Majorization criterion

This article deals with the separation criterion of quantum mechanics; for the convergence criterion for series see majorant criterion .

The majorization criterion is a feature in quantum mechanics that serves to distinguish entangled from separable states on the basis of their density matrix . It is fulfilled for separable states, but separability does not follow from its fulfillment. It is therefore the weakest of the three operator separation criteria in quantum mechanics, which also include the reduction criterion and the Peres-Horodecki criterion .

The criterion was published in 1999 by Michael Nielsen and in 2001 together with Julia Kempe, and is therefore also called Nielsen's majorization theorem or Nielsen-Kempe majorization criterion . Put clearly, the majorization criterion means that separable states show greater disorder globally than locally.

definition

The majorization criterion is fulfilled when the eigenvalues ​​of the density matrix of a quantum mechanical state are majorized by the eigenvalues ​​of its partial traces and with respect to the Hilbert spaces and : ${\displaystyle \lambda }$ ${\displaystyle \rho _{AB}}$ ${\displaystyle \rho _{A}}$ ${\displaystyle \rho _{B}}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle \lambda _{\rho _{AB}}\prec \lambda _{\rho _{A}}\ {\text{und}}\ \lambda _{\rho _{AB}}\prec \lambda _{\rho _{B}}}$

It says:

• ${\displaystyle \lambda _{\rho _{AB}}}$for the eigenvalue vector of the density matrix of the overall system consisting of Alice and Bob , ${\displaystyle \rho _{AB}}$
• ${\displaystyle \lambda _{\rho _{A}}}$for the eigenvalue vector of Alice's density matrix , ${\displaystyle \rho _{A}}$
• ${\displaystyle \lambda _{\rho _{B}}}$for the eigenvalue vector of Bob's density matrix . ${\displaystyle \rho _{B}}$

The operator is spoken as "... is majorized by ..." (see majorization ) and means for two eigenvalue vectors and the dimension : ${\displaystyle \prec }$ ${\displaystyle x\prec y}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle d}$

${\displaystyle x\prec y\Leftrightarrow {\begin{cases}\sum _{j=1}^{k}x_{j}^{\downarrow }\leq \sum _{j=1}^{k}y_{j}^{\downarrow }&{\text{wenn }}1\leq k\leq d-1\\\sum _{j=1}^{d}x_{j}^{\downarrow }=\sum _{j=1}^{d}y_{j}^{\downarrow }&k=d\end{cases}}}$

The marking indicates that the eigenvalues ​​in the respective eigenvalue vector are sorted in descending order according to their size. If the dimension of an eigenvalue vector is smaller than the other, it is adjusted to the larger one by adding zeros. ${\displaystyle \,\downarrow }$ ${\displaystyle x_{i},y_{i}}$

use

The criterion is met for separable density matrices. If the criterion is not met, the density matrix is ​​not separable. The reverse conclusion that the separability of the density matrix follows from the fulfillment of the majorization criterion or that the majorization criterion is not fulfilled for a non-separable density matrix does not apply.

There are, for example, isospectal states that meet the majorization criterion but are not separable (see below).

The majorization criterion is weaker than the reduction criterion or the Peres-Horodecki criterion , because a fulfilled Peres-Horodecki criterion in 2x2 or 2x3 dimensions (i.e. a state consisting of two qubits or one qubit and one qutrite, i.e. a particle with three possible states, is composed) the separability of follows, while the reverse direction for the majorization criterion is generally not fulfilled. ${\displaystyle \rho }$

proof

In terms of content, the evidence is based on relevant literature. To prove this criterion, it is necessary, a bistochastische matrix to find, so that the eigenvalues of the matrix density (in any order) follow from the eigenvalues of the reduced density matrix: . With the help of this equation it can be shown that all Schur convex functions are inequality shown above ${\displaystyle B}$ ${\displaystyle \lambda _{\rho _{AB}}=B\cdot \lambda _{\rho _{A}}}$

${\displaystyle x^{\downarrow }\prec y^{\downarrow }\Rightarrow \sum _{j=1}^{k}x_{j}^{\downarrow }\leq \sum _{j=1}^{k}y_{j}^{\downarrow }}$

satisfy, for Schur concave function the same inequality follows, in which the direction of the inequality sign changes. The majorization criterion can thus be traced back to the entropy criterion, in which conclusions can be drawn about the separability from the Von Neumann entropy : In a pure state , the Von Neumann entropy disappears, but if it does not disappear for its partial traces , the state is entangled.

Examples

Isospectral states

Be

${\displaystyle \rho _{AB}={\frac {1}{3}}{\begin{pmatrix}1&0&0&0\\0&1&1&0\\0&1&1&0\\0&0&0&0\end{pmatrix}}}$

a state entangled with the eigenvalues ​​of the density matrix (according to the PPT criterion ), then its partial traces are given as ${\displaystyle \textstyle \lambda _{\rho _{AB}}^{\downarrow }=\left\{{\frac {2}{3}},{\frac {1}{3}},0,0\right\}}$

${\displaystyle \rho _{A}=\rho _{B}={\frac {1}{3}}{\begin{pmatrix}2&0\\0&1\end{pmatrix}}}$

with the eigenvalues . The majorization criterion is obviously fulfilled here, since the sums over the eigenvalues ​​are identical. If the reverse direction of the majorization criterion were met, one would be subject to the fallacy that it would be separable, although it is an entangled state. ${\displaystyle \textstyle \lambda _{\rho _{A},\rho _{B}}^{\downarrow }=\left\{{\frac {2}{3}},{\frac {1}{3}}\right\}}$ ${\displaystyle \rho _{AB}}$

