We study macroscopic entanglement of various pure states of a one-dimensional
N-spin system with
N⪢1. Here, a quantum state is said to be macroscopically entangled if it is a superposition of macroscopically distinct states. To judge whether such superposition is hidden in a general state, we use an essentially unique index
p: A pure state is macroscopically entangled if
p=2, whereas it may be entangled but not macroscopically if
p<2. This index is directly related to fundamental stabilities of many-body states. We calculate the index
p for various states in which magnons are excited with various densities and wave numbers. We find macroscopically entangled states
(p=2) as well as states with
p=1. The former states are unstable in the sense that they are unstable against some local measurements. On the other hand, the latter states are stable in the senses that they are stable against any local measurements and that their decoherence rates never exceed
O(N) in any weak classical noises. For comparison, we also calculate the von Neumann entropy
SN∕2(N) of a subsystem composed of
N∕2 spins as a measure of bipartite entanglement. We find that
SN∕2(N) of some states with
p=1 is of the same order of magnitude as the maximum value
N∕2. On the other hand,
SN∕2(N) of the macroscopically entangled states with
p=2 is as small as
O(logN)⪡N∕2. Therefore larger
SN∕2(N) does not mean more instability. We also point out that these results are partly analogous to those for interacting many bosons. Furthermore, the origin of the huge entanglement, as measured either by
p or
SN∕2(N), is discussed to be due to spatial propagation of magnons.
Theory and experiment both demonstrate that an entangled quantum state of two subsystems is neither a superposition of states of its subsystems nor a superposition of composite states but rather a coherent superposition of nonlocal correlations between incoherently mixed local states of the two subsystems. Thus, even if one subsystem happens to be macroscopic as in the entangled "Schrodinger's cat" state resulting from an ideal measurement, this state is not the paradoxical macroscopic superposition it is generally presumed to be. It is, instead, a "macroscopic correlation," a coherent quantum correlation in which one of the two correlated sub-systems happens to be macroscopic. This clarifies the physical meaning of entanglement: When a superposed quantum system A is unitarily entangled with a second quantum system B, the coherence of the original superposition of different states of A is transferred to different correlations between states of A and B, so the entangled state becomes a superposition of correlations rather than a superposition of states. This transfer preserves unitary evolution while permitting B to be macroscopic without entailing a macroscopic superposition. This resolves the "problem of outcomes" but is not a complete resolution of the measurement problem because the entangled state is still reversible.
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.
We investigate macroscopic entanglement of quantum states in quantum computers, where we say a quantum state is entangled macroscopically if the state has superposition of macroscopically distinct states. The index
p of the macroscopic entanglement is calculated as a function of the step of the computation, for Grover's quantum search algorithm and Shor's factoring algorithm. It is found that whether macroscopically entangled states are used or not depends on the numbers and properties of the solutions to the problem to be solved. When the solutions are such that the problem becomes hard in the sense that classical algorithms take more than polynomial steps to find a solution, macroscopically entangled states are always used in Grover's algorithm and almost always used in Shor's algorithm. Since they are representative algorithms for unstructured and structured problems, respectively, our results support strongly the conjecture that quantum computers utilize macroscopically entangled states when they solve hard problems much faster than any classical algorithms.
The possibility to observe quantum features at the macroscopic level stands out as one of the most challenging yet fascinating open questions in modern quantum physics. The difficulties inherent in such a quest are manifold. First, the unavoidable interaction with the surrounding environment lies deeply at the roots of the loss of coherence and corruption of quantum correlations in a quantum state, being more severe as the size of the system being studied grows. Second, one faces the debated problem of achieving a measurement-precision that is sufficient to observe quantum effects at such macro-scales. Recent studies have shown that, depending on the Hamiltonian models, classicality may emerge from the coarse-graininess of measurements performed over a quantum system.
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.