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arXiv:quant-ph/0003002 · Physical Review A

Optimal local preparation of an arbitrary mixed state of two qubits: Closed expression for the single-copy case

Guifré Vidal

Published Nov 14, 2000• 26 citations
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Abstract

We consider the problem of preparing a single copy of an arbitrary two-qubit mixed state $\ensuremath{\rho},$ starting from an entangled pure state $\ensuremath{\psi}$ and using only local operations assisted with classical communication. We present an analytical expression for the minimal amount of pure-state entanglement required, and describe the corresponding local strategy. We also examine optimal probabilistic generalizations of the previous process.

DOI
10.1103/physreva.62.062315

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We consider the problem of preparing a single copy of an arbitrary two-qubit mixed state \ensuremathρ,\ensuremath{\rho},\ensuremathρ, starting from an entangled pure state \ensuremathψ\ensuremath{\psi}\ensuremathψ and using only local operations assisted with classical communication. We present an analytical expression for the minimal amount of pure-state entanglement required, and describe the corresponding local strategy. We also examine optimal probabilistic generalizations of the previous process.
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 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.
We consider one single copy of a mixed state of two qubits and investigate how its entanglement changes under local quantum operations and classical communications (LQCC) of the type ${\ensuremath{\rho}}^{\ensuremath{'}}\ensuremath{\sim}(A\ensuremath{\bigotimes}B)\ensuremath{\rho}(A\ensuremath{\bigotimes}{B)}^{\ifmmode\dagger\else\textdagger\fi{}}.$ We consider a real matrix parametrization of the set of density matrices and show that these LQCC operations correspond to left and right multiplication by a Lorentz matrix, followed by normalization. A constructive way of bringing this matrix into a normal form is derived. This allows us to calculate explicitly the optimal local filtering operations for concentrating entanglement. Furthermore, we give a complete characterization of the mixed states that can be purified arbitrarily close to a Bell state. Finally, we obtain a new way of calculating the entanglement of formation.
We determine the preshared entanglement required for spatially separated parties, restricted to local operations and classical communication, to attain globally optimal probabilistic two-copy purification of arbitrary bipartite pure states under depolarizing noise. In every local dimension d≥2d\ge 2d≥2, one shared maximally entangled qubit pair suffices: local controlled-SWAP operations exactly reproduce the globally optimal successful transformation. Conversely, any finite-dimensional preshared resource state that attains the same benchmark, even under a positive-partial-transpose relaxation, must have entanglement of formation at least one ebit. For pure resources with one ebit, or for two-qubit resources including mixed states, attaining the benchmark is possible only for states with exactly two nonzero Schmidt weights, both equal to 1/21/21/2, namely those locally unitarily equivalent to a maximally entangled qubit pair. These findings provide a benchmark for evaluating the entanglement demands of noise management in quantum networks and modular quantum computers.