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Macroscopic objects in quantum mechanics: A combinatorial approach
Itamar Pitowsky
abstract: Why we do not see large macroscopic objects in entangled states? There are
two ways to approach this question. The first is dynamic: the coupling of a
large object to its environment cause any entanglement to decrease
considerably. The second approach, which is discussed in this paper, puts the
stress on the difficulty to observe a large scale entanglement. As the number
of particles n grows we need an ever more precise knowledge of the state, and
an ever more carefully designed experiment, in order to recognize entanglement.
To develop this point we consider a family of observables, called witnesses,
which are designed to detect entanglement. A witness W distinguishes all the
separable (unentangled) states from some entangled states. If we normalize the
witness W to satisfy |tr(W\rho)| \leq 1 for all separable states \rho, then the
efficiency of W depends on the size of its maximal eigenvalue in absolute
value; that is, its operator norm ||W||. It is known that there are witnesses
on the space of n qbits for which ||W|| is exponential in n. However, we
conjecture that for a large majority of n-qbit witnesses ||W|| \leq O(\sqrt{n
logn}). Thus, in a non ideal measurement, which includes errors, the largest
eigenvalue of a typical witness lies below the threshold of detection. We prove
this conjecture for the family of extremal witnesses introduced by Werner and
Wolf (Phys. Rev. A 64, 032112 (2001)).
- oai_identifier:
- oai:arXiv.org:quant-ph/0404051
- categories:
- quant-ph cond-mat.stat-mech
- comments:
- RevTeX, 14 pages, some additions to the published version: A second
conjecture added, discussion expanded, and references added
- doi:
- 10.1103/PhysRevA.70.022103
- arxiv_id:
- quant-ph/0404051
- journal_ref:
- Physical Review A 70, 022103-1-6 (2004)
- created:
- 2004-04-08
- updated:
- 2006-03-18
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