State selective protocols, like entanglement purification, lead to an essentially non-linear quantum evolution, unusual in naturally occurring quantum processes. Sensitivity to initial states in quantum systems, stemming from such non-linear dynamics, is a promising perspective for applications. Here we demonstrate that chaotic behaviour is a rather generic feature in state selective protocols: exponential sensitivity can exist for all initial states in an experimentally realisable optical scheme. Moreover, any complex rational polynomial map, including the example of the Mandelbrot set, can be directly realised. In state selective protocols, one needs an ensemble of initial states, the size of which decreases with each iteration. We prove that exponential sensitivity to initial states in any quantum system has to be related to downsizing the initial ensemble also exponentially. Our results show that magnifying initial differences of quantum states (a Schrödinger microscope) is possible; however, there is a strict bound on the number of copies needed.

Department of Theoretical Physics Budapest University of Technology and Economics Budapest H 1111 Hungary

Faculty of Nuclear Sciences and Physical Engineering Czech Technical University Prague Prague 115 19 Praha 1 Czech Republic

Institute for Solid State Physics and Optics Wigner Research Centre for Physics Hungarian Academy of Sciences Budapest H 1121 Hungary

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Rabitz H. Focus on Quantum Control. New J. Phys. 11, 105030 (2009).

Knill E., Laflamme R. & Milburn G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 46–52 (2001). PubMed

Gendra B., Ronco-Bonvehi E., Calsamiglia J., Muñoz-Tapia R. & Bagan E. Quantum Metrology Assisted by Abstention. Phys. Rev. Lett. 110, 100501 (2013). PubMed

Duan L. M., Lukin M. D., Cirac J. I. & Zoller P. Long-distance quantum communication with atomic ensembles and linear optics. Nature 414, 413–418 (2001). PubMed

Bennett C. H., Brassard G., Popescu S., Schumacher B., Smolin J. A. & Wootters W. K. Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels. Phys. Rev. Lett. 76, 722 (1996). PubMed

Bennett C. H., DiVincenzo D. P., Smolin J. A. & Wootters W. K. Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996). PubMed

Deutsch D., Ekert A., Jozsa R., Macchiavello C., Popescu S. & Sanpera A. Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels. Phys. Rev. Lett. 77, 2818 (1996). PubMed

Bechmann-Pasquinucci H., Huttner B. & Gisin N. Non-linear quantum state transformation of spin-1/2. Phys. Lett. A 242, 198–204 (1998).

Scott A. J. & Milburn G. J. Quantum nonlinear dynamics of continuously measured systems. Phys. Rev. A 63, 042101 (2001).

Habib S., Jacobs K. & Shizume K. Emergence of Chaos in Quantum Systems Far from the Classical Limit. Phys. Rev. Lett. 96, 010403 (2006). PubMed

Everitt M. J. On the correspondence principle: implications from a study of the nonlinear dynamics of a macroscopic quantum device. New J. Phys. 11, 013014 (2009).

Cvitanović P., Artuso R., Mainieri R., Tanner G. & Vattay G. Chaos: Classical and Quantum, ChaosBook.org (Niels Bohr Institute, Copenhagen, 2012).

Kiss T., Jex I., Alber G. & Vymĕtal S. Complex chaos in the conditional dynamics of qubits. Phys. Rev. A 74, 040301(R) (2006).

Kiss T., Vymĕtal S., Tóth L. D., Gábris A., Jex I. & Alber G. Measurement induced chaos with entangled states. Phys. Rev. Lett. 107, 100501 (2011). PubMed

Guan Y., Nguyen D. Q., Xu J. & Gong J. Reexamination of measurement-induced chaos in entanglement-purification protocols. Phys. Rev. A 87, 052316 (2013).

Devaney R. L. An Introduction to Chaotic Dynamical Systems (Westview Press, 2003).

Pan J. W., Simon C., Brukner S. & Zeilinger A. Entanglement purification for quantum communication. Nature 410, 1067–1070 (2001). PubMed

Pan J. W., Gasparoni S., Ursin R., Weihs G. & Zeilinger A. Experimental entanglement purification of arbitrary unknown states. Nature 423, 417–422 (2003). PubMed

Milnor J. W. Dynamics in One Complex Variable (Princeton Univ. Press, 2006).

Milnor J. W. On Lattès maps. in Dynamics on the Riemann Sphere [ Hjorth P. & Petersen C. L. (eds.)] (Eur. Math. Soc., Zürich, 2006).

Bolsinov A. V. & Fomenko A. T. Integrable Hamiltonian Systems: Geometry, Topology, Classification (CRC Press, 2004).

Mahadev U. & de Wolf R. Rational approximations and quantum algorithms with postselection. Quant. Inf. & Comp. 15, 295–307 (2015).

Lloyd S. & Slotine J. E. Quantum feedback with weak measurements. Phys. Rev. A 62, 012307 (2000).

Aaronson S. Quantum computing, postselection, and probabilistic polynomial-time. Proc. R. Soc. A 461, 3473–3482 (2005).

Helstrom C. W. Quantum Detection and Estimation Theory (Academic Press, 1976).

Bagan E., Muñoz-Tapia R., Olivares-Rentería G. A. & Bergou J. A. Optimal discrimination of quantum states with a fixed rate of inconclusive outcomes. Phys. Rev. A 86, 040303(R) (2012).

Herzog U. Optimal state discrimination with a fixed rate of inconclusive results: Analytical solutions and relation to state discrimination with a fixed error rate. Phys. Rev. A 86, 032314 (2012).

da Luz M. G. E. & Anteneodo C. Nonlinear dynamics in meso and nano scales: fundamental aspects and applications. Phil. Trans. R. Soc. A 369, 245–259 (2011). PubMed

Madhok V., Riofrío C. A., Ghose S. & Deutsch I. H. Information Gain in Tomography – A Quantum Signature of Chaos. Phys. Rev. Lett. 112, 014102 (2014). PubMed

Douady A. Does a Julia set depend continuously on the polynomial? in Complex dynamical systems [ Devaney R. L. (ed.)] Proc. Sympos. Appl. Math. 49, 91–138 (1994).