Abstract
For many completely positive maps repeated compositions will eventually become entanglement breaking. To quantify this behaviour we develop a technique based on the Schmidt number: If a completely positive map breaks the entanglement with respect to any qubit ancilla, then applying it to part of a bipartite quantum state will result in a Schmidt number bounded away from the maximum possible value. Iterating this result puts a successively decreasing upper bound on the Schmidt number arising in this way from compositions of such a map. By applying this technique to completely positive maps in dimension three that are also completely copositive we prove the so-called PPT squared conjecture in this dimension. We then give more examples of completely positive maps where our technique can be applied, e.g. maps close to the completely depolarizing map, and maps of low rank. Finally, we study the PPT squared conjecture in more detail, establishing equivalent conjectures related to other parts of quantum information theory, and we prove the conjecture for Gaussian quantum channels.
Similar content being viewed by others
References
Horodecki, M., Shor, P.W., Ruskai, M.B.: Entanglement breaking channels. Rev. Math. Phys. 15(06), 629–641 (2003)
Holevo, A.S., Werner, R.F.: Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63(3), 032312 (2001)
Horodecki, K., Horodecki, M., Horodecki, P., Oppenheim, J.: Secure key from bound entanglement. Phys. Rev. Lett. 94(16), 160502 (2005)
Lami, L., Giovannetti, V.: Entanglement-breaking indices. J. Math. Phys. 56(9), 092201 (2015)
De Pasquale, A., Giovannetti, V.: Quantifying the noise of a quantum channel by noise addition. Phys. Rev. A 86(5), 052302 (2012)
De Pasquale, A., Mari, A., Porzio, A., Giovannetti, V.: Amendable Gaussian channels: restoring entanglement via a unitary filter. Phys. Rev. A 87(6), 062307 (2013)
Lami, L., Giovannetti, V.: Entanglement-saving channels. J. Math. Phys. 57(3), 032201 (2016)
Christandl, M.: PPT square conjecture. In: Banff International Research Station Workshop: Operator Structures in Quantum Information Theory (2012)
Bäuml, S., Christandl, M., Horodecki, K., Winter, A.: Limitations on quantum key repeaters. Nat. Commun. 6, 6908 (2015)
Christandl, M., Ferrara, R.: Private states, quantum data hiding, and the swapping of perfect secrecy. Phys. Rev. Lett. 119(22), 220506 (2017)
Kennedy, M., Manor, N.A., Paulsen, V.I.: Composition of PPT maps. Quantum Inf. Comput. 18(5 & 6), 0472–0480 (2018)
Rahaman, M., Jaques, S., Paulsen, V.I.: Eventually entanglement breaking maps. J. Math. Phys. 59(6), 062201 (2018)
Terhal, B.M., Horodecki, P.: Schmidt number for density matrices. Phys. Rev. A 61(4), 040301 (2000)
Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10(3), 285–290 (1975)
Skowronek, Ł., Størmer, E., Życzkowski, K.: Cones of positive maps and their duality relations. J. Math. Phys. 50(6), 062106 (2009)
Chruściński, D., Kossakowski, A.: On partially entanglement breaking channels. Open Syst. Inf. Dyn. 13(1), 17–26 (2006)
Woronowicz, S.L.: Positive maps of low dimensional matrix algebras. Rep. Math. Phys. 10(2), 165–183 (1976)
Tang, W.-S.: On positive linear maps between matrix algebras. Linear Algebra Appl. 79, 33–44 (1986)
Horodecki, P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232(5), 333–339 (1997)
Sanpera, A., Bruß, D., Lewenstein, M.: Schmidt-number witnesses and bound entanglement. Phys. Rev. A 63(5), 050301 (2001)
