a state: the maximum overhead of the common randomness over the one-way classical communication if the latter is unbounded. It is an operational measure of (total) correlation in a quantum state. For classical-quantum correlations it is given by the Holevo mutual information of its associated ensemble, for pure states it is the entropy of entanglement. In general, it is given by an optimization problem over measurements and regularization; for the case of separable states we show that this can be single-letterized.Comment: 22 pages, LaTe

Relating quantum privacy and quantum coherence: an operational approach

We describe how to achieve optimal entanglement generation and one-way entanglement distillation rates by coherent implementation of a class of secret key generation and secret key distillation protocols, respectively. This short paper is a high-level descrioption of our detailed papers [8] and [10].Comment: 4 pages, revtex

A Resource Framework for Quantum Shannon Theory

Quantum Shannon theory is loosely defined as a collection of coding theorems, such as classical and quantum source compression, noisy channel coding theorems, entanglement distillation, etc., which characterize asymptotic properties of quantum and classical channels and states. In this paper we advocate a unified approach to an important class of problems in quantum Shannon theory, consisting of those that are bipartite, unidirectional and memoryless. We formalize two principles that have long been tacitly understood. First, we describe how the Church of the larger Hilbert space allows us to move flexibly between states, channels, ensembles and their purifications. Second, we introduce finite and asymptotic (quantum) information processing resources as the basic objects of quantum Shannon theory and recast the protocols used in direct coding theorems as inequalities between resources. We develop the rules of a resource calculus which allows us to manipulate and combine resource inequalities. This framework simplifies many coding theorem proofs and provides structural insights into the logical dependencies among coding theorems. We review the above-mentioned basic coding results and show how a subset of them can be unified into a family of related resource inequalities. Finally, we use this family to find optimal trade-off curves for all protocols involving one noisy quantum resource and two noiseless ones.Comment: 60 page

A family of quantum protocols

We introduce two dual, purely quantum protocols: for entanglement distillation assisted by quantum communication (``mother'' protocol) and for entanglement assisted quantum communication (``father'' protocol). We show how a large class of ``children'' protocols (including many previously known ones) can be derived from the two by direct application of teleportation or super-dense coding. Furthermore, the parent may be recovered from most of the children protocols by making them ``coherent''. We also summarize the various resource trade-offs these protocols give rise to.Comment: 5 pages, 1 figur

Quantum information can be negative

Given an unknown quantum state distributed over two systems, we determine how much quantum communication is needed to transfer the full state to one system. This communication measures the "partial information" one system needs conditioned on it's prior information. It turns out to be given by an extremely simple formula, the conditional entropy. In the classical case, partial information must always be positive, but we find that in the quantum world this physical quantity can be negative. If the partial information is positive, its sender needs to communicate this number of quantum bits to the receiver; if it is negative, the sender and receiver instead gain the corresponding potential for future quantum communication. We introduce a primitive "quantum state merging" which optimally transfers partial information. We show how it enables a systematic understanding of quantum network theory, and discuss several important applications including distributed compression, multiple access channels and multipartite assisted entanglement distillation (localizable entanglement). Negative channel capacities also receive a natural interpretation

On the quantum, classical and total amount of correlations in a quantum state

We give an operational definition of the quantum, classical and total amount of correlations in a bipartite quantum state. We argue that these quantities can be defined via the amount of work (noise) that is required to erase (destroy) the correlations: for the total correlation, we have to erase completely, for the quantum correlation one has to erase until a separable state is obtained, and the classical correlation is the maximal correlation left after erasing the quantum correlations. In particular, we show that the total amount of correlations is equal to the quantum mutual information, thus providing it with a direct operational interpretation for the first time. As a byproduct, we obtain a direct, operational and elementary proof of strong subadditivity of quantum entropy.Comment: 12 pages ReVTeX4, 2 eps figures. v2 has some arguments clarified and references update

Tema Con Variazioni: Quantum Channel Capacity

Channel capacity describes the size of the nearly ideal channels, which can be obtained from many uses of a given channel, using an optimal error correcting code. In this paper we collect and compare minor and major variations in the mathematically precise statements of this idea which have been put forward in the literature. We show that all the variations considered lead to equivalent capacity definitions. In particular, it makes no difference whether one requires mean or maximal errors to go to zero, and it makes no difference whether errors are required to vanish for any sequence of block sizes compatible with the rate, or only for one infinite sequence.Comment: 32 pages, uses iopart.cl

Classical information deficit and monotonicity on local operations

We investigate classical information deficit: a candidate for measure of classical correlations emerging from thermodynamical approach initiated in [Phys. Rev. Lett 89, 180402]. It is defined as a difference between amount of information that can be concentrated by use of LOCC and the information contained in subsystems. We show nonintuitive fact, that one way version of this quantity can increase under local operation, hence it does not possess property required for a good measure of classical correlations. Recently it was shown by Igor Devetak, that regularised version of this quantity is monotonic under LO. In this context, our result implies that regularization plays a role of "monotoniser".Comment: 6 pages, revte

Entanglement transmission and generation under channel uncertainty: Universal quantum channel coding

We determine the optimal rates of universal quantum codes for entanglement transmission and generation under channel uncertainty. In the simplest scenario the sender and receiver are provided merely with the information that the channel they use belongs to a given set of channels, so that they are forced to use quantum codes that are reliable for the whole set of channels. This is precisely the quantum analog of the compound channel coding problem. We determine the entanglement transmission and entanglement-generating capacities of compound quantum channels and show that they are equal. Moreover, we investigate two variants of that basic scenario, namely the cases of informed decoder or informed encoder, and derive corresponding capacity results.Comment: 45 pages, no figures. Section 6.2 rewritten due to an error in equation (72) of the old version. Added table of contents, added section 'Conclusions and further remarks'. Accepted for publication in 'Communications in Mathematical Physics