Communications in Mathematical Physics

, Volume 295, Issue 3, pp 791–828 | Cite as

A Generalization of Quantum Stein’s Lemma

  • Fernando G. S. L. Brandão
  • Martin B. Plenio


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.


Entangle State Relative Entropy Entanglement Measure Direct Part Positive Partial Transpose 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Fernando G. S. L. Brandão
    • 1
    • 2
  • Martin B. Plenio
    • 1
    • 3
  1. 1.Institute of Mathematical Sciences and QOLS, Blackett LaboratoryImperial College LondonLondonUK
  2. 2.Departamento de FisicaUniversidade Federal de Minas GeraisBelo HorizonteBrazil
  3. 3.Institut für Theoretische PhysikUniversität UlmUlmGermany

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