Abstract
By using a geometrical approach, we investigate the behavior of the quantum Toffoli gate in connection to quantum correlations. Special attention is paid to states with maximally mixed marginals. Finally and in the same vein, we scrutinize the Hadamard gate.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Cf. [28] for a discussion of some problems with the geometric measure of discord.
Or more generally, an arbitrary quantum operation \({\mathcal {E}}\).
Remember that both definitions of discord yield the same family of null-discordant states.
References
Aharonov, D.: A simple proof that Toffoli and Hadamard are quantum universal. arXiv:quant-ph/0301040v1 (2003)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)
Stojanović, V.M., Fedorov, A., Wallraff, A., Bruder, C.: Quantum-control approach to realizing a Toffoli gate in circuit QED. Phys. Rev. B 85, 054504 (2012)
Guo, Y., Zhao, Z., Wang, Y., Wang, P., Huang, D., Ho Lee, M.: On implementing nondestructive triplet Toffoli gate with entanglement swapping operations via the GHZ state analysis. Quantum Inf. Process 13, 2039–2047 (2014)
Aharanov, D., Kitaev, A., Nisan, N.: Quantum circuits with mixed states. 13th Annual ACM Symposium on Theory of Computation, STOC, pp. 20–30 (1998)
Tarasov, V.: Quantum computer with mixed states and four-valued logic. J. Phys. A Math. Gen. 35, 5207 (2002)
Dalla Chiara, M.L., Giuntini, R., Sergioli, G.: Probability in quantum computation and in quantum computational logics. Math. Struct. Comput. Sci. 14, 1–14 (2013)
Deutsch, D.: Quantum theory, the Church–Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A 400, 97–117 (1985)
Freytes, H., Sergioli, G.: Fuzzy approach for Toffoli gate in quantum computation with mixed states. Rep. Math. Phys. 74, 154–180 (2014)
Dakić, B., Vedral, V., Brukner, Č.: Necessary and sufficient condition for nonzero quantum discord. Phys. Rev. Lett. 105, 190502 (2010)
Werner, R.F.: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40(8), 4277–4281 (1989)
Feynman, R.P.: Simulating physics with computers. International Journal of Theoretical Physics 21(6/7), 467–488 (1982)
Modi, K., Brodutch, A., Cable, H., Paterek, T., Vedral, V.: The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655–1707 (2012)
Datta, A., Shaji, A., Caves, C.M.: Quantum discord and the power of one qubit. Phys. Rev. Lett. 100, 050502 (2008)
Lanyon, B.P., Barbieri, M., Almeida, M.P., White, A.G.: Experimental quantum computing without entanglement. Phys. Rev. Lett. 101, 200501 (2008)
Knill, E., Laflamme, R.: Power of one bit of quantum information. Phys. Rev. Lett. 81, 5672 (1998)
Brodutch, A.: Discord and quantum computational resources. Phys. Rev. A 88, 022307 (2013)
Brodutch, A., Gilchrist, A., Terno, D.R., Wood, C.J.: Quantum discord in quantum computation. J. Phys. Conf. Ser. 306, 012030 (2011)
Zhang, C., Yu, S., Chen, Q., Oh, C.H.: Quantum discord of two-qubit X states. Phys. Rev. A 84, 032122 (2011)
Moqadam, J.K., Portugal, R., Svaiter, N.F., de Oliveira Corrêa, G.: Analyzing the Toffoli gate in disordered circuit QED. Phys. Rev. A 87, 042324 (2013)
Song, L.-C., Xia, Y., Song, J.: Noise resistance of Toffoli gate in an array of coupled cavities. J. Mod. Opt. 61(16), 1290–1297 (2014)
Singh, H., Chakraborty, T., Panigrahi, P.K., Mitra, C.: Experimental estimation of discord in an antiferromagnetic Heisenberg compound \(\text{ Cu }(\text{ NO }_{3})_{2} \cdot 2.5\,\text{ H }_{2}\text{ O }\). Quantum Inf. Process. 14, 951–961 (2015)
Hayashi, M.: Quantum Information. Springer, Berlin (2006)
Datta, A.: A condition for the nullity of quantum discord. arXiv preprint, arXiv:1003.5256 (2010)
Ferraro, A., Aolita, L., Cavalcanti, D., Cucchietti, F.M., Acin, A.: Almost all quantum states have nonclassical correlations. Phys. Rev. A 81, 052318 (2010)
Luo, S., Fu, S.: Geometric measure of quantum discord. Phys. Rev. A 82, 034302 (2010)
Wei, H.-R., Ren, B.-C., Deng, F.-G.: Geometric measure of quantum discord for a two-parameter class of states in a qubit–qutrit system under various dissipative channels. Quantum Inf. Process. 12, 1109–1124 (2013)
Piani, M.: Problem with geometric discord. Phys. Rev. A 86, 034101 (2012)
Toffoli, T.: Reversible computing. In: Proceedings of the 7th Colloquium on Automata, Languages and Programming. Springer, London, pp. 632–644 (1980)
Freytes, H., Domenech, G.: Quantum computational logic with mixed states. Math. Logic Q. 59, 27–50 (2013)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, London (1998)
Goldblatt, R.: Topoi, The Categorial Analysis of Logic. Dover Publications Inc., Mineola (2006)
Husemoller, D.: Fibre bundles, 3rd edn. Springer, Berlin (1994)
Kraus, K.: States, Effects and Operations. Springer, Berlin (1983)
Freytes, H., Sergioli, G., Aricó, A.: Representing continuous t-norms in quantum computation with mixed states. J. Phys. A 43(46), 465306 (2010)
Cereceda, J.: Three-particle entanglement versus three-particle nonlocality. Phys. Rev. A 66, 024102 (2002)
Luo, S., Fu, S.: Measurement-induced nonlocality. Phys. Rev. Lett. 106, 120401 (2011)
Ming-Liang, H., Fan, H.: Measurement-induced nonlocality based on trace norm. New J. Phys. 17, 033004 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Holik, F., Sergioli, G., Freytes, H. et al. Toffoli gate and quantum correlations: a geometrical approach. Quantum Inf Process 16, 55 (2017). https://doi.org/10.1007/s11128-016-1509-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11128-016-1509-3