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arXiv:0912.3338 · Physical Review A

Superposition of macroscopically distinct states means large multipartite entanglement

Tomoyuki Morimae

Published Jan 25, 2010• 21 citations
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Abstract

I show relations between superposition of macroscopically distinct states and entanglement. These relations lead to the important conclusion that if a state includes superposition of macroscopically distinct states, the state also includes large multipartite entanglement in terms of several measures. Such a multipartite entanglement property also suggests that if a state includes superposition of macroscopically distinct states, a measurement on a single particle drastically changes the state of macroscopically many other particles, as in the case of the $N$-qubit Greenberger- Horne-Zeilinger state.

DOI
10.1103/physreva.81.010101

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I show relations between superposition of macroscopically distinct states and entanglement. These relations lead to the important conclusion that if a state includes superposition of macroscopically distinct states, the state also includes large multipartite entanglement in terms of several measures. Such a multipartite entanglement property also suggests that if a state includes superposition of macroscopically distinct states, a measurement on a single particle drastically changes the state of macroscopically many other particles, as in the case of the NNN-qubit Greenberger- Horne-Zeilinger state.
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.
The entanglement of superpositions [Linden et al., Phys. Rev. Lett. 97, 100502 (2006)]is generalized to the multipartite scenario: an upper bound to the multipartite entanglement of a superposition is given in terms of the entanglement of the superposed states and the superposition coefficients. This bound is proven to be tight for a class of states composed of an arbitrary number of qubits. We also extend the result to a large family of quantifiers, which includes the negativity, the robustness of entanglement, and the best separable approximation measure.
The notion of a macroscopic quantum state must be pinned down in order to assess how well experiments probe the large-scale limits of quantum mechanics. However, the issue of quantifying so-called quantum macroscopicity is fraught with multiple approaches having varying interpretations and levels of computability and measurability. Here, we introduce two measures that capture independent macroscopic properties: i) extensive size, measuring the degree of coherence in position or momentum observables relative to atomic-scale units, and ii) entangled size, quantifying the number of entangled subsystems. These measures unify many desirable features of past proposals while having rigorously justified interpretations from quantum information theory in terms of coherence and multipartite entanglement. We demonstrate how to estimate them and obtain lower-bounds using experimental data from micro-mechanical oscillators and diffracting molecules. Notably, we find evidence for genuine multipartite entanglement of 10 5 to 10 7 atoms in recent mechanical superposition states.
We propose a measure for the ``size'' of a quantum superposition of two many-body states with (supposedly) macroscopically distinct properties by counting how many single-particle operations are needed to map one state onto the other. This definition gives sensible results for simple, analytically tractable cases and is consistent with a previous definition restricted to Greenberger-Horne-Zeilinger-like states. We apply our measure to the experimentally relevant, nontrivial example of a superconducting three-junction flux qubit put into a superposition of left- and right-circulating supercurrent states, and we find the size of this superposition to be surprisingly small.
Superposition in quantum states of large systems plays a crucial role in quantum theory and has important fundamental as well as technological implications, ranging from quantum measurement theory to quantum computers. The power of the potential implications of such states lies in the quantum coherence, as measured by the amount of entanglement and other quantum correlations, between its microscopic and macroscopic sectors. We show that in contrast to other macroscopic superposition states, it is possible to choose the states of the macroscopic sector in a way that the resulting state, which we term as the HC{H}_{C}HC​ state, has quantum coherence that is resistant to the twin effects of environmental noise: (a) local decoherence on all the particles and (b) loss of a finite fraction of its particles. The states of the macroscopic sector of the HC{H}_{C}HC​ state are macroscopically distinct in terms of their violation of Bell inequality.