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arXiv:quant-ph/0105046 · Physical Review A

Finding a state among a complete set of orthogonal states

Niko Donath, Karl Svozil

Published Mar 21, 2002• 15 citations
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Abstract

We consider the problem to single out a particular state among ${2}^{n}$ orthogonal pure states. As it turns out, in general the optimal strategy is not to measure the particles separately, but to consider joint properties of the n-particle system. The required number of propositions is n. There exist ${2}^{n}!$ equivalent operational procedures to do so. We enumerate some configurations for three particles, in particular the Greenberger-Horne-Zeilinger (GHZ) and W states, which are specific cases of a unitary transformation. For the GHZ case, an explicit physical meaning of the projection operators is discussed.

DOI
10.1103/physreva.65.044302

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We consider the problem to single out a particular state among 2n{2}^{n}2n orthogonal pure states. As it turns out, in general the optimal strategy is not to measure the particles separately, but to consider joint properties of the n-particle system. The required number of propositions is n. There exist 2n!{2}^{n}!2n! equivalent operational procedures to do so. We enumerate some configurations for three particles, in particular the Greenberger-Horne-Zeilinger (GHZ) and W states, which are specific cases of a unitary transformation. For the GHZ case, an explicit physical meaning of the projection operators is discussed.
We propose a measure of entanglement between two subsystems of arbitrary (but finite) number of levels. This measure is invariant under unitary transformations of the subsystems and varies between 0 (product states) and 1 (maximum entangled states). The case of two two-level systems is worked out explicitly, where the known complementarity between one- and two-particle interference results in a very natural way from a sum rule. Generalizations to more than two subsystems are discussed. Here we show that for three particles there cannot exist a pure state that is completely characterized by three-particle entanglement alone. For the example of the Greenberger-Horne-Zeilinger (GHZ) state, an experimental setup is proposed, in which its corresponding two-particle entanglement would show up.
We give a complete solution for the three-tangle of mixed three-qubit states composed of a generalized Greenberger–Horne–Zeilinger (GHZ) state, a |000⟩+ b |111⟩, and a generalized W state, c |001⟩+ d |010⟩+ f |100⟩. Using the methods introduced by Lohmayer et al (2006 Phys. Rev. Lett. 97 260502), we provide explicit expressions for the mixed-state three-tangle and the corresponding optimal decompositions for this more general case. Moreover, as a special case, we obtain a general solution for a family of states consisting of a generalized GHZ state and an orthogonal product state.
The generalized nnn-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of nnn qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of $n\ensuremath{-}1$ qubits. Thus, the generalized GHZ states are the only ones containing information at the nnn-party level.
Only nnn-qubit Greenberger-Horne-Zeilinger states contain nnn-partite information
The generalized nnn-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of nnn qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of $n\ensuremath{-}1$ qubits. Thus, the generalized GHZ states are the only ones containing information at the nnn-party level.
Using just polarizing beam splitters and half-wave plates, we present a practical experimental scheme, by which one can conveniently identify two of the three-particle Greenberger-Horne-Zeilinger (GHZ) states. The scheme can easily be generalized to the NNN-particle case. The GHZ-state analyzer is essential for multiparticle generalizations of quantum dense coding, quantum teleportation, and entanglement swapping.
Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for detecting class-specific entanglement in arbitrary three-qubit states.