A block matrix $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ is positive partial transpose (PPT) if both $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ and $\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right]$ are positive semi-definite. This class is significant in studying the separability criterion for density matrices. The current paper presents new relations for such matrices. This includes some equivalent forms and new related inequalities that extend some results from the literature. In the end of the paper, we present some related results for positive semi-definite block matrices, which have similar forms as those presented for PPT matrices, with applications that include significant improvement of numerical radius inequalities.A block matrix $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ is positive partial transpose (PPT) if both $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ and $\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right]$ are positive semi-definite. This class is significant in studying the separability criterion for density matrices. The current paper presents new relations for such matrices. This includes some equivalent forms and new related inequalities that extend some results from the literature. In the end of the paper, we present some related results for positive semi-definite block matrices, which have similar forms as those presented for PPT matrices, with applications that include significant improvement of numerical radius inequalities.

Electronic Journal of Linear AlgebraDec 15, 20226 citations

Inequalities related to partial transpose and partial trace

Daeshik Choi

Linear Algebra and its ApplicationsNov 23, 201610 citations

Partial transpose criteria for symmetric states

Fabian Bohnet-Waldraff, Daniel Braun, O. Giraud

We express the positive-partial-transpose (PPT) separability criterion for symmetric states of multiqubit systems in terms of matrix inequalities based on the recently introduced tensor representation for spin states. We construct a matrix from the tensor representation of the state and show that it is similar to the partial transpose of the density matrix written in the computational basis. Furthermore, the positivity of this matrix is equivalent to the positivity of a correlation matrix constructed from tensor products of Pauli operators. This allows for a more transparent experimental interpretation of the PPT criteria for an arbitrary spin- $j$ state. The unitary matrices connecting our matrix to the partial transpose of the state generalize the so-called magic basis that plays a central role in Wootters' explicit formula for the concurrence of a two-qubit system and the Bell bases used for the teleportation of a one- or two-qubit state.We express the positive-partial-transpose (PPT) separability criterion for symmetric states of multiqubit systems in terms of matrix inequalities based on the recently introduced tensor representation for spin states. We construct a matrix from the tensor representation of the state and show that it is similar to the partial transpose of the density matrix written in the computational basis. Furthermore, the positivity of this matrix is equivalent to the positivity of a correlation matrix constructed from tensor products of Pauli operators. This allows for a more transparent experimental interpretation of the PPT criteria for an arbitrary spin-$j$ state. The unitary matrices connecting our matrix to the partial transpose of the state generalize the so-called magic basis that plays a central role in Wootters' explicit formula for the concurrence of a two-qubit system and the Bell bases used for the teleportation of a one- or two-qubit state.

arXiv:1606.07635Physical Review AOct 31, 201628 citations

Multivariate Trace Inequalities

David Sutter

SpringerBriefs in mathematical physicsJan 1, 20188 citations

Positive partial transpose from spectra

Roland Hildebrand

In this paper we solve the following problem. Let ${\mathcal{H}}_{nm}$ be a Hilbert space of dimension $nm$ , and let $A$ be a positive semidefinite self-adjoint linear operator on ${\mathcal{H}}_{nm}$ . Under which conditions on the spectrum has $A$ a positive partial transpose (is PPT) with respect to any partition ${\mathcal{H}}_{n}\ensuremath{\bigotimes}{\mathcal{H}}_{m}$ of the space ${\mathcal{H}}_{nm}$ as a tensor product of an $n$ -dimensional and an $m$ -dimensional Hilbert space? We show that the necessary and sufficient conditions can be expressed as a set of linear matrix inequalities (LMIs) on the eigenvalues of $A$ .In this paper we solve the following problem. Let ${\mathcal{H}}_{nm}$ be a Hilbert space of dimension $nm$, and let $A$ be a positive semidefinite self-adjoint linear operator on ${\mathcal{H}}_{nm}$. Under which conditions on the spectrum has $A$ a positive partial transpose (is PPT) with respect to any partition ${\mathcal{H}}_{n}\ensuremath{\bigotimes}{\mathcal{H}}_{m}$ of the space ${\mathcal{H}}_{nm}$ as a tensor product of an $n$-dimensional and an $m$-dimensional Hilbert space? We show that the necessary and sufficient conditions can be expressed as a set of linear matrix inequalities (LMIs) on the eigenvalues of $A$.

arXiv:quant-ph/0502170Physical Review ANov 28, 200759 citations

A block matrix

\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]

is positive partial transpose (PPT) if both

\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]

and

\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right]

are positive semi-definite. This class is significant in studying the separability criterion for density matrices. The current paper presents new relations for such matrices. This includes some equivalent forms and new related inequalities that extend some results from the literature. In the end of the paper, we present some related results for positive semi-definite block matrices, which have similar forms as those presented for PPT matrices, with applications that include significant improvement of numerical radius inequalities.

j

state. The unitary matrices connecting our matrix to the partial transpose of the state generalize the so-called magic basis that plays a central role in Wootters' explicit formula for the concurrence of a two-qubit system and the Bell bases used for the teleportation of a one- or two-qubit state.

In this paper we solve the following problem. Let

{\mathcal{H}}_{nm}

be a Hilbert space of dimension

nm

, and let

A

be a positive semidefinite self-adjoint linear operator on

{\mathcal{H}}_{nm}

. Under which conditions on the spectrum has

A

a positive partial transpose (is PPT) with respect to any partition ${\mathcal{H}}_{n}\ensuremath{\bigotimes}{\mathcal{H}}_{m}$ of the space

{\mathcal{H}}_{nm}

as a tensor product of an

n

-dimensional and an

m

-dimensional Hilbert space? We show that the necessary and sufficient conditions can be expressed as a set of linear matrix inequalities (LMIs) on the eigenvalues of

A