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arXiv:quant-ph/0102040 · Physical Review A

Universal state inversion and concurrence in arbitrary dimensions

Pranaw Rungta, Vladimír Bužek, Carlton M. Caves, Mark S. Hillery, Gerard J Milburn

Published Sep 18, 2001• 848 citations
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Abstract

Wootters [Phys. Rev. Lett. 80, 2245 (1998)] has given an explicit formula for the entanglement of formation of two qubits in terms of what he calls the concurrence of the joint density operator. Wootters's concurrence is defined with the help of the superoperator that flips the spin of a qubit. We generalize the spin-flip superoperator to a ``universal inverter,'' which acts on quantum systems of arbitrary dimension, and we introduce the corresponding generalized concurrence for joint pure states of ${D}_{1}\ifmmode\times\else\texttimes\fi{}{D}_{2}$ bipartite quantum systems. We call this generalized concurrence the I concurrence to emphasize its relation to the universal inverter. The universal inverter, which is a positive, but not completely positive superoperator, is closely related to the completely positive universal-NOT superoperator, the quantum analogue of a classical NOT gate. We present a physical realization of the universal-NOT superoperator.

DOI
10.1103/physreva.64.042315

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Wootters [Phys. Rev. Lett. 80, 2245 (1998)] has given an explicit formula for the entanglement of formation of two qubits in terms of what he calls the concurrence of the joint density operator. Wootters's concurrence is defined with the help of the superoperator that flips the spin of a qubit. We generalize the spin-flip superoperator to a ``universal inverter,'' which acts on quantum systems of arbitrary dimension, and we introduce the corresponding generalized concurrence for joint pure states of D1\ifmmode×\else\texttimes\fiD2{D}_{1}\ifmmode\times\else\texttimes\fi{}{D}_{2}D1​\ifmmode×\else\texttimes\fiD2​ bipartite quantum systems. We call this generalized concurrence the I concurrence to emphasize its relation to the universal inverter. The universal inverter, which is a positive, but not completely positive superoperator, is closely related to the completely positive universal-NOT superoperator, the quantum analogue of a classical NOT gate. We present a physical realization of the universal-NOT superoperator.
Concurrence provided an important measure of entanglement for general pair of qubits, which is strictly positive for entangled states and vanishing all for separable states. We present an extension of concurrence to multiparticle pure states using the Lagrange’s identity and wedge product representation, which coincides with ‘I-concurrence’ by extending the spin-flip operator to a so-called universal inverter superoperator. This framework may be useful for the extension to mixed and continuous variable systems.
Wootters concurrence is an entanglement measure for bipartite qubit systems.It is defined in terms of a spin inverter.Here, we try to generalize Wootters concurrence for qutrit systems by defining a naïve inverter analogous to the (2) qubit inverter given in terms of Pauli matrix .For qutrits, the corresponding group is (3), so the inverter has to be given in terms of Gell-Mann matrices.The naïve inverter proposed here is given in terms of the Gell-Mann matrices analogous to and we show that it can deliver equivalent information to that given by the formal inverter in just in certain cases.
Entanglement monotones, such as the concurrence, are useful tools to characterize quantum correlations in various physical systems. The computation of the concurrence involves, however, difficult optimizations and only for the simplest case of two qubits a closed formula was found by Wootters [Phys. Rev. Lett. 80, 2245 (1998)]. We show how this approach can be generalized, resulting in lower bounds on the concurrence for higher dimensional systems as well as for multipartite systems. We demonstrate that for certain families of states our results constitute the strongest bipartite entanglement criterion so far; moreover, they allow us to recognize novel families of multiparticle bound entangled states.
Concurrence, introduced by Hill and Wootters [Hill and Wootters, Phys. Rev. Lett. 78, 5022 (1997)], provides an important measure of entanglement for a general pair of qubits that is strictly positive for entangled states and vanishes for all separable states. We present an extension of the entanglement measure to general pure continuous variable states of multiple degrees of freedom by generalizing the Lagrange's identity and wedge product framework proposed by Bhaskara and Panigrahi [Bhaskara and Panigrahi, Quantum Inf. Process. 16, 118 (2017)] for pure discrete-variable systems in arbitrary dimensions and extending the concept to mixed continuous-variable states. A family of faithful entanglement measures is constructed that admits necessary and sufficient conditions for separability across arbitrary bipartitions presented by Vedral et al. [Vedral, Plenio, Rippin, and Knight, Phys. Rev. Lett. 78, 2275 (1997)]. The computed entanglement measure in the present approach for general Gaussian states, pair-coherent states, and non-Gaussian continuous-variable Bell states matches with known results. We also quantify entanglement of phase-randomized squeezed states and superposition of squeezed states. Our results also simplify several results in quantum entanglement theory.
The entanglement content of superpositions of quantum states is investigated based on a measure called concurrence. Given a bipartite pure state in arbitrary dimension written as the quantum superposition of two other such states, we find simple inequalities relating the concurrence of the state to that of its components. We derive an exact expression for the concurrence when the component states are biorthogonal and provide elegant upper and lower bounds in all other cases. For quantum bits, our upper bound is tighter than the previously derived bound [N. Linden et al., Phys. Rev. Lett. 97, 100502 (2006)].
We discuss properties of entanglement measures called I-concurrence and tangle. For a bipartite pure state, I-concurrence and tangle are simply related to the purity of the marginal density operators. The I-concurrence (tangle) of a bipartite mixed state is the minimum average I-concurrence (tangle) of ensemble decompositions of pure states of the joint density operator. Terhal and Vollbrecht [Phys. Rev. Lett. 85, 2625 (2000)] have given an explicit formula for the entanglement of formation of isotropic states in arbitrary dimensions. We use their formalism to derive comparable expressions for the I-concurrence and tangle of isotropic states.