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New Journal of Physics

Entropy and information causality in general probabilistic theories

Howard Barnum, Jonathan Barrett, Lisa Orloff Clark, Matthew Leifer, Robert W. Spekkens, Nicholas Stepanik, Alexander Wilce, Robin Wilke

Published Dec 5, 2012• 16 citations
Publisher

Abstract

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.
DOI
10.1088/1367-2630/14/12/129401

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Addendum Entropy and information causality in general probabilistic theories

Howard Barnum, Jonathan Barrett, Lisa Orloff Clark +5 more

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/)

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We investigate the concept of entropy in probabilistic theories more general than quantum mechanics, with particular reference to the notion of information causality (IC) recently proposed by Pawlowski et al (2009 arXiv:0905.2292). We consider two entropic quantities, which we term measurement and mixing entropy. In the context of classical and quantum theory, these coincide, being given by the Shannon and von Neumann entropies, respectively; in general, however, they are very different. In particular, while measurement entropy is easily seen to be concave, mixing entropy need not be. In fact, as we show, mixing entropy is not concave whenever the state space is a non-simplicial polytope. Thus, the condition that measurement and mixing entropies coincide is a strong constraint on possible theories. We call theories with this property monoentropic . Measurement entropy is subadditive, but not in general strongly subadditive. Equivalently, if we define the mutual information between two systems A and B by the usual formula I ( A : B )= H ( A )+ H ( B )- H ( AB ), where H denotes the measurement entropy and AB is a non-signaling composite of A and B , then it can happen that I ( A : BC )< I ( A : B ). This is relevant to IC in the sense of Pawlowski et al : we show that any monoentropic non-signaling theory in which measurement entropy is strongly subadditive, and also satisfies a version of the Holevo bound, is informationally causal, and on the other hand we observe that Popescu–Rohrlich boxes, which violate IC, also violate strong subadditivity. We also explore the interplay between measurement and mixing entropy and various natural conditions on theories that arise in quantum axiomatics.

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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/)
We investigate the concept of entropy in probabilistic theories more general than quantum mechanics, with particular reference to the notion of information causality (IC) recently proposed by Pawlowski et al (2009 arXiv:0905.2292). We consider two entropic quantities, which we term measurement and mixing entropy. In the context of classical and quantum theory, these coincide, being given by the Shannon and von Neumann entropies, respectively; in general, however, they are very different. In particular, while measurement entropy is easily seen to be concave, mixing entropy need not be. In fact, as we show, mixing entropy is not concave whenever the state space is a non-simplicial polytope. Thus, the condition that measurement and mixing entropies coincide is a strong constraint on possible theories. We call theories with this property monoentropic . Measurement entropy is subadditive, but not in general strongly subadditive. Equivalently, if we define the mutual information between two systems A and B by the usual formula I ( A : B )= H ( A )+ H ( B )- H ( AB ), where H denotes the measurement entropy and AB is a non-signaling composite of A and B , then it can happen that I ( A : BC )< I ( A : B ). This is relevant to IC in the sense of Pawlowski et al : we show that any monoentropic non-signaling theory in which measurement entropy is strongly subadditive, and also satisfies a version of the Holevo bound, is informationally causal, and on the other hand we observe that Popescu–Rohrlich boxes, which violate IC, also violate strong subadditivity. We also explore the interplay between measurement and mixing entropy and various natural conditions on theories that arise in quantum axiomatics.