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).
An important example of
separability
criteria via PnCP maps is the PPT criterion, as
stated in the following theorem:
If a state πβββSAβ
ββ
SB
is separable, then πTBβ:=β(idAβ
ββ
TB)[π]ββ₯β0,
where β
TB
is the transposition on the second subsystem of the compound system
SAβ
ββ
SB.
Therefore if πTBβ:=β(idAβ
ββ
TB)[π]
is nonpositive, the state π 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 dAdBββ€β6,
i.e. the bipartite system is of the type 2β
ββ
2 or 2β
ββ
3 or 3β
ββ
2, then the PPT criterion provides a
necessary and sufficient condition for separability, as follows.
Theorem: A state πABβββSAβ
ββ
SB
with dAdBββ€β6
is separable if and only if πABTBβ:=β(idAβ
ββ
TB)[πAB]ββ₯β0,
i.e. if and only if πAB 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:
Monday, October 26, 2015 - 17:56
