In this paper we examine a generalization of the symmetric informationally complete POVMs. SIC-POVMs are the optimal measurements for full quantum tomography, but if some parameters of the density matrix are known, then the optimal SIC POVM should be
orthogonal to a subspace. This gives the concept of the conditional SIC-POVM. The existence is not known in general, but we give a result in the special cases when the diagonal is known of the density matrix.
Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz quasi-entropies. Many well known distinguishability measures of quantum states are given by, or derived from, f-divergences; s
pecial examples include the quantum relative entropy, the Renyi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under the dual of Schwarz maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Holder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.
The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $phi$ in the form $K_D^phi(H,K)=sum_{i,j}phi(lambda_i,lambda_j)^{-1} Tr P_iHP_jK$ when $sum_ilambda_iP_i$ is the spectral decomposition of the foot po
int $D$ and the Hermitian matrices $H,K$ are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping $Dmapsto G(D)$ is a kernel metric as well. Several Riemannian geometries of the literature are particular cases, for example, the Fisher-Rao metric for multivariate Gaussian distributions and the quantum Fisher information. In the paper the case $phi(x,y)=M(x,y)^theta$ is mostly studied when $M(x,y)$ is a mean of the positive numbers $x$ and $y$. There are results about the geodesic curves and geodesic distances. The geometric mean, the logarithmic mean and the root mean are important cases.
The history of complementary observables and mutual unbiased bases is reviewed. A characterization is given in terms of conditional entropy of subalgebras. The concept of complementarity is extended to non-commutative subalgebras. Complementary decom
positions of a 4-level quantum system are described and a characterization of the Bell basis is obtained.
A new expression as a certain asymptotic limit via discrete micro-states of permutations is provided to the mutual information of both continuous and discrete random variables.
In a quantum mechanical model, Diosi, Feldmann and Kosloff arrived at a conjecture stating that the limit of the entropy of certain mixtures is the relative entropy as system size goes to infinity. The conjecture is proven in this paper for density m
atrices. The first proof is analytic and uses the quantum law of large numbers. The second one clarifies the relation to channel capacity per unit cost for classical-quantum channels. Both proofs lead to generalization of the conjecture.
The estimation of the density matrix of a $k$-level quantum system is studied when the parametrization is given by the real and imaginary part of the entries and they are estimated by independent measurements. It is established that the properties of
the estimation procedure depend very much on the invertibility of the true state. In particular, in case of a pure state the estimation is less efficient. Moreover, several estimation schemes are compared for the unknown state of a qubit when one copy is measured at a time. It is shown that the average mean quadratic error matrix is the smallest if the applied observables are complementary. The results are illustrated by computer simulations.
The optimal state determination (or tomography) is studied for a composite system of two qubits when measurements can be performed on one of the qubits and interactions of the two qubits can be implemented. The goal is to minimize the number of inter
actions to be used. The algebraic method applied in the paper leads to an extension of the concept of mutually unbiased measurements.
In this paper two independent and unitarily invariant projection matrices P(N) and Q(N) are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size $N$ converges to infinity. The result is
formulated on the tracial state space $TS({cal A})$ of the universal $C^*$-algebra ${cal A}$ generated by two selfadjoint projections. The random pair $(P(N),Q(N))$ determines a random tracial state $tau_N in TS({cal A})$ and $tau_N$ satisfies the large deviation. The rate function is in close connection with Voiculescus free entropy defined for pairs of projections.
We give a new proof of the free transportation cost inequality for measures on the circle following M. Ledouxs idea.
