Monotone Riemannian metrics and relative entropy on noncommutative probability spaces

Abstract

We use the relative modular operator to define a generalized relative entropy for any convex operator function g on (0, ∞) satisfying g(1) = 0. We show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy, a unique family (parameterized by density matrices) of continuous monotone Riemannian metrics, a unique geodesic distance on the space of density matrices, and a unique monotone operator function satisfying certain symmetry and normalization conditions. We describe these objects explicitly in several important special cases, including g(w) = − log w which yields the familiar logarithmic relative entropy. The relative entropies, Riemannian

Bures volume of the set of mixed quantum states

“…They are distinguished above by the appropriate labels. The maximal concave function f max (t), the straight line t/2 + 1/2, induces the Bures metric and the Bures distance [1,37]. If f (0) = 1/2 the metric induces the natural Riemannian metric on the subspace CP N −1 of pure states and is called Fubini-Study adjusted [1,8].…”

Another short and elementary proof of strong subadditivity of quantum entropy

“…This only changes the weight function in the integral; see [9,19] for details. Replacing 1 (1+t) 2 by δ(1 − t) in (7) yields (Q − P ) 1 L P +R Q (Q − P ) which is the generalized relative entropy whose Hessian yields the Riemmanian metric associated with the Bures metric…”

“…However, without the additional ingredient of L P and R Q , which are motivated by Araki's subsequent introduction [1] of the relative modular operator, the results in [11] are not sufficient to prove SSA. The recognition that the argument in [11] could be modified to prove SSA took another 25 years [9].…”

“…In this note, we give a self-contained proof of SSA, valid for finite dimensional systems, using only basic linear algebra and an easily verified integral representation. The basic strategy was used in [9]. However, the presentation here, unlike that in [9] and [15], does not explicitly use the relative modular operator.…”

“…The basic strategy was used in [9]. However, the presentation here, unlike that in [9] and [15], does not explicitly use the relative modular operator. Instead, the simple left and right multiplication operations explained in Section 2.1 suffice.…”