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AIP conference proceedings

Visualization of Superposition of Macroscopically Distinct States by Joint Probability Density of Macroscopic Observables

Tomoyuki Morimae, Akira Shimizu, Guillaume Adenier, Andrei Yu. Khrennikov, PEKKA J. LAHTI, Vladimir I. Man’ko, Theo M. Nieuwenhuizen

Published Jan 1, 2007• 0 citations
Publisher

Abstract

We consider a coarse‐grained quasi joint probability density Ξ of two macroscopic observables. If a quantum many‐body state has a superposition of macroscopically distinct states, the structure of the superposition can be visualized by plotting Ξ for appropriate macroscopic observables. For example, the ground state of the XY model, that of the Heisenberg antiferromagnet, and a state in Grover's quantum search algorithm, which have superpositions of macroscopically distinct states, can be visualized. Like the Wigner distribution function, Ξ can take negative values. However, if the characteristic width W of the coarse‐graining is sufficiently large, Ξ becomes nonnegative. We study a sufficient magnitude of such W. We see that W = O(N) is not sufficient, but W = O(N) would be sufficient, where N is the size of the system.

DOI
10.1063/1.2827320

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We consider a coarse‐grained quasi joint probability density Ξ of two macroscopic observables. If a quantum many‐body state has a superposition of macroscopically distinct states, the structure of the superposition can be visualized by plotting Ξ for appropriate macroscopic observables. For example, the ground state of the XY model, that of the Heisenberg antiferromagnet, and a state in Grover's quantum search algorithm, which have superpositions of macroscopically distinct states, can be visualized. Like the Wigner distribution function, Ξ can take negative values. However, if the characteristic width W of the coarse‐graining is sufficiently large, Ξ becomes nonnegative. We study a sufficient magnitude of such W. We see that W = O(N) is not sufficient, but W = O(N) would be sufficient, where N is the size of the system.
We propose a method of visualizing superpositions of macroscopically distinct states in many-body pure states. We introduce a visualization function, which is a coarse-grained quasi joint probability density for two or more Hermitian additive operators. If a state contains superpositions of macroscopically distinct states, one can visualize them by plotting the visualization function for appropriately taken operators. We also explain how to efficiently find appropriate operators for a given state. As examples, we visualize four states containing superpositions of macroscopically distinct states: the ground state of the XYXYXY model, that of the Heisenberg antiferromagnet, a state in Shor's factoring algorithm, and a state in Grover's quantum search algorithm. Although the visualization function can take negative values, it becomes nonnegative (hence, becomes a coarse-grained joint probability density) if the characteristic width of the coarse-graining function used in the visualization function is sufficiently large.
What are the conditions for adiabatic quantum computation (AQC) to outperform classical computation? Although there exist several quantum adiabatic algorithms achieving the strong quantum speedup, the essential keys to their speedups are still unclear. Here, we investigate the connection between superpositions of macroscopically distinct states and known examples of the speedup in AQC. To formalize this notion we consider an index ppp that quantifies a superposition of macroscopically distinct states from the asymptotic behaviors of fluctuations of additive observables. We determine this index for five examples of adiabatic algorithms exhibiting various degrees of the speedup. The results suggest that the superposition of macroscopically distinct states is an appropriate indicator of entanglement crucial to the strong quantum speedup in AQC.
We propose a criterion which defines whether a superposition of two photonic components is macroscopic. It is based on the ability to discriminate these components with a particular class of ``classical'' detectors, namely, a photon number measurement with a resolution coarse-grained by noise. We show how our criterion can be extended to a measure of the size of macroscopic superpositions by quantifying the amount of noise that can be tolerated and taking the distinctness of two Fock states differing by N photons as a reference. After applying our measure to several well-known examples, we demonstrate that the superpositions which meet our criterion are very sensitive to phase fluctuations. This suggests that quantifying the macroscopicity of a superposition state through the distinguishability of its components with ``classical'' detectors not only is a natural measure but also explains why it is difficult to observe superpositions at the macroscopic scale.
What are the conditions for adiabatic quantum computation (AQC) to outperform classical computation? Although there exist several quantum adiabatic algorithms achieving the strong quantum speedup, the essential keys to their speedups are still unclear. Here, we investigate the connection between superpositions of macroscopically distinct states and known examples of the speedup in AQC. To formalize this notion we consider an index ppp that quantifies a superposition of macroscopically distinct states from the asymptotic behaviors of fluctuations of additive observables. We determine this index for five examples of adiabatic algorithms exhibiting various degrees of the speedup. The results suggest that the superposition of macroscopically distinct states is an appropriate indicator of entanglement crucial to the strong quantum speedup in AQC.
For quantum computation, we investigate the conjecture that the superposition of macroscopically distinct states is necessary for a large quantum speedup. Although this conjecture was supported for a circuit-based quantum computer performing Shor's factoring algorithm [A. Ukena and A. Shimizu, Phys. Rev. A69 (2004) 022301], it needs to be generalized for it to be applicable to a large class of algorithms and/or other models such as measurement-based quantum computers. To treat such general cases, we first generalize the indices for the superposition of macroscopically distinct states. We then generalize the conjecture, using the generalized indices, in such a way that it is unambiguously applicable to general models if a quantum algorithm achieves exponential speedup. On the basis of this generalized conjecture, we further extend the conjecture to Grover's quantum search algorithm, whose speedup is large but quadratic. It is shown that this extended conjecture is also correct. Since Grover's algorithm is a representative algorithm for unstructured problems, the present result further supports the conjecture.