Properties of q‐entropies Available to Purchase

Z.

Daróczy

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Here, the prefactor is not $(q−1)−1$ but $(1−21−q)−1$ for reasons of normalization.

A.

Wehrl

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The quantum version (our Eq. (2)) of this formula is displayed on p. 247 of Wehrl’s excellent 1978 review who, of course, cites Ref. 1.

C.

Tsallis

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E. M. F.

Curado

and

C.

Tsallis

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The present state of affairs is reviewed in: C. Tsallis “Some Comments on Boltzmann-Gibbs Statistical Mechanics,” in “Chaos, Solitons and Fractals,” edited by G. Marshall (Pergamon, Oxford, 1994).

$SαR$ is additive but not subadditive; and not concave for $α>1$

For negative q and in finite dimension $Sq[⋅]$ is given by (1) if ρ is non-degenerate (i.e., $ρj>0$ for all j) otherwise it is $∞;Sq[⋅]$ is then strictly convex on the interior of the simplex of probability distributions. In infinite dimensions the congergence of $Σn = 1N ρn$ to 1 as $N→∞,$ implies that $Σn = 1Nρnq$ is divergent.

M. Ohya and D. Petz, Quantum entropy and its use (Springer-Verlag, Berlin-Heidelberg, 1993).

$ηq(ρ): = Σnηq(ρn)pn,$ where ${pn$ are the spectral projections to the eigenvalues ${pn$ of ρ.

The equality for disjoint probability measures is observed in Ref. 4, Eq. (7).

C. A.

McCarthy

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B. Simon, Trace ideals and their applications, London Math. Soc. Lecture Note Series 35 (Cambridge University, Cambridge, 1979).

G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (Cambridge University, Cambridge, 1934).

The argument of Ref. 2 works. Assume $0<q<1,$ and suppose for simplicity that ρ is a pure state. Then $Sq[ρ] = 0,$ and ρ is a one-dimensional projection. Let ${pn:n = 1,2…}$ be a family of pairwise orthogonal one-dimensional projections such that ${npn = 1−ρ;$ and put $λn = cn−1/q$ for $n = 1,2,…$ where c is a positive real to be specified shortly. Let $φ = (1−cζ(1/q))ρ+Σnλnpn,$ where $ζ(1/q) = σnn−1/q.$ If $cζ(l/q)⩽1$ then φ is a density operator with eigenvalues ${(l−cζ(l/q)),λ1,λ2,…{$ and, since $Σnλnq∼Σnn−1,$ we have $Sq[φ] = ∞.$ Moreover, $|ρ−φ| = cζ(1/q)ρΣnλnPn,$ so that $‖ρ−φ‖ = 2cζ(1/q)$ which, for given $ε>0,$ is not larger than ε as soon as $c⩽(2ζ(1/q))−1ε.$

This inequality provides us with an alternative proof of Lemma $2: |tr((ρ−φ)φq−11)|⩽tr(|ρ−φ|)‖φq−1‖∞⩽‖ρ−φ‖,$ and similarly for the other bound, because for $q>1$ the operator norm $‖φq−1‖∞$ does not exceed 1.

The referee points out that this equation appears in C. Tsallis, “Extensive versus Nonextensive Physics,” in New Trends in Magnetic Materials and Their Applications, edited by J. L. Moran-Lopez and J. M. Sanchez (Plenum, New York, 1994).

Doing Problem 1 on page 182 of Jauch’s book [J. M. Jauch, Foundations of Quantum Mechanics (Addison-Wesley, Reading, MA, 1968)] you will see that if $ω1$ or $ω2$ is pure, then $ω1 = ω1⊗ω2.$

C. Tsallis, Seminar given at FaMAF (Córdoba), December 1993.

This follows also directly from the fact that $0<q⟼ρq$ is non-increasing (in fact strictly decreasing when ρ is not pure). One can give simple examples for states ρ such that $Sq[ρ] = ∞$ for every $0<q⩽1,$ or alternatively, such that there is a $qo$ (necessarily $qo⩽1)$ with $Sq[ρ] = ∞$ for every q such that $0<q⩽qo$ and $Sq[ρ]<∞$ for all $q>qo$