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2018

Quantum version of Wielandt's Inequality revisited

Michałek, Mateusz, Shitov, Yaroslav

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Consider a linear space L of complex D-dimensional linear operators, and assume that some power L^k of L is the whole space of DxD matrices.

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).

Quantum version of Wielandt's Inequality revisited · Around

Built on

Unzerlegbare, nicht negative matrizen
H. Wielandt · 1950
Earlier work this paper cites.
O počtu kladnỳch prvku v mocninách nezáporné matice
Zbyněk Šidák · 1964
Earlier work this paper cites.
An application of the cayley-hamilton theorem to matrix polynomials in several variables
Azaria Paz · 1984
Earlier work this paper cites.
Linear spaces of nilpotent matrices
Ben Mathes, Matjaž Omladič, and Heydar Radjavi · 1991
Earlier work this paper cites.

Similar

An upper bound for the length of a finite-dimensional algebra
Christopher Pappacena · 1997
Cited alongside, same era.
Matrix product state representations
D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac · 2007
Cited alongside, same era.
A quantum version of Wielandt’s inequality
Mikel Sanz, David Perez-Garcia, Michael M. Wolf, and Juan I. Cirac · 2010
Cited alongside, same era.

Then

Linear Algebra and its Applications
Alexander Guterman, Thomas Laffey, Olga Markova, and Helena Šmigoc · 2018
Closest in time.
A new bound on quantum Wielandt inequality
Original
Mizanur Rahaman · 2018
Closest in time.
An improved bound for the length of matrix algebras
Original
Yaroslav Shitov · 2018
Closest in time.

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