Yang, Y., Leung, D.H., Tang, W.-S.: All 2-positive linear maps from M3(C) to M3(C) are decomposable. Linear Algebra Appl. 503, 233–247 (2016)
Huber, M., Lami, L., Lancien, C., Müller-Hermes, A.: High-dimensional entanglement in states with positive partial transposition. Phys. Rev. Lett. 121, 200503 (2018)
Gurvits, L., Barnum, H.: Largest separable balls around the maximally mixed bipartite quantum state. Phys. Rev. A 66(6), 062311 (2002)
Johnston, N.: Separability from spectrum for qubit–qudit states. Phys. Rev. A 88(6), 062330 (2013)
Lami, L., Huber, M.: Bipartite depolarizing maps. J. Math. Phys. 57(9), 092201 (2016)
Cariello, D.: Separability for weakly irreducible matrices. Quantum Inf. Comput. 14(15–16), 1308–1337 (2014)
Cariello, D.: Does symmetry imply PPT property? Quantum Inf. Comput. 15(9–10), 812–824 (2015)
Heinosaari, T., Jivulescu, M.A., Reeb, D., Wolf, M.M.: Extending quantum operations. J. Math. Phys. 53(10), 102208 (2012)
Moravčíková, L., Ziman, M.: Entanglement-annihilating and entanglement-breaking channels. J. Phys. A Math. Theor. 43(27), 275306 (2010)
Müller-Hermes, A., Reeb, D., Wolf, M.M.: Positivity of linear maps under tensor powers. J. Math. Phys. 57(1), 015202 (2016)
Filippov, S.N., Rybár, T., Ziman, M.: Local two-qubit entanglement-annihilating channels. Phys. Rev. A 85(1), 012303 (2012)
Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223(1), 1–8 (1996)
Størmer, E.: Duality of cones of positive maps. Preprint arXiv:0810.4253 (2008)
Müller-Hermes, A.: Decomposability of linear maps under tensor powers. J. Math. Phys. 59(10), 102203 (2018)
Collins, B., Yin, Z., Zhong, P.: The PPT square conjecture holds generically for some classes of independent states. J. Phys. A Math. Theor. 51(42), 425301 (2018)
Vollbrecht, K.G.H., Wolf, M.M.: Activating distillation with an infinitesimal amount of bound entanglement. Phys. Rev. Lett. 88(24), 247901 (2002)
Vollbrecht, K.G.H., Werner, R.F.: Entanglement measures under symmetry. Phys. Rev. A 64(6), 062307 (2001)
Audenaert, K., Eisert, J., Jané, E., Plenio, M., Virmani, S., De Moor, B.: Asymptotic relative entropy of entanglement. Phys. Rev. Lett. 87(21), 217902 (2001)
Christandl, M., Schuch, N., Winter, A.: Entanglement of the antisymmetric state. Commun. Math. Phys. 311(2), 397–422 (2012)
Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81(2), 865 (2009)
Holevo, A.S.: Quantum Systems, Channels, Information: A Mathematical Introduction, vol. 16. Walter de Gruyter, Berlin (2013)
Chen, L., Yang, Y., Tang, W.-S.: Schmidt number of bipartite and multipartite states under local projections. Quantum Inf. Process. 16(3), 75 (2017)
Kraus, B., Cirac, J., Karnas, S., Lewenstein, M.: Separability in \(2\times \) N composite quantum systems. Phys. Rev. A 61(6), 062302 (2000)
Acknowledgements
We thank Daniel Cariello for pointing out how his results [26, 27] together with the techniques from [22] imply Corollary 3.5. We also thank Ion Nechita for pointing out the different version of Theorem A.1. MC and AMH acknowledge financial support from the European Research Council (ERC Grant Agreement no 337603) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). MW acknowledges the hospitality of the QMATH Centre.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by David Pérez-García.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Christandl, M., Müller-Hermes, A. & Wolf, M.M. When Do Composed Maps Become Entanglement Breaking?. Ann. Henri Poincaré 20, 2295–2322 (2019). https://doi.org/10.1007/s00023-019-00774-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-019-00774-7