arXiv:1710.08475 · Quantum Information and Computation

Composition of PPT maps

Mathew Kennedy, Nicholas Manor, Vern I. Paulsen

Published May 1, 2018• 15 citations

Abstract

M. Christandl conjectured that the composition of any trace preserving PPT map with itself is entanglement breaking. We prove that Christandl's conjecture holds asymptotically by showing that the distance between the iterates of any unital or trace preserving PPT map and the set of entanglement breaking maps tends to zero. Finally, for every graph we define a one-parameter family of maps on matrices and determine the least value of the parameter such that the map is variously, positive, completely positive, PPT and entanglement breaking in terms of properties of the graph. Our estimates are sharp enough to conclude that Christandl's conjecture holds for these families.

DOI: 10.26421/qic18.5-6-4

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arXiv:1710.08475 · Quantum Information and Computation

Composition of PPT maps

Mathew Kennedy, Nicholas Manor, Vern I. Paulsen

Published May 1, 2018• 15 citations

arXiv HTML PDF

Abstract

DOI: 10.26421/qic18.5-6-4

Related papers

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1

-undistillable with any CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in either order. Moreover, we show that the cone of

1

-undistillable CP maps is exactly the largest qutrit cone of CP maps whose composition with every CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in both orders. Finally, although map composition captures only the standard maximally entangled outcome in entanglement swapping, we prove that any

1

-undistillable two-qutrit state and any state of Schmidt number at most two cannot generate terminal entanglement under an arbitrary selective measurement on the intermediate systems.

Φ_{A,B,C}

and

Φ_{D,E,F}

M_d

, the entrywise bound

(AD)_{ij}\ge 2\lfloor\sqrt{d}\rfloor\sqrt{A_{ii}A_{jj}D_{ii}D_{jj}}

for all

i,j

guarantees that

Φ_{A,B,C}\circΦ_{D,E,F}

is EB. In dimension four the universal coefficient becomes four. A continuous family of bistochastic PPT channels is then constructed; every member with

a\ne1

is not EB, while every pairwise composition within the family is proved to be EB. A rank-sensitive refinement shows explicitly how low coherence rank can improve the universal coefficient. The resulting certificate is sufficient rather than necessary and is complementary to factor-width and semidefinite-hierarchy approaches.

The PPT2 conjecture asserts that the composition of any two PPT maps is entanglement breaking. It is proven for maps on matrices of size up to

3 times 3

and for several structured families, but the general case remains open