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arXiv:quant-ph/9902022 · Physical Review Letters

Optimal Entanglement Enhancement for Mixed States

Adrian Kent, Noah Linden, Serge Massar

Published Sep 27, 1999• 70 citations
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Abstract

We consider the actions of protocols involving local quantum operations and classical communication (LQCC) on a single system consisting of two separated qubits. We give a complete description of the orbits of the space of states under LQCC and characterize the representatives with maximal entanglement of formation. We thus obtain a LQCC entanglement concentration protocol for a single given state (pure or mixed) of two qubits which is optimal in the sense that the protocol produces, with nonzero probability, a state of maximal possible entanglement of formation. This defines a new entanglement measure, the maximum extractable entanglement.

DOI
10.1103/physrevlett.83.2656

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We consider the actions of protocols involving local quantum operations and classical communication (LQCC) on a single system consisting of two separated qubits. We give a complete description of the orbits of the space of states under LQCC and characterize the representatives with maximal entanglement of formation. We thus obtain a LQCC entanglement concentration protocol for a single given state (pure or mixed) of two qubits which is optimal in the sense that the protocol produces, with nonzero probability, a state of maximal possible entanglement of formation. This defines a new entanglement measure, the maximum extractable entanglement.
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Quantum entanglement is an indispensable resource for many significant quantum information processing tasks. However, in practice, it is difficult to distribute quantum entanglement over a long distance, due to the absorption and noise in quantum channels. A solution to this challenge is a quantum repeater, which can extend the distance of entanglement distribution. In this scheme, the time consumption of classical communication and local operations takes an important place with respect to time efficiency. Motivated by this observation, we consider a basic quantum repeater scheme that focuses on not only the optimal rate of entanglement concentration but also the complexity of local operations and classical communication. First, we consider the case where two different two-qubit pure states are initially distributed in the scenario. We construct a protocol with the optimal entanglement-concentration rate and less consumption of local operations and classical communication. We also find a criterion for the projective measurements to achieve the optimal probability of creating a maximally entangled state between the two ends. Second, we consider the case in which two general pure states are prepared and general measurements are allowed. We get an upper bound on the probability for a successful measurement operation to produce a maximally entangled state without any further local operations.
We consider a single copy of a mixed state of two qubits and derive the optimal trace-preserving local operations assisted by classical communication such as to maximize the fidelity of teleportation that can be achieved with this state. These optimal local operations turn out to be implementable by one-way communication and always yield a teleportation fidelity larger than 2/3 if the original state is entangled. This maximal achievable fidelity is an entanglement measure and turns out quantifying the minimal amount of mixing required to destroy the entanglement in a quantum state.
We consider mixed states of two qubits and show under which global unitary operations their entanglement is maximized. This leads to a class of states that is a generalization of the Bell states. Three measures of entanglement are considered: entanglement of formation, negativity, and relative entropy of entanglement. Surprisingly all states that maximize one measure also maximize the others. We give a complete characterization of these generalized Bell states and prove that these states for fixed eigenvalues are all equivalent under local unitary transformations. Furthermore we characterize all nearly entangled states closest to the maximally mixed state and derive a lower bound on the volume of separable mixed states.
We consider a single copy of a mixed state of two qubits and prove tight lower and upper bounds for its fidelity in terms of its concurrence or negativity. Next we derive the optimal filtering operations and the optimal trace-preserving local operations assisted by classical communication (LOCC) such as to maximize the fidelity of a given copy of a mixed state of two qubits. The optimal LOCC operations turn out to be very simple and implementable by 1-LOCC protocols, and always yield a fidelity larger then 1/2 if the original state was entangled.
We analytically obtain the maximum probability of converting a finite number of copies of an arbitrary two-qubit pure state to a single copy of a maximally entangled two-qubit pure state via entanglement assisted local operations and classical communications using a two-qubit catalyst state. We show that the optimal catalyst for this transformation is always more entangled than the initial state but any two-qubit state can act as a (non-optimal) catalyst. Interestingly, the entanglement of the optimal two-qubit catalyst state is shown to decrease with that of the initial state. Entanglement assisted strategies for obtaining multiple Bell states are discussed.