The PPT square conjecture holds generically for some classes of independent states

Collins, Benoı̂t; Yin, Zhi; Zhong, Ping

doi:10.1088/1751-8121/aadd52

Cited by 16 publications

(10 citation statements)

References 33 publications

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“…For the next dimension d = 3, the conjecture has been proven independently in [10,12]. For higher dimensions however, the validity of the conjecture still remains ambiguous [13,20]. In infinite dimensional systems, the set of Gaussian maps has been shown to satisfy the conjecture [10].…”

Section: Conjecture 11 the Composition Of Two Arbitrary Ppt Linear Ma...mentioning

confidence: 98%

The PPT$$^2$$ Conjecture Holds for All Choi-Type Maps

Singh

Nechita

2022

Ann. Henri Poincaré

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We prove that the PPT$$^2$$ 2 conjecture holds for linear maps between matrix algebras which are covariant under the action of the diagonal unitary group. Many salient examples, like the Choi-type maps, depolarizing maps, dephasing maps, amplitude damping maps, and mixtures thereof, lie in this class. Our proof relies on a generalization of the matrix-theoretic notion of factor width for pairwise completely positive matrices, and a complete characterization in the case of factor width two.

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Section: Conjecture 11 the Composition Of Two Arbitrary Ppt Linear Ma...mentioning

confidence: 98%

The PPT$$^2$$ Conjecture Holds for All Choi-Type Maps

Singh

Nechita

2022

Ann. Henri Poincaré

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“…However, in light of the PPT squared conjecture [6], it is also desirable to obtain quantitative bounds to when a channel becomes entanglement breaking. So far, the conjecture was only proved for low-dimensional cases [16,18] or for some particular families of quantum channels [20,35]. In [18], the authors obtain upper bounds on the number of iterations in terms of the Schmidt number of a channel.…”

Section: Introductionmentioning

confidence: 99%

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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“…We conclude this section with a brief side remark regarding the so-called PPT squared conjecture [29], whether for linear maps T 1 , T 2 that are both completely positive and completely copositive the composition T 1 • T 2 is entanglement breaking (see [30] for details). Recently, this conjecture has received much attention [31,32,33,34,35] and Pauli diagonal maps might be a natural candidate for finding a counterexample. However, we can show that no such counterexample can be found among ququart Pauli diagonal maps.…”

Section: If πmentioning

confidence: 99%

Decomposable Pauli diagonal maps and Tensor Squares of Qubit Maps

Müller-Hermes

2020

Preprint

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It is a well-known result due to E. Størmer that every positive qubit map is decomposable into a sum of a completely positive map and a completely copositive map. Here, we generalize this result to tensor squares of qubit maps. Specifically, we show that any positive tensor product of a qubit map with itself is decomposable. This solves a recent conjecture by S. Fillipov and K. Magadov. We contrast this result with examples of non-decomposable positive maps arising as the tensor product of two distinct qubit maps or as the tensor square of a decomposable map from a qubit to a ququart. To show our main result, we reduce the problem to Pauli diagonal maps. We then characterize the cone of decomposable ququart Pauli diagonal maps by determining all 252 extremal rays of ququart Pauli diagonal maps that are both completely positive and completely copositive. These extremal rays split into three disjoint orbits under a natural symmetry group, and two of these orbits contain only entanglement breaking maps. Finally, we develop a general combinatorial method to determine the extremal rays of Pauli diagonal maps that are both completely positive and completely copositive between multiqubit systems using the ordered spectra of their Choi matrices. Classifying these extremal rays beyond ququarts is left as an open problem.

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The PPT square conjecture holds generically for some classes of independent states

Cited by 16 publications

References 33 publications

The PPT$$^2$$ Conjecture Holds for All Choi-Type Maps

The PPT$$^2$$ Conjecture Holds for All Choi-Type Maps

Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times

Decomposable Pauli diagonal maps and Tensor Squares of Qubit Maps

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