Peres-Horodecki criterion

The Peres-Horodecki criterion (or English positive partial transpose criterion (PPT criterion)) for the composite density matrix of two quantum mechanical systems and is a necessary condition for determining their separability. In the case of 2x2 and 2x3 dimensions, the condition is also sufficient. The PPT criterion can be used to determine whether a mixed quantum state is entangled when the Schmidt decomposition cannot be applied. ${\displaystyle \rho }$ ${\displaystyle A}$ ${\displaystyle B}$

In higher-dimensional spaces, the criterion is not clear and other methods have to be used.

definition

Let there be a general, mixed state that acts on. Then applies ${\displaystyle \rho _{AB}}$ ${\displaystyle {\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}}$

${\displaystyle \rho _{AB}=\sum _{ijkl}p_{kl}^{ij}|i\rangle \langle j|\otimes |k\rangle \langle l|}$.

The partially transposed matrix is defined as ${\displaystyle B}$

${\displaystyle \rho _{AB}^{T_{B}}:=I\otimes T(\rho _{AB})=\sum _{ijkl}p_{kl}^{ij}|i\rangle \langle j|\otimes (|k\rangle \langle l|)^{T}=\sum _{ijkl}p_{kl}^{ij}|i\rangle \langle j|\otimes |l\rangle \langle k|=\sum _{ijkl}p_{lk}^{ij}|i\rangle \langle j|\otimes |k\rangle \langle l|}$.

Partial in this case means that only part of the state is transposed. In the expression is apparent that the identity operator on acts and thus leaves it unchanged, and the transposition of acts. ${\displaystyle I\otimes T(\rho )}$ ${\displaystyle I}$ ${\displaystyle A}$ ${\displaystyle T}$ ${\displaystyle B}$

The Peres-Horodecki criterion says that if is separable, then it will be positive semidefinite , ie only have non-negative eigenvalues . Conversely, this means that if it has negative eigenvalues, the system is entangled . In general, it does not matter whether the system is as transposed in the example, or system with . ${\displaystyle \rho _{AB}}$ ${\displaystyle \rho _{AB}^{T_{B}}}$ ${\displaystyle \rho _{AB}^{T_{B}}}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle \rho _{AB}^{T_{A}}}$

example

The qubit family of Werner states are considered:

${\displaystyle \rho _{AB}=p|\Psi ^{-}\rangle \langle \Psi ^{-}|+(1-p){\frac {I}{4}}}$

The density matrix is

${\displaystyle \rho ={\frac {1}{4}}{\begin{pmatrix}1-p&0&0&0\\0&p+1&-2p&0\\0&-2p&p+1&0\\0&0&0&1-p\end{pmatrix}}}$

and the partially transposed matrix

${\displaystyle \rho ^{T_{B}}={\frac {1}{4}}{\begin{pmatrix}1-p&0&0&-2p\\0&p+1&0&0\\0&0&p+1&0\\-2p&0&0&1-p\end{pmatrix}}}$.

The lowest eigenvalue is . It follows from this that the state for is entangled. ${\displaystyle (1-3p)/4}$ ${\displaystyle p>1/3}$

Individual evidence

• Asher Peres , Separability Criterion for Density Matrices , Phys. Rev. Lett. 77: 1413-1415 (1996)
• Michał Horodecki, Paweł Horodecki, Ryszard Horodecki: Separability of mixed states: necessary and sufficient conditions . In: Physics Letters A . 223, September, pp. 1-8. doi : 10.1016 / s0375-9601 (96) 00706-2 .
• Karol Życzkowski and Ingemar Bengtsson, Geometry of Quantum States , Cambridge University Press, 2006
• SL Woronowicz: Positive maps of low dimensional matrix algebras . In: Rep. Math. Phys. . 10, 1976, pp. 165-183. doi : 10.1016 / 0034-4877 (76) 90038-0 .