Peres–Horodecki criterion について

, to be separable. It is also called the PPT criterion, for ''positive partial transpose''. In the 2x2 and 2x3 dimensional cases the condition is also sufficient. It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply.
In higher dimensions, the test is inconclusive, and one should supplement it with more advanced tests, such as those based on entanglement witnesses.
==Definition==

If we have a general state

\rho

which acts on

\mathcal_A \otimes \mathcal_B

\rho = \sum_ p^_ |i\rangle \langle j | \otimes |k\rangle \langle l|

Its partial transpose (with respect to the B party) is defined as
:

\rho^ := I \otimes T (\rho) = \sum_ p^ _ |i\rangle \langle j | \otimes (|k\rangle \langle l|)^T = \sum_ p^ _ |i\rangle \langle j | \otimes |l\rangle \langle k|

Note that the ''partial'' in the name implies that only part of the state is transposed. More precisely,

I \otimes T (\rho)

is the identity map applied to the A party and the transposition map applied to the B party.
This definition can be seen more clearly if we write the state as a block matrix:
:

\rho = \begin A_ & A_ & \dots & A_ \\ A_ & A_ & & \\ \vdots & & \ddots & \\ A_ & & & A_ \end

Where

n = \dim \mathcal_A

, and each block is a square matrix of dimension

m = \dim \mathcal_B

. Then the partial transpose is
:

\rho^ = \begin A_^T & A_^T & \dots & A_^T \\ A_^T & A_^T & & \\ \vdots & & \ddots & \\ A_^T & & & A_^T \end

The criterion states that if

\rho\;\!

is separable,

\rho^

has non-negative eigenvalues. In other words, if

\rho^

has a negative eigenvalue,

\rho\;\!

is guaranteed to be entangled. If the eigenvalues are non-negative, and the dimension is larger than 6, the test is inconclusive.
The result is independent of the party that was transposed, because

\rho^ = (\rho^)^T

.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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