18
votes
0answers
520 views

conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
3
votes
1answer
77 views

Set of orthogonal simplexes or partial mutually unbiased bases

I am interested in the existence of a set of vectors $\{ v_{ij} \}_{ij} \subseteq \mathbb{C}^N$ for $i \in \{1,\dots,N\}$, $j \in \{1,\dots,N+1\}$ such that $\left\vert v^*_{ij} v_{ij'} \right\vert = ...
2
votes
2answers
213 views

Existence of a projection operator onto a classical set of density matrices

I have a Hilbert space of quantum density matrices written in the Glauber-Sudarshan P representation - ie. we have coherent states $|\alpha \rangle$ and we write density matrices as $$ \rho = \int ...
2
votes
1answer
586 views

Bounding the von Neumann entropy of a density matrix with the Hilbert-Schmidt norm

Question Suppose I have a $D$-dimensional density matrix $\rho_0$ $\rho_0^\dagger = \rho_0 \quad, \quad \mathrm{Tr} \rho_0 = 1 \quad, \quad \rho_0 > 0,$ with a known spectrum $\{\lambda_i^0\}$ ...
6
votes
1answer
477 views

Can the von Neumann entropy of a positive, positive semi-definite, and unit-trace density matrix with equal on-diagonal terms be bounded by equalizing all off-diagonal elements to their highest/lowest value?

Statement of problem Consider the density matrix $M = (m_{i,j})$ in $d$-dimensions with all positive elements: $m_{i,j} > 0$. From physics, a density matrix is Hermitian, positive semi-definite, ...
5
votes
3answers
857 views

Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces?

I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation here. Background A simple consequence of the singular value decomposition is that any vector ...
8
votes
1answer
166 views

Operator compression preserving lowest energy eigenspace.

I have a large ($10^6$ by $10^6$) sparse ($0.4$% nonzero) hermitian matrix $H$ arising from the discretization of an elliptic PDE. I would like to approximate $H$ with a smaller matrix $H'$ in such a ...
3
votes
1answer
456 views

What is the entropy of a density matrix which is the sum of two unitarily equivalent projectors?

Construction Suppose I have a density matrix $\rho$ which is proportional to a projector $P$ formed by tensoring together $N$ small projectors $P^{(i)}$ of rank 2: $P^{(i)} = |a\rangle_i\langle a| + ...
2
votes
3answers
523 views

Are the Gell-Mann matrices extremal when used as Kraus operators for a quantum channel?

Landau and Streater proved that a set of Kraus operators, Ai, is extremal if and only if the set $\{A_{k}^{\dagger}A_{l}\}_{k,l \ldots N}$ are linearly independent. I have seen very convincing ...