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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.
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We provide a partial solution to the problem of constructing mutually unbiased bases (MUBs) and symmetric informationally complete POVMs (SIC-POVMs) in non-prime-power dimensions. An algebraic description of a SIC-POVM in dimension six is given. Furthermore it is shown that several sets of three mutually unbiased bases in dimension six are maximal, i.e., cannot be extended.
The existence of a set of d^2 pairwise equiangular complex lines (equivalently, a SIC-POVM) in d-dimensional Hilbert space is currently known only for a finite set of dimensions d. We prove that, if there exists a set of real units in a certain ray class field (depending on d) satisfying certain congruence conditions and algebraic properties, a SIC-POVM may be constructed when d is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s=0 and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact solution to the SIC-POVM problem in dimension 23.
We introduce and study the entanglement breaking rank of an entanglement breaking channel. We show that the entanglement breaking rank of the channel $mathfrak Z: M_d to M_d$ defined by begin{align*} mathfrak Z(X) = frac{1}{d+1}(X+text{Tr}(X)mathbb I_d) end{align*} is $d^2$ if and only if there exists a symmetric informationally-complete POVM in dimension $d$.
In this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. The latter inequality, which we call approximate tensorization of the relative entropy, can be expressed as a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.
For a Pfaffian point process we show that its Palm measures, its normalised compositions with multiplicative functionals, and its conditional measures with respect to fixing the configuration in a bounded subset are Pfaffian point processes whose kernels we find explicitly.
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