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arXiv:quant-ph/9912033 · Physical Review A

Mixedness and teleportation

Sougato Bose, Vlatko Vedral

Published Mar 6, 2000• 122 citations
arXivHTMLPDF

Abstract

We show that on exceeding a certain degree of mixedness (as quantified by the von Neumann entropy), entangled states become useless for teleportation. By increasing the dimension of the entangled systems, this entropy threshold can be made arbitrarily close to maximal. This entropy is found to exceed the entropy threshold sufficient to ensure the failure of dense coding.

DOI
10.1103/physreva.61.040101

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We show that on exceeding a certain degree of mixedness (as quantified by the von Neumann entropy), entangled states become useless for teleportation. By increasing the dimension of the entangled systems, this entropy threshold can be made arbitrarily close to maximal. This entropy is found to exceed the entropy threshold sufficient to ensure the failure of dense coding.
Entanglement and mixedness of a bipartite mixed state resource are crucial for the success of quantum teleportation. Upper bounds on measures of mixedness, namely, von Neumann entropy and linear entropy beyond which the bipartite state ceases to be useful for quantum teleportation are known in the literature. In this work, we generalize these bounds and obtain rank dependent upper bounds on von Nuemann entropy and linear entropy for an arbitrary bipartite mixed state resource. We observe that the upper bounds on measures of mixedness increase with increase in the rank. For two qubit mixed states, we obtain rank dependent lower bound on the concurrence, a measure of entanglement, below which the state is useless for quantum teleportation. Werner state, which is a fourth rank state, exhibits the theoretical upper bound on mixedness among two qubit mixed states. We construct Werner like states of lower ranks and show that these states possess the theoretical rank dependent upper bounds obtained on the measures of mixedness and theoretical rank dependent lower bounds on the concurrence.
The ability to teleport entanglement through maximally entangled mixed states as defined by concurrence and linear entropy is studied. We show how the teleported entanglement depends on the quality of the quantum channel used, as defined through its entanglement and mixedness, as well as the form of the target state to be teleported. We present new results based on the fidelity of the teleported state as well as an experimental setup that is immediately implementable with currently available technology.
Maximally entangled mixed states are those states that, for a given mixedness, achieve the greatest possible entanglement. For two-qubit systems and for various combinations of entanglement and mixedness measures, the form of the corresponding maximally entangled mixed states is determined primarily analytically. As measures of entanglement, we consider entanglement of formation, relative entropy of entanglement, and negativity; as measures of mixedness, we consider linear and von Neumann entropies. We show that the forms of the maximally entangled mixed states can vary with the combination of (entanglement and mixedness) measures chosen. Moreover, for certain combinations, the forms of the maximally entangled mixed states can change discontinuously at a specific value of the entropy. Along the way, we determine the states that, for a given value of entropy, achieve maximal violation of Bell's inequality.
The single qubit quantum teleportation (sender and receiver are Alice and Bob respectively) is analyzed from the aspect of the quantum information theories. The various quantum entropies are computed at each stage, which ensures the emergence of the entangled states in the intermediate step. The mutual information S(B:C)S(B:C)S(B:C) becomes non-zero before performing quantum measurement, which seems to be consistent to the original purpose of the quantum teleportation. It is shown that if the teleported state ∣ψ>|ψ>∣ψ> is near the computational basis, the quantum measurement in CCC-system is dominantly responsible for the joint entropy S(A,C)S(A,C)S(A,C) at the final stage. If, however, ∣ψ>|ψ>∣ψ> is far from the computational basis, this dominant responsibility is moved into the quantum measurement of system AAA. A possible extension of our results are briefly discussed.
Abstract This chapter describes entanglement in bipartite systems. We define entanglement for pure states introducing the concept of Schmidt decomposition, and we show that entanglement can be quantified through the von Neumann entropy. We then present dense coding and teleportation, two quantum information protocols that use entanglement as a resource. We present the notions of locality and realism and derive the Clauser-Horne-Shimony-Holt inequality, an extended version of Bell’s inequalities. We then focus on mixed states and we present different techniques to solve the separability problem
Recently, Munro, Nemoto, and White (The Bell Inequality: A Measure of Entanglement?, quant-ph/0102119) tried to indicate that the reason behind a state $\ensuremath{\rho}$ having a higher amount of entanglement (as quantified by the entanglement of formation) than a state ${\ensuremath{\rho}}^{\ensuremath{'}},$ but producing the same amount of Bell violation, is due to the fact that the amount of mixedness (as quantified by the linearized entropy) in $\ensuremath{\rho}$ is higher than that in ${\ensuremath{\rho}}^{\ensuremath{'}}.$ We counter their argument with examples. We extend these considerations to the von Neumann entropy. Our results suggest that the reason as to why equal amount of Bell violation requires different amounts of entanglement cannot, at least, be explained by mixedness alone.