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arXiv:0812.3772 · Quantum Information and Computation

Teleportation via maximally and non-maximally entangled mixed states

Satyabrata Adhikari, A. S. Majumdar, Sovik Roy, Biswajit Ghosh, N. Nayak

Published May 1, 2010• 33 citations
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Abstract

We study the efficiency of two-qubit mixed entangled states as resources for quantum teleportation. We first consider two maximally entangled mixed states, viz., the Werner state\cite{werner}, and a class of states introduced by Munro {\it et al.} \cite{munro}. We show that the Werner state when used as teleportation channel, gives rise to better average teleportation fidelity compared to the latter class of states for any finite value of mixedness. We then introduce a non-maximally entangled mixed state obtained as a convex combination of a two-qubit entangled mixed state and a two-qubit separable mixed state. It is shown that such a teleportation channel can outperform another non-maximally entangled channel, viz., the Werner derivative for a certain range of mixedness. Further, there exists a range of parameter values where the former state satisfies a Bell-CHSH type inequality and still performs better as a teleportation channel compared to the Werner derivative even though the latter violates the inequality.

DOI
10.26421/qic10.5-6-3

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We study the efficiency of two-qubit mixed entangled states as resources for quantum teleportation. We first consider two maximally entangled mixed states, viz., the Werner state\cite{werner}, and a class of states introduced by Munro {\it et al.} \cite{munro}. We show that the Werner state when used as teleportation channel, gives rise to better average teleportation fidelity compared to the latter class of states for any finite value of mixedness. We then introduce a non-maximally entangled mixed state obtained as a convex combination of a two-qubit entangled mixed state and a two-qubit separable mixed state. It is shown that such a teleportation channel can outperform another non-maximally entangled channel, viz., the Werner derivative for a certain range of mixedness. Further, there exists a range of parameter values where the former state satisfies a Bell-CHSH type inequality and still performs better as a teleportation channel compared to the Werner derivative even though the latter violates the inequality.
Three-qubit mixed states are used as a channel for controlled quantum teleportation (CQT) of single-qubit states. The connection between different channel parameters to achieve maximum controlled teleportation fidelity is investigated. We show that for a given multipartite entanglement and mixedness a class of non-maximally-entangled mixed XXX states (XXX-NMEMS) achieves optimum controlled quantum teleportation fidelity; interestingly a class of maximally entangled mixed XXX states (XXX-MEMS) fails to do so. This demonstrates, for a given spectrum and mixedness, that XXX-MEMS are not sufficient to attain optimum controlled quantum teleportation fidelity, which is in contradiction with the traditional quantum teleportation of single qubits. In addition, we show that biseparable XXX-NMEMS, for a certain range of mixedness, are useful as a resource to attain high controlled quantum teleportation fidelity, which essentially lowers the requirements of quantum channels for CQT.
In this paper we extend our recent results for quantum teleportation of single qubit pure information state using non-maximally entangled pure state resource [Quantum Inf Process, 11 (2012) 1251] to the case of a general mixed information state and mixed Werner states resource. Fidelity is found dependent on information state in general. We minimize it over all possible information states and maximize it over possible unitary transformations which can be done by the receiver to obtain a value dependent on purity of information state and mixing parameter of Werner states. For pure information state, this value of fidelity is independent of information and for mixed information states it is less than unity even for perfect entangled resource. The latter result is in contrast to the case of pure states.
We study quantum teleportation with the resource of non-orthogonal qubit states. We first extend the standard teleportation protocol to the case of such states. We investigate how the loss of teleportation fidelity resulting from the use of non-orthogonal states compares to a similar loss of fidelity when noisy or non-maximally entangled states are used as the teleportation resource. Our analysis leads to some interesting results on the teleportation efficiency of both pure and mixed non-orthogonal states compared to that of non-maximally entangled and mixed states.
The violation of Bell’s inequality in quantum mechanics implies that there exist nonlocality and entanglement. When the density matrix of a composite system cannot be written as a convex combination of the product of the density matrices of its subsystems, we say there exists entanglement. For pure states, the existence of entanglement always leads to the violation of Bell’s inequality. However, in the case of the mixed states, there may be entanglement, but Bell's inequality is not violated and in other words, the nonlocality is not manifested. In addition to Bell's inequality, quantum teleportation is also a manifestation of nonlocality. Quantum teleportation using entangled states is more successful than quantum teleportation with separable states. Therefore, the corresponding fidelity of teleported state with the initial state (in short, the fidelity) of the former is always greater than the fidelity of the latter. In this paper, for Werner's state, we will show that in a range of the related parameter, while the CHSH inequality is violated, the fidelity, which indicates the amount of success of quantum teleportation, is lower than the upper bound of the corresponding fidelity for states that can be simulated with a local hidden variable theory. Meanwhile, we will see that for Gisin's state with hidden nonlocality, filtering, which leads to the appearance of nonlocality and more specifically leads to the violation of the CHSH inequality, also increases the fidelity.
In our work we consider the following problem in the context of teleportation: an unknown pure state have to be teleported and there are two laboratories which can perform the task. One laboratory uses a pure non maximally entangled channel but has a capability of performing the joint measurement on bases with a constrained degree of entanglement; the other lab makes use of a mixed X-state channel but can perform a joint measurement on bases with higher entanglement degrees. We compare the average teleportation fidelity achieved in both cases, finding that the fidelity achieved with the X-state can surpass the obtained with a pure channel, even though the X-state is less entangled than the latter. We find the conditions under which this effect occurs. Our results evidence that the entanglement of the joint measurement plays a role as important as the entanglement of the channel in order to optimize the teleportation process. We include an example showing that the average fidelity of teleportation obtained with a Werner state channel can be grater than that obtained with a Bell state channel.
We investigate teleportation of bipartite pure entanglement through two equally noisy quantum channel states, namely, a mixture of Bell states. There is a particular mixed-state channel for which all pure entangled states in a known biorthogonal product basis remain entangled after this inexact teleportation, which occurs for the duration of the entanglement. The Werner state channel lacks both these features. The relation of these noisy channels to the violation of Bell's inequality and quantum two-entropy inequality is studied.