Entropic trade-off relations for quantum operations

Wojciech Roga, Zbigniew Puchala, Lukasz Rudnicki, Karol Życzkowski

Research output: Contribution to journalArticlepeer-review

19 Citations (Scopus)


Spectral properties of an arbitrary matrix can be characterized by the entropy of its rescaled singular values. Any quantum operation can be described by the associated dynamical matrix or by the corresponding superoperator. The entropy of the dynamical matrix describes the degree of decoherence introduced by the map, while the entropy of the superoperator characterizes the a priori knowledge of the receiver of the outcome of a quantum channel Φ. We prove that for any map acting on an N-dimensional quantum system the sum of both entropies is not smaller than lnN. For any bistochastic map this lower bound reads 2lnN. We investigate also the corresponding Rényi entropies, providing an upper bound for their sum, and analyze the entanglement of the bi-partite quantum state associated with the channel.
Original languageEnglish
Article number032308
Number of pages13
JournalPhysical Review A
Issue number3
Publication statusPublished - 7 Mar 2013


  • entropy
  • quantum operation
  • superoperator
  • matrix
  • one-qubit


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