Monotonicity Under Local Operations: Linear Entropic Formulas

Abstract

All correlation measures, classical and quantum, must be monotonic under local operations. In this paper, we characterize monotonic formulas that are linear combinations of the von Neumann entropies associated with the quantum state of a physical system which has n parts. We show that these formulas form a polyhedral convex cone, which we call the monotonicity cone, and enumerate its facets. We illustrate its structure and prove that it is equivalent to the cone of monotonic formulas implied by strong subadditivity. We explicitly compute its extremal rays for n ≤ 5. We also consider the symmetric monotonicity cone, in which the formulas are required to be invariant under subsystem permutations. We describe this cone fully for all n. In addition, we show that these results hold even when states and operations are constrained to be classical.

Optimized Measures of Bipartite Quantum Correlation

“…that E P is monotonically non-increasing under local processing of both subsystems of a bipartite state 2 . The proofs for monotocity under A-processing and B-processing are different, and point towards a method for identifying instances of (2) which satisfy (1). First we show that E P (ρ AB ) is monotone under Bprocessing.…”

“…for a correlation measure E. In this work, we will take (1) to be the defining property of a bipartite correlation measure. This property has been studied by [1] for linear entropic quantities, but here we wish to identify bipartite correlation measures formed by minimizing a linear entropic quantity over all purifications of a state ρ AB . More formally, we will study the space of quantities of the form…”

“…where α ∈ R 15 and f α (|ψ ψ| AA ′ BB ′ ) = ∅ =J ⊆{A,B,A ′ ,B ′ } α J S J (each entry of α corresponds to a non-empty subset of {A, B, A ′ , B ′ }), and identify instances which satisfy (1). Quantities of this form are of particular interest, since they often admit operational interpretations, usually in the form of bounds on performance in information theoretic tasks.…”

“…Note that the minimization is now over all extensions ρ ABV , not only purifications |ψ ψ| AA ′ BB ′ . By first examining the entanglement of purification, a wellknown instance of (2) which satisfies (1), we are led to the construction of four convex polyhedral cones in R 7 . These four cones consist of α vectors which give rise to optimized bipartite correlation measures which all satisfy (1).…”

“…By first examining the entanglement of purification, a wellknown instance of (2) which satisfies (1), we are led to the construction of four convex polyhedral cones in R 7 . These four cones consist of α vectors which give rise to optimized bipartite correlation measures which all satisfy (1). We examine the extreme rays of these cones and find four nontrivial rays, two of which are new.…”