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Register a new userEntanglement Breaking Rank and the existence of SIC POVMs
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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 $d$.
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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.
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 revisit the problem of finding the Naimark extension of a probability operator-valued measure (POVM), i.e. its implementation as a projective measurement in a larger Hilbert space. In particular, we suggest an iterative method to build the projective measurement from the sole requirements of orthogonality and positivity. Our method improves existing ones, as it may be employed also to extend POVMs containing elements with rank larger than one. It is also more effective in terms of computational steps.
Van Dam and Hayden introduced a concept commonly referred to as embezzlement, where, for any entangled quantum state $phi$, there is an entangled catalyst state $psi$, from which a high fidelity approximation of $phi otimes psi$ can be produced using only local operations. We investigate a version of this where the embezzlement is perfect (i.e., the fidelity is 1). We prove that perfect embezzlement is impossible in a tensor product framework, even with infinite-dimensional Hilbert spaces and infinite entanglement entropy. Then we prove that perfect embezzlement is possible in a commuting operator framework. We prove this using the theory of C*-algebras and we also provide an explicit construction. Next, we apply our results to analyze perfe
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