Simplifying additivity problems using direct sum constructions

We study the additivity problems for the classical capacity of quantum channels, the minimal output entropy and its convex closure. We show for each of them that additivity for arbitrary pairs of channels holds iff it holds for arbitrary equal pairs, which in turn can be taken to be unital. In a similar sense, weak additivity is shown to imply strong additivity for any convex entanglement monotone. The implications are obtained by considering direct sums of channels (or states) for which we show how to obtain several information theoretic quantities from their values on the summands. This provides a simple and general tool for lifting additivity results.Comment: 5 page

Notes on multiplicativity of maximal output purity for completely positive qubit maps

A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. This quantity is defined as the maximum value of the purity one can get at the output of a channel by varying over all physical input states, when purity is measured by the Schatten $q$-norm, and is denoted by $\nu_q$. The multiplicativity problem is the question whether two channels used in parallel have a combined $\nu_q$ that is the product of the $\nu_q$ of the two channels. A positive answer would imply a number of other additivity results in QIT. Very recently, P. Hayden has found counterexamples for every value of $q>1$. Nevertheless, these counterexamples require that the dimension of these channels increases with $1-q$ and therefore do not rule out multiplicativity for $q$ in intervals $[1,q_0)$ with $q_0$ depending on the channel dimension. I argue that this would be enough to prove additivity of entanglement of formation and of the classical capacity of quantum channels. More importantly, no counterexamples have as yet been found in the important special case where one of the channels is a qubit-channel, i.e. its input states are 2-dimensional. In this paper I focus attention to this qubit case and I rephrase the multiplicativity conjecture in the language of block matrices and prove the conjecture in a number of special cases.Comment: Manuscript for a talk presented at the SSPCM07 conference in Myczkowce, Poland, 10/09/2007. 12 page

On Hastings' counterexamples to the minimum output entropy additivity conjecture

Hastings recently reported a randomized construction of channels violating the minimum output entropy additivity conjecture. Here we revisit his argument, presenting a simplified proof. In particular, we do not resort to the exact probability distribution of the Schmidt coefficients of a random bipartite pure state, as in the original proof, but rather derive the necessary large deviation bounds by a concentration of measure argument. Furthermore, we prove non-additivity for the overwhelming majority of channels consisting of a Haar random isometry followed by partial trace over the environment, for an environment dimension much bigger than the output dimension. This makes Hastings' original reasoning clearer and extends the class of channels for which additivity can be shown to be violated.Comment: 17 pages + 1 lin

On Strong Superadditivity of the Entanglement of Formation

We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation $E_F$ and the conjugate function $E^*$ of the entanglement function E(\rho)=S(\trace_A\rho). We then consider the conjectured strong superadditivity of the entanglement of formation $E_F(\rho) \ge E_F(\rho_I)+E_F(\rho_{II})$, where $\rho_I$ and $\rho_{II}$ are the reductions of $\rho$ to the different Hilbert space copies, and prove that it is equivalent with subadditivity of $E^*$. As an application, we show that strong superadditivity would follow from multiplicativity of the maximal channel output purity for all non-trace-preserving quantum channels, when purity is measured by Schatten $p$-norms for $p$ tending to 1.Comment: 11 pages; refs added, explanatory improvement