A simple proof of the strong subadditivity inequality

A short and elementary proof of the joint convexity of relative entropy is presented, using nothing beyond linear algebra. The key ingredients are an easily verified integral representation and the strategy used to prove the Cauchy-Schwarz inequality in elementary courses. Several consequences are proved in a way which allow an elementary proof of strong subadditivity in a few more lines. Some expository material on Schwarz inequalities for operators and the Holevo bound fo...

We derive the strong subadditivity of the von Neumann entropy with a strict lower bound dependent on the distribution of quantum correlation in the system. We investigate the structure of states saturating the bounded subadditivity and explore its consequences for the quantum data processing inequality. The quantum data processing achieves a lower bound associated with the locally inaccessible information.

It is well known that the strong subadditivity theorem is hold for classical system, but it is very difficult to prove that it is hold for quantum system. The first proof of this theorem is due to Lieb by using the Lieb's theorem. Here we use the conditions obtained in our previous work of matrix analysis method to give a new proof of this famous theorem. This new proof is very elementary, it only needs to carefully analyse the minimal value of a function. This proof also sho...

The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy S[\rho]= - Tr (\rho \ln \rho) of a density matrix \rho_{123} on the product of three Hilbert spaces satisfies S[\rho_{123}] - S[\rho_{23}] \leq S[\rho_{12}]- S[\rho_2]. We strengthen this to S[\rho_{123}] - S[\rho_{12}] \leq \sum_\alpha n^\alpha (S[\rho_{23}^\alpha ] - S[\rho_2^\alpha ]), where the n^\alpha are weights and the \rho_{23}^\alpha are parti...

In this short note, we give a simple derivation of the strong subadditivity inequalities: S(13) + S(23) \ge S(1) + S(2) and S(12) + S(23) \ge S(2) + S(123). The simplicity is due to the way we represent the quantum systems. We make a few remarks about such a representation.

Incremental information, as measured by the quantum entropy, is increasing when two ensembles are united. This result was proved by Lieb and Ruskai, and it is the foundation for the proof of strong subadditivity of quantum entropy. We present a truly elementary proof of this fact in the context of the broader family of matrix entropies introduced by Chen and Tropp.

We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so-called short quantum Markov chain, which in turn implies that two of the systems are independent conditioned on the third, in a physically mea...

This paper presents self-contained proofs of the strong subadditivity inequality for quantum entropy and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein's inequality and one of Lieb's less well-known concave trace functions, allows one to obtain conditions for equality. Using the fact that the Holevo bound on the accessible informat...

We prove an entropic uncertainty relation for two quantum channels, extending the work of Frank and Lieb for quantum measurements. This is obtained via a generalized strong super-additivity (SSA) of quantum entropy. Motivated by Petz's algebraic SSA inequality, we also obtain a generalized SSA for quantum relative entropy. As a special case, it gives an improved data processing inequality.

Strong subadditivity of von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone of quantum coding theory. All other known inequalities for entropies of quantum systems may be derived from it. Here we prove a new inequality for the von Neumann entropy which we prove is independent of strong subadditivity: it is an inequality which is true for any four party quantum state, provided that it satisfies three linear relations (constraints) on the entropies of certa...