Sinkhorn–Knopp theorem for PPT states

Cariello, Daniel

doi:10.1007/s11005-019-01169-9

Cited by 2 publications

(2 citation statements)

References 39 publications

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“…Using this notion, the following result shows that PPT channels which are block preserving in the above sense must annihilate off-diagonal blocks. We note that results of a similar flavor were shown in [12,13]. The eigenvalues of this matrix are the eigenvalues of Φ(|0 0|), together with the eigenvalues of Φ(|1 1|), and the eigenvalues of the block matrix…”

Section: Definition 35 ( {Psupporting

confidence: 60%

Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times

Hanson

Rouzé

França

2020

Ann. Henri Poincaré

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We investigate entanglement breaking times of Markovian evolutions in discrete and continuous time. In continuous time, we characterize which Markovian evolutions are eventually entanglement breaking, that is, evolutions for which there is a finite time after which any entanglement initially present has been destroyed by the noisy evolution. In the discrete-time framework, we consider the entanglement breaking index, that is, the number of times a quantum channel has to be composed with itself before it becomes entanglement breaking. The PPT 2 conjecture is that every PPT quantum channel has an entanglement breaking index of at most 2; we prove that every faithful PPT quantum channel has a finite entanglement breaking index, and more generally, any faithful PPT CP map whose Hilbert-Schmidt adjoint is also faithful is eventually entanglement breaking. We also provide a method to obtain concrete bounds on this index for any faithful quantum channel. To obtain these estimates, we use a notion of robustness of separability to obtain bounds on the radius of the largest separable ball around faithful product states. We also extend the framework of Poincaré inequalities for non-primitive semigroups to the discrete setting to quantify the convergence of quantum semigroups in discrete time, which is of independent interest.

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Section: Definition 35 ( {Psupporting

confidence: 60%

Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times

Hanson

Rouzé

França

2020

Ann. Henri Poincaré

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“…The PPT counterpart of this result is an algorithm that determines whether a PPT state can be put in the filter normal form or not already found in [7]. This algorithm is based on the complete reducibility property.…”

Section: Introductionmentioning

confidence: 99%

A triality pattern in entanglement theory

Cariello¹

2022

Preprint

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In this work, we present new connections between three types of quantum states: positive under partial transpose states, symmetric with positive coefficients states and invariant under realignment states. First, we obtain a common upper bound for their spectral radii and a result on their filter normal forms. Then we prove the existence of a lower bound for their ranks and the fact that whenever this bound is attained the states are separable. These connections add new evidence to the pattern that for every proven result for one of these types, there are counterparts for the other two, which is a potential source of information for entanglement theory.

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Sinkhorn–Knopp theorem for PPT states

Cited by 2 publications

References 39 publications

Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times

Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times

A triality pattern in entanglement theory

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