Quantum Version of Wielandt’s Inequality Revisited

Abstract

Consider a linear space L of complex D-dimensional linear operators, and assume that some power L k of L is the whole set End(C D ). Perez-Garcia, Verstraete, Wolf and Cirac conjectured that the sequence L 1 , L 2 , . . . stablilizes after O(D 2 ) terms; we prove that this happens after O(D 2 log D) terms, improving the previously known bound of O(D 4 ).

Mathematical open problems in projected entangled pair states

“…In [39] it is shown that if A is normal then i(A) ≤ (D 2 − d + 1)D 2 . This result has been recently improved [27] to p(T ) ≤ 2D 2 (6 + log 2 (D)). Up to a logarithmic factor, the order O(D 2 log(D)) is optimal just by invoking the optimality of the classical Wielandt inequality.…”

Tensor network representations from the geometry of entangled states

“…For completeness we will give the argument below. In contrast to this border bond dimension 2 representation, the results from [37] on the quantum Wielandt inequality imply that the bond dimension of a translation-invariant restriction has to grow as exp 1 3 ω(3L) , with ω(x) the product logarithm or Lambert function and it has been conjectured that the growth should be of the order L 1/3 [6]. We note however that without the restriction to the translation invariant setting one can also find a bond dimension 2 representation of the W-state.…”

A Tensor Version of the Quantum Wielandt Theorem

“…. , A d ) be a d-tuple of D × D-matrices, and assume that there is an N such that the linear span of [4] and is conjectured to be O(D 2 ) [5].…”

“…The bounds for quantum Wielandt theorem in [8,4,7] were obtained using explicit methods from linear algebra. Our main new insight is the application of nonconstructive Noetherian arguments from nonlinear algebra.…”

“…This is known as the quantum Wielandt theorem; see [5,Conjecture 1]. An effective version was proved in [8,4].…”