dorsal/arxiv
View SchemaMacroscopic objects in quantum mechanics: A combinatorial approach
| Authors | Itamar Pitowsky |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0404051 |
| URL | https://arxiv.org/abs/quant-ph/0404051 |
| DOI | 10.1103/PhysRevA.70.022103 |
| Journal | Physical Review A 70, 022103-1-6 (2004) |
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)).
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"abstract": "Why we do not see large macroscopic objects in entangled states? There are\ntwo ways to approach this question. The first is dynamic: the coupling of a\nlarge object to its environment cause any entanglement to decrease\nconsiderably. The second approach, which is discussed in this paper, puts the\nstress on the difficulty to observe a large scale entanglement. As the number\nof particles n grows we need an ever more precise knowledge of the state, and\nan ever more carefully designed experiment, in order to recognize entanglement.\nTo develop this point we consider a family of observables, called witnesses,\nwhich are designed to detect entanglement. A witness W distinguishes all the\nseparable (unentangled) states from some entangled states. If we normalize the\nwitness W to satisfy |tr(W\\rho)| \\leq 1 for all separable states \\rho, then the\nefficiency of W depends on the size of its maximal eigenvalue in absolute\nvalue; that is, its operator norm ||W||. It is known that there are witnesses\non the space of n qbits for which ||W|| is exponential in n. However, we\nconjecture that for a large majority of n-qbit witnesses ||W|| \\leq O(\\sqrt{n\nlogn}). Thus, in a non ideal measurement, which includes errors, the largest\neigenvalue of a typical witness lies below the threshold of detection. We prove\nthis conjecture for the family of extremal witnesses introduced by Werner and\nWolf (Phys. Rev. A 64, 032112 (2001)).",
"arxiv_id": "quant-ph/0404051",
"authors": [
"Itamar Pitowsky"
],
"categories": [
"quant-ph",
"cond-mat.stat-mech"
],
"doi": "10.1103/PhysRevA.70.022103",
"journal_ref": "Physical Review A 70, 022103-1-6 (2004)",
"title": "Macroscopic objects in quantum mechanics: A combinatorial approach",
"url": "https://arxiv.org/abs/quant-ph/0404051"
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