bibkey: krugerwerner2005openproblems
authors: O. Krüger; R. F. Werner
year: 2005
title: "Some Open Problems in Quantum Information Theory"
doi: null
url: https://arxiv.org/abs/quant-ph/0504166v1
claim: "Problem 3 of the open-problem list asks whether polynomial entanglement invariants are complete and, as a closely related problem, whether the set of separable states can be described by a single polynomial invariant f with f(ρ) ≥ 0 equivalent to separability, possibly depending on the dimensions."
strata_touched:
  - D5/S3/Quantum/Information/SeparabilitySinglePolynomialRefutation
license: citation-only
triage: anchor

Some Open Problems in Quantum Information Theory

O. Krüger and R. F. Werner, arXiv:quant-ph/0504166v1 (2005), the printed snapshot of the open-problem list maintained by the authors. Quotations are from the v1 TeX source, Problem 3, “Polynomial entanglement invariants”.

The definitions:

We say that two bipartite quantum states $\rho$ and $\sigma$ are ``equally entangled’’ if they differ only by a choice of bases in Alice’s and Bob’s subspaces, i.,e., if we can find unitaries $U_A$, $U_B$, such that $\rho=(U_A\otimes U_B) \sigma (U_A\otimes U_B)^*.$

An \emph{entanglement invariant} is by definition any real valued function on the space of bipartite density operators, which assigns the same value to equally entangled density operators. A \emph{polynomial invariant} is an entanglement invariant, which can be computed as a polynomial in the matrix elements of $\rho$. Note that because we only consider hermitian operators, allowing polynomials in the matrix elements and their complex conjugates does not enlarge this class.

The separability question, the third of the “closely related problems”:

Decide whether the set of separable states can be described in terms of a polynomial invariant $f$, such that $f(\rho )\ge 0$ is equivalent to separability. There are many weaker versions of this statement, which may be of interest. For example, we might merely ask for a sufficient or a necessary separability criterion, and we might allow $f$ to depend on the dimensions.

The solution section of the problem answers the main completeness question by the Onishchik–Vinberg theorem, “The orbits of a compact linear group acting in a real vector space are separated by the polynomial invariants.”

Verified locator

Source: https://arxiv.org/abs/quant-ph/0504166v1

• URL: https://arxiv.org/abs/quant-ph/0504166v1 (source quantph.tex, md5 4b39c408ab408b5fb289c5167ccc1e96):
  • equally entangled states and polynomial invariants, l. 1215–1231;
  • the completeness question and the related problems, l. 1233–1253;
  • the solution of the completeness question, l. 1304–1314.