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arXiv:math-ph/0412009 · Physical Review A

Stronger subadditivity of entropy

Élliott H. Lieb, Robert Seiringer

Published Jun 23, 2005• 27 citations
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Abstract

The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy $S[\ensuremath{\varrho}]=\ensuremath{-}\mathrm{Tr}(\ensuremath{\varrho}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varrho})$ of a density matrix ${\ensuremath{\varrho}}_{123}$ on the product of three Hilbert spaces satisfies $S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}S[{\ensuremath{\varrho}}_{23}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}]$. We strengthen this to $S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}{\ensuremath{\sum}}_{\ensuremath{\alpha}}{n}^{\ensuremath{\alpha}}(S[{\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}^{\ensuremath{\alpha}}])$, where the ${n}^{\ensuremath{\alpha}}$ are weights and the ${\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}$ are partitions of ${\ensuremath{\varrho}}_{23}$. Correspondingly, there is a strengthening of the theorem that the map $A\ensuremath{\mapsto}\mathrm{Tr}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[L+\mathrm{ln}\phantom{\rule{0.2em}{0ex}}A]$ is concave. As applications we prove some monotonicity and convexity properties of the Wehrl coherent state entropy and entropy inequalities for quantum gases.

DOI
10.1103/physreva.71.062329

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The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy S[\ensuremathϱ]=\ensuremath−Tr(\ensuremathϱln\ensuremathϱ)S[\ensuremath{\varrho}]=\ensuremath{-}\mathrm{Tr}(\ensuremath{\varrho}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varrho})S[\ensuremathϱ]=\ensuremath−Tr(\ensuremathϱln\ensuremathϱ) of a density matrix \ensuremathϱ123{\ensuremath{\varrho}}_{123}\ensuremathϱ123​ on the product of three Hilbert spaces satisfies S[\ensuremathϱ123]\ensuremath−S[\ensuremathϱ12]\ensuremath⩽S[\ensuremathϱ23]\ensuremath−S[\ensuremathϱ2]S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}S[{\ensuremath{\varrho}}_{23}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}]S[\ensuremathϱ123​]\ensuremath−S[\ensuremathϱ12​]\ensuremath⩽S[\ensuremathϱ23​]\ensuremath−S[\ensuremathϱ2​]. We strengthen this to S[\ensuremathϱ123]\ensuremath−S[\ensuremathϱ12]\ensuremath⩽\ensuremath∑\ensuremathαn\ensuremathα(S[\ensuremathϱ23\ensuremathα]\ensuremath−S[\ensuremathϱ2\ensuremathα])S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}{\ensuremath{\sum}}_{\ensuremath{\alpha}}{n}^{\ensuremath{\alpha}}(S[{\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}^{\ensuremath{\alpha}}])S[\ensuremathϱ123​]\ensuremath−S[\ensuremathϱ12​]\ensuremath⩽\ensuremath∑\ensuremathα​n\ensuremathα(S[\ensuremathϱ23\ensuremathα​]\ensuremath−S[\ensuremathϱ2\ensuremathα​]), where the n\ensuremathα{n}^{\ensuremath{\alpha}}n\ensuremathα are weights and the \ensuremathϱ23\ensuremathα{\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}\ensuremathϱ23\ensuremathα​ are partitions of \ensuremathϱ23{\ensuremath{\varrho}}_{23}\ensuremathϱ23​. Correspondingly, there is a strengthening of the theorem that the map A\ensuremath↦Trexp[L+lnA]A\ensuremath{\mapsto}\mathrm{Tr}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[L+\mathrm{ln}\phantom{\rule{0.2em}{0ex}}A]A\ensuremath↦Trexp[L+lnA] is concave. As applications we prove some monotonicity and convexity properties of the Wehrl coherent state entropy and entropy inequalities for quantum gases.
In this addendum to our paper (2010 New J. Phys. 12 033024), we point out that an elementary consequence of the strong subadditivity inequality allows us to strengthen one of the main conclusions of that paper. In this addendum to our paper [1], we point out that an elementary consequence of the strong subadditivity inequality allows us to strengthen one of the main conclusions of that paper. In [1], we defined notions of measurement and mixing entropy for a broad class of probabilistic theories (which includes finite-dimensional quantum theory). We defined a theory to be monoentropic if for every state of a system in the theory, the mixing entropy (denoted S) is equal to the measurement entropy (denoted H ). This broad class of theories has a notion of composite system, and of the ‘marginal state’ of a subsystem given a state for the composite system, so that the strong subadditivity inequality may be formulated for real-valued functions of states, including the