Abstract
We address the issue of one-side local broadcasting for correlations in a quantum bipartite state, and conjecture that the correlations can be broadcast if and only if they are classical–quantum, or equivalently, the quantum discord, as defined by Ollivier and Zurek (Phys Rev Lett 88:017901, 2002), vanishes. We prove this conjecture when the reduced state is maximally mixed and further provide various plausible arguments supporting this conjecture. Moreover, we demonstrate that the conjecture implies the following two elegant and fundamental no-broadcasting theorems: (1) The original no-broadcasting theorem by Barnum et al. (Phys Rev Lett 76:2818, 1996), which states that a family of quantum states can be broadcast if and only if the quantum states commute. (2) The no-local-broadcasting theorem for quantum correlations by Piani et al. (Phys Rev Lett 100:090502, 2008), which states that the correlations in a single bipartite state can be locally broadcast if and only if they are classical. The results provide an informational interpretation for classical–quantum states from an operational perspective and shed new lights on the intrinsic relation between non-commutativity and quantumness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barnum H., Barrett J., Leifer M., Wilce A.: Generalized no-broadcasting theorem. Phys. Rev. Lett. 99, 240501 (2007)
Barnum H., Caves C.M., Fuchs C.A., Jozsa R., Schumacher B.: Noncommuting mixed states cannot be broadcast. Phys. Rev. Lett. 76, 2818–2821 (1996)
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)
Biham E., Brassard G., Kenigsberg D., Mor T.: Quantum computing without entanglement. Theoret. Comput. Sci. 320, 15–33 (2004)
Braunstein S.L., Caves C.M., Jozsa R., Linden N., Popescu S., Schack R.: Separability of very noisy mixed states and implications for NMR quantum computing. Phys. Rev. Lett. 83, 1054–1057 (1999)
Christandl M., Winter A.: “Squashed entanglement”—an additive entanglement measure. J. Math. Phys. 45, 829–840 (2004)
Datta A., Flammia S.T., Caves C.M.: Entanglement and the power of one qubit. Phys. Rev. A 72, 042316 (2005)
Datta A., Vidal G.: Role of entanglement and correlations in mixed-state quantum computation. Phys. Rev. A 75, 042310 (2007)
Devetak I., Winter A.: Distilling common randomness from bipartite quantum states. IEEE Trans. Inf. Theory 50, 3183–3196 (2004)
Dieks D.: Communications by EPR devices. Phys. Lett. A 92, 271–272 (1982)
Groisman B., Popescu S., Winter A.: Quantum, classical, and total amount of correlations in a quantum state. Phys. Rev. A 72, 032317 (2005)
Hayden P., Jozsa R., Petz D., Winter A.: Structure of states which satisfy strong subadditivity of quantum entropy with equality. Commun. Math. Phys. 246, 359–374 (2004)
Henderson L., Vedral V.: Classical, quantum and total correlations. J. Phys. A 34, 6899–6905 (2001)
Horodecki M.: Entanglement measures. Quantum Inf. Comput. 1, 3–26 (2001)
Kalev A., Hen I.: No-broadcasting theorem and its classical counterpart. Phys. Rev. Lett. 100, 210502 (2008)
Li N., Luo S.: Total versus quantum correlations in quantum states. Phys. Rev. A 76, 032327 (2007)
Li N., Luo S.: Classical states versus separable states. Phys. Rev. A 78, 024303 (2008)
Lindblad G.: Completely positive maps and entropy inequalities. Commun. Math. Phys. 40, 147–151 (1975)
Lindblad G.: A general no-cloning theorem. Lett. Math. Phys. 47, 189–196 (1999)
Luo S.: From quantum no-cloning to wave-packet collapse. Phys. Lett. A 374, 1350–1353 (2010)
Luo S.: Using measurement-induced disturbance to characterize correlations as classical or quantum. Phys. Rev. A 77, 022301 (2008)
Luo S.: Quantum discord for two-qubit systems. Phys. Rev. A 77, 042303 (2008)
Luo S., Li N., Cao X.: Relation between “no broadcasting” for noncommuting states and “no local broadcasting” for quantum correlations. Phys. Rev. A 79, 054305 (2009)
Meyer D.A.: Sophisticated quantum search without entanglement. Phys. Rev. Lett. 85, 2014–2017 (2000)
Ollivier H., Zurek W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2002)
Piani M., Horodecki P., Horodecki R.: No-local-broadcasting theorem for multipartite quantum correlations. Phys. Rev. Lett. 100, 090502 (2008)
Scarani V., Iblisdir S., Gisin N., Acín A.: Quantum cloning. Rev. Mod. Phys. 77, 1225–1256 (2005)
Schumacher B., Westmoreland M.D.: Quantum mutual information and the one-time pad. Phys. Rev. A 74, 042305 (2006)
Shabani A., Lidar D.A.: Vanishing quantum discord is necessary and sufficient for completely positive maps. Phys. Rev. Lett. 102, 100402 (2009)
Uhlmann A.: Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory. Commun. Math. Phys. 54, 21–32 (1977)
Umegaki H.: Conditional expectation in an operator algebra, IV. Entropy and information. Kōdai Math. Sem. Rep. 14, 59–85 (1962)
Vedral V.: The role of relative entropy in quantum information theory. Rev. Mod. Phys. 74, 197–234 (2002)
Werner R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277–4281 (1989)
Wootters W.K.: Entanglement of formation and concurrence. Quantum Inf. Comput. 1, 27–44 (2001)
Wootters W.K., Zurek W.H.: A single quantum cannot be cloned. Nature (London) 299, 802–803 (1982)
Yuen H.P.: Amplification of quantum states and noiseless photon amplifiers. Phys. Lett. A 113, 405–407 (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Luo, S. On Quantum No-Broadcasting. Lett Math Phys 92, 143–153 (2010). https://doi.org/10.1007/s11005-010-0389-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-010-0389-1