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arXiv:1201.2184 · Physical Review A

Stability of encoded macroscopic quantum superpositions

Florian Fröwis, Wolfgang Dür

Published May 29, 2012• 31 citations
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Abstract

The multipartite Greenberger-Horne-Zeilinger (GHZ) state is a paradigmatic example of a highly entangled multipartite state with distinct quantum features. However, the GHZ state is very sensitive to generic decoherence processes, where its quantum features and, in particular, its entanglement diminish rapidly, thereby hindering possible practical applications. In this paper, we discuss GHZ-like quantum states with a block-local structure and show that they exhibit a drastically increased stability against noise for certain choices of block-encoding, thereby extending results of [Phys. Rev. Lett. 106, 110402 (2011)]. We analyze in detail the decay of the interference terms, the entanglement in terms of distillable entanglement, and negativity as well as the notion of macroscopicity as measured by the so-called index $q$ and provide general bounds on these quantities. We focus on an encoding where logical qubits are themselves encoded as GHZ states, which leads to so-called concatenated GHZ (C-GHZ) states. We compare the stability of C-GHZ states with other types of encodings, thereby showing the superior stability of the C-GHZ states. Analytic results are complemented by numerical studies, where tensor network techniques are used to investigate the quantum properties of multipartite entangled states under the influence of decoherence.

DOI
10.1103/physreva.85.052329

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The multipartite Greenberger-Horne-Zeilinger (GHZ) state is a paradigmatic example of a highly entangled multipartite state with distinct quantum features. However, the GHZ state is very sensitive to generic decoherence processes, where its quantum features and, in particular, its entanglement diminish rapidly, thereby hindering possible practical applications. In this paper, we discuss GHZ-like quantum states with a block-local structure and show that they exhibit a drastically increased stability against noise for certain choices of block-encoding, thereby extending results of [Phys. Rev. Lett. 106, 110402 (2011)]. We analyze in detail the decay of the interference terms, the entanglement in terms of distillable entanglement, and negativity as well as the notion of macroscopicity as measured by the so-called index qqq and provide general bounds on these quantities. We focus on an encoding where logical qubits are themselves encoded as GHZ states, which leads to so-called concatenated GHZ (C-GHZ) states. We compare the stability of C-GHZ states with other types of encodings, thereby showing the superior stability of the C-GHZ states. Analytic results are complemented by numerical studies, where tensor network techniques are used to investigate the quantum properties of multipartite entangled states under the influence of decoherence.
We study the stability of superpositions of macroscopically distinct quantum states under decoherence. We introduce a class of quantum states with entanglement features similar to Greenberger-Horne-Zeilinger (GHZ) states, but with an inherent stability against noise and decoherence. We show that in contrast to GHZ states, these so-called concatenated GHZ states remain multipartite entangled even for macroscopic numbers of particles and can be used for quantum metrology in noisy environments. We also propose a scalable experimental realization of these states using existing ion-trap setups.
Multipartite concatenated Greenberger–Horne–Zeilinger (C-GHZ) states, a class of multipartite entangled states, have superior stability under the influence of decoherence. We propose two means of constructing multiphoton C-GHZ states based on entanglers of multiphoton GHZ states that are both experimentally realizable and scalable. Given a km-photon GHZ state as an input state, if m is odd, one can create a km-photon C-GHZ state. In addition, we have designed a scheme to prepare km-photon C-GHZ states from km single-photon states by using k entanglers of m-photon GHZ states and km-control Toffoli gates.
We investigate entanglement properties of multipartite states under the influence of decoherence. We show that the lifetime of (distillable) entanglement for Greenberger-Horne-Zeilinger (GHZ) -type superposition states decreases with the size of the system, while for a class of other states---namely, all graph states with constant degree---the lifetime is independent of the system size. We show that these results are largely independent of the specific decoherence model and are in particular valid for all models which deal with individual couplings of particles to independent environments, described by some quantum optical master equation of Lindblad form. For GHZ states, we derive analytic expressions for the lifetime of distillable entanglement and determine when the state becomes fully separable. For all graph states, we derive lower and upper bounds on the lifetime of entanglement. The lower bound is based on a specific distillation protocol, while upper bounds are obtained by showing that states resulting from decoherence in general become nondistillable or even separable after a finite time. This is done using different methods, namely, (i) the map describing the decoherence process (e.g., the action of a thermal bath on the system) becomes entanglement breaking, (ii) the resulting state becomes separable, and (iii) the partial transposition with respect to certain partitions becomes positive. To this aim, we establish a method to calculate the spectrum of the partial transposition for all mixed states which are diagonal in a graph-state basis. We also consider entanglement between different groups of particles and determine the corresponding lifetimes as well as the change of the kind of entanglement with time. This enables us to investigate the behavior of entanglement under rescaling and in the limit of large number of particles $N\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}$. Finally we investigate the lifetime of encoded quantum superposition states and show that one can define an effective time in the encoded system which can be orders of magnitude smaller than the physical time. This provides an alternative view on quantum error correction and examples of states whose lifetime of entanglement (between groups of particles) in fact increases with the size of the system.
We consider the collision model of Ziman et al. and study the robustness of NNN-qubit Greenberger-Horne-Zeilinger (GHZ), WWW, and linear cluster states. Our results show that NNN-qubit entanglement of GHZ states would be extremely fragile under collisional decoherence, and that of WWW states could be more robust than that of linear cluster states. We indicate that the collision model of Ziman et al. could provide a physical mechanism for some known results in this area of investigation. More importantly, we show that it could give a clue as to how NNN-partite distillable entanglement would be relatively rare in our macroscopic classical world.
Decoherence is one of the main obstacles in long-distance quantum communication. Recently, the theoretical work of Fröwis and W. Dür (Phys. Rev. Lett. \textbf{106}, 110402 (2011)) and the experiment of Lu \emph{et al.} (Nat. Photon. \textbf{8}, 364 (2014)) both showed that the logic qubits entanglement say the concatenated Greenberger-Horne-Zeilinger (C-GHZ) state is more robust under decoherence. In this paper, we describe a protocol for Bell-state analysis for this logic qubits entanglement. This protocol can also be extended to the multipartite C-GHZ state analysis. Also, we discuss its application in the quantum teleportation of a unknown logic qubit and in the entanglement swapping of logic Bell states. As the logic qubits entanglement is more robust under decoherence, our protocol shows that it is possible to realize the long-distance quantum communication based on logic qubits entanglement.
Macroscopic entanglement, or say the Schrödinger's Cat state has attracted much attention for a long time. Recently, the first theoretical work of Fröwis and Dür ( Phys. Rev. Lett. , 106 (2011) 110402) and the first experiment of Lu et al. ( Nat. Photon. , 8 (2014) 364) both showed that, a new type of Schrödinger's Cat state, the logic-qubit entanglement (concatenated Greenberger-Horne-Zeilinger (C-GHZ) state) is immune and robust to the noise, and is possible to be applied in future large-scale quantum networks. In this paper, we describe a protocol of entanglement analysis for this kind of Schrödinger's Cat state. Both the Bell-state type of logic-qubit entanglement and multipartite C-GHZ state can be completely distinguished. Based on the entanglement analysis, an arbitrary unknown macroscopic Schrödinger's Cat superposed state can be teleportated and we can also perform the macroscopic entanglement swapping. Our protocol shows that it is possible to realize long-distance quantum communication and large-scale quantum network based on logic-qubit entanglement.