Page 24 - Textos de Matemática Vol. 44

14 A. KOVACˇEC
can be used to get a lower bound for the Helmholtz free energy function F = −θlogtr(e−H/θ), where θ = kT, k is the Boltzmann constant, and T the temperature.
For hermitian A assume the eigenvalue ordering [λ(A)] = λ1(A) ≥ ··· ≥ λn(A) and write A ≽w B if [λ(A)] ≽w [λ(B)], where ≽w means weak majoriza- tion. Also write A ≼log B if logA ≼ logB. Lenard and Thompson indepen- dently extended the Golden-Thompson inequality to
eA+B ≺w eB/2eAeB/2.
Araki proved
(Aq/2BqAq/2)1/q ≺log (Ap/2BpAp/2)1/p if 0 < q ≤ p.
Via this Hiai and Petz showed
tr(eA+B) ≤ tr(epA/2epBepA/2)1/p,p > 0.
Letting X(k) denote the k-th compound of X ∈ Mn(C), the authors point to work of Cohen who had introduced partial traces by
j n tr(k)(X) = λi(X(k)), k = 1,...,n, j = 1,...,
jk i=1
and proved a strengthened version
tr(k)[(eT/peT∗/p)p] ≤ tr(k)(eT+T∗),
jj
of the Bernstein inequality. In it p ∈ Z≥1, while in the classical case k = 1 and p = 1.
Let D be a psd matrix such that tr(D) > 1. In the context of statistical mechanics, this is a density matrix. The entropy is S(D) = − tr(D log D). It is an important problem to determine
max ψ(D) = tr(H) + θ tr(D log D)
which is an approximation to Helmholtz’ free energy function, subject to D > 0,
tr(D) = 1. It is proved that for hermitian H we have max{tr(HD) + S(D) : D > 0, tr(D) = 1} = log tr(eH ),
which implies the thermodynamic inequality
tr(HD) + S(D) ≤ log tr(eH ).
Let Hn be the real vector space of n × n hermitian matrices. The authors also prove via the Legendre transform of a real valued function f on Hn, defined by f∗(Y ) = sup{tr(XY ) − f(X) : X hermitian } a result of Hiai-Petz on the