Abstract
We prove an analytic positive lower bound for the geometric distance between entangled positive partial transpose (PPT) states of a broad class and any private state that delivers one secure key bit. Our proof holds for any Hilbert space of finite dimension. Although our result is proven for a specific class of PPT states, we show that our bound nonetheless holds for all known entangled PPT states with non-zero distillable key rates, irrespective of whether they are in our special class or not. Thus, our result naturally leads to the conjecture of impossibility of using PPT-bound entangled state in physical implementation of quantum key distribution.
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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, 1895–1899 (1993)
Bennett C.H., Wiesner S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881–2884 (1992)
Horodecki M., Horodecki P., Horodecki R.: Mixed-state entanglement and distillation: Is there a “bound” entanglement in nature?. Phys. Rev. Lett 80, 5239–5242 (1998)
Shor P.W., Preskill J.: Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett 85, 441–444 (2000)
Curty M., Lewenstein M., Lütkenhaus N.: Entanglement as a precondition for secure quantum key distribution. Phys. Rev. Lett. 92, 217903 (2004)
Curty M., Gühen O., Lewenstein M., Lütkenhaus N.: Detecting two-party quantum correlations in quantum-key-distribution protocols. Phys. Rev. A 71, 022306 (2005)
Horodecki P., Horodecki M., Horodecki R.: Bound entanglement can be activated. Phys. Rev. Lett. 82, 1056 (1999)
Masanes L.: All bipartite entangled states are useful for information processing. Phys. Rev. Lett. 96, 150501 (2006)
Peres A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)
Horodecki M., Horodecki P., Horodecki R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996)
Horodecki K., Horodecki M., Horodecki P., Oppenheim J.: Secure key from bound entanglement. Phys. Rev. Lett 94, 160502 (2005)
Horodecki K., Horodecki M., Horodecki P., Oppenheim J.: General paradigm for distilling classical key from quantum states. IEEE Trans. Inf. Theory 55, 1898 (2009)
Chi D.P., Choi J.W., Kim J.S., Kim T., Lee S.: Bound entangled states with nonzero distillable key rate. Phys. Rev. A 75, 032306 (2007)
Horodecki, P., Augusiak, R.: On quantum cryptography with bipartite bound entangled states. In: Angelakis, D.G., et al. (eds.) Quantum Information Processing: From Theory to Experiment, NATO Science Series III, vol. 199, pp. 19–29. IOS Press, Amsterdam (2006) arXiv:0712.3999
Horodecki K., Pankowski L., Horodecki M., Horodecki P.: Low dimensional bound entanglement with one-way distillable cryptographic key. IEEE Trans. Inf. Theory 54, 2621 (2008)
Kim J.S., Das A., Sanders B.C.: Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity. Phys. Rev. A 79, 012329 (2009)
Vidal G., Werner R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002)
Dür D., Cirac J.I., Lewenstein M., Bruß D.: Distillabillty and partial transposition in bipartite system. Phys. Rev. A 61, 062313 (2000)
Bennett C.H., DiVincenzo D.P., Smolin J.A., Wootters W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996)
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Kim, J.S., Sanders, B.C. On the Geometric Distance Between Quantum States with Positive Partial Transposition and Private States. Lett Math Phys 92, 67–79 (2010). https://doi.org/10.1007/s11005-010-0376-6
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DOI: https://doi.org/10.1007/s11005-010-0376-6