# The Inequalities of Quantum Information Theory

## Document Type

Article

## Department

Mathematics (HMC)

## Publication Date

4-2003

## Abstract

Let ρ denote the density matrix of a quantum state having n parts 1, ..., n. For I⊆N={1, ..., n}, let ρ_{I}=Tr_{N}I/(ρ) denote the density matrix of the state comprising those parts i such that i∈I, and let S(ρ_{I}) denote the von Neumann (1927) entropy of the state ρ_{I}. The collection of ν=2^{n} numbers {S(ρ_{I})}_{I⊆N} may be regarded as a point, called the allocation of entropy for ρ, in the vector space R^{ν}. Let A_{n} denote the set of points in R^{ν} that are allocations of entropy for n-part quantum states. We show that A~_{n}~ (the topological closure of A_{n}) is a closed convex cone in R^{ν}. This implies that the approximate achievability of a point as an allocation of entropy is determined by the linear inequalities that it satisfies. Lieb and Ruskai (1973) have established a number of inequalities for multipartite quantum states (strong subadditivity and weak monotonicity). We give a finite set of instances of these inequalities that is complete (in the sense that any valid linear inequality for allocations of entropy can be deduced from them by taking positive linear combinations) and independent (in the sense that none of them can be deduced from the others by taking positive linear combinations). Let B_{n} denote the polyhedral cone in R^{ν} determined by these inequalities. We show that A~_{n}~=B_{n} for n≤3. The status of this equality is open for n≥4. We also consider a symmetric version of this situation, in which S(ρ_{I}) depends on I only through the number i=≠I of indexes in I and can thus be denoted S(ρ_{i}). In this case, we give for each n a finite complete and independent set of inequalities governing the symmetric allocations of entropy {S(ρ_{i})}_{0≤i≤n} in R^{n+1}.

## Rights Information

© 2003 IEEE

## DOI

10.1109/TIT.2003.809569

## Recommended Citation

Pippenger, N.; , "The inequalities of quantum information theory," Information Theory, IEEE Transactions on , vol.49, no.4, pp. 773- 789, April 2003.