measurement and mixing entropies. We proved the following theorem. Theorem 1 (theorem 4 of [1]). Suppose that a theory has the following properties: (1) it is monoentropic, meaning that measurement entropy equals mixing entropy for all systems, (2) its measurement entropy is strongly subadditive and (3) its measurement entropy satisfies the Holevo bound. Then the theory satisfies information causality. It follows that any theory satisfying these conditions cannot violate Tsirel’son’s bound. In [1] we noted that monoentropicity was only used to establish the technical condition of positivity of conditional information, H(A|C), defined as H(AC)− H(C), when A is classical. This condition does not hold for all theories in the broad class we consider. Shortly after submitting the final version of [1], but too late for inclusion therein, we noticed that it does, however, hold for all theories satisfying strong subadditivity, enabling us to strengthen our theorem as follows. Theorem 2. Suppose that a theory’s measurement entropy is strongly subadditive and satisfies the Holevo bound. Then the theory satisfies information causality. It follows that any theory satisfying these conditions cannot violate Tsirel’son’s bound. As noted in our earlier paper, one can replace the two premises of this theorem with the single premise that the measurement entropy satisfies the data processing inequality. To see that H(A|C)> 0 for classical A follows from strong subadditivity, we recall (from e.g. lemma 1 of [1]), that strong subadditivity is equivalent to the positivity of conditional mutual information, I (A : B|C)> 0. Letting A and B be perfectly correlated classical systems in this inequality, we have I (A : B|C)= H(A|C)+ H(B|C)− H(AB|C) = H(AC)− H(C)+ H(BC) − H(C)− H(ABC)+ H(C) = H(AC)+ H(BC)− H(ABC)− H(C). New Journal of Physics 14 (2012) 129401 (http://www.njp.org/)
The strong subadditivity inequality of von Neumann entropy relates the entropy of subsystems of a tripartite state ${\ensuremath{\rho}}_{ABC}$ to that of the composite system. Here, we define ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$ as the extent to which ${\ensuremath{\rho}}_{ABC}$ fails to satisfy the strong subadditivity inequality $S({\ensuremath{\rho}}_{B})+S({\ensuremath{\rho}}_{C})\ensuremath{\le}S({\ensuremath{\rho}}_{AB})+S({\ensuremath{\rho}}_{AC})$ with equality and investigate its properties. In particular, by introducing auxiliary subsystem EEE, we consider any purification $|{\ensuremath{\psi}}_{ABCE}\ensuremath{\rangle}$ of ${\ensuremath{\rho}}_{ABC}$ and formulate ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$ as the extent to which the bipartite quantum correlations of ${\ensuremath{\rho}}_{AB}$ and ${\ensuremath{\rho}}_{AC}$, measured by entanglement of formation and quantum discord, change under the transformation $B\ensuremath{\rightarrow}BE$ and $C\ensuremath{\rightarrow}CE$. Invariance of quantum correlations of ${\ensuremath{\rho}}_{AB}$ and ${\ensuremath{\rho}}_{AC}$ under such transformation is shown to be a necessary and sufficient condition for vanishing ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$. Our approach allows one to characterize, intuitively, the structure of states for which the strong subadditivity is saturated. Moreover, along with providing a conservation law for quantum correlations of states for which the strong subadditivity inequality is satisfied with equality, we find that such states coincide with those that the Koashi-Winter monogamy relation is saturated.
Abstract Chapter 20 provides a discussion of entropies and entropic quantities in quantum information theory, briefly considering the generalizations of the Rényi entropy to the quantum regime, but mainly focusing on the von Neumann entropy and its properties such as concavity, the Araki-Lieb inequality, and subadditivity. We introduce the quantum relative entropy, quantum joint entropy, quantum conditional entropy, and quantum mutual information, and prove key properties and results for these quantities, including Klein’s inequality, joint convexity, additivity, and monotonicity under completely positive and trace-preserving maps of the relative entropy, as well as weak monotonicity and strong subadditivity of the von Neumann entropy. We then analyse the relation of (negative) conditional entropy and entanglement. In this context we discuss the conditional amplitude operator and the mutual amplitude operator, as well as conditional Rényi entropies and their role for entanglement detection.