arXiv:1807.03636 · Physical Review A

Positive-partial-transpose square conjecture for n=3

Lin Chen, Yang Yu, Wai-Shing Tang

Published Jan 22, 2019• 20 citations

Abstract

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 $n=4$.

DOI: 10.1103/physreva.99.012337

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arXiv:1807.03636 · Physical Review A

Positive-partial-transpose square conjecture for n=3

Lin Chen, Yang Yu, Wai-Shing Tang

Published Jan 22, 2019• 20 citations

arXiv HTML PDF

Abstract

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

DOI: 10.1103/physreva.99.012337

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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.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.

arXiv:2010.15554Oct 27, 20201 citation

Beyond the Positive Partial Transpose Squared Conjecture: The Qutrit Case

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

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The positive-partial-transpose-squared (PPT-squared) conjecture asks whether the composition of two positive-partial-transpose (PPT) quantum channels must be entanglement breaking (EB). The problem remains open in general and, within the diagonal orthogonal covariant (DOC) class, the first unresolved deterministic dimension is four. A sufficient condition for PPT-squared composition is derived for DOC maps in arbitrary finite dimension. The proof isolates the quantum coherence data into two correlation matrices, mixes each with the identity until it lies in the convex hull of rank-one correlation matrices, and converts the resulting objects into an explicit triplewise completely positive (TCP) core. The remaining contribution is purely classical and is TCP whenever a simple entrywise mixing inequality is satisfied. For PPT DOC maps $Φ_{A,B,C}$ and $Φ_{D,E,F}$ on $M_d$ , 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 positive-partial-transpose-squared (PPT-squared) conjecture asks whether the composition of two positive-partial-transpose (PPT) quantum channels must be entanglement breaking (EB). The problem remains open in general and, within the diagonal orthogonal covariant (DOC) class, the first unresolved deterministic dimension is four. A sufficient condition for PPT-squared composition is derived for DOC maps in arbitrary finite dimension. The proof isolates the quantum coherence data into two correlation matrices, mixes each with the identity until it lies in the convex hull of rank-one correlation matrices, and converts the resulting objects into an explicit triplewise completely positive (TCP) core. The remaining contribution is purely classical and is TCP whenever a simple entrywise mixing inequality is satisfied. For PPT DOC maps $Φ_{A,B,C}$ and $Φ_{D,E,F}$ on $M_d$, 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.

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Entangled states with a positive partial transpose (so-called PPT states) are central to many interesting problems in quantum theory. On the one hand, they are considered to be weakly entangled, since no pure state entanglement can be distilled from them. On the other hand, it has been shown recently that some of these PPT states exhibit genuinely high-dimensional entanglement, i.e., they have a high Schmidt number. Here we investigate the ($d\ifmmode\times\else\texttimes\fi{}d$)-dimensional PPT states for $d\ensuremath{\ge}4$ discussed recently by E. Sindici and M. Piani [Phys. Rev. A 97, 032319 (2018)]. By generalizing their methods to the calculation of Schmidt numbers, we show that a linear $d/2$ scaling of its Schmidt number in the local dimension $d$ can be attained.Entangled states with a positive partial transpose (so-called PPT states) are central to many interesting problems in quantum theory. On the one hand, they are considered to be weakly entangled, since no pure state entanglement can be distilled from them. On the other hand, it has been shown recently that some of these PPT states exhibit genuinely high-dimensional entanglement, i.e., they have a high Schmidt number. Here we investigate the ($d\ifmmode\times\else\texttimes\fi{}d$)-dimensional PPT states for $d\ensuremath{\ge}4$ discussed recently by E. Sindici and M. Piani [Phys. Rev. A 97, 032319 (2018)]. By generalizing their methods to the calculation of Schmidt numbers, we show that a linear $d/2$ scaling of its Schmidt number in the local dimension $d$ can be attained.

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Construction of three-qubit genuine entanglement with bipartite positive partial transposes

Kil-Chan Ha, Seung-Hyeok Kye

arXiv:1512.04693Physical Review AMar 11, 20169 citations

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

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.

d/2

scaling of its Schmidt number in the local dimension

d

can be attained.