Local Information as a Resource in Distributed Quantum Systems

Abstract

A new paradigm for distributed quantum systems where information is a valuable resource is developed. After finding a unique measure for information, we construct a scheme for it's manipulation in analogy with entanglement theory. In this scheme instead of maximally entangled states, two parties distill local states. We show that, surprisingly, the main tools of entanglement theory are general enough to work in this opposite scheme. Up to plausible assumptions, we show that the amount of information that must be lost during the protocol of concentration of local information can be expressed as the relative entropy distance from some special set of states.The notion of quantum correlations is a more general than entanglement [1,2]. A formal measure of quantum correlations in measurements (quantum discord) [2] was found, based primarily on an entropy-like function. Recently, the first operational approach to quantify quantum correlations was introduced in [3]. Subsequently, a similar approach was used to justify a physical interpretation of (the optimized) quantum discord [4]. The results of [3] was based on the idea that using a system in a pure state one can draw work from a single heat bath. This scenario was used in the case of distributed quantum systems: Alice and Bob share a state, have local heat baths, and can use only local operations and classical communication (LOCC), to concentrate the information contained in the state, in order to draw work. The amount of work drawn by LOCC is usually smaller than the one extractable if Alice and Bob can use global operations. The resulting difference denoted by the deficit ∆ accounts for the part of correlations that must be lost during classical communication, thus describing purely quantum correlations. In the case of ∆ for pure states, it was argued to be exactly equal to the entanglement while for mixed states it is supposed to be an independent quantity. In this context, it is clear that understanding the problem of concentration of information will provide valuable insight into the nature of quantum correlations. Yet the early development of these ideas [3,5] indicated that the proposed scenario is completely different that anything we had in quantum information theory so far. In particular the serious difficulty (which shall be removed in this paper) was that one is not even able to obtain (without additional assumptions) the value of ∆ in the simplest case of a two-qubit Bell state.In this context basic questions arise: (i) What is the connection between the above thermodynamical quantification of quantum correlations and the main concepts of quantum information theory? (ii) Can we formulate the concentration of information within a framework of manipulating resources like in entanglement theory? An even more basic question is: (iii) can we give up thermodynamics, and formulate the problem solely in terms of quantum information? To see the importance of the latter, let us note that the discovery of teleportation and Shor's algorithm was possible by restrict...

Quantum channel capacities: Multiparty communication

“…The most sophisticated case is the two-way communication scenario C =↔ (sometimes called ping-pong protocol) [25].…”

Bounds on localizable information via semidefinite programming

“…Or, equivalently, How large is quantum deficit for a given state? For pure state the answer is known [2]: the amount of information that cannot be localized is precisely entanglement of formation of the state [6], given by entropy of subsystem. However, for mixed states even separable states can have nonzero deficit.…”

“…In recent development [1,2] an idea of localizing information (or concentrating) in paradigm of distant laboratories was devised. It originated from the concept of drawing thermodynamical work form heat bath and a source of negentropy (see e.g.…”

Probabilistic implementation of Hadamard and unitary gates

“…We also derive the best transformation efficiencies. [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] which operate either in a deterministic or probabilistic way. Corresponding to the quantum no-cloning theorem, Pati and Braunstein [23] demonstrated that the linearity of quantum mechanics also forbids one to delete one unknown state ideally against a copy [23], which is called the quantum no-deleting principle.…”