Werner states

Consider the states of the qubit family of Werner states , with the identity matrix . ${\displaystyle \textstyle \rho _{AB}=p\,|\Psi ^{-}\rangle \langle \Psi ^{-}|+{\frac {1-p}{4}}I}$ ${\displaystyle I}$

The density matrix is

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

The eigenvalues ​​of this density matrix are given as

${\displaystyle \lambda _{\rho _{AB}}^{\downarrow }{\begin{pmatrix}{\frac {1+3p}{4}}\\{\frac {1-p}{4}}\\{\frac {1-p}{4}}\\{\frac {1-p}{4}}\end{pmatrix}}:={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\\a_{4}\end{pmatrix}}}$.

The partial trace over and is given as , so the eigenvalues ​​are given as . Now fill this with zeros until the same dimension as for is reached and received ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \textstyle \rho _{A}={\text{tr}}_{B}\left(\rho _{AB}\right)={\frac {1}{2}}I}$ ${\displaystyle \textstyle \left\{{\frac {1}{2}},{\frac {1}{2}}\right\}}$ ${\displaystyle \lambda _{\rho _{AB}}^{\downarrow }}$

${\displaystyle \lambda _{\rho _{A}}^{\downarrow }=\lambda _{\rho _{B}}^{\downarrow }={\begin{pmatrix}{\frac {1}{2}}\\{\frac {1}{2}}\\0\\0\end{pmatrix}}:={\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\\b_{4}\end{pmatrix}}}$.

Now calculate the majorization criterion:

${\displaystyle {\begin{aligned}k=1\colon &\quad b_{1}\leq a_{1}\Rightarrow {\frac {1+3p}{4}}\leq {\frac {1}{2}}\Rightarrow p\leq {\frac {1}{3}}\\k=2\colon &\quad b_{1}+b_{2}\leq a_{1}+a_{2}\Rightarrow {\frac {1+3p}{4}}+{\frac {1-p}{4}}\leq {\frac {1}{2}}+{\frac {1}{2}}\Rightarrow p\leq 1\\k=3\colon &\quad \sum _{i=1}^{3}b_{i}\leq \sum _{i=1}^{3}a_{i}\Rightarrow {\frac {3}{4}}+{\frac {p}{4}}\leq 1\Rightarrow p\leq 1\\k=4\colon &\quad \sum _{i=1}^{4}b_{i}\leq \sum _{i=1}^{4}a_{i}\Rightarrow 1\leq 1\end{aligned}}}$

It follows from this that the Werner state is for entangled, which is the same result, which is also evident from the Peres-Horodecki criterion . ${\displaystyle p>{\tfrac {1}{3}}}$

literature

• Tohya Hiroshima: Majorization Criterion for Distillability of a Bipartite Quantum State . In: Physical Review Letters . tape 91 , no. 5 , 2003, p. 057902 , doi : 10.1103 / PhysRevLett.91.057902 , arxiv : quant-ph / 0303057 .

Individual evidence

1. ↑ Sinisa Karnas: On the structure of entangled states . Dissertation. University of Hanover, 2001, p. 26 ( d-nb.info ).
2. ↑ Michael Nielsen: Conditions for a Class of Entanglement Transformations . In: Physical Review Letter . tape 83 , 1999, pp. 436-139 , doi : 10.1103 / PhysRevLett.83.436 .
3. ↑ Teiko Heinosaari, Mário Ziman: The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement . Cambridge University Press, 2001, pp. 285 ( limited preview in Google Book search).
4. ↑ ^{a b c} Ingemar Bengtsson, Karol Zyczkowski: Geometry of Quantum States by Ingemar Bengtsson . Cambridge University Press, Cambridge December 6, 2007, pp. 386 , doi : 10.1017 / CBO9780511535048 ( limited preview in Google Book search).
5. ↑ Remigiusz Augusiak, Julia Stasińska: Positive maps, majorization, entropic inequalities and detection of entanglement . In: New Journal of Physics . tape 11 , no. 5 , p. 53018 , doi : 10.1088 / 1367-2630 / 11/5/053018 , arxiv : 0811.3604v3 .
6. ↑ Michael Nielsen, Julia Kempe: Separable States Are More Disordered Globally than Locally . In: Physical Review Letters . tape 86 , no. 22 , 2001, p. 5184-5187 , doi : 10.1103 / PhysRevLett.86.5184 .
7. ↑ Michael Nielsen: Characterizing mixing and measurement in quantum mechanics . In: Physical Review A . tape 63 , no. 2 , 2011, p. 22114-22125 , doi : 10.1103 / PhysRevA.63.022114 .
8. ^ Asher Peres: Separability Criterion for Density Matrices . In: Physical Review Letters . tape 77 , 1996, pp. 1413 , doi : 10.1103 / PhysRevLett.77.1413 .
9. ^ Vlatko Vedral, Martin Plenio, Michael Rippin, Peter Knight: Quantifying Entanglement . In: Physical Review Letters . tape 78 , no. 12 , March 24, 1997, p. 2275-2279 , doi : 10.1103 / PhysRevLett.78.2275 .