TY - JOUR

T1 - Inequalities for the ranks of multipartite quantum states

AU - Cadney, Josh

AU - Huber, Marcus

AU - Linden, Noah

AU - Winter, Andreas

PY - 2014/7/1

Y1 - 2014/7/1

N2 - We investigate relations between the ranks of marginals of multipartite quantum states. We show that there exist inequalities constraining the possible distribution of ranks. This is, perhaps, surprising since it was recently discovered that the α-Rényi entropies for α(0,1)(1,∞) satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for α(0,1) is completely unconstrained beyond non-negativity. Our results resolve an important open question by showing that the case of α=0 (logarithm of the rank) is restricted by nontrivial linear relations and thus the cases of von Neumann entropy (i.e., α=1) and 0-Rényi entropy are exceptionally interesting measures of entanglement in the multipartite setting. We close the paper with an intriguing open problem, which has a simple statement, but is seemingly difficult to resolve. © 2014 Elsevier Inc.

AB - We investigate relations between the ranks of marginals of multipartite quantum states. We show that there exist inequalities constraining the possible distribution of ranks. This is, perhaps, surprising since it was recently discovered that the α-Rényi entropies for α(0,1)(1,∞) satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for α(0,1) is completely unconstrained beyond non-negativity. Our results resolve an important open question by showing that the case of α=0 (logarithm of the rank) is restricted by nontrivial linear relations and thus the cases of von Neumann entropy (i.e., α=1) and 0-Rényi entropy are exceptionally interesting measures of entanglement in the multipartite setting. We close the paper with an intriguing open problem, which has a simple statement, but is seemingly difficult to resolve. © 2014 Elsevier Inc.

KW - Entropy inequalities

KW - Marginals

KW - Matrix rank

KW - Quantum states

U2 - https://doi.org/10.1016/j.laa.2014.03.035

DO - https://doi.org/10.1016/j.laa.2014.03.035

M3 - Article

VL - 452

SP - 153

EP - 171

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -