Abstract
In this paper, I will point out some curious connections between entangled quantum states and classical knot configurations. In particular, I will show that the entanglement of the particles in a Greenberger-Horne-Zeilingerl (GHZ) state is modelled by a set of interlinked rings known as the Borromean rings. It is widely acknowledged that the non-local properties of multiparticle quantum states (such as the GHZ state) derive from their entanglement. By the entanglement of a multiparticle state, I mean simply that the wave function of the state cannot be written as a product of wave functions of the individual particles. Now one of the images conjured up by the term “entanglement” is that of a tangled collection of strings. This led me to enquire whether there might be any similarities between the entanglement of quantum particles and the entanglement of loops of string, or whether the expectation of such a connection is completely far-fetched.
It is a pleasure to contribute this essay to this volume honouring Abner Shimony. It was my good fortune to become acquainted with Abner a few years back and I have benefited in many ways from my interaction with him ever since.
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Notes and References
D.M. Greenberger, M.A. Home and A. Zeilinger, “Going beyond Bell’s Theorem”, in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, M. Kafatos, ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989, p. 73; D.M. Greenberger, M.A. Home, A. Shimony and A. Zeilinger, `Bell’s Theorem without Inequalities“, Am. J. Phys. 58, 1131 (1990); N.D. Mermin, ”Quantum Mysteries Revisited“, Am. J. Phys. 58, 731 (1990).
The z-direction need not be the same for all the particles, but we assume for simplicity that it is.
The history of the Borromean rings recounted here has been taken from Supplement to Not Knot by D. Epstein and C.Gunn (Jones and Bartlett Publishers, Boston, MA, 1991), p. 7.
J.S. Bell, Physics (Long Island City, N.Y.) 1, 195 (1964); J.F. Clauser, M.A. Home, A.Shimony and R.Holt, Phys. Rev. Lett. 23, 880 (1969).
A. Zeilinger, M.A. Home and D.M. Greenberger, “Higher-order Quantum Entanglement”, in Workshop on Squeezed States and Uncertainty Relations, D. Han, Y.S. Kim and W.W. Zachary, eds., NASA Conference Publication 3135, 1992.
I have in mind only configurations in which each ring is either simply linked or else unlinked with every other ring. By a pair of “simply linked” rings, I mean one for which each ring goes over (and under) the other ring exactly once.
L.H. Kauffman, Knots and Physics (World Scientific, New Jersey, 1991); Figure 4 of this paper has been taken from p. 38 of this book.
C. Livingston, Knot Theory (The Mathematical Association of America, Washington D.C., 1993); see exercise 6 on p. 10.
In making this remark, I have in mind only finite dimensional quantum systems. For the infinite dimensional case see the essay by Wayne Myrvold in this volume.
A. Shimony in Fundamental Problems in Quantum Theory, D. Greenberger, ed (Annals of the New York Academy of Sciences, 1995); A. Shimony, “Measures of Entanglement”, presented at the EPR Meeting at the Technion, 20–23 March 1995.
P.K. Aravind, unpublished.
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Aravind, P.K. (1997). Borromean Entanglement of the GHZ State. In: Cohen, R.S., Horne, M., Stachel, J. (eds) Potentiality, Entanglement and Passion-at-a-Distance. Boston Studies in the Philosophy of Science, vol 194. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2732-7_4
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