Subadditivity of q-entropies for q>1

TL;DR

This paper proves a fundamental inequality for Schatten q-norms of quantum states, establishing the subadditivity of q-entropies for q>1 in finite-dimensional bipartite systems, confirming Raggio's conjecture.

Contribution

It introduces a new inequality for Schatten q-norms and proves the subadditivity of q-entropies for q>1, confirming a longstanding conjecture.

Findings

Proved a basic inequality for Schatten q-norms of quantum states.

Established the subadditivity of q-entropies for q>1.

Confirmed Raggio's conjecture in the finite-dimensional case.

Abstract

I prove a basic inequality for Schatten q-norms of quantum states on a finite-dimensional bipartite Hilbert space H_1\otimes H_2: 1+||\rho||_q \ge ||\trace_1\rho||_q + ||\trace_2\rho||_q. This leads to a proof--in the finite dimensional case--of Raggio's conjecture (G.A. Raggio, J. Math. Phys.\ \textbf{36}, 4785--4791 (1995)) that the q-entropies S_q(\rho)=(1-\trace[\rho^q])/(q-1) are subadditive for q > 1; that is, for any state \rho, S_q(\rho) is not greater than the sum of the S_q of its reductions, S_q(\rho) \le S_q(\trace_1\rho)+S_q(\trace_2\rho).