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Approximate Quantum Markov Chains

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Approximate Quantum Markov Chains

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 28))

Abstract

In this chapter we discuss the concept of quantum Markov chains with a particular focus on the robustness of their properties. This results in a new class of states called approximate quantum Markov chains. To understand the properties of these states the mathematical tools introduced in the preceding chapters will be helpful. As it happens, the key result that justifies the definition of approximate quantum Markov chains (see Theorem 5.5) is closely related to various celebrated entropy inequalities. We explain this connection and show how the new insights about approximate quantum Markov chains can be used to prove strengthened versions of different famous entropy inequalities.

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Notes

  1. 1.

    The max-relative entropy and its properties are discussed in more detail in Sect. 2.5.4. It is the largest sensible relative entropy measure.

  2. 2.

    Recall that for recovery maps that leave the B system invariant the \(\varLambda _{\max }\)-term vanishes as explained above.

  3. 3.

    Equivalent means that every statement can be derived from every other one by simple manipulations only.

  4. 4.

    In case \(\mathscr {E}(\sigma ) \in \mathsf {P}_{\!\!\!+}(B)\) the recovery map \(\bar{\mathscr {T}}_{\sigma ,\mathscr {E}}\) is trace-preserving.

  5. 5.

    Choosing \(\rho =\rho _{ABC}\), \(\sigma =\mathrm {id}_A \otimes \rho _{BC}\), and \(\mathscr {E}=\mathrm {tr}_C\) we obtain that \(I(A:C|B)_{\rho }=0\) implies \(\mathscr {T}^{[t]}_{B\rightarrow BC}(\rho _{AB}) = \rho _{ABC}\) for \(\mathscr {T}^{[t]}_{B\rightarrow BC}\) defined in (5.2).

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  1. Institute for Theoretical Physics, ETH Zurich, Zürich, Switzerland

    David Sutter

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Sutter, D. (2018). Approximate Quantum Markov Chains. In: Approximate Quantum Markov Chains. SpringerBriefs in Mathematical Physics, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-319-78732-9_5

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