Given many independent and identically-distributed (i.i.d.) copies of a quantum system described either by the state ρ or σ (called null and alternative hypotheses, respectively), what is the optimal measurement to learn the identity of the true state? In asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying ρ instead of σ, while requiring that the probability that σ is identified in the place of ρ is bounded by a small fixed number. Quantum Stein’s Lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of ρ and σ.
We present a generalization of quantum Stein’s Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in a very similar fashion to quantum Stein’s Lemma, in terms of the quantum relative entropy.
Our result has two applications to entanglement theory. First it gives an operational meaning to an entanglement measure known as regularized relative entropy of entanglement. Second, it shows that this measure is faithful, being strictly positive on every entangled state. This implies, in particular, that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition.
Bjelakovic I., Deuschel J.-D., Krueger T., Seiler R., Siegmund-Schultze Ra., Szkola A. (2008) Typical support and Sanov large deviations of correlated states. Commun. Math. Phys. 279: 559MATHCrossRefADSGoogle Scholar
Mosonyi M., Hiai F., Ogawa T., Fannes M. (2008) Asymptotic distinguishability measures for shift-invariant quasi-free states of fermionic lattice systems. J. Math. Phys. 49: 032112CrossRefMathSciNetADSGoogle Scholar
Bjelaković I., Deuschel J.D., Krüger T., Seiler R., Siegmund-Schultze Ra., Szola A. (2005) A quantum version of Sanov’s theorem. Commun. Math. Phys. 260: 659MATHCrossRefADSGoogle Scholar
Renner, R.: Security of quantum key distribution. PhD thesis ETH, Zurich 2005Google Scholar
Boyd S., Vandenberghe L. (2000) Convex Optimization. Cambridge University Press, CambridgeGoogle Scholar
Bathia R. Matrix Analysis. Graduate Texts in Mathematics. Berlin-Heidelberg-New York: Springer, 1996Google Scholar
Horodecki K., Horodecki M., Horodecki P., Oppenheim J. (2005) Locking entanglement measures with a single qubit. Locking entanglement measures with a single qubit. Phys. Rev. Lett. 94: 200501CrossRefMathSciNetADSGoogle Scholar
Fulton W., Harris J. (1991) Representation Theory: A First Course. Springer, New YorkMATHGoogle Scholar
Horodecki K., Horodecki M., Horodecki P., Leung D., Oppenheim J. (2008) Quantum key distribution based on private states: unconditional security over untrusted channels with zero quantum capacity. IEEE Trans. Inf. Theory 54: 2604CrossRefMathSciNetGoogle Scholar
Horodecki K., Horodecki M., Horodecki P., Leung D., Oppenheim J. (2008) Unconditional privacy over channels which cannot convey quantum information. Phys. Rev. Lett. 100: 110502CrossRefADSGoogle Scholar
Dembo A., Zeitouni O. (1998) Large Deviations Techniques and Applications. Springer-Verlag, Berlin-Heidelberg-New YorkMATHGoogle Scholar
Christandl, M.: The structure of bipartite quantum states - insights from group theory and cryptography. PhD thesis, February 2006, University of Cambridge, available at http://arxiv.org/abs/quant-ph/0604183v1, 2006
Datta N., Renner R. (2009) Smooth Renyi entropies and the quantum information spectrum. IEEE Trans. Inf. Theory 55: 2807CrossRefGoogle Scholar
Han T.S. (2003) Information-spectrum Methods in Information Theory. Springer, Berlin-Heidelberg-New YorkMATHGoogle Scholar