arXiv:2010.15554

Investigation of the PPT Squared Conjecture for High Dimensions

Jin, Ryan

Published Oct 27, 2020• 1 citation

Abstract

We present the positive-partial-transpose squared conjecture introduced by M. Christandl at Banff International Research Station Workshop: Operator Structures in Quantum Information Theory (Banff International Research Station, Alberta, 2012). We investigate the conjecture in higher dimensions and offer two novel approaches (decomposition and composition of quantum channels) and correspondingly, several schemes for finding counterexamples to this conjecture. One of the schemes involving the composition of PPT quantum channels in unsolved dimensions yields a potential counterexample.

DOI: 10.48550/arxiv.2010.15554

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arXiv:2010.15554

Investigation of the PPT Squared Conjecture for High Dimensions

Jin, Ryan

Published Oct 27, 2020• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2010.15554

Related papers

Positive-partial-transpose square conjecture for n=3

Lin Chen, Yang Yu, Wai-Shing Tang

We present the positive-partial-transpose (PPT) square conjecture introduced by M. Christandl Banff International Research Station Workshop: Operator Structures in Quantum Information Theory (Banff International Research Station, Alberta, 2012). We prove the conjecture in the case $n=3$ as a consequence of the fact that two-qutrit PPT states have Schmidt number of at most 2. The PPT square conjecture in the case of $n\ensuremath{\ge}4$ is still open. We present an example to support the conjecture for $n=4$ .We present the positive-partial-transpose (PPT) square conjecture introduced by M. Christandl Banff International Research Station Workshop: Operator Structures in Quantum Information Theory (Banff International Research Station, Alberta, 2012). We prove the conjecture in the case $n=3$ as a consequence of the fact that two-qutrit PPT states have Schmidt number of at most 2. The PPT square conjecture in the case of $n\ensuremath{\ge}4$ is still open. We present an example to support the conjecture for $n=4$.

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Toroidal-Core Certificates for PPT-Squared Diagonal Orthogonal Covariant Channels

Samuel Marquez Gonzalez

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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.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.

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We prove that the composition of two PPT maps through an intermediate qubit system is entanglement breaking. Specifically, for arbitrary finite dimensions m and n, if Φ: Mₘ → M₂ and Ψ: M₂ → Mₙ are completely positive and completely copositive, then Ψ ∘ Φ is entanglement breaking. The proof uses the special 2 × 2 block structure imposed by the qubit intermediate system, Schur-complement inequalities, an operator interpolation argument, and an explicit separability decomposition. As a consequence, no counterexample to the PPT-squared conjecture can occur through the 4 → 2 → 4 composition pattern.We prove that the composition of two PPT maps through an intermediate qubit system is entanglement breaking. Specifically, for arbitrary finite dimensions m and n, if Φ: Mₘ → M₂ and Ψ: M₂ → Mₙ are completely positive and completely copositive, then Ψ ∘ Φ is entanglement breaking. The proof uses the special 2 × 2 block structure imposed by the qubit intermediate system, Schur-complement inequalities, an operator interpolation argument, and an explicit separability decomposition. As a consequence, no counterexample to the PPT-squared conjecture can occur through the 4 → 2 → 4 composition pattern.

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n=3

as a consequence of the fact that two-qutrit PPT states have Schmidt number of at most 2. The PPT square conjecture in the case of $n\ensuremath{\ge}4$ is still open. We present an example to support the conjecture for

n=4

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.

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.