2010
Journal article
Open Access
Not all pure entangled states are useful for sub-shot-noise interferometry
Hyllus P, Guhne O, Smerzi A
Atomic and Molecular Physics entanglement and Optics quantum interferometry Quantum Physics (quant-ph) Quantum Physics Condensed Matter - Quantum Gases Quantum Gases (cond-mat.quant-gas) FOS: Physical sciences
We investigate the connection between the shot-noise limit in linear interferometers and particle entanglement. In particular, we ask whether sub-shot-noise sensitivity can be reached with all pure entangled input states of N particles if they can be optimized with local operations. Results on the optimal local transformations allow us to show that for N=2 all pure entangled states can be made useful for sub-shot-noise interferometry while for N>2 this is not the case. We completely classify the useful entangled states available in a bosonic two-mode interferometer. We apply our results to several states, in particular to multiparticle singlet states and to cluster states. The latter turn out to be practically useless for sub-shot-noise interferometry. Our results are based on the Cramer-Rao bound and the Fisher information.
Source: PHYSICAL REVIEW. A, vol. 82, p. 012337
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[44] R. H. Dicke, Phys. Rev. 93, 99 (1954).
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[54] M. L. Cohen, IEEE Transactions on Information Theory 14, 591 (1968).
[55] A. Fujiwara, Phys. Rev. A 63, 042304 (2001).
[56] R. F. Werner, Phys. Rev. A 40, 4277 (1989).
[57] S.L. Braunstein, A.S. Lane, and C.M. Caves, Phys. Rev. Lett. 69, 2153 (1992); A.S. Lane, S.L. Braunstein, and C.M. Caves, Phys. Rev. A 47, 1667 (1993).
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[65] To see this, note that Eq. (30d) in Ref. [64] implies that for separable states λmax(N γ) ≤ (N − 1)T r(γ) − N (N − 2)/4 since X = N γ + hJ~ihJ~ˆiT ≥ N γ. Using Eq. (30b) ˆ implies then that N λmax(γ) ≤ N (N − 1)/2 − N (N − 2)/4 hence λmax(γ) ≤ N/4. We used the notations of Ref. [64] here.
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[75] D. M. Greenberger, M. Horne, and A. Zeilinger, Am. J. Phys. 58, 1131 (1990).
[76] D. Gottesman, Phys. Rev. A 54, 1862 (1996).
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[78] W. Du¨r, H. Aschauer, and H.-J. Briegel, Phys. Rev. Lett. 91, 107903 (2003).
[2] C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[3] B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 (1986).
[4] M. J. Holland and K. Burnett, Phys. Rev. Lett. 71, 1355 (1993).
[5] J. P. Dowling, Phys. Rev. A 57, 4736 (1998).
[6] A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Nature 409, 63 (2001).
[7] R. A. Campos, C. C. Gerry, and A. Benmoussa, Phys. Rev. A 68, 023810 (2003).
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[9] L. Pezz´e and A. Smerzi, Phys. Rev. A 73, 011801(R) (2006).
[10] L. Pezz´e and A. Smerzi, Phys. Rev. Lett. 100, 073601; L. Pezz´e and A. Smerzi, arXiv:1004.5486.
[11] H. Uys and P. Meystre, Phys. Rev. A 76, 013804 (2007).
[12] H. Cable and G. A. Durkin, Phys. Rev. Lett. 105, 013603 (2010).
[13] D.W. Berry and H.M. Wiseman, Phys. Rev. Lett. 85 5098 (2000).
[14] B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, Nature 450, 393 (2007).
[15] D.W. Berry, B. L. Higgins, S. D. Bartlett, M. W. Mitchell, G. J. Pryde, and H. M. Wiseman, Phys. Rev. A 80 052114 (2009).
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[17] U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Phys. Rev. Lett. 102, 040403 (2009).
[18] R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Phys. Rev. A 80, 013825 (2009).
[19] M. Rosenkranz and D. Jaksch, Phys. Rev. A 79, 022103 (2009).
[20] M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, Nature 429, 161 (2004).
[21] P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, Nature 429, 158 (2004).
[22] T. Nagata, R. Okamoto, J. L. O'Brien, K. Sasaki, and S. Takeuchi, Science 316, 726 (2007).
[23] J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O'Brien, Nat. Phot. 3, 346 (2009).
[24] W.-B. Gao et al., C.-Y. Lu, X.-C. Yao, P. Xu, O. Gu¨hne, A. Goebel, Y.-A. Chen, C.-Z. Peng, Z.-B. Chen, and J.- W. Pan, Nature Phys. 6, 331 (2010).
[25] D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. Jost, C. Langer, et al., Nature (London) 438, 639 (2005).
[26] C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, and M. A. Kasevich, Science 291, 2386 (2001).
[27] G.-B. Jo, Y. Shin, S. Will, T. A. Pasquini, M. Saba, W. Ketterle, D. E. Pritchard, M. Vengalattore, and M. Prentiss, Phys. Rev. Lett. 98, 030407 (2007).
[28] J. Est`eve, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, Nature 455, 1216 (2008).
[29] C. Gross, T. Zibold, E. Nicklas, J. Est`eve, and M. K. Oberthaler, Nature 464, 1165 (2010).
[30] M. F. Riedel, P. B¨ohi, Y. Li, T. W. H¨ansch, A. Sinatra and P. Treutlein, Nature 464, 1170 (2010).
[31] T. Fernholz, H. Krauter, K. Jensen, J. F. Sherson, A. S. Sørensen, and E. S. Polzik, Phys. Rev. Lett. 101, 073601 (2008).
[32] K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nature Phys. 4, 472 (2008).
[33] H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
[34] A. Luis, Phys. Lett. A 329, 8 (2004).
[35] S.M. Roy and S.L. Braunstein, Phys. Rev. Lett. 100, 220501 (2008).
[36] S. Boixo, S. T. Flammia, C. M. Caves, and J. M. Geremia, Phys. Rev. Lett. 98, 090401 (2007).
[37] S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, Phys. Rev. Lett. 101, 040403 (2008).
[38] S. Choi and B. Sundaram, Phys. Rev. A 77, 053613 (2008).
[39] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006).
[40] L. Pezz´e and A. Smerzi, Phys. Rev. Lett. 102, 100401 (2009).
[41] The connection between shot-noise limit and separability for a non-fixed number of particles has been discussed in P. Hyllus, L. Pezz´e, and A. Smerzi, arXiv:1003.0649.
[42] H. H¨affner, C.F. Roos, and R. Blatt, Phys. Rep. 469, 155 (2008).
[43] A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1995).
[44] R. H. Dicke, Phys. Rev. 93, 99 (1954).
[45] G. T´oth and O. Gu¨hne, Phys. Rev. Lett 102, 170503 (2009).
[46] C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
[47] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982).
[48] M. G. A. Paris, Int. J. Quant. Inf. 7, 125 (2009).
[49] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
[50] H. Cramer, Mathematical methods of Statistics (Princeton University Press, Princeton, NJ, 1946), esp. pp. 500- 504.
[51] W.K. Wootters, Phys. Rev. D 23, 357 (1981).
[52] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
[53] S. L. Braunstein, C. M. Caves, and G. J. Milburn, Ann. Phys. 247, 135 (1996).
[54] M. L. Cohen, IEEE Transactions on Information Theory 14, 591 (1968).
[55] A. Fujiwara, Phys. Rev. A 63, 042304 (2001).
[56] R. F. Werner, Phys. Rev. A 40, 4277 (1989).
[57] S.L. Braunstein, A.S. Lane, and C.M. Caves, Phys. Rev. Lett. 69, 2153 (1992); A.S. Lane, S.L. Braunstein, and C.M. Caves, Phys. Rev. A 47, 1667 (1993).
[58] It has been argued that for indistinguishable particles, the entanglement defined in this way is not physical since the individual particles cannot be adressed [59, 60]. Nontheless, in the context of interferometry, it is a useful resource for sub shot-noise sensitivity.
[59] P. Zanardi, Phys. Rev. A 65, 042101 (2002).
[60] F. Benatti, R. Floreanini, and U. Marzolino, Annals of Physics 325, 924 (2010).
[61] J. J. Sakurai, Modern Quantum Mechanics (AddisonWesley, 1994).
[62] The conversion from ~nmax to UCopLt = ⊗kN=1U opt can be done as follows. If ~nmax = OT yˆ has been found, then from yˆ · ~nmax = cos(φ) and yˆ × ~nmax = sin(φ) nˆ the angle φ and the axis of rotation nˆ can be extracted, and U opt = exp(iφ~σˆ · nˆ/2) is (up to a phase) the unitary matrix such that (U opt)†σˆyˆU opt = σˆ~nmax [61]. The positive sign in the exponent is due to the fact that OT appears in the definition of ~nmax.
[63] A´ . Rivas and A. Luis, Phys. Rev. A 77, 022105 (2008).
[64] G. T´oth, C. Knapp, O. Gu¨hne, and H. J. Briegel, Phys. Rev. A 79, 042334 (2009).
[65] To see this, note that Eq. (30d) in Ref. [64] implies that for separable states λmax(N γ) ≤ (N − 1)T r(γ) − N (N − 2)/4 since X = N γ + hJ~ihJ~ˆiT ≥ N γ. Using Eq. (30b) ˆ implies then that N λmax(γ) ≤ N (N − 1)/2 − N (N − 2)/4 hence λmax(γ) ≤ N/4. We used the notations of Ref. [64] here.
[66] H. Lee, P. Kok, and J. P. Dowling, J. Mod. Opt. 49, 2325 (2002).
[67] A. Shimizu and T. Miyadera, Phys. Rev. Lett. 89, 270403 (2002).
[68] T. Morimae, A. Sugita, and A. Shimizu, Phys. Rev. A 71, 032317 (2005).
[69] E.P. Wigner and M.M. Yakanse, Proc. Natl. Acad. Sci. U.S.A. 49, 910 (1963).
[70] Z. Chen, Phys. Rev. A 71, 052302 (2005).
[71] M.B. Plenio and S. Virmani, Quant. Inf. Comp. 7, 1 (2007).
[72] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
[73] A. Cabello, Phys. Rev. A 68, 012304 (2003).
[74] M. Hein, W. Du¨r, J. Eisert, R. Raussendorf, M. V. den Nest, and H.-J. Briegel, in Proceedings of the International School of Physics “Enrico Fermi” on Quantum Computers, Algorithms and Chaos, edited by P. Casati, D. Shepelyansky, and G. Beneti (IOS Press, Varenna, Italy, 2005), see also quant-ph/0602096.
[75] D. M. Greenberger, M. Horne, and A. Zeilinger, Am. J. Phys. 58, 1131 (1990).
[76] D. Gottesman, Phys. Rev. A 54, 1862 (1996).
[77] M. Hein, J. Eisert, and H. J. Briegel, Phys. Rev. A 69, 062311 (2004).
[78] W. Du¨r, H. Aschauer, and H.-J. Briegel, Phys. Rev. Lett. 91, 107903 (2003).
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BibTeX entry
@article{oai:it.cnr:prodotti:58262,
title = {Not all pure entangled states are useful for sub-shot-noise interferometry},
author = {Hyllus P and Guhne O and Smerzi A},
doi = {10.1103/physreva.82.012337 and 10.48550/arxiv.0912.4349},
year = {2010}
}
0000-0002-4967-6939
