Error message
- Deprecated function: TYPO3\PharStreamWrapper\Manager::initialize(): Implicitly marking parameter $resolver as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 19 of includes/file.phar.inc).
- Deprecated function: TYPO3\PharStreamWrapper\Manager::initialize(): Implicitly marking parameter $collection as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 19 of includes/file.phar.inc).
- Deprecated function: TYPO3\PharStreamWrapper\Manager::__construct(): Implicitly marking parameter $resolver as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 19 of includes/file.phar.inc).
- Deprecated function: TYPO3\PharStreamWrapper\Manager::__construct(): Implicitly marking parameter $collection as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 19 of includes/file.phar.inc).
- Deprecated function: UpdateQuery::expression(): Implicitly marking parameter $arguments as nullable is deprecated, the explicit nullable type must be used instead in require_once() (line 1884 of includes/database/database.inc).
- Deprecated function: MergeQuery::expression(): Implicitly marking parameter $arguments as nullable is deprecated, the explicit nullable type must be used instead in require_once() (line 1884 of includes/database/database.inc).
- Deprecated function: SelectQueryInterface::getArguments(): Implicitly marking parameter $queryPlaceholder as nullable is deprecated, the explicit nullable type must be used instead in require_once() (line 1884 of includes/database/database.inc).
- Deprecated function: SelectQueryInterface::preExecute(): Implicitly marking parameter $query as nullable is deprecated, the explicit nullable type must be used instead in require_once() (line 1884 of includes/database/database.inc).
- Deprecated function: SelectQueryExtender::getArguments(): Implicitly marking parameter $queryPlaceholder as nullable is deprecated, the explicit nullable type must be used instead in require_once() (line 1884 of includes/database/database.inc).
- Deprecated function: SelectQueryExtender::preExecute(): Implicitly marking parameter $query as nullable is deprecated, the explicit nullable type must be used instead in require_once() (line 1884 of includes/database/database.inc).
- Deprecated function: SelectQuery::getArguments(): Implicitly marking parameter $queryPlaceholder as nullable is deprecated, the explicit nullable type must be used instead in require_once() (line 1884 of includes/database/database.inc).
- Deprecated function: SelectQuery::preExecute(): Implicitly marking parameter $query as nullable is deprecated, the explicit nullable type must be used instead in require_once() (line 1884 of includes/database/database.inc).
PPT separability criterion
Theorem [PPT separability criterion]: If a state 𝜚 ∈ SA ⊗ SB
is separable, then 𝜚TB := (idA ⊗ TB)[𝜚] ≥ 0,
where $T_{B}$, is the transposition on the second subsystem of the
compound system $S_A \otimes S_B$.
Therefore if 𝜚TB := (idA ⊗ TB)[𝜚]
is nonpositive, the state 𝜚 is
entangled.
In arbitrary dimensions of the two subsystems the PPT criterion
provides only a necessary but not sufficient condition for separability,
i.e. in "high" dimensions there exist states that remain positive under
partial transposition (PPT states) even if they are entangled.
If instead the bipartite system has dimension $d_A d_B \leq 6$, i.e.,
the bipartite system is of the type $2 \otimes 2$ or $2 \otimes 3$ or $3
\otimes 2$, then the PPT criterion provides a necessary and sufficient
condition for separability, as follows.
Theorem: A state $\varrho_{AB} \in S_A \otimes S_B$ with $d_A
d_B \leq 6$ is separable if and only if $\varrho_{AB}^{T_B} :=
(id_A \otimes T_B)[\varrho_{AB}] \geq 0$, i.e. if and only if
$\varrho_{AB}$ is PPT.
- A. Peres, Separability criterion for density matrices,
Phys. Rev. Lett. 77(8), 1413 (1996).
- M. Horodecki, P. Horodecki, R. Horodecki, Separability of mixed
states: necessary and sufficient conditions, Phys. Lett. A
223, 1 (1996).
Category:Entanglement
Category:Handbook
of Quantum Information
Last modified:
Friday, July 7, 2017 - 21:06
