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arXiv:1504.04856 · Physical Review A

Experimental construction of a W superposition state and its equivalence to the Greenberger-Horne-Zeilinger state under local filtration

Debmalya Das, Shruti Dogra, Kavita Dorai, Arvind

Published Aug 5, 2015• 28 citations
arXivHTMLPDF

Abstract

We experimentally construct a three-qubit entangled $W$ superposition $(W\overline{W})$ state on an NMR quantum information processor. We give a measurement-based filtration protocol for the invertible local operation (ILO) that converts the $W\overline{W}$ state to the Greenberger-Horne-Zeilinger (GHZ) state, using a register of three ancilla qubits. Further we implement an experimental protocol to reconstruct full information about the three-party $W\overline{W}$ state using only two-party reduced density matrices. An intriguing fact unearthed recently is that the $W\overline{W}$ state, which is equivalent to the GHZ state under an ILO, is in fact reconstructible from its two-party reduced density matrices, unlike the GHZ state. We hence demonstrate that, although the $W\overline{W}$ state is interconvertible with the GHZ state, it stores entanglement very differently.

DOI
10.1103/physreva.92.022307

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We experimentally construct a three-qubit entangled WWW superposition (WW‾)(W\overline{W})(WW) state on an NMR quantum information processor. We give a measurement-based filtration protocol for the invertible local operation (ILO) that converts the WW‾W\overline{W}WW state to the Greenberger-Horne-Zeilinger (GHZ) state, using a register of three ancilla qubits. Further we implement an experimental protocol to reconstruct full information about the three-party WW‾W\overline{W}WW state using only two-party reduced density matrices. An intriguing fact unearthed recently is that the WW‾W\overline{W}WW state, which is equivalent to the GHZ state under an ILO, is in fact reconstructible from its two-party reduced density matrices, unlike the GHZ state. We hence demonstrate that, although the WW‾W\overline{W}WW state is interconvertible with the GHZ state, it stores entanglement very differently.
We experimentally explore the state space of three qubits on a nuclear magnetic resonance (NMR) quantum-information processor. We construct a scheme to experimentally realize a canonical form for general three-qubit states up to single-qubit unitaries. This form involves a nontrivial combination of Greenberger-Horne-Zeilinger (GHZ) and WWW-type maximally entangled states of three qubits. The general circuit that we have constructed for the generic state reduces to those for GHZ and WWW states as special cases. The experimental construction of a generic state is carried out for a nontrivial set of parameters and the good fidelity of preparation is confirmed by complete state tomography. The GHZ and WWW states are constructed as special cases of the general experimental scheme. Further, we experimentally demonstrate a curious fact about three-qubit states, where for almost all pure states, the two-qubit reduced states can be used to reconstruct the full three-qubit state. For the case of a generic state and for the WWW state, we demonstrate this method of reconstruction by comparing it to the directly tomographed three-qubit state.
Genuine 3-qubit entanglement comes in two different inconvertible types represented by the Greenberger-Horne-Zeilinger (GHZ) state and the WWW state. We describe a specific method based on local positive operator valued measures and classical communication that can convert the ideal NNN-qubit GHZ state to a state arbitrarily close to the ideal NNN-qubit WWW state. We then experimentally implement this scheme in the 3-qubit case and characterize the input and the final state using 3-photon quantum state tomography.
A novel use of Majorana geometric representation brings out distinct entanglement families of permutation symmetric states of qubits. The paradigmatic W and GHZ (Greenberger-Horne-Zeilinger) states of three qubits respectively contain two and three independent Majorana spinors. Another unique state with three distinct Majorana spinors -- constructed through a permutation symmetric superposition of two up qubits and one down qubit (W state) and its obverse state Wbar) exhibits genuine three-party entanglement, which is robust under loss of a qubit. While the GHZ state has irreducible correlations and cannot be determined from its parts, we show here that the correlation information of the W-superposition state is imprinted uniquely in its two party reduced states. This striking example sheds light on the contrasting irreducibility features of interconvertible states.
We study the interconversion of multipartite symmetric NNN-qubit states under stochastic local operations and classical communication (SLOCC). We demonstrate that if two symmetric states can be connected with a nonsymmetric invertible local operation (ILO), then they belong necessarily to the separable, W, or Greenberger-Horne-Zeilinger (GHZ) entanglement class, establishing a practical method of discriminating subsets of entanglement classes. Furthermore, we prove that there always exists a symmetric ILO connecting any pair of symmetric NNN-qubit states equivalent under SLOCC, simplifying the requirements for experimental implementations of local interconversion of those states.
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
The paradox of Greenberger-Horne-Zeilinger (GHZ) disproves directly the concept of EPR elements of reality, based on the EPR correlations, in an all-versus-nothing way. A three-qubit experimental demonstration of the GHZ paradox was achieved nearly 20 years ago, followed by demonstrations for more qubits. Still, the GHZ contradictions underlying the tests can be reduced to a three-qubit one. We show an irreducible four-qubit GHZ paradox, and report its experimental demonstration. The bound of a three-setting-per-party Bell-GHZ inequality is violated by $7\ensuremath{\sigma}$. The fidelity of the GHZ state was around 8181%81, and an entanglement witness reveals a violation of the separability threshold by $19\ensuremath{\sigma}$.