by being correlated with the entropy of the other. The left-hand inequality can be roughly interpreted as saying that entropy can only be cancelled by an equal amount of entropy. If system and system have different amounts of entropy, the smaller can only partially cancel the greater, and some entropy must be left over. Likewise, the right-hand inequality can be interpreted as saying that the entropy of a composite system is maximized when its components are uncorrelated, in which case the total entropy is just a sum of the sub-entropies. This may be more intuitive in the phase space formulation, instead of Hilbert space one, where the Von Neumann entropy amounts to minus the expected value of the ★-logarithm of the Wigner function, , up to an offset shift. Up to this normalization offset shift, the entropy is majorized by that of its classical limit. Strong subadditivity The von Neumann entropy is also strongly subadditive. Given three Hilbert spaces, A, B, C, This is a more difficult theorem and was proved first by J. Kiefer in 1959 and independently by Elliott H. Lieb and Mary Beth Ruskai in 1973, using a matrix inequality of Elliott H. Lieb proved in 1973. By using the proof technique that establishes the left side of the triangle inequality above, one can show that the strong subadditivity inequality is equivalent to the following inequality. when , etc. are the reduced density matrices of a density matrix . If we apply ordinary subadditivity to the left side of this inequality, and consider all permutations of A, B, C, we obtain the triangle inequality for : Each of the three numbers is less than or equal to the sum of the other two. See also Entropy (information theory) Linear entropy Partition function (mathematics) Quantum conditional entropy Quantum mutual information Quantum entanglement Strong subadditivity of quantum entropy Wehrl entropy References Quantum mechanical entropy John von Neumann