Global versus local quantum correlations in the Grover search algorithm. (English) Zbl 1333.81097
Summary: Quantum correlations are thought to be the reason why certain quantum algorithms overcome their classical counterparts. Since the nature of this resource is still not fully understood, we shall investigate how entanglement and nonlocality among register qubits vary as the Grover search algorithm is run. We shall encounter pronounced differences between the measures employed as far as bipartite and global correlations are concerned.
References:
| [1] | Lo, H.-K., Popescu, S., Spiller, T.: Introduction to Quantum Computation and Information. World Scientific, River-Edge (1998) · Zbl 0957.68045 · doi:10.1142/9789812385253 |
| [2] | Galindo, A., Martín-Delgado, M.A.: Information and computation: classical and quantum aspects. Rev. Mod. Phys. 74, 347 (2002) · Zbl 1205.94004 · doi:10.1103/RevModPhys.74.347 |
| [3] | Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) · Zbl 1049.81015 |
| [4] | Williams, C.P., Clearwater, S.H.: Explorations in Quantum Computing. Springer, New York (1997) · Zbl 0906.68069 |
| [5] | Williams, C.P.: Quantum Computing and Quantum Communications. Springer, Berlin (1998) |
| [6] | Bennett, C.H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1993) · Zbl 1051.81505 · doi:10.1103/PhysRevLett.70.1895 |
| [7] | Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881 (1993) · Zbl 0968.81506 · doi:10.1103/PhysRevLett.69.2881 |
| [8] | Ekert, A., Jozsa, R.: Quantum computation and Shors factoring algorithm. Rev. Mod. Phys. 68, 773 (1996) · doi:10.1103/RevModPhys.68.733 |
| [9] | Berman, G.P., Doolen, G.D., Mainieri, R., Tsifrinovich, V.I.: Introduction to Quantum Computers. World Scientific, Singapore (1998) · Zbl 1010.81007 · doi:10.1142/3808 |
| [10] | Batle, J., Casas, M.: Nonlocality and entanglement in the XY-model. Phys. Rev. A 82, 062101 (2010) · Zbl 1231.81010 · doi:10.1103/PhysRevA.82.062101 |
| [11] | Batle, J., Casas, M.: Nonlocality and entanglement in qubit systems. J. Phys. A Math. Theor. 44, 445304 (2011) · Zbl 1231.81010 · doi:10.1088/1751-8113/44/44/445304 |
| [12] | Gottesman, D.: Theory of fault-tolerant quantum computation. Phys. Rev. A 57, 127 (1998) · doi:10.1103/PhysRevA.57.127 |
| [13] | Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Sci Stat. Comput. 26, 1484-1509 (1997) · Zbl 1005.11065 · doi:10.1137/S0097539795293172 |
| [14] | Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325 (1997) · doi:10.1103/PhysRevLett.79.325 |
| [15] | Fang, Y.Y., Kaszlikowski, D., Chin, C., Tay, K., Kwek, L.C., Oh, C.H.: Correlations in Grover search. Phys. Lett. A 345, 265 (2005) · Zbl 1345.81010 · doi:10.1016/j.physleta.2005.07.017 |
| [16] | Cui, J., Fan, H.: Correlations in the Grover search. J. Phys. A Math. Theor. 43, 045305 (2010) · Zbl 1186.81039 · doi:10.1088/1751-8113/43/4/045305 |
| [17] | Qu, R., et al.: Multipartite entanglement and Grover’s search algorithm. arXiv:1210.3418 (2012) · Zbl 1186.81039 |
| [18] | Chakraborty, S., et al.: Entanglement in the Grover’s Search Algorithm. arXiv:1305.4454 (2013) |
| [19] | Vidal, G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003) · doi:10.1103/PhysRevLett.91.147902 |
| [20] | Orus, R., Latorre, J.I.: Universality of entanglement and quantum-computation complexity. Phys. Rev. A 69, 052308 (2004) · doi:10.1103/PhysRevA.69.052308 |
| [21] | Ukena, A., Shimizu, A.: Macroscopic entanglement in quantum computation. quant-ph/0505057 [quant-ph] (2005) |
| [22] | Shimoni, Y., Shapira, D., Biham, O.: Entangled quantum states generated by Shor’s factoring algorithm. Phys. Rev. A. 72, 062308 (2005) · doi:10.1103/PhysRevA.72.062308 |
| [23] | Brown, I., Stepney, S., Sudbery, A., Braunstein, S.L.: Searching for highly entangled multi-qubit states. J. Phys. A Math. Gen. 38, 1119 (2005) · Zbl 1062.81007 · doi:10.1088/0305-4470/38/5/013 |
| [24] | Meyer, D.A., Wallach, N.R.: Global entanglement in multiparticle systems. J. Math. Phys. 43, 4273 (2002) · Zbl 1060.81506 · doi:10.1063/1.1497700 |
| [25] | Brennen, G.K.: An observable measure of entanglement for pure states of multi-qubit systems. Quantum Inf. Comput. 3, 619-626 (2003) · Zbl 1152.81683 |
| [26] | Weinstein, Y.S., Hellberg, C.S.: Entanglement generation of nearly random operators. Phys. Rev. Lett. 95, 030501 (2005) · Zbl 1255.82028 · doi:10.1103/PhysRevLett.95.030501 |
| [27] | Weinstein, Y.S., Hellberg, C.S.: Matrix-element distributions as a signature of entanglement generation. Phys. Rev. A 72, 022331 (2005) · doi:10.1103/PhysRevA.72.022331 |
| [28] | Cao, M., Zhu, S.: Thermal entanglement between alternate qubits of a four-qubit Heisenberg XX chain in a magnetic field. Phys. Rev. A 71, 034311 (2005) · Zbl 1227.81021 · doi:10.1103/PhysRevA.71.034311 |
| [29] | Lakshminarayan, A., Subrahmanyam, W.: Multipartite entanglement in a one-dimensional time-dependent Ising model. Phys. Rev. A 71, 062334 (2005) · doi:10.1103/PhysRevA.71.062334 |
| [30] | Scott, A.J.: Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions. Phys. Rev. A 69, 052330 (2004) · doi:10.1103/PhysRevA.69.052330 |
| [31] | Carvalho, A., Mintert, F., Buchleitner, A.: Decoherence and multipartite entanglement. Phys. Rev. Lett. 93, 230501 (2004) · doi:10.1103/PhysRevLett.93.230501 |
| [32] | Aolita, M., Mintert, F.: Measuring multipartite concurrence with a single factorizable observable. Phys. Rev. Lett. 97, 050501 (2006) · doi:10.1103/PhysRevLett.97.050501 |
| [33] | Calsamiglia, J., Hartmann, L., Durr, W., Briegel, H.: Spin gases: quantum entanglement driven by classical kinematics. Phys. Rev. Lett. 95, 180502 (2005) · doi:10.1103/PhysRevLett.95.180502 |
| [34] | Briegel, H.J., Raussendorf, R.: Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86, 910 (2001) · doi:10.1103/PhysRevLett.86.910 |
| [35] | Cleve, R., Gottesman, D., Lo, H.-K.: How to share a quantum secret. Phys. Rev. Lett. 83, 648 (1999) · doi:10.1103/PhysRevLett.83.648 |
| [36] | Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000) · doi:10.1103/PhysRevA.61.052306 |
| [37] | Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880 (1969) · Zbl 1371.81014 · doi:10.1103/PhysRevLett.23.880 |
| [38] | Collins, D., Gisin, N.: A relevant two qubit Bell inequality inequivalent to the CHSH inequality. J. Phys. A Math. Gen. 37, 1775 (2004) · Zbl 1069.81507 · doi:10.1088/0305-4470/37/5/021 |
| [39] | Tsirelson, B.S.: Quantum generalizations of Bells inequality. Lett. Math. Phys. 4, 93100 (1980) |
| [40] | Tsirelson, B.S.: Quantum analogues of the Bell inequalities. The case of two spatially separated domains. J. Sov. Math. 36, 557570 (1987) · Zbl 0617.46066 |
| [41] | Tsirelson, B.S.: Some results and problems on quantum Bell-type inequalities. Hadron. J. Suppl. 8, 329345 (1993) · Zbl 0788.15008 |
| [42] | Toner, B.: Monogamy of nonlocal quantum correlations. Proc. R. Soc. A 465, 5969 (2009) · Zbl 1186.81012 · doi:10.1098/rspa.2008.0149 |
| [43] | Mermin, N.D.: Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett. 65, 1838 (1990) · Zbl 0971.81507 · doi:10.1103/PhysRevLett.65.1838 |
| [44] | Ardehali, M.: Bell inequalities with a magnitude of violation that grows exponentially with the number of particles. Phys. Rev. A 46, 5375 (1992) · doi:10.1103/PhysRevA.46.5375 |
| [45] | Belinskii, A.V., Klyshko, D.N.: Interference of light and Bells theorem. Phys. Usp. 36, 653 (1993) · doi:10.1070/PU1993v036n08ABEH002299 |
| [46] | Gisin, N., Bechmann-Pasquinucci, H.: Bell inequality, Bell states and maximally entangled states for n qubits. Phys. Lett. A 246, 1 (1998) · Zbl 0941.81006 · doi:10.1016/S0375-9601(98)00516-7 |
| [47] | Scarani, V., Acín, A., Schenck, E., Aspelmeyer, M.: Nonlocality of cluster states of qubits. Phys. Rev. A 71, 042325 (2005) · Zbl 1227.81128 · doi:10.1103/PhysRevA.71.042325 |
| [48] | Svetlichny, G.: Distinguishing three-body from two-body nonseparability by a Bell-type inequality. Phys. Rev. D. 35, 3066 (1987) · doi:10.1103/PhysRevD.35.3066 |
| [49] | Collins, D., Gisin, N., Popescu, S., Roberts, D., Scarani, V.: Bell-type inequalities to detect true n-body nonseparability. Phys. Rev. Lett. 88, 170405 (2002) · doi:10.1103/PhysRevLett.88.170405 |
| [50] | Seevinck, M., Svetlichny, G.: Bell-type inequalities for partial separability in N-particle systems and quantum mechanical violations. Phys. Rev. Lett. 89, 060401 (2002) · Zbl 1267.81047 · doi:10.1103/PhysRevLett.89.060401 |
| [51] | Rossi, M., Bruss, D., Macchiavello, C.: Scale invariance of entanglement dynamics in Grover’s quantum search algorithm. Phys. Rev. A 87, 022331 (2013) · doi:10.1103/PhysRevA.87.022331 |
| [52] | Kendon, V.M., Munro, W.J.: Entanglement and its role in Shor’s algorithm. Quantum Inf. Comput. 6(7), 630 (2006) · Zbl 1152.81750 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
