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arXiv:quant-ph/0512247 · Communications in Mathematical Physics

Quantum State Merging and Negative Information

Michał Horodecki, Jonathan Oppenheim, Andreas Winter

Published Oct 20, 2006• 338 citations
arXivHTMLPDF
DOI
10.1007/s00220-006-0118-x

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In this brief report we revisit the concept of dissension, which was introduced as a natural extension of quantum discord for three qubit system. Here we bring in new expression for quantum dissensions and more interestingly we name one such expression as \textit{residual correlation}. The basic objective behind the introduction of such a quantity is to capture the extra amount of correlation generated by doing measurement in a correlated system from a situation where we do not bring in a correlated system in the measurement process. Apart from this we also present an operational interpretation of this correlation in context of state merging. Remarkably, we find that for three qubit system if one discards relevant prior information, the change in the cost of state merging ( merging the quantum information of two parties into one) is captured by the \textit{residual correlation}. In addition to that we found that this quantity can be negative for mixed states. This indeed opens up a new dimension in the tripartite scenario where we can observe situations where there is a decrease in the cost of state merging on discarding relevant prior information. We claim that this result establishes a re conceptualization of information processing tasks in tripartite situations where we can use suitable measurement and states to bring down the cost of the protocol.
Recently, it was discovered that the `quantum partial information' needed to merge one party's state with another party's state is given by the conditional entropy, which can be negative [Horodecki, Oppenheim, and Winter, Nature 436, 673 (2005)]. Here we find a classical analogue of this, based on a long known relationship between entanglement and shared private correlations: namely, we consider a private distribution held between two parties, and correlated to a reference system, and ask how much secret communication is needed for one party to send her distribution to the other. We give optimal protocols for this task, and find that private information can be negative - the sender's distribution can be transferred and the potential to send future distributions in secret is gained through the distillation of a secret key. An analogue of `quantum state exchange' is also discussed and one finds cases where exchanging a distribution costs less than for one party to send it. The results give new classical protocols, and also clarify the various relationships between entanglement and privacy.
Conditional von Neumann entropy is an intriguing concept in quantum information theory. In the present work, we examine the effect of global unitary operations on the conditional entropy of the system. We start with a set containing states with a non-negative conditional entropy and find that some states preserve the non-negativity under unitary operations on the composite system. We call this class of states the absolute conditional von Neumann entropy non-negative (ACVENN) class. We characterize such states for $2\ensuremath{\bigotimes}2$--dimensional systems. From a different perspective the characterization accentuates the detection of states whose conditional entropy becomes negative after the global unitary action. Interestingly, we show that this ACVENN class of states forms a set which is convex and compact. This feature enables the existence of Hermitian witness operators. With these we can distinguish the unknown states which will have a negative conditional entropy after the global unitary operation. We also show that this has immediate application to superdense coding and state merging, as the negativity of the conditional entropy plays a key role in both these information processing tasks. Some illustrations followed by analysis are also provided to probe the connection of such states with absolutely separable states and absolutely